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Lectures on the
Geometry of
Manifolds
Third Edition
Other World Scientific Titles by the Author
Introduction to Real Analysis

ISBN: 978-981-121-038-9
ISBN: 978-981-121-075-4 (pack)
Lectures on the
Geometry of
Manifolds
Third Edition
Liviu I Nicolaescu
University of Notre Dame, USA
World Scientific
NEW JERSEY . LONDON . SINGAPORE . BEIJING . SHANGHAI . HONG KONG . TAIPEI . CHENNAI . TOKYO
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Library of Congress Cataloging-in-Publication Data

Names: Nicolaescu, Liviu I., author.
Title: Lectures on the geometry of manifolds / Liviu I. Nicolaescu, University of Notre Dame, USA.
Description: Third edition. | New Jersey : World Scientific Publishing Co., [2021] |
Includes bibliographical references and index.
Identifiers: LCCN 2020037906 | ISBN 9789811214813 (hardcover) | ISBN 9789811215957 (paperback) |
ISBN 9789811214820 (ebook) I ISBN 9789811214837 ( b o o k other)
Subjects: LCSH: Geometry, Differential. | Manifolds (Mathematics)
Classification: LCC QA649 .N53 2021 | DDC 516.3/62--dc23
LC record available at https://lccn.loc.gov/2020037906
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To the magical summer nights of my hometown
This page intentionally left blank
Preface
Shape is a fascinating and intriguing subject which has stimulated the imagination of many
people. It suffices to look around to become curious. Euclid did just that and came up
with the first pure creation. Relying on the common experience, he created an abstract
world that had a life of its own. As the human knowledge progressed so did the ability
of formulating and answering penetrating questions. In particular, mathematicians started
wondering whether Euclid's "obvious" absolute postulates were indeed obvious and/or ab-
solute. Scientists realized that Shape and Space are two closely related concepts and asked
whether they really look the way our senses tell us. As Felix Klein pointed out in his Er-
langen Program, there are many ways of looking at Shape and Space so that various points
of view may produce different images. In particular, the most basic issue of "measuring
the Shape" cannot have a clear cut answer. This is a book about Shape, Space and some
particular ways of studying them.
Since its inception, the differential and integral calculus proved to be a very versatile
tool in dealing with previously untouchable problems. It did not take long until it found
uses in geometry in the hands of the Great Masters. This is the path we want to follow in
the present book.
In the early days of geometry nobody wonted about the natural context in which the
methods of calculus "feel at home". There was no need to address this aspect since for the
particular problems studied this was a non-issue. As mathematics progressed as a whole
the "natural context" mentioned above crystallized in the minds of mathematicians and it
was a notion so important that it had to be given a name. The geometric objects which can
be studied using the methods of calculus were called smooth manifolds. Special cases of
manifolds are the curves and the surfaces and these were quite well understood. B. Riemann
was the first to note that the low dimensional ideas of his time were particular aspects of a
higher dimensional world.
The first chapter of this book introduces the reader to the concept of smooth mani-
fold through abstract definitions and, more importantly, through many we believe relevant
examples. In particular, we introduce at this early stage the notion of Lie group. The
main geometric and algebraic properties of these objects will be gradually described as we
progress with our study of the geometry of manifolds. Besides their obvious usefulness in
geometry, the Lie groups are academically very friendly. They provide a marvelous testing
vii
viii Lectures on the Geometry of Manzfolds
ground for abstract results. We have consistently taken advantage of this feature through-
out this book. As a bonus, by the end of these lectures the reader will feel comfortable
manipulating basic Lie theoretic concepts.
To apply the techniques of calculus we need things to derivate and integrate. These
"things" are introduced in Chapter 2. The reason why smooth manifolds have many dif-
ferentiable objects attached to them is that they can be locally very well approximated
by linear spaces called tangent spaces. Locally, everything looks like traditional calculus.
Each point has a tangent space attached to it so that we obtain a "bunch of tangent spaces"
called the tangent bundle. We found it appropriate to introduce at this early point the notion
of vector bundle. It helps in structuring both the language and the thinking.
Once we have "things to derivate and integrate" we need to know how to explicitly
perform these operations. We devote the Chapter 3 to this purpose. This is perhaps one of
the most unattractive aspects of differential geometry but is crucial for all further develop-
ments. To spice up the presentation, we have included many examples which will found
applications in later chapters. In particular, we have included a whole section devoted
to the representation theory of compact Lie groups essentially describing the equivalence
between representations and their characters.
The study of Shape begins in earnest in Chapter 4 which deals with Riemann manifolds.
We approach these objects gradually. The first section introduces the reader to the notion of
geodesics which are defined using the Levi-Civita connection. Locally, the geodesics play
the same role as the straight lines in an Euclidian space but globally new phenomena arise.
We illustrate these aspects with many concrete examples. In the final part of this section
we show how the Euclidian vector calculus generalizes to Riemann manifolds.
The second section of this chapter initiates the local study of Riemann manifolds. Up
to first order these manifolds look like Euclidian spaces. The novelty arises when we study
"second order approximations" of these spaces. The Riemann tensor provides the complete
measure of how far is a Riemann manifold from being flat. This is a very involved object
and, to enhance its understanding, we compute it in several instances: on surfaces (which
can be easily visualized) and on Lie groups (which can be easily formalized). We have also
included Caftan's moving frame technique which is extremely useful in concrete computa-
tions. As an application of this technique we prove the celebrated Theorem Egregium of
Gauss. This section concludes with the first global result of the book, namely the Gauss-
Bonnet theorem. We present a proof inspired from [38] relying on the fact that all Riemann
surfaces are Einstein manifolds. The Gauss-Bonnet theorem will be a recurring theme in
this book and we will provide several other proofs and generalizations.
One of the most fascinating aspects of Riemann geometry is the intimate correlation
"local-global". The Riemann tensor is a local object with global effects. There are cur-
rently many techniques of capturing this correlation. We have already described one in the
proof of Gauss-Bonnet theorem. In Chapter 5 we describe another such technique which
relies on the study of the global behavior of geodesics. We felt we had the moral obliga-
tion to present the natural setting of this technique and we briefly introduce the reader to
the wonderful world of the calculus of variations. The ideas of the calculus of variations
Preface ix
produce remarkable results when applied to Riemann manifolds. For example, we explain
in rigorous terms why "very curved manifolds" cannot be "too long".
In Chapter 6 we leave for a while the "differentiable realm" and we briefly discuss the
fundamental group and covering spaces. These notions shed a new light on the results
of Chapter 5. As a simple application we prove Weyl's theorem that the semisimple Lie
groups with definite Killing form are compact and have finite fundamental group.
Chapter 7 is the topological core of the book. We discuss in detail the cohomology of
smooth manifolds relying entirely on the methods of calculus. In writing this chapter we
could not, and would not escape the influence of the beautiful monograph [22], and this
explains the frequent overlaps. In the first section we introduce the DeRham cohomology
and the Mayer-Vietoris technique. Section 2 is devoted to the Poincaré duality, a feature
which sets the manifolds apart from many other types of topological spaces. The third sec-
tion offers a glimpse at homology theory. We introduce the notion of (smooth) cycle and
then present some applications: intersection theory, degree theory, Thom isomorphism and
we prove a higher dimensional version of the Gauss-Bonnet theorem at the cohomologi-
cal level. The fourth section analyzes the role of symmetry in restricting the topological
type of a manifold. We prove Elie Cartan's old result that the cosmology of a symmetric
space is given by the linear space of its bi-invariant forms. We use this technique to com-
pute the lower degree cohomology of compact semisimple Lie groups. We conclude this
section by computing the cosmology of complex Grassmannians relying on Weyl's inte-
gration formula and Schur polynomials. The chapter ends with a fifth section containing a
concentrated description of tech cohomology.
Chapter 8 is a natural extension of the previous one. We describe the Chern-Weil con-
struction for arbitrary principal bundles and then we concretely describe the most important
examples: Chern classes, Pontryagin classes and the Euler class. In the process, we com-
pute the ring of invariant polynomials of many classical groups. Usually, the connections in
principal bundles are defined in a global manner, as horizontal distributions. This approach
is geometrically very intuitive but, at a first contact, it may look a bit unfriendly in concrete
computations. We chose a local approach build on the reader's experience with connec-
tions on vector bundles which we hope will attenuate the formalism shock. In proving the
various identities involving characteristic classes we adopt an invariant theoretic point of
view. The chapter concludes with the general Gauss-Bonnet-Chern theorem. Our proof is
a variation of Chern's proof.
Chapter 9 is the analytical core of the book.' Many objects in differential geometry
are defined by differential equations and, among these, the elliptic ones play an important
role. This chapter represents a minimal introduction to this subject. After presenting some
basic notions concerning arbitrary partial differential operators we introduce the Sobolev
spaces and describe their main functional analytic features. We then go straight to the core
of elliptic theory. We provide an almost complete proof of the elliptic a priori estimates
(we left out only the proof of the Calderon-Zygmund inequality). The regularity results are
then deduced from the a priori estimates via a simple approximation technique. As a first
lin the current edition, this is Chapter 10.

X Lectures on the Geometry of Manzfolds
application of these results we consider a Kazhdan-Warner type equation which recently

found applications in solving the Seiberg-Witten equations on a Kohler manifold. We adopt
a variational approach. The uniformization theorem for compact Riemann surfaces is then
a nice bonus. This may not be the most direct proof but it has an academic advantage. It
builds a circle of ideas with a wide range of applications. The last section of this chapter
is devoted to Fredholm theory. We prove that the elliptic operators on compact manifolds
are Fredholm and establish the homotopy invariance of the index. These are very gen-
eral Hodge type theorems. The classical one follows immediately from these results. We
conclude with a few facts about the spectral properties of elliptic operators.
The last chapter is entirely devoted to a very important class of elliptic operators namely
the Dirac operators. The important role played by these operators was singled out in the
works of Atiyah and Singer and, since then, they continue to be involved in the most dra-
matic advances of modern geometry. We begin by first describing a general notion of Dirac
operators and their natural geometric environment, much like in [14]. We then isolate a
special subclass we called geometric Dirac operators. Associated to each such operator is
a very concrete WeitzenbOck formula which can be viewed as a bridge between geometry
and analysis, and which is often the source of many interesting applications. The abstract
considerations are backed by a full section describing many important concrete examples.
In writing this book we had in mind the beginning graduate student who wants to spe-
cialize in global geometric analysis in general and gauge theory in particular. The second
half of the book is an extended version of a graduate course in differential geometry we
taught at the University of Michigan during the winter semester of 1996.
The minimal background needed to successfully go through this book is a good knowl-
edge of vector calculus and real analysis, some basic elements of point set topology and
linear algebra. A familiarity with some basic facts about the differential geometry of curves
of surfaces would ease the understanding of the general theory, but this is not a must. Some
parts of the chapter on elliptic equations may require a more advanced background in func-
tional analysis.
The theory is complemented by a large list of exercises. Quite a few of them contain
technical results we did not prove so we would not obscure the main arguments. There
are however many non-technical results which contain additional information about the
subjects discussed in a particular section. We left hints whenever we believed the solution
is not straightforward.
Personal note It has been a great personal experience writing this book, and I sincerely
hope I could convey some of the magic of the subject. Having access to the remarkable
science library of the University of Michigan and its computer facilities certainly made my
job a lot easier and improved the quality of the final product.
I learned differential equations from Professor Viorel Barbu, a very generous and en-
thusiastic person who guided my first steps in this field of research. He stimulated my
curiosity by his remarkable ability of unveiling the hidden beauty of this highly technical
subject. My thesis advisor, Professor Tom Parker, introduced me to more than the funda-
mentals of modern geometry. He played a key role in shaping the manner in which I regard
Preface xi
mathematics. In particular, he convinced me that behind each formalism there must be a

picture, and uncovering it, is a very important part of the creation process. Although I did
not directly acknowledge it, their influence is present throughout this book. I only hope the
filter of my mind captured the full richness of the ideas they so generously shared with me.
My friends Louis Funar and Gheorghe Ionesez read parts of the manuscript. I am
grateful to them for their effort, their suggestions and for their friendship. I want to thank
Arthur Greenspoon for his advice, enthusiasm and relentless curiosity which boosted my
spirits when I most needed it. Also, I appreciate very much the input I received from the
graduate students of my "Special topics in differential geometry" course at the University
of Michigan which had a beneficial impact on the style and content of this book.
At last, but not the least, I want to thank my family who supported me from the begin-
ning to the completion of this project.
Ann Arbor, 1996.
Preface to the second edition
Rarely in life is a man given the chance to revisit his "youthful indiscretions". With this
second edition I have been given this opportunity, and I have tried to make the best of it.
The first edition was generously sprinkled with many typos, which I can only attribute
to the impatience of youth. In spite of this problem, I have received very good feedback
from a very indulgent and helpful audience, from all over the world.
In preparing the new edition, I have been engaged on a massive typo hunting, supported
by the wisdom of time, and the useful comments that I have received over the years from
many readers. I can only say that the number of typos is substantially reduced. However,
experience tells me that Murphy's Law is still at work, and there are still typos out there
which will become obvious only in the printed version.
The passage of time has only strengthened my conviction that, in the words of Isaac
Newton, "in learning the sciences examples are of more use than precepts". The new
edition continues to be guided by this principle. I have not changed the old examples, but
I have polished many of my old arguments, and I have added quite a large number of new
examples and exercises.
The only major addition to the contents is a new chapter (Chapter 9) on classical integral
geometry. This is a subject that captured my imagination over the last few years, and since
the first edition developed all the tools needed to understand some of the juiciest results
in this area of geometry, I could not pass the chance to share with a curious reader my
excitement about this line of thought.
One novel feature in our presentation of the classical results of integral geometry is the
use of tame geometry. This is a recent extension of the better know area of real algebraic
geometry which allowed us to avoid many heavy analytical arguments, and present the
geometric ideas in as clear a light as possible.
Notre Dame, 2007.
2He passed away while I was preparing the second edition. He was the ultimate poet of mathematics.
xii Lectures on the Geometry of Manifolds
Preface to the third edition
I started writing the first edition 25 years ago as a fresh PhD and I was warned by
many "adults" that this was a wrong career move. I embarked in this project with all the
energy, enthusiasm, inexperience and confidence of young age. It was meant to be an
honest presentation of the basic elements of differential geometry used in global analysis.
By "honest" presentation I understood that I should include clear and detailed explanations
for many of the folklore results, examples and points of view that are harder to trace in the
literature and had helped in my research.
It was an exciting experience writing the first edition and I was rewarded for my effort it
in many ways. There was an immediate reward for the time spent immersed in the minutia
of many examples and proofs. The detailed understanding I achieved allowed me to push
my own research to new directions and in such a depth that I did not believe I was capable
of at that age.
There was a long term reward since it turns out that the useful facts and examples and
computations that were harder to trace are still useful and harder to trace, but now I can
always turn to this book for details. Personal bias aside, I keep a copy of my book on my
desk since I frequently need to look up something in it.
There is a personal reward, as an author. I have been receiving input and acknowledg-
ments from many readers from all the corners of the world. I have implemented all the
corrections and suggestions I have received. A book lives through its readers and appar-
ently the present one is still alive.
So what is new in the third edition? There are some obvious additions reflecting my
current research interests. There is the new Chapter 11 on spectral geometry leading to the
original results in Subsection 11.4.5. This is the first place where they appear in published
form.
Describing new results in a monograph rather than in a research journal has allowed me
to go to a level of detail and provide perspective that is not possible in a journal. In particu-
lar, I have added two new Subsections 4.2.5 and 5.2.3 on Riemannian geometry containing
facts that, surprisingly, are not familiar to many geometers. These play an important role
in Subsection 11.4.5.
There are less conspicuous changes. I have been using various parts of the book for
graduate courses I taught over the years. The new edition contains the useful feedback
I have received from my students. I have enhanced and cleaned many proofs and I have
added new examples. Of course I have corrected the typos pointed out to me by readers. It
is likely I have introduced new ones.
Notre Dame, 2019.
Contents
Preface vii
1. Manifolds 1
1.1 Preliminaries 1
1.1.1 Space and Coordinatization 1
1.1.2 The implicit function theorem 3
1.2 Smooth manifolds . 6
1.2.1 Basic definitions . . . . . . . . 6
1.2.2 Partitions of unity 9
1.2.3 Examples . 10
1 2 4 How many manifolds are there? 20
2 • Natural Constructions on Manifolds 23

2.1 The tangent bundle 23
2.1.1 Tangent spaces. • . . . . o f o f 23
2.1.2 The tangent bundle . . . . . . . 27
2.1.3 Transversality 29
2.1.4 Vector bundles 33
2.1.5 Some examples of vector bundles 37
2.2 A linear algebra interlude 41
2.2.1 Tensor products 41
2.2.2 Symmetric and skew-symmetric tensors 46
2.2.3 The "super" slang 53
2.2.4 Duality................ 56
2.2.5 Some complex linear algebra . . . . . 64
2.3 Tensor fields 67
2.3.1 Operations with vector bundles 67
2.3.2 Tensor fields 69
2.3.3 Fiber bundles . . . . . . . . . . 73
xiii
xiv Lectures on the Geometry of Manifolds
3. Calculus on Manifolds 79
3.1 The Lie d e r i v a t i v e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
3.1.1 Flows on manifolds 79
3.1.2 The Lie d e r i v a t i v e . . . . . . . . . . . . . . . . . . . . . . . . . 81
3.1.3 Examples . 86
3.2 Derivations of Q' (M) 88
3.2.1 The exterior derivative . . . . . . . . . . . . . . . . . . . . . . . 88
3.2.2 E x a m p l e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
3.3 Connections on vector bundles . . . . . . . . . . . . . . . . . . . . . . . 94
3.3.1 Covariant derivatives . . . . . . . . . . . . . . . . . . . . . . . . 94
3.3.2 Parallel t r a n s p o r t . . . . . . . . . . . . . . . . . . . 100
3.3.3 The curvature of a connection . . . . . . . . . . . . . 101
3.3.4 Holonomy....................... 105
3.3.5 The Bianchi identities . . . . . . . . . . . . . . . . . 108
3.3.6 Connections on tangent bundles . . . . . . . . . . . . 109
3.4 Integration on manifolds . . . . . . . . . . . . . . . . . . . . 111
3.4.1 Integration of 1-densities . . . . . . . . . . . . . . . 111
3.4.2 Orientability and integration of differential forms . . . 115
3.4.3 Stokes'formula 123
3.4.4 Representations and characters of compact Lie groups . . . . . . 127
3.4.5 Fiberedca1culus..........................133
4. Riemannian Geometry 139

4.1 Metric properties 139
4.1.1 Definitions and examples . . . . . . . . . . 139
4.1.2 The Levi-Civita connection 143
4. 1.3 The exponential map and normal coordinates 147
4. l .4 The length minimizing property of geodesics 150
4.1.5 Calculus on Riemann manifolds . . . . . . 155
4.2 The Riemann curvature 165
4.2.1 Definitions and properties . . . . . . . . . . . . 165
4.2.2 E x a m p l e s . . . . . . . . . . . . . . . . . . . . . 169
4.2.3 Ca1"tan's moving frame method . . . . . . . . . 172
4.2.4 The geometry of submanifolds . . . . . . . . . 175
4.2.5 Correlators and their geometry 182
4.2.6 The Gauss-Bonnet theorem for oriented surfaces 192
5. Elements of the Calculus of Variations 201

5.1 The least action principle 201
5.1.1 The 1-dimensional Euler-Lagrange equations 201
5.1.2 Noether's conservation principle 207
5.2 The variational theory of geodesics . . . 210
Contents xv
5.2.1 Variational formulae 211

5.2.2 Jacobi fields 214
5.2.3 The Hamilton-Jacobi equations 220
6. The Fundamental Group and Covering Spaces 227

6.1 The fundamental group . . ..228
6.1.1 Basic n o t i o n s . . . . . . . . . . . . . . ..228
6.1.2 Of categories and functors . . . ... o n .232
6.2 Covering Spaces . . . . . . . . . . . . . .. .. .. 233
6.2.1 Definitions and examples . . . . . . . ..233
6.2.2 Unique lifting property . . . . . . . . ..235
6.2.3 Homotopy lifting property . . . t o o . . .236
6.2.4 On the existence of lifts .. . . . . 237
6.2.5 The universal cover and the fundamental group ..239
7. Cohomology 241
7.1 DeRham c o h o m o l o g y . . . . . . . . . . . . . . . . . . . . . . . . . . 241
7.1.1 Speculations around the Poincaré lemma . . . . . . . 241
7.1.2 (tech vs. DeRham . . . . . . . . . . . 245
7.1.3 Very little homological algebra . . . . . . . . . . . . . . . 247
7. 1.4 Functorial properties of the DeRham cohomology . . 254
7.1.5 Some simple examples . . . . . . . . . . . . . . . . . . . . . . . 257
7.1.6 The Mayer-Vietoris principle . . 259
7.1.7 The Kenneth formula . . . . . . . . . .. . 262
7.2 The Poincaré duality . . . . . . . . . . . . . . . . 265
7.2.1 Cohomology with compact supports • . . 265
7.2.2 The Poincaré duality . . . . . . • 268
7.3 Intersection theory . . . . . . . . . . . . . . . • • 272
7.3.1 Cycles and their duals . . . . . • 272
7.3.2 Intersection theory . . . . . . . . . . . • • 277
7.3.3 The topological degree . . . . . . . . . • • 282
7.3.4 The Thom isomorphism theorem 284
7.3.5 Gauss-Bonnet revisited . . . . 287
7.4 Symmetry and topology . . 291
7.4.1 Symmetric spaces . . . . . . . . . . . 291
7.4.2 Symmetry and cosmology . . . . . . . . . . . . . . 294
7.4.3 The cosmology of compact Lie groups 298
7.4.4 Invariant forms on Grassmannians and Weyl's integral formula 299
7.4.5 The Poincaré polynomial of a complex Grassmannian 306
7.5 éechcohomology................ 312
7.5.1 Sheaves and presheaves . . . . 313
7.5.2 tech cosmology . . . . . . . 317
xvi Lectures on the Geometry of Manzfolds
8. Characteristic Classes 329

8.1 Chern-Weil theory 329
8.1.1 Connections in principal G-bundles 329
8.1.2 G-vector bundles . 335
8.1.3 Invariant polynomials 336
8. 1.4 The Chern-Weil Theory 339
8.2 Important examples 343
8.2.1 The invariants of the torus T" 343
8.2.2 Chern classes . 343
8.2.3 Pontryagin classes 346
8.2.4 The Euler class . 348
8.2.5 Universal classes . 351
8.3 Computing characteristic classes 357
8.3.1 Reductions 358
8.3.2 The Gauss-Bonnet-Chern theorem 363
9. Classical Integral Geometry 373

9.1 The integral geometry of real Grassmannians . ..n o n o o.373
f n o
9.1.1 C o - a r e a f o r m u l a e . . . . . . . . . . . . . . . . . . . . . . . . . . 373
9. 1.2 Invariant measures on linear Grassmannians . . . . . o f . . 386 O .
9. 1.3 Affine Grassmannians . 395

9.2 G a u s s - B o n n e t a g a i n ? ! ' ? . . . . . . . . . . . . . . . . . . . . . . . . . . . 398
9.2.1 The shape operator and the second fundamental form . . . . . . 398
9.2.2 The Gauss-Bonnet theorem for hypersurfaces of an Euclidean
space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401
9.2.3 Gauss-Bonnet theorem for domains in an Euclidean space . 405
9.3 Curvature m e a s u r e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409
9.3.1 Tame geometry . . . . . . . . . 409
9.3.2 Invariants of the orthogonal group . . . . . . . . . . . . . . . . 414
9.3.3 The tube formula and curvature measures . . . . . . . . . . . . 418
9.3.4 Tube formula => Gauss-Bonnet formula for arbitrary
submanifolds of an Euclidean space . . . . . . . . . . . . . . . 429
9.3.5 Curvature measures of domains in an Euclidean space 431
9.3.6 Crofton formulae for domains of an Euclidean space . 433
9.3.7 Crofton formulae for submanifolds of an Euclidean space 443
10. Elliptic Equations on Manifolds 451

10.1 Partial differential operators: algebraic aspects 451
10.1.1 Basic notions . 451
10.1.2 Examples . 457
10.1.3 Formal adjoints 459
10.2 Functional framework . 464
Contents xvii
10.2.1 Sobolev spaces in RN . . . . . . . . . . . . . 464

10.2.2 Embedding theorems: integrability properties. . . . 471
10.2.3 Embedding theorems: differentiability properties . 476
10.2.4 Functional spaces on manifolds . . . . . . 480
10.3 Elliptic partial differential operators: analytic aspects 484
10.3.1 Elliptic estimates in RN 485
10.3.2 Elliptic regularity 489
10.3.3 An application: prescribing the curvature of surfaces 494
10.4 Elliptic operators on compact manifolds 504
10.4.1 Fredholm theory . 504
10.4.2 Spectral t h e o r y . . . . . . . . . . . . . . . . . . . . 513
10.4.3 Hodge theory . 518
11. Spectral Geometry 523

11.1 Generalized functions and currents . . . . . . . . . . . . .. . 523
11.1.1 Generalized functions and operations withy them . .. o f 0 . . . 523
11.1.2 C u r r e n t s . . . . . . . . . . . . . . . . . . . . . 528
11.1.3 Temperate distributions and the Fourier transform . . 529
11.1.4 Linear differential equations with distributional data 531
11.2 Important families of generalized functions 536
11.2.1 Some classical generalized functions on the real axis . 536
11.2.2 Homogeneous generalized functions . . . . . . . . . 542
11.3 The wave equation 548
11.3.1 Fundamental solutions of the wave 548
11.3.2 The wave family 550
11.3.3 Local parametrices for the wave equation with variable
coefficients....................... 567
I I .4 Spectral geometry 574
l 1.4.1 The spectral function of the Laplacian on a compact manifold 574
11.4.2 Short time asymptotic for the wave kernel 580
11.4.3 Spectral function asymptotics . . . . . . . . . . . . . . . . . 588
11.4.4 Spectral estimates of smoothing operators 593
11.4.5 Spectral perestroika . . . . . . . . . . . . . . . . . . . . . . 598
12. Dirac Operators 611

12.1 The structure of Dirac operators . . . . 611
12.1.1 Basic definitions and examples 611
12.1.2 Clifford algebras 614
12.1.3 Clifford modules: the even case 617
12.1.4 Clifford modules: the odd case 622
12.1.5 A look ahead . 623
12.1.6 The spin group . 625
xviii Lectures on the Geometry of Manifolds
12.1.7 The complex spin group 633

12.1.8 Low dimensional examples . . 635
12.1.9 Dirac bundles 640
12.2 Fundamental examples 644
12.2.1 The Hodge-DeRham operator . 644
12.2.2 The Hodge-Dolbeault operator 648
12.2.3 The spin Dirac operator . . . . 654
12.2.4 The spine Dirac operator 660
Bibliog raphy 669
Index 675
Chapter 1
Manifolds
1.1 Preliminaries
1.1.1 Space and Coordinatization

Mathematics is a natural science with a special modus operandi. It replaces concrete natural
objects with mental abstractions which serve as intermediaries. One studies the properties
of these abstractions in the hope they reflect facts of life. So far, this approach proved to be
very productive.
The most visible natural object is the Space, the place where all things happen. The first
and most important mathematical abstraction is the notion of number. Loosely speaking,
the aim of this book is to illustrate how these two concepts, Space and Number, fit together.
It is safe to say that geometry as a rigorous science is a creation of ancient Greeks.
Euclid proposed a method of research that was later adopted by the entire mathematics.
We refer of course to the axiomatic method. He viewed the Space as a collection of points,
and he distinguished some basic objects in the space such as lines, planes etc. He then
postulated certain (natural) relations between them. All the other properties were derived
from these simple axioms.
Euclid's work is a masterpiece of mathematics, and it has produced many interesting
results, but it has its own limitations. For example, the most complicated shapes one could
reasonably study using this method are the conics and/or quadrics, and the Greeks certainly
did this. A major breakthrough in geometry was the discovery of coordinates by René
Descartes in the 17th century. Numbers were put to work in the study of Space.
Descartes' idea of producing what is now commonly referred to as Cartesian coordi-
nates is familiar to any undergraduate. These coordinates are obtained using a very special
method (in this case using three concurrent, pairwise perpendicular lines, each one en-
dowed with an orientation and a unit length standard. What is important here is that they
produced a one-to-one mapping
Euclidian Space -> R3 7 P I

> (;v(P)?y(Pl»2(P))-
We call such a process coordinatization. The corresponding map is called (in this case)
Cartesian system of coordinates. A line or a plane becomes via coordinatization an alge-
braic object, more precisely, an equation.
1
2 Lectures on the Geometry of Manzfolds
In general, any coordinatization replaces geometry by algebra and we get a two-way

correspondence
Study of Space < > Study of Equations.

The shift from geometry to numbers is beneficial to geometry as long as one has efficient
tools do deal with numbers and equations. Fortunately, about the same time with the in-
troduction of coordinates, Isaac Newton created the differential and integral calculus and
opened new horizons in the study of equations.
The Cartesian system of coordinates is by no means the unique, or the most useful coor-
dinatization. Concrete problems dictate other choices. For example, the polar coordinates
represent another coordinatization of (a piece of the plane) (see Figure 1.1).
Fig. 1.1 Polar coordinates.
P >(1~(p),9(p))@(0,oo) X ( or, fl-

This choice is related to the Cartesian choice by the well known formulae
at rosH y=rsin9. (1.1.1)
A remarkable feature of (l.1.l) is that ;L°(Pl and y(P) depend smoothly upon r(Pl and
@(p).
As science progressed, so did the notion of Space. One can think of Space as a config-
uration set, i.e., the collection of all possible states of a certain phenomenon. For example,
we know from the principles of Newtonian mechanics that the motion of a particle in the
ambient space can be completely described if we know the position and the velocity of the
particle at a given moment. The space associated with this problem consists of all pairs
(position, velocity) a particle can possibly have. We can coordinatize this space using six
Manifolds 3
functions: three of them will describe the position, and the other three of them will de-
scribe the velocity. We say the configuration space is 6-dimensional. We cannot visualize
this space, but it helps to think of it as an Euclidian space, only "roomier".
There are many ways to coordinatize the configuration space of a motion of a particle,
and for each choice of coordinates we get a different description of the motion. Clearly, all
these descriptions must "agree" in some sense, since they all reflect the same phenomenon.
In other words, these descriptions should be independent of coordinates. Differential ge-
ometry studies the objects which are independent of coordinates.
The coordinatization process had been used by people centuries before mathematicians
accepted it as a method. For example, sailors used it to travel from one point to another
on Earth. Each point has a latitude and a longitude that completely determines its position
on Earth. This coordinatization is not a global one. There exist four domains delimited
by the Equator and the Greenwich meridian, and each of them is then naturally coordina-
tized. Note that the points on the Equator or the Greenwich meridian admit two different
coordinatizations which are smoothly related.
The manifolds are precisely those spaces which can be piecewise coordinatized, with
smooth correspondence on overlaps, and the intention of this book is to introduce the reader
to the problems and the methods which arise in the study of manifolds. The next section
is a technical interlude. We will review the implicit function theorem which will be one of
the basic tools for detecting manifolds.
1.1.2 The implicit function theorem

We gather here, with only sketchy proofs, a collection of classical analytical facts. For
more details one can consult [34].
Let X and Y be two Banach spaces and denote by L ( X , Y) the space of bounded linear
operators X -> Y. For example, if X = R", Y = RT", then L ( X , Y) can be identified
with the space of m X n matrices with real entries. For any set S we will denote by k g the
identity map S -> S.
Definition 1.1.1. Let F U C X -> Y be a continuous function (U is an open subset of
X). The map F is said to be (Fréchet) differentiable at u 6 U if there exists T e L ( X , Y )
such that
lIF<~@+ h) - F(u@) - T h e y I <»(llhllx) as h -> 0,
i.e.,
l
IIHIIx num0
lim h) ' F(Uol - Telly = 0.
h,->0
Loosely speaking, a continuous function is differentiable at a point if, near that point, it
admits a "best approximation" by a linear map.
When F is differentiable at U0 6 U, the operator T in the above definition is uniquely
determined by
d . 1
Th i - F(u0 -|- t h ) nm - (F(u0 -|- th) - F(u0l)
dt t->0 t
We will use the notation T = DU0 F and we will call T the Fréchet derivative of F at U0 .
Assume that the map F : U -> Y is differentiable at each point u 6 U. Then F is said
to be of class C1, i f the map u +-> D'LLF 6 L ( X , Y) is continuous. F is said to be of class
C2 if u +-> D'LLF is of class C1. One can define inductively Ck and coo (or smooth) maps.
Example 1.1.2. Consider F : U C R" -> Run. Using Cartesian coordinates

QUO .CE
1
7 . .. ,$ n
in R" and
1 m
'LL 'LL 7 ° .
. ,U
in Run, we can think of F as a collection of m functions on U

1 1 l
'LL 'u,(;13,...,a3"), ,um U'771(;U17°°'7a:n)'
The map F is differentiable at a point p = (p1, . . . ,1>") 6 U if and only if the functions u'll
are differentiable at p in the usual sense of calculus. The Fréchet derivative of F at p is the
linear operator DpF : R" -> RM given by the Jacobian matrix
31/,1
(al 5':1:2 (al (p)
3(7u,1 um
7 7 31/,2 nu2 51/,2
D,,F in 3:151 (p) 31/32 (al 3515" (p)
3(a21 7 7
(p)
sum
31/32 (p)
sum
3113" (p) _
The map F is smooth if and only if the functions ) are smooth.
Exercise 1.1.3. (a) Let 'LL C LGR" ,WI denote the set of invertible n X n matrices. Show
that U is an open subset of L(]R"', Rn).
(b) Let F : 'LL -> U be defined as A -> A-1_ Show that DAF(H) = -A-1HA-1 for any
n X n matrix H .
(c) Show that the Fréchet derivative of the map det : L(]Rn*,R") -> R, A »-> det A, at
A = Ilia" 6 L(Rn*, Rn) is given by tr H, i.e.,
d
L 0
det(J1Rn + t H ) = t H , oH € L(]R",]R").
Theorem 1.1.4 (Inverse function theorem). Let X , Y be two Banach spaces, U C X

open and F : U C X -> Y a smooth map. If at a point U 0 6 U the derivative DU0 F E
L(X,Y) is invertible, then there exits an open neighborhood U1 of U 0 in U such that
F(U1) is an open neighborhood of Uo = F(u0) in Y and F : U1 -> F(U1) is bijective,
with smooth inverse.
The spirit of the theorem is very clear: the convertibility of the derivative DU0 F "propa-
gates" locally to F because DU0 F is a very good local approximation for F.
Manifolds 5
More formally, if we set T DuoF, then

F(u0 + h) = F(u0) + Th + ¢~(h),
where r(h) = o(llhll) as h -> 0. The theorem states that, for every 'u s u f i ciently close to
U0, the equation F('u.) = 'U has a unique solution 'LL = up -|- h, with h very small. To prove
the theorem one has to show that, for llv - 'U0 fly sufficiently small, the equation below
t)0+Th+v°(h) ='u
has a unique solution. We can rewrite the above equation as Th = 'U - U0 - r(h) or,
equivalently, as h = T"1('v - U0 - r(h)). This last equation is a fixed point problem that
can be approached successfully via the Banach fixed point theorem.
Theorem 1.1.5 (Implicit function theorem). Let X, Y , Z be Banach spaces, W C X ,

"//c Y open sets and
F :W X "If -> Z
a smooth map. Let (330, ygl 6 W X "//, and Z0 := F(:z;0, y0l. Set
$2 : "// -> Z, F2(Y) = FllimYl-
Assume that D90 F2 G L(Y, Z ) is invertible. Then there exist open neighborhoods U C W
of £170in X , V C 7/ of Yo in Y , and o smooth map G : U -> V such that the set S of
solution (as, y ) of the equation F(:c, y ) = Z0 which lie inside U X V can be identified with
the graph of G, i.e.,
{(a:,y)€UxV; F(w,y)=20}={(w,G(w)l G U x V ; a2€U}.

In pre-Bourbaki times, the classics regarded the coordinate y as a function of :u defined
implicitly by the equality F(a2, yl = Z0»
Proof. Consider the map

H:X X Y->X X Z, § = ( a : , y ) »-> (:c,F(;z3,y)).
The map H is a smooth map, and at £0 = (1U073/0) its derivative D€01-I : X X Y -> X X Z
has the block decomposition
D H=
so -D€oF1 D€0 F2
Above, DF1 (respectively DF2) denotes the derivative of ac »-> F(:L°, 3/ol (respectively the
derivative of y »-> F(x0, y)). The linear operator Do0 H is invertible, and its inverse has
the block decomposition
Hx 0
(D50H)
(D1oe)-1 o (D§0F1) (D 0F2)*
Thus, by the inverse function theorem, the equation (as, F(;u, y)) = (in, zo) has a unique
solution (M) = H-1(x, Zeal in a neighborhood of (5130, yo). It obviously satisfies 5: = as
and F(i;,g) = 20. Hence, the set {(x,y); F(ac,y) = z0} is locally the graph of ac +->
H-1 (ac, 20).
1.2 Smooth manifolds
1.2.1 Basic definitions
We now introduce the object which will be the main focus of this book, namely, the concept
of (smooth) manifold. It formalizes the general principles outlined in Subsection 1.1.1.
Definition 1.2.1. A smooth manifold of dimension m is a locally compact, second count-

able' Hausdorff space M together with the following collection of data (henceforth called
atlas or smooth structure) consisting of the following.
(a) An open cover {U'i}i@I of M,

(b) A collection of homeomorphisms
\Pi u -> \p(u-> C Run; i 6 I}

(called charts or local coordinates) such that, 'Pt (Up) is open in RM and, if UZ-O(/'j ¢ (Z),
then the transition map
\IJjo 'Pi 1 \P¢(U¢ M Url C RM \Ifjlu¢ O Url C RM
is smooth. (We say that the various Cha s are smoothly compatible, see Figure 1.2).
J
Ui J
'Vi W.
J
/
"Ii 'Vi-1
Run
Fig. 1.2 Transition maps.
l A second countable space is a topological space that admits a countable basis of open sets.
Manifolds 7
Remark 1.2.2. (a) Each chart 'Pi : Uz- -> RM can be viewed as a collection of m functions
( X 1 7 ° ° ° 7 a 3 M 7 on Up,
l 1-1
1,2
(p)
(p) 1
We (p)
(p)
at771
Similarly, we can view another chart 'PJ as another collection of functions (y1, - - JW)-
•
The transition map 'Pa o 'Pi l can then be interpreted as a collection of maps
(x1,...,;u"") +-> l3/1(a31,...,a:""),...,yM(ac1,...,a:M)l.
(b) Since a manifold is a second countable space we can always work with atlases that are
at most countable.
The first and the most important example of manifold is R" itself. The natural smooth
structure consists of an atlas with a single chart, Ilia" : R" -> R". To construct more
examples we will use the implicit function theorem.
Definition 1.2.3. (a) Let M, N be two smooth manifolds of dimensions m and respectively
n. A continuous map f : M -> N is said to be smooth if, for any local charts QS on M
w w
and on N, the composition o f o (V1 (whenever this makes sense) is a smooth map
Run -> Rn_
(b) A smooth map f : M -> N is called a dyffeomorphism if it is invertible and its inverse
is also a smooth map.
Example 1.2.4. The map t »-> et is a diffeomorphism ( -oo, oo) -> (0,oo). The map
t +-> t3 is a homeomorphism R -> R, but it is not a diffeomorphism!
If M is a smooth m-dimensional manifold, we will denote by cOO (M) the linear space
of all smooth functions f : M -> R. Let us point out a simple procedure that we will use
frequently in the sequel. Suppose that f : M -> R is a smooth function. If (U, \If) is a
local chart on M so \P(Ul is an open subset of Run, then, by definition, the composition
f o \I/-1 : \P(Ul -> R is a smooth function on the open set \If(U) C R". If we denote
w
by x1, . . . ,513771 the canonical Euclidean coordinates on Run, then o \I/-1 is a function
depending on the m variables .CU1 . . m and we will use the notation f lx, . . . ,LU m
7 '
when referring to this function.
Remark 1.2.5. Let U be an open subset of the smooth manifold M (dim M m ) and
\If:U-HRM
a smooth, one-to-one map with open image and smooth inverse. Then \If defines local
coordinates over U compatible with the existing atlas of M. Thus (U, \I/) can be added to
the original atlas and the new smooth structure is diffeomorphic with the initial one. Using
Zermelo's Axiom we can produce a maximal atlas (no more compatible local chart can be
added to it).
Our next result is a general recipe for producing manifolds. Historically, this is how
manifolds entered mathematics.
Proposition 1.2.6. Let M be a smooth manzfold of dimension m and f l , 7 fk 6

cOO ( M ) . Define
Z Z»(f1, 7 fol {ppM; MP) = =f1=(p) =0}-

Assume that the functions fl, . . . , fk are functionally independent along Z, i.e., for each
p 6 Z, there exist local coordinates (331, . . . I dejined in a neighborhood of p in M
such that so (p) = 0, i = 1, . . . ,m, and the matrix
6f1 $f1 3f1
3x1 3x2 hmm
. a
95;
p ask 6 f k ask
3x1 3x2 hmm :n 1 :1;'"=0
has rank k. Then Z has a natural structure of smooth manifold of dimension m - k.
Proof. Step 1: Constructing the charts. Let Po E Z., and denote by ( X 1 7 ' ° ° 7 X M 7 local
coordinates near Po such that x'll (pg) = 0. One of the A: X k minors of the matrix
fifl
hmm
..
3:5 3fk
p
31/31 31/32 hmm 11:1 :nm 0
is nonzero. Assume this minor is determined by the last k columns (and all the k lines).
We can think of the functions fl, . . . , fk as defined on an open subset U of Run. Split
RM as R m - k X Rk, and set
al 1
(:z3,...,a:M -k
),
ml/
.. ("-'°+1, . . . , mM).
We are now in the setting of the implicit function theorem with
x=Rm-'" 7 y=1Re.'" 7 z=R'* 7
and F X X Y - > Z given by

f1l£ul
an-> 6 R/*
fk(x))
OF
In this case, DF2 = ( 5933" ) : Re' -> Rk is invertible since its determinant corresponds to
our nonzero minor. Thus, in a product neighborhood UP0 = UPo X U" Po
of Po, the set Z, is
the graph of some function
g _
. UPo) C RM- k >U/I
Po
CRk*,
i.e.,
Z O UPo (;u/,g(xI) ) 6 Rm-k X Re. co' 67 UPo) 7 at / small .
Manifolds 9
We now define MY I Z FW UP() -> ]RM"* by
(22', 9(11') ) +¢P0> 513I E R""-k.
The map 1% is a local chart of Z near Po.
Step 2. The transition maps for the charts constructed above are smooth. The details are
left to the reader.
Exercise 1.2.7. Complete Step 2 in the proof of Proposition 1.2.6.
Definition 1.2.8. Let M be a m-dimensional manifold. A codimension k submamfold

of M is a subset S C M locally defined as the common zero locus of k functionally
independent functions fl, . . . , fk 6 COO(M). More precisely, this means that, for any
Po 6 S, there exists an open neighborhood U of Po e M and k functionally independent
smooth functions
f17...7fk¢2U- R
such that p 6 U m S if and only if fl (p) fA¢(Pl=0-
Proposition 1.2.6 shows that any submanifold N C M has a natural smooth structure
so it becomes a manifold per se. Moreover, the inclusion map z' : N <-> M is smooth.
Exercise 1.2.9. Suppose that M is a smooth m-dimensional manifold. Prive that S C M is

a codimension k-submanifold of M if and only if, for any Po 6 M, there exists a coordinate
chart (U, \I/) with local coordinates (x1, . . . ,arM) such that Po 6 U and
urns { p a U ; 1v1(p) 1vI"(p) 0 }~
1.2.2 Partitions of unity
This is a very brief technical subsection describing a trick we will extensively use in this
book. Recall that manifolds are locally compact, second countable topological spaces. As
such, they are paracompact so they admit continuous partitions of unity, see [30, §3.7]. A
much more precise result is in fact true.
Definition 1.2.10. Let M be a smooth manifold and (UJQQA an open cover of M. A

(smooth) partition of unity subordinated to this cover is a family (fB)Be8 C COOIMI
satisfying the following conditions.
(i) 0 S f B S 1.
(ii) EIQS : 8 -> A such that supp fa C U¢(/3)-
(iii) The family (supp f ii) is locally finite, i.e., any point as 6 M admits an open neighbor-
hood intersecting only finitely many of the supports supp fa .
(iv) EB f g l i ) = 1 for allay 6 M .
10 Lectures on the Geometry of Manifolds
We include here for the reader's convenience the basic existence result concerning par-
titions of unity. For a proof we refer to [128].
Proposition 1.2.11. (a) For any open cover U = ( U ) o € A of a smooth manifold M there
exists at least one smooth partition of unity IJBIBQ8 subordinated to U such that supp f B
is compact for any B.
(b) If we do not require compact supports, then we can _lind a partition of unity in which
'B = A and QUO = HA.
Exercise 1.2.12. Let M be a smooth manifold and S C M a closed submanifold, i.e., S is

a closed subset of M. Prove that the restriction map
7:Croo(M)->CYOO(S) f *->fls
is surjective. Deduce that for any finite set X C M and any function g : F -> R there exists
a smooth, compactly supported function f : M -> R such that f(:z:) = glatl, Van 6 X .
1.2.3 Examples
Manifolds are everywhere, and in fact, to many physical phenomena which can be modelled
mathematically one can naturally associate a manifold. On the other hand, many problems
in mathematics find their most natural presentation using the language of manifolds. To
give the reader an idea of the scope and extent of modern geometry, we present here a she
list of examples of manifolds. This list will be enlarged as we enter deeper into the study
of manifolds.
Example 1.2.13. (The n-dimensional sphere) This is the codimension l submanifold of

Rn+l given by the equation
n
I I
:u 2
(W T'
2
, at
0
.CU 7 7 so'n) 6 R'n+l
'i 0
One checks that, along the sphere, the differential of laval2 is nowhere zero, so by Proposi-
tion 1.2.6, S" is indeed a smooth manifold. In this case one can explicitly construct an atlas
(consisting of two charts) which is useful in many applications. The construction relies on
stereographic projections.
Let N and S denote the North and resp. South pole of S" (N = (0, . . . ,0, l) 6 lR"+1,
S = (0, . . . ,0, -1) 6 R"+1). Consider the open sets Un = S"\{N} and Us = S"\{S}.
They form an open cover of S". The stereographic projection from the North pole is the
map uN : Un -> R" such that, for any P 6 Un, the point oN (P) is the intersection of the
line NP with the hyperplane {;u" = 0} 2 R" .
The stereographic projection from the South pole is defined similarly.
.. For P Q U n we denote by lY1lPl)

For Q GUs we denote by (zwQ),
7 y"(P) ) the coordinates of uN (P).
z"(Q) ) the coordinates of us (Q).
7
Manifolds 11
A simple argument shows the map
(y1(p),...,y"(p)) »-> (21(P)7...72"(Pl)7 P G Un VW Us,

is smooth (see the exercise below). Hence {(UN, oN), (US, 0g)} defines a smooth struc-
tune on S".
Exercise 1.2.14. Show that the functions y Zi constructed in the above example satisfy
ni
Zi
y vi = 1, ,n.
lE]=1lY]l2
Example 1.2.15. (The n-dimensional torus) This is the codimension n submanifold of the
vector space R211 with Cartesian coordinates (al , y1, , can, yn), defined by the equalities
2 2 2
£I31+?/1 Mn 2
-l-yn = 1.
Note that T1 is diffeomorphic with the 1-dimensional sphere Sl (unit circle). As a set T"
is a direct product of n circles (see Figure 1.3)
T"={=v21+yI=1} X {:u2 2
Yn 1} S1 X X 51.
I I
Fig. 1.3 The 2-dimensional torus.
The above example suggests the following general construction.
Example 1.2.16. Let M and N be smooth manifolds of dimension m and respectively

n. Then their topological direct product has a natural structure of smooth manifold of
dimension m - i - n .
Example 1.2.17. (The connected sum of two manifolds) Let M1 and M2 be two manifolds
of the same dimension m. Pick Pi 6 Mt (i = 1, 2), choose small open neighborhoods Up
of Pt in M and then local charts QM identifying each of these neighborhoods with 8220),
the ball of radius 2 in RM.
Let Vi C Up correspond (via ) to the annulus {1/2 < lac°| < 2} C Run. Consider
at
go: {1/2<lxl <2}->{l/2<l33l <2}, ( )
I II2
The action of <l> is clear: it switches the two boundary components of {1/2 < laval < 2},
and reverses the orientation of the radial directions.
Now "glue" V1 to VI using the "prescription" given by $2 1 o QUO o zp1 : V1 -> VI. In
this way we obtain a new topological space with a natural smooth structure induced by
the smooth structures on Mi. Up to a diffeomeorphism, the new manifold thus obtained is
independent of the choices of local coordinates ([24]), and it is called the connected sum of
M1 and M2 and is denoted by M1#M2 (see Figure 1.4).
V V2
1
l
\
I
I
I
WI
'T
l l
l l
I
I1 I
Fig. 1.4 Connected sum of tori.
Example 1.2.18. (The real projective space ]RIP'") As a topological space RIP" is the quo-
tient of Rn-I-1 modulo the equivalence relation
as y ~:de6> EA E R* : at Ay.
The equivalence class of as = (xO, . . . , in 6 R \ {0} is usually denoted by

ac0,...,;z:" . Alternatively, RIP" is the set of all lines (directions) in lR"+1. Tradition-
ally, one attaches a point to each direction in ]R"+1, the so called "point at infinity" along
that direction, so that RIP" can be thought as the collection of all points at infinity along all
the directions in lRTL+1.
The space RIP" has a natural structure of smooth manifold. To describe it consider the
sets
Up: [ a c 0 , . . . , a 3 " ] € R P " ; m k ¢ 0}, If 0, ,n.
Now define
0 wk: k
] » - > (az / 7 . ° 7 ; U k
-1 /
'l,Dk :Up->Rn [110 as
, ...,x"
Manifolds 13
The maps wk define local coordinates on the projective space. The transition map on the
overlap region Up O Um = {[;z:0 . . . , asn] , ;z:k;u"" 96 0} can be easily described. Set
7
¢k([£U07 . . . ,we
_ (€1,..-,€nl7 ¢m(llv0 7 ° ° ° 7 ; U n D = (m,---n77)-
The equality
[¢v°». • ' ,iv"] I [§1, ' • • 7g/<7-17 l a , 7 in] [7117 7 77771-17 17 U m ) 7 Un1
immediately implies (assume Is < m)
61 = u1/Dk, 7 Uk-1/Uk, §k+1 Up

fk = U k + 1 / U k , 7 17m-1/nk, €m-1 I 1/*71<: (1.2.1)
am I 77'rn77ka 7 Un/Wk
This shows the map Wk, o ism-1 is smooth and proves that RIP" is a smooth manifold. Note
that when n = 1, RIP1 is diffeomorphic with Sl. One way to see this is to observe that
the projective space can be alternatively described as the quotient space of S" modulo the
equivalence relation which identifies antipodal points.
Example 1.2.19. (The complex projective space ClP'") The definition is formally identical
to that of RIP". CIP" is the quotient space of C"--1 \ {0} modulo the equivalence relation
def
at f\J
y '\
E1/\€lC*: at Ay.
The open sets Up are defined similarly and so are the local charts Wk : Up -> CC". They
satisfy transition rules similar to (1.2.l) so that CIP" is a smooth manifold of dimension 271.
Exercise 1.2.20. Prove that (CIP1 is diffeomorphic to S2 .
In the above example we encountered a special (and very pleasant) situation: the gluing
maps not only are smooth, they are also holomorphic as maps W o ism - 1 : U -> V where
U and V are open sets in lcn This type of gluing induces a "rigidity" in the underlying
manifold and it is worth distinguishing this situation.
Definition 1.2.21 (Complex manifolds). A complex manifold is a smooth, 2n-

dimensional manifold M which admits an atlas UmM : Up -> @"} such that all transi-
tion maps are holomorphic.
The complex projective space is a complex manifold. Our next example naturally general-
izes the projective spaces described above.
Example 1.2.22. (The real and complex Grassmannians Gr 1<(R"l» Gr k(<Q").)

Suppose V is a real vector space of dimension n. For every 0 < lc < n we denote by
Gr k(Vl the set of Is-dimensional vector subspaces of V. We will say that Grk(V) is the
linear Grassmannian of It-planes in E. When V Rn we will write Grk (R) instead of
Gr a l l n ) .
We would like to give several equivalent descriptions of the natural structure of smooth
manifold on Grk (V). To do this it is very convenient to fix an Euclidean metric on V.
Any k-dimensional subspace L C V is uniquely determined by the orthogonal projec-
tion onto L which we will denote by PL. Thus we can identify Gr MV) with the set of
rank k projectors
ProjkM {p= V - > V ; P* 1}.
Q_*
.. P2 rankP
II
II
7
Let use observe that the rank of an orthogonal projector is determined by the equality
rank P tr P.
We have a natural map
P Gr k(V ) -> Projk(vl, L I-) PL
with inverse P *-> Range (P).
The set Proj k l ) is a subset of the vector space of symmetric endomorphisms
End+(v) == {A E EndlVl, A* = A }.
The space End+ (V) is equipped with a natural inner product
( B ) := 5l tr(AB), vA, B E End-*(V). (1.2.2)
We denote by II II the norm on End+ (V) induced by this inner product,

l
11A112 = 5 ttlA2l.
Note that the subset Proj J V ) C End+ (V) can alternatively be described by the equalities
P2 = p, t r P = l .
This proves that Projm / ) is a closed subset of End+ (V). From the equality
1 k
lIPII 2 - t I P 2 = § t r P = §» 'v'P G Projm / )
2
we deduce that Projm / ) is also bounded subset of End+ (V). The bijection
P : Gr k l ) -> Projk(V), L »-> PL
induces a topology on Grk (V), and with this topology Grk (V) is a compact metric space.
We want to show that Grk,(V) has a natural structure of smooth manifold compatible
with this topology. To see this, we define for every L C Gr k(V) the set
Gr : = { U € G r MV); U M L J ' = 0 } .
Lemma 1.2.23. (a) Let L G Gr /K7(V)- Then

U F) L; = 0 ~< >» II - PL -|- PU : V -> V is an isomorphism. (1.2.3)
(b) The set Grk(V, L ) is an open subset ofGrk(V).

Manifolds 15
Proof. (a) Note first that a dimension count implies that

UmL_L i 0 `\I \, P
U+Li i V `\I \, P
U re L 0.
Let us show that U O L_L = 0 implies that ]1 - PL -|- PL is an isomorphism. It suffices to
show that
Ker(]l - PL -|- Pu) = 0.
Suppose 'u E ker(]1 - PL -|- PU). Then
0=pL(11-pL 'l'PU)'U=PLPU'U=0--)>PU'U Q Urwke1~pL =Ural = 0 .

Hence PU?) = 0, so thats E U I . From the equality (11 - pa - Pu)V 0 we also deduce
(II - PL)V = 0 so thats E L. Hence
uQU1'-LM L 0.
Conversely, we will show that if II - PL + PU = PL_L + PU is onto, then U -|- LJ- = V.

Indeed, let 'u E V. Then there exists :IJ E V such that
FU=PLX+Pua;eLl+U.
(b) We have to show that, for every K 6 Grk(V, L), there exists 6 > 0 such that any U
satisfying
lIP - PK I < e
intersects LJ- trivially. Since K e Gr k(V, L) we deduce from (a) that the map It - PL
PK : V -> V is an isomorphism. Note that
I (11 PL PK) - (11 - PL - PUlll = lInK - PU||-

The space of isomorphisms of V is an open subset of End(Vl. Hence there exists 6 > 0
such that, for any subspace U satisfying I PU - PK I < 5, the endomorphism (it - PL - PU )
is an isomorphism. We now conclude using pa (a).
Since L 6 Gr k l , L l , v L @ Gr h e ) , the collection
Gr k(V§Ll; L Q Gr I<(V)
is an open cover of G1°k1J(V). Note that for every L E Gr k l v ) we have a natural map
F :Hom(L, Ll) -> Gr k(V, L), (1 .2.4)
that associates to each linear map S : L -> L-L its graph (see Figure 1.5)
Fs = {;u-i-S566 L + Ll V; $ 6 L}.
The map (1.2.4) is obviously injective. We claim that it is in fact a bijection.

Indeed, if U E Gr k:(V, L), then the restriction of PL to U is injective since
Unke1~pL = U n L l = 0 .
LJ_
v
vL
L 1 rS
\
\
Sx "
I
I
I
I
v x L
L
Fig. 1.5 Subspaces as graphs o f linear operators.
Thus, the linear map PLlu : U -> L, U 9 Up »-> PLU is a linear isomorphism because
dim U dimL
i k. Denote by HU its inverse, HU : L -> U. It is easy to see that U is
Z
the graph of the linear map

5 : L -> L l , Sac = PLiHUX.
We will show that the bijection (1.2.4) is a homeomorphism. We first prove that it is
continuous by providing an explicit description of the orthogonal projection PpS
Observe first that the orthogonal complement of FS is the graph of -S* : L-L -> L.
More precisely,
1"SJ.=1*_S*={y-s*yeL_L+L=v; y Q L - L } .
Let ' U = P L ' U + P L J . ' U = ' U L - l - U L + E V (see Figure 1.5). Then
'U
P1*S'U=LU+SCC, :UE L \
(;z:+Sa9l€II;9§
/
\
\.
I §1xEL, Z / € L J - such that

at + S*y UL
Sac y 'UL_L
Consider the operator S : L ® LJ- -> L ® L_L which has the block decomposition
Then the above linear system can be rewritten as
Now observe that

8.
H
:u
y
'UL
'UL-L ]
S2 ]lL+S*S 0
0 HL +SS*
Hence S is invertible, and
S-12 (]lL-I-S*S) 0
S
0 l:l]_L_l_ -l-SS*)'1
Manifolds 17
l]l L -I-S*S)'1 (HL + S*S)'1S*

u +ss*)-1s -(EL -|- SS*l-1
We deduce
QUO l]lL + S*S)-1'UL -|- (HL -|- S*Sl-1S*'ULJ_
and,
QUO
Hence PpS has the block decomposition

Pos?)
[ ] S33
IlL
Pos
S ] [lm + s*s)-1 (aL + s * 5 > - l s *
l]l L -I-S*Sl-1 (aL -|- S*S)'1S*

(l.2.5)
so + 5*s) S(J1L +S *s) -15 *
This proves that the map
Hom(L, Ll) 9 S »-> Pos E Grklv, L)
is continuous. Note that if U 6 Grk(V, L), then with respect to the decomposition V
L + L-L the projector PU has the block form
PLPuIL PLPUIL
PU
A
C
B
D
PLJ.PUIL PLLPUILJ.
where for every subspace K <-> V we denoted by IK : K -> V the canonical inclusion,
] 7
then U = Fs. If U = Fs, then (l.2.5) implies

PLPuIL A(U) = (HL +5* 5>- 1 0(U) = S(11L + s*5> -1 7
7 PLPUILJ.
so S = C A 1 Since A and C depend continuously on PU we deduce shows that the
inverse of the graph map
H o m ( L , L l 9 S »-> FS 6 Gr 1<(Vl
is also continuous. Moreover, the above formulae show that if U E Gr k(V, Low
Gr k (V, Ll), then we can represent U in two ways,
U = FSO = FS1, Si G H0>mlL1,Ll), i = 0, 1,
and the correspondence So -> $1 is smooth. This shows that Grk(V) has a natural struc-
ture of smooth manifold of dimension
dim Grklvl = dim Hom(L, Ll) k ( n - k).
Gr k(@") is defined as the space of complex k-dimensional subspaces of Q". It can be
structured as above as a smooth manifold of dimension 2k(n - k). Note that Gp1 (RTLI =
~
~
RP7n-1, and Gp1 (ml = <clpn-1. The Grassmannians have important applications in many
classification problems.
Exercise 1.2.24. Show that Gr llCTL) is a complex manifold of complex dimension k(n
1<)).
Example 1.2.25. (Lie groups). A Lie group is a smooth manifold G together with a group
structure on it such that the map
G X G->G (9,I2)>9-1V1
is smooth. These structures provide an excellent way to formalize the notion of symmetry.
(a) (Rn, -I-) is a commutative Lie group.
(b) The unit circle S1 can be alternatively described as the set of complex numbers of
norm one and the complex multiplication defines a Lie group structure on it. This is a
commutative group. More generally, the torus T" is a Lie group as a direct product of n
circles.2
(c) The general linear group GL(n, K) defined as the group of invertible n X n matrices
with entries in the field K = R, (C is a Lie group. Indeed, GL(n, K) is an open subset (see
Exercise l.1.3) in the linear space of n X n matrices with entries in K. It has dimension
den2, where do is the dimension of K as a linear space over R. We will often use the
alternate notation GL(K'") when referring to GL(n, K).
(d) The orthogonal group O(n) is the group of real n X n matrices satisfying
T.Tt Jl.
To describe its smooth structure we will use the Cayley transform trick as in [113] (see also
the classical [133]). Set
m,,(1R()# {T E MAR); det(l +Tl ¢ 0 }.

o f
The matrices in m,,(1R()# are called non exceptional. Clearly II E O(n)# = O(n) VI
Mn (lR)# so that O(n)# is a nonempty open subset of O(nl. The Cayley transform is the
map # : My(Rl# -> m(02a) defined by
A »-> A# <11 A)(]1 + A )
The Cayley transform has some very nice properties.
(i) A# E m,,(nl# for every A E Mn(R)#.
(ii) # is involutive, i.e., ( A # l # = A for any A E m,,(n)#.
(iii) For every T 6 O(n)# the matrix T# is skew-symmetric, and conversely, if A E
m,,(n)# is skew-symmetric then A# E O(n).
Thus the Cayley transform is a homeomorphism from O(n)# to the space of non-
exceptional, skew-symmetric, matrices. The latter space is an open subset in the linear
space of real n X n skew-symmetric matrices, Q(n).
Any T 6 O(n) defines a self-homeomorphism of O(n) by left translation in the group
LT : O(n) -> O(n) S »-> LT(S) = T S. -
20ne can show that any connected commutative Lie group has the from Tn X R "
Manifolds 19
We obtain an open cover of O(n):
O(n) = U T - O(n)#.
T€O(n)
Define QT : T - O ( n ) # -) Q(n) by S »-> (T-1 - S)#. One can show that the collection
IT . o<n>#,@T ) T€0(n)
defines a smooth structure on O(n). In particular, we deduce
dim O(n) n(n - 1)/2.
Inside O(n) lies a normal subgroup (the special orthogonal group)
SO(n) = {T 6 O(n) det T = 1}.
The group SO(n) is a Lie group as well and dim SO(n) = dim O(n).
(e) The unitary group U (in) is defined as
U(n) = {T 6 GL (n,(Cl; T - T * = ] l } ,
where T* denotes the conjugate transpose (adjoint) of T. To prove that U (n) is a manifold
one uses again the Cayley transform trick. This time, we coordinatize the group using the
space u(n) of skew-adjoint (skew-Hermitian) n X n complex matrices (A = -A*). Thus
U (n) is a smooth manifold of dimension
dim U(n) = dim_u(n) n2
Inside U(n) sits the normal subgroup SU(n), the kernel of the group homomorphism det :
U(n) -> S1. SU(nl is also called the special unitary group. This a smooth manifold of
dimension 112 - 1. In fact the Cayley transform trick allows one to coordinatize SU(n)
using the space
W) {A E _u(nl , teA 0}.
Exercise 1.2.26. (a) Prove the properties (i)-(iii) of the Cayley transform, and then show
-
that (T 0(n)#, \ P T ) T € 0 ( n ) defines a smooth structure on O(nl.
(b) Prove that U(n) and SU(n) are manifolds.
(c) Show that O(n), SO(n), U(n), SU(n) are compact spaces.
(d) Prove that SU(2) is diffeomorphic with S3. (Hint: think of S3 as the group of unit
quaternions.)
Exercise 1.2.27. Let SL(n; K) denote the group of n X n matrices of determinant 1 with
entries in the field K = K, C Use the implicit function theorem to show that SL(n; K) is
a smooth manifold of dimension dK(77,2 - 1), where did = dim K.
Exercise 1.2.28. (Quillen). Suppose Vo, V1 are two real, finite dimensional Euclidean
space, and T : V0 -> V1 is a linear map. We denote by T* is adjoint, T* : V1 -> Vo,
and by FT the graph of T,
FT = { ( U 0 7 U1l E V0 ® V13 'U1 I T'U0)}.
We form the skew-symmetric operator

0 T*
[ ] [ N ]
00 U0
XzV0QEV1->V0@BV1, X
U1 T 0 U1
We denote by CT the Cayley transform of X ,

C'T=(]1-Xl(]l-i-X) 7
and by R0:V0€BV1->V0€BV1 the reflection
Ro
Show that RT = CTRL is an orthogonal involution, i.e.,

R§~=1, R T = R T ,
and ker(]1 - RT) FT. In other words, RT is the orthogonal reflection in the subspace
FT,
RT = 2PPT 11,
where PPT denotes the orthogonal projection onto FT.
Exercise 1.2.29. Suppose G is a Lie group, and H is an abstract subgroup of G. Prove that
the closure of H is also a subgroup of G.
Exercise 1.2.30. (a) Let G be a connected Lie group and denote by U a neighborhood of
1 6 G. If H is the subgroup algebraically generated by U show that H is dense in G.
(b) Let G be a compact Lie group and g 6 G. Show that l 6 G lies in the closure of
{9" ; n G Z \ {0}}-
Remark 1.2.31. If G is a Lie group, and H is a closed subgroup of G, then H is in fact a
smooth submanifold of G, and with respect to this smooth structure H is a Lie group. For a
proof we refer to [61, 128]. In view of Exercise 1.2.29, this fact allows us to produce many
examples of Lie groups.
1.2.4 How many manifolds are there?

The list of examples in the previous subsection can go on for ever, so one may ask whether
there is any coherent way to organize the collection of all possible manifolds. This is too
general a question to expect a clear cut answer. We have to be more specific. For example,
we can ask
Manifolds 21
Question 1: Which are the compact, connected manifolds of a given dimension d?

For d = 1 the answer is very simple: the only compact connected 1-dimensional mani-
fold is the circle S1. (Can you prove this?)
We can raise the stakes and try the same problem for d = 2. Already the situation
is more elaborate. We know at least two surfaces: the sphere S2 and the torus T2. They
clearly look different but we have not yet proved rigorously that they are indeed not diffeo-
morphic. This is not the end of the story. We can connect sum two tori, three tori or any
number g of tori. We obtain doughnut-shaped surface as in Figure 1.6
Fig. 1.6 Connected sum o f 3 tori.
Again we face the same question: do we get non-diffeomorphic surfaces for different
choices of g? Figure 1.6 suggests that this may be the case but this is no rigorous argument.
We know another example of compact surface, the projective plane RIP2, and we nat-
urally ask whether it looks like one of the surfaces constructed above. Unfortunately, we
cannot visualize the real projective plane (one can prove rigorously it does not have enough
room to exist inside our 3-dimensional Universe). We have to decide this question using a
little more than the raw geometric intuition provided by a picture. To kill the suspense, we
mention that RIP2 does not belong to the family of donuts. One reason is that, for example,
a torus has two faces: an inside face and an outside face (think of a car rubber tube). RP2
has a weird behavior: it has "no inside" and "no outside". It has only one side! One says
the torus is orientable while the projective plane is not.
We can now connect sum any numbers of lRlPD2's to any donut an thus obtain more and
more surfaces, which we cannot visualize and we have yet no idea if they are pairwise
distinct. A classical result in topology says that all compact surfaces can be obtained in
this way (see [90]), but in the above list some manifolds are diffeomorphic, and we have
to describe which. In dimension 3 things are not yet settled3 and, to make things look
hopeless, in dimension > _ 4 Question 1 is algorithmically undecidable .
We can reconsider our goals, and look for all the manifolds with a given property X . In
many instances one can give fairly accurate answers. Property X may refer to more than
the (differential) topology of a manifold. Real life situations suggest the study of manifolds
with additional structure. The following problem may give the reader a taste of the types
of problems we will be concerned with in this book.
3 Things are still not settled in 2007, but there has been considerable progress due to G. Perelman's proof of the
Poincaré's and Thursron's conjectures.
Question 2: Can we wrap a planar piece of canvas around a metal sphere in a one-to-one
fashion? (The canvas is flexible but not elastic).
A simple do-it-yourself experiment is enough to convince anyone that this is not pos-
sible. Naturally, one asks for a rigorous explanation of what goes wrong. The best ex-
planation of this phenomenon is contained in the celebrated Theorem Egregium (Golden
Theorem) of Gauss. Canvas surfaces have additional structure (they are made of a special
material), and for such objects there is a rigorous way to measure "how curved" are they.
One then realizes that the problem in Question 2 is impossible, since a (canvas) sphere is
curved in a different way than a plane canvas.
There are many other structures Nature forced us into studying them, but they may not
be so easily described in elementary terms.
A word to the reader. The next two chapters are probably the most arid in geometry but,
keep in mind that, behind each construction lies a natural motivation and, even if we do
not always have the time to show it to the reader, it is there, and it may take a while to
reveal itself. Most of the constructions the reader will have to "endure" in the next two
chapters constitute not just some difficult to "swallow" formalism, but the basic language
of geometry. It might comfort the reader during this less than glamorous journey to carry
in the back of his mind Hermann Weyl's elegantly phrased advise.
"It is certainly regrettable that we have to enter into the purely formal aspect in such detail
and to give it so much space but, nevertheless, it cannot be avoided. Just as anyone who
wishes to give expressions to his thoughts with ease must spend laborious hours learning
language and writing, so here too the only way we can lessen the burden of formulae is to
master the technique of tensor analysis to such a degree that we can turn to real problems
that concern us without feeling any encumbrance, our object being to get an insight into
the nature of space [...]. Whoever sets out in quest of these goals must possess a perfect
mathematical equipment from the outset."
H. Weyl: Space, Time, Mattel:
Chapter 2
Natural Constructions on Manifolds
The goal of this chapter is to introduce the basic terminology used in differential geometry.
The key concept is that of tangent space at a point which is a first order approximation of
the manifold near that point. We will be able to transport many notions in linear analysis
to manifolds via the tangent space.
2.1 The tangent bundle
2.1.1 Tangent spaces

We begin with a simple example which will serve as a motivation for the abstract defini-
dons
Example 2.1.1. Consider the sphere 2
S2) 2 + Z 2 l in RB.
( I +
We want to find the plane passing through the North pole N (0, 0, 1) that is "closest" to the
sphere. The classics would refer to such a plane as an oscillator plane.
The natural candidate for this oscillator plane would be a plane given by a linear equa-
tion that best approximates the defining equation 3132 +3/2 -l- Z2 = 1 in a neighborhood of the
North pole. The linear approximation of 232 -1- y2 -l- Z2 near N seems like the best candidate.
We have
1v2+y2+22 l 2(z 1) + O(2),
where 0(2) denotes a quadratic error. Hence, the oscillator plane is Z = 1. Geometrically,
it is the horizontal amine plane through the North pole. The linear subspace {z = 0} C $3
is called the tangent space to S2 at N.
The above construction has one deficiency: it is not intrinsic, i.e., it relies on objects
"outside" the manifold S2. There is one natural way to fix this problem. Look at a smooth
path 'y(t) on S2 passing through N at t = 0. Hence, t »-> 'y(t) 6 R3, and
Iv(t)l 2 = 1. (2.1.1)
If we differentiate (2.1.1) at t = 0 we get 07(0), 'y(0)) = 0, i.e., (0) J_ 'y(0), so that (0)
lies in the linear subspace Z = 0. We deduce that the tangent space consists of the tangents
to the curves on S2 passing through N.
23
This is apparently no major conceptual gain since we still regard the tangent space
as a subspace of R3, and this is still an extrinsic description. However, if we use the
stereographic projection from the South pole we get local coordinates (u, Fu) near N, and
any curve 'y(t) as above can be viewed as a curve t »-> (u(t), v(t)) in the (u, UI plane. If
(WI is another curve through N given in local coordinates by t »-> (_
'u,(t),y(t)), then
(U°(0l»'ul(0) )
`\I \, P
(0l (/5(0)
The right-hand side of the above equality defines an equivalence relation on the set of
smooth curves passing trough (0, 0). Thus, there is a bijective correspondence between the
tangents to the curves through N, and the equivalence classes of "~". This equivalence
relation is now intrinsic modulo one problem: "~" may depend on the choice of the local
coordinates. Fortunately, as we are going to see, this is a non-issue.
Definition 2.1.2. Let M771 be a smooth manifold and Po a point in M. Two smooth paths
a, B : (-e, 5 -> M such that a(0) = B(0) = Po are said to have a ]?rst order contact at
Po if there exist local coordinates (x) = (xl, . . . ,:z:m*) near Po such that
w'o(0) i £i35(0l,
where
a(t) = ( m ) (al (t), • • • ,x;"(t) ) 7
and
it) = ( t ) = (we), . . »w2"(t))-•
We write this O d ~ 1
Lemma 2.1.3. ~1 is an equivalence relation.
Sketch of proof. The binary relation -1 is obviously reflexive and symmetric, so we only
have to check the transitivity. Let a -1 B and B -1 fy. Thus there exist local coordinates
(w)=(x% ,xm) and (y) = Ly1, . 7 ym) near Po such that
(mum) = (am) and (y°@(0) ) IzM0II-
The transitivity follows from the equality
<93/i 33/i
9(0) Z yaw) 3307
in) 3307
°j
11» (0) i y°j (0)-
j J
Definition 2.1.4. A tangent vector to M at p is a first-order-contact equivalence class of

curves through p. The equivalence class of a curve a(t) such that a(0) = p will be tem-
porarily denoted by [a(0)]. The set of these equivalence classes is denoted by TpM, and is
called the tangent space to M at p.
Lemma 2.1.5. TpM has a natural structure of vector space.

Natural Constructions on Manifolds 25
Proof. Choose local coordinates (x1, . . . , a s ) near p such that xi (p) = 0, W, and let a
and 5 be two smooth curves through p. In the above local coordinates the curves a, B
become (w;(t)) (w;<t))~ Construct a new curve 'Y through p whose as coordinates are
given by
( ) = (wav) +w;(t))-
We want to emphasize that the above curve depends on a, 3 and the choice of local coor-
dinates (x1, . . . , a s ) . We will indicate this using the notation a +x B when referring to by.
Set
[Ci(0)] -|- [B(0)1 ii Mon.

The zero vector in TpM is described by a curve a such that go; (0) 0, W. For this
operation to be well defined one has to check several things.
(a) The equivalence class ['y°(0)] is independent of coordinates. In other words, if

x 1 , . . . , x M and (y1, . . . , am) are local coordinates near p such that
t'(p)==y"(p)= 0, am,
then
(b) If a curve represents the zero vector in some local coordinates (x'i) at p, then it repre-
sents the zero vector in any other choice of local coordinates at p.
(c) If [o21(0)] = [o22(0)] and [B1(0)] = [B2(0)], then
[021(0)] + [B1(0)] [022(0)] + [82(0)]-
We let the reader supply the routine details.
Exercise 2.1.6. Finish the proof of the Emma 2. l .5.
From this point on we will omit the brackets [ - ] in the notation of a tangent vector.
Thus, [ci(0)1 will be written simply as 0;(0).
As one expects, all the above notions admit a nice description using local coordinates.
Let (1131, . . . ,as"") be coordinates near p E M such that xi (al = 0, Vi. Consider the curves
1 7 . . . 7 MY
where 6; denotes Kronecker's delta symbol. For example
€1ltl=lt,0,0,...,0< €2ltl=l0,t,0,...,0l,
We set
@xk(p) Z= éklol I . .7Mk ) . (2.1.2)
Often, when the point p is clear from the context, we will omit it in the above notation.
" N o t e that the vectors a (p) depend on the choice of local coordinates (5131 7 7 :um .
Lemma 2.1.7. The collection (3wk ( p ) llgkgtm is a basis ofT pM.

26 Lectures on the Geometry of Mamfolds
Proof. Given a smooth path

a(t) Z ( l (t), 7 xi*(t))» 0z(Ul=p,
we have
do) = (f1 (0), 7 ;'(0)) = f1 (0)(1,0, ,0)+ +12gl(0)(0,...,0,1)
TTI
(2.l.3)
531 (0)31 + -|- 52T(0)<9'm i:;(0l<9Q3
i 1
Exercise 2.1.8. Suppose that M C R" is an m-dimensional smooth submanifold of R" .

Let Po E M and suppose that a, 5 : (-et, 5) -> M are two smooth paths such that a(0) =
5(0) P0- Prove that the following statements are equivalent.
(1) The paths a have first order contact at Po .
(2)
day) = d,B(O)
as vectors in Rn .
dt dt
Deduce from the above that Thor can be identified with the vector subspace of R"
spanned by the vectors U e R" with the property that there exists a smooth path a :
(6, 5) -> M C R" such that
da(0)
04(0) Po, 17.
dt
Exercise 2.1.9. Let F : RN -> Rkbe a smooth map. Assume that
(a) M = F-1(0) # V);
(b) rank D11:F = k, for all zz: 6 M.
Then M is a smooth manifold of dimension N - k and
Lm r her Dx3 F, Vcc E M.
Example 2.1.10. We want to describe T11 G, where G is one of the Lie groups discussed in
Section 1.2.2.
(a) G = O(n). Let (-6, 6) 9 .s »-> T(.s) be a smooth path of orthogonal matrices such that
T(0l = ]1. Then Tt(s) T(sl = JL Differentiating this equality at .s = 0 we get
Tt(0) -|- T(0) = 0.
The matrix T ( 0 ) defines a vector in T1O(nl, so the above equality states that this tangent
space lies inside the space of skew-symmetric matrices, i.e., To O(n) C Q(n). On the other
hand, we proved in Section 1.2.2 that dim G = dim Q(n) so that
T]1O(nl = Q(n).
(b) G = SL(n;]R). Let ( - e , 5 9 s »-> T(s) be a smooth path in SL (n;]R.) such that
T(0l = 1. Then det T ( s ) = 1 and differentiating this equality at S = 0 we get (see
Exercise 1.1.3)
tr T(0) = 0.
Thus, the tangent space at ]1 lies inside the space of traceless matrices, i.e., To SL(n; R) C
QW; R). Since (according to Exercise 1.2.27) dim SL(n; R) = d i m ( n ; R) we deduce
T]1SL(n;R) = M; R).
Exercise 2.1.11. Show that T]1U(n) = g(n) and To SU(n) =
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Caput decimum octāvum.
Suspīciō in laetitiam et admīrātiōnem versa. — Cāsus quī rīsum

legentī movēbit. — Rēbus secundīs adversae levantur.
P osterō diē Rōbinson cum sociō statim ad eum locum sē

contulit, ubi hesternō barbarōs atrōcibus epulīs accumbentēs
vīderat. Inter eundum eō dēvēnērunt, ubi ambō barbarī ā Rōbinsōne
interfectī harēnā obrutī jacēbant. Vendredi, locum dominō
dēmōnstrāns, apertē significāvit quantum gestīret cadāvera ista
ēruere, ut improbam carnis aviditātem explēret. Quem Rōbinson
intuēns torvō vultū, eī dēclārat quantum ā tālī facinore ipse
abhorreat, hastāque ēlātā, īnfēstō gestū, mortem eī dēnūntiat, eī
umquam ille ejusmodī cibum [145]attingeret. Quō quidem intellēctō,
Vendredi statim dominō pāruit, incertus tamen quam ob causam sibi
epulīs interdīcerētur, quās ā tenerīs nōn mediocriter appetierat.
Tunc ad locum convīviī dēvēnērunt. Heu ! quālis adspectus !

terra cruōre tīncta ! disjecta passim ossa ! Rōbinson oculōs āvertit,
sociumque jubet terram statim fodere, trīstēsque barbarōrum
helluōnum reliquiās condere.
Dum ille mandāta exsequitur, Rōbinson cinerēs sōpītōs attentē

suscitābat, spērāns sē aliquam ignis particulam inventūrum. Sed
frūstrā. Erat ignis omnīnō exstīnctus ; quō quidem Rōbinson
magnopere doluit. Ex quō enim dīvīnō mūnere socium sibi
adjūnxerat, nihil ipsī ferē optandum supererat praeter ignem. Dum
autem ille, capite inclīnātō, maestōque vultū exstīnctōs cinerēs
adspicit, ecce Vendredi, cum dominum cōgitātiōne dēfīxum
animadvertisset, nōnnūlla signa dedit, quae Rōbinson nōn
[146]intellēxit ; tum subitō arreptā secūrī, citātissimō cursū silvam
petit intimam, Rōbinsōnemque relinquit obstupēscentem.
« Quid hoc sibi vult ? ait sēcum Rōbinson, dum euntem ānxiīs
prōsequitur oculīs. Mēne homō dēserat, et ablātā secūrī aufugiat ?
an ille sit tam perfidus, ut meum habitāculum occupet, ut mē ipsum
vī inde arceat, et populāribus suīs inhūmānīs prōdat. Prōh scelus ! »
Statimque īnflammātus īrā hastam corripit, ut prōditōrem
persequātur, nefāriaque cōnsilia et pūniat et praevertat. Dum sīc dē
fide barbarī timet, videt hominem citātissimō cursū redeuntem.
Rōbinson sistit gradum, mīrāturque eum, quem prōditiōnem
māchinārī suspicātus fuerat, sublātum grāminis āridī manipulum
tenēre unde fūmus oriēbātur. Jam in flammam ērumpit ; et Vendredi,
manipulō in terram prōjectō, addit dīligenter majōrem grāminis āridī
atque sarmentōrum cōpiam, clārumque et ārdentem ignem
succendit : quod [147]quidem nōn minōrem Rōbinsōnī laetitiam quam
admīrātiōnem mōvit.
Tum compertā causā propter quam Vendredi subitō excurrerat,

cum ipse tantam laetitiam capere nōn posset, hic illum amplexus
ōsculātusque tacitē rogat praeconceptae suspīciōnis veniam.
Scīlicet Vendredi dīligenter silvam petierat, ut ē truncō āridō duo

lignī fragmenta excīderet. Quā ille scītē alterum alterō collīserat tantā
celeritāte, ut ignem conciperent. Tum citius lignīs āridō grāmine
involūtīs, cum istō manipulō prōcurrerat, quā motūs vēlōcitāte
exārserat fēnum.
Tantopere agitātiōne flammārum Rōbinson dēlectātus est, ut

illārum adspectū satiārī nōn posset. Tandem arreptā taedā, comite
Vendredi, in habitāculum properat.
Tum igne accēnsō, nōnnūllīsque sōlānōrum tūberibus circā

focum positīs, ad gregem festīnat ; lamae pullum ēligit, mac [148]tat,
dissecat, quārtamque ejus partem verū affīgit, quod sociō
versandum mandat.
Intereā dum ille hōc mūnere fungitur, Rōbinson segmentum

pectoris amputat ; tum nōnnūlla tūbera probē lavat, manipulumque
zēae duōbus saxīs adhibitīs molit ; haec omnia ōllae committit, in
quā, additō sale, idōneam aquae partem īnfundit, ac dēmum ōlla ignī
appōnitur.
Vendredi istum omnem apparātum cernēbat, nec intellegēbat

quō rēs spectāret. Nōverat quidem ūsum assandī, sed numquam dē
arte coquendī audīverat. Quīn etiam ignōrābat, quaenam esset vīs
ignis in aquam ōllae īnfūsam. Quae quidem fervere coepit, cum
Rōbinson paululum in spēluncam sēcessisset. Vendredi obstupuit,
mīrātus quid esset quod sīc aquam movēret.
Cum vērō eam vīdit exaestuantem et undequāque exundantem,

stultē putāvit animal in ōllā esse, quod istum dērepente aestum
excitāret : ut autem impedīret, nē omnis ab ollā aqua effunderētur,
manum [149]celeriter immersit, ut animal noxium caperet. Tum vērō
clāmōrem et ululātum ēdidit, quō tōta personuit spēlunca. Hōc audītō
Rōbinson valdē exterritus est, exīstimāns ā barbarīs
supervenientibus socium opprimī. Itaque timor et īnsitus in animīs
propriae salūtis amor suādēbant, ut per subterrāneum cunīculum
effugeret. Mox autem cōnsilium abjēcit, turpe ratus subditum, aut
potius amīcum, in ancipitī perīculō dēserere. Sine morā igitur ē
spēluncā prōrūpit armīs īnstrūctus, parātusque vītam ipse suam
profundere, ut comitem ē manibus barbarōrum iterum ēriperet.
Ut vērō obstupuit, cum hominem sōlum, āmentis īnstar,

ululantem gestūque īnsolitō circum trepidantem vīdit ! Diū quoque
Rōbinson dubius ānxiusque haesit. Rē dēmum explānātā, intellēxit
omne malum ex eō esse, quod manum sibi Vendredi leviter usserit.
Numquam ille neque audiendō, neque experiendō cognōverat aquae
fervōrem addī posse ; numquam manū tetigerat [150]aquam
fervidam : itaque nōn potuit intellegere, quae causa esset dolōris
ejus quem manū in aquam immersā sēnsit. Sīc igitur magicā
quādam arte hoc fierī, dominumque magum esse exīstimāvit,
Rōbinson aegrē animum sociī sēdāre, eīque persuādēre potuit, ut
dēnuō carnī verū versandae assidēret. Ille mandātō tandem
obsecūtus, ōllam nōn sine horrōris quōdam sēnsū, dominum vērō,
quem hūmānā condiciōne majōrem nunc putāvit, timidā cum
reverentiā contemplātus est. Quam quidem opīniōnem albus color
prōmissaque barba etiam cōnfirmābant : haec enim efficiēbant, ut
Rōbinson speciē ōris longē differret ā sociō ejusque populāribus,
fuscum colōrem et imberbem vultum praeferentibus.
Jam vērō prandium parātum erat. Quantopere Rōbinson calidīs

et pinguibus cibīs dēlectātus sit, facile est intellegere. Nunc quoque
calamitātum praeteritārum oblītus sibi in animō fingit jam nōn dēsertā
in īnsulā, sed in regiōne frequentissimā sē [151]versārī. Sīc animī
vulnera īnspērātō quōdam gaudiō sānārī solent, etsī illa plērumque
īnsānābilia putāmus.
Prandiō cōnfectō, Rōbinson sēcessit, ut dē prōsperā rērum
suārum commūtātiōne sēcum ipse meditārētur. Nunc omnia eī
arrīdēre vīsa sunt. Quippe quī jam nōn esset homō sōlitārius,
sociōque gaudēret quīcum nōndum quidem colloquī, sed ex cujus
cōnsuētūdine multum sōlāciī opisque licēbat exspectāre.
Et vērō cōgitantī vēnit quidem in mentem vītam mollem atque

ōtiōsam agere, dum intereā Vendredi, juvenis rōbustusque, dē quō
praetereā tam bene meritus fuerat, ut eum sibi famulum jūre quōdam
vindicāret, necessāriīs mūneribus labōribusque perfungerētur. Sed
cum intus reputāsset fierī posse ut aliquandō tam fēlīcī ipse
condiciōne excideret ; tunc vērō, sī ōtiō atque inertiā corrumpī sē
paterētur, molestum sibi fore ad dūritiem paupertātemque priōris
vītae redīre ; statuit in labōre [152]aequē nāviter ac strēnuē
persevērāre : atque, hōc dēcrētō, ē strātō exsilit, citātōque gressū in
vestibulō domiciliī obambulat. Intereā Vendredi cibōrum reliquiīs in
cellā sēpositīs, Rōbinsōnis jussū, ad lamās mulgendās abit.
[153]
Caput decimum nōnum.
Rōbinson habitāculum fossā et pālīs mūnit. — Docet socium

Germānicē loquī. — Ambō scapham fabricāre statuunt.
N umquam sānē, ex quō hanc in īnsulam advēnerat, laetior fuerat

Rōbinsōnis condiciō. Hoc ūnum erat illī metuendum, nē barbarī
reverterentur ad sociōs repetendōs, unde novīs cruentīsque proeliīs
sibi cum iīs certandum foret. Quīn ille cohorruit, dum cōgitāret in hōc
ancipitī discrīmine posse sē versārī, in quō sibi pereundum esset,
aut sanguis hominum profundendus.
Rēs igitur postulābat ut ad suam ipsīus dēfēnsiōnem nihil

intentātum relinqueret. [154]Jam diū habitāculum suum castellī īnstar
mūnīre optāverat ; hūc ūsque vērō, quamdiū vīxerat sōlitārius, hanc
spem penitus abjēcerat. Nunc autem, cum adesset socius labōris,
facilius hoc aggredī potuit. Itaque montis cacūmen cōnscendit, ut
meditārētur quōmodo sē tūtius firmiusque mūnīret ; atque hoc brevī
excōgitāvit, cum inde tōtam regiōnem oculīs complecterētur. Statuit
prīmum fossā lātā altāque domicilium cingere, pālīs validiōribus
saeptā, ac deinde rīvum nōn procul ab habitāculō scatentem ita
dīvidere, ut altera pars in fossā, altera per medium ātrium flueret, nē
umquam obsidiōne cīnctus aquae inopiā labōrāret.
Haud facile erat haec omnia sociō signīs indicāre. Hic vērō cum
rem aliquā ex parte intellēxisset, ad lītus prōcurrit, atque inde retulit
varia īnstrūmenta fodiendō apta, conchās scīlicet magnās,
lapidēsque plānōs et acūtōs. Tum ambō opus inchoant.
[155]
Jam intellegere satis in prōmptū est hoc quoque arduum fuisse.

Fossa enim, ut esset idōnea, sex pedum altitūdinem, octo vērō
lātitūdinem requīrēbat. Ejus autem longitūdō octōgintā vel centum
passūs esse vidēbātur. Adde illōs īnstrūmentīs ferreīs, ligōne, spathā
penitus carēre. Pālīs quoque quadringentīs ferē opus erat, quōs ūnā
secūrī siliceā coaptāre et praeacuere rēs erat profectō plēna operae
et labōris. Praetereā, ut rīvum in fossam dēdūcerent, canālis erat per
tumulum quemdam interpositum fodiendus.
Neque tamen, licet omnibus hīs difficultātibus circumventus,

Rōbinson ā prōpositō dēterritus est. Sōbriē quoque vīvendī genus, et
corpus labōre exercendī cōnsuētūdō animum eī addiderat, quō
carēre iī solent quī, in ōtiō ēducātī, dēliciīs atque mollitiē diffluunt.
Ambō nīmīrum sociī ā summō māne ūsque ad vesperam nōn

alacrī minus quam [156]strēnuō animō operī incumbunt. Mīrum itaque
quantum vel īnstrūmentīs quam minimē aptīs adjūtī in diem
prōfēcerint. Nec per duōs mēnsēs continuōs, ventō obstante, barbarī
īnsulam invīsēre. Itaque licuit sociīs continuam operam mūnīmentīs
dare, nec illīs opus fuit sibi contrā repentīnam incursiōnem
praecavēre.
Rōbinson comitem, inter opus faciendum, germānicam linguam

docuit, cujus ope animī cōgitāta cum sociō commūnicāre
ārdentissimē cupiēbat. Hic autem tam docilem et attentum sē
praebuit, ut brevī tempore māximōs prōgressūs fēcerit. In quō
Rōbinson eādem arte ūsus est, quā sollers magister ut linguam
Latīnam Gallicamve discipulōs doceat. Quotiēs nempe fierī potuit, eā
rē, dē quā sermō erat, positā ante oculōs, distīnctā vōce nōmen ejus
prōnūntiābat. Cum vērō dē ejusmodī rēbus agerētur, quae oculīs
subjicī nōn poterant, vultū gestūque ita exprimere stu [157]duit, ut
Vendredi annō dīmidiō nōndum exāctō Rōbinsōnem intellegere,
cōgitātiōnumque participem eum facere posset.
Quod quidem multum Rōbinsōnis fēlīcitātī adjēcit. Hūc ūsque

enim socium quidem, sed mūtum eum habuerat ; tum vērō amīcum
sibi comparāvit. Vendredi enim semper honestum, candidum,
fidēlem grātumque dominō sē praestitit. Itaque Rōbinson cāriōrem
eum in diēs habuit, ac brevī tempore ēlāpsō nōn dubitāvit hoc eī
concēdere, ut in eādem spēluncā sēcum pernoctāret.
Duōrum ferē mēnsium spatiō fossa absolūta fuit. Quō factō, ita
mūnītī erant, ut jam barbarōrum impetum nōn modo nōn
extimēscere, sed eōs etiam, sī umquam oppugnārent arcem,
possent repellere. Prius enim quam ūllus fossam trānsīret aut pālōs
superāret, ab obsessīs aut sagittīs interficī, aut hastīs trānsfodī
impūne poterat. Itaque sēcūritātī suae nunc satis cōnsuluisse
vidēbantur.
[158]
Cum nōn ita multō post Rōbinson sociusque collem lītorī

vīcīnum forte cōnscendissent, unde maris adspectus longē et lātē
patēbat, Vendredi oculōs in quamdam ejus partem intendit, ubi
īnsulae aliquot ēminentēs cōnspicī ē longinquō poterant. Tum ille
subitō prae laetitiā exsultāre atque mīrum in modum gestīre coepit.
Quā dē rē interrogātus ā Rōbinsōne laetus ille exclāmat ! « Ēn ego
patriam adspiciō ! ibi gēns mea habitat. » Vultū vērō, oculīs gestūque
significābat quam cāra sibi esset patria, ac quantopere illam revīsere
optāret. Quod quidem Rōbinsōnī minimē placuit. Erat sānē affectus
ille animī in virō laudābilis, quō dēclārābat cāram esse patriam,
cārōs amīcōs, cārōs parentēs. Nāscitur enim nōbīscum sēnsus ille,
et, quaecumque illa sit quam prīmō quisque cognōvit intuitū, vel
signāvit vestīgiō, nūlla magis regiō arrīdet. Sed veritus nē
populārium amantior ipsum aliquandō dēsereret, animum ejus
pertentāre voluit, et sīc colloquium cum eō [159]iniit, ex quō optima
ejus indolēs patuit apertius.
Rōbinson. Num tū ad populārēs tuōs redīre et cum iīs habitāre

māllēs ?
Vendredi. Libenter equidem eōs revīserem.
Rōb. Scīlicet optārēs carne hūmānā cum iīs vēscī ?
Vendr. Minimē ! Eōs potius ad mōrēs hūmāniōrēs trādūcerem,

docēremque vēscī lacte et carne animālium, et imprīmīs ab hūmānā
abstinēre.
Rōb. Quid ? sī tē ipsum dēvorārent ?
Vendr. Hoc illī nōn facient.
Rōb. Attamen illī vēscuntur carne hūmānā.
Vendr. Sānē illī quidem ; sed nōnnisi carne hostium.
Rōb. An tū scapham cōnficere potes, quā ad eōs trānsvehāris ?
Vendr. Sānē quidem !

Rōb. Eugē ! cōnfice tibi scapham, atque [160]ad illōs revertere.
Quid ! dējicis oculōs ! unde sīc dolēs ?
Vendr. Doleō quippe, quod dominus mihi cārissimus īrāscātur.
Rōb. Cūr vērō tibi īrāscar ?
Vendr. Ita rēs est ; quippe ille mē ā sē relēgātum velit.
Rōb. Anteā vērō optābās redīre in patriam ?
Vendr. Optāvī equidem : nisi vērō dominus meus ibi mēcum

versētur, ibi quoque ego versārī nōlim.
Rōb. Mē quidem populārēs tuī hostem exīstimantēs interficient

dēvorābuntque. Tū igitur sōlus proficīscere.
Quibus audītīs, Vendredi arreptam secūrim dominō reddit

porrēctā cervīce.
Rōb. Quid vīs ego faciam ?
Vendr. Ut mē interficiās. Mālim ego ā tē interficī quam relēgārī.
Hīs dictīs, Vendredi vim lacrimārum profundit. Rōbinson autem

vehementissimē commōtus eum amplectitur, exclā [161]māns : « Nōlī
timēre, ō bone. Ego quoque optō ut numquam ā tē dīvellar ; ex
animō enim tē dīligō. Quae anteā dīxī, haec ad fidem tuam
explōrandam dicta sunt, ut intellegerem an tuus meō amōrī amor pār
esset. Quās vidēs lacrimās, eae testēs sunt cāritātis meae. Ego tē
iterum amplectar ; flētum teneāmus, nec alter alterum umquam
dēserat. »
Itaque ut animum amīcī ā maerōre āverteret, init sermōnem dē
scaphā quādam fabricandā, plūraque eā dē rē scīscitātur. Cum
Vendredi respōnsīs Rōbinsōnī abundē satisfēcisset, hic illum manū
prehēnsum sēcum abdūxit, ut ostenderet cymbam, cui fabricandae
ipse permultōs jam annōs impenderat. Vendredi, truncum intuēns,
vix tertiā parte excavātum, subrīsit. Cum autem Rōbinson ex eō
quaesiisset, quidnam illī in hōc opere minus probārētur, Vendredi
respondit minimē opus fuisse tantō istō labōre, ejusmodī truncum
intrā paucōs diēs melius excavārī posse.
[162]
Quibus audītīs, Rōbinson vehementer laetātus est. Jam

cymbam omnīnō cōnfectam sibi fingēbat animō, et nāvigātiōne
fēlīciter perāctā ad continentem terram appellere sibi vidēbātur. Ō
quanta illī laetitia, cum spēs lībertātis recuperandae arrīsit ! Et subitō
cōnstituunt opus proximō māne aggredī.
[163]
Caput vīcēsimum.
Pluviārum tempus. — Sociī nectunt strāgulās, rētia. — Cymba

cōnficitur.
I psā diē operī dēstinātā, aderat pluviārum tempus : quod bis per
annum ingruere Rōbinson nōn ūnīus annī experientiā didicerat.
Quō tempore per duōs mēnsēs perpetuōs nūllī negōtiō vacāre extrā
domum licēbat ; tantā vī continuus imber ruēbat dē caelō !
Animadverterat quoque Rōbinson esse valētūdinī omnīnō contrārium
illā tempestāte forās exīre. Quid igitur faciendum erat ? Dīlātā nāvis
cōnfectiōne, tempus in labōribus domesticīs cōnsūmendum fuit.
Quantum prōfuit Rōbinsōnī per diēs illōs pluviōs, atque hōrās

vespertīnās lon [164]gās eāsdem obscūrāsque igne ūtī et lūmine,
amīcumque habēre, quīcum tempus inter opera domestica jūcundīs
cōnfābulātiōnibus trādūceret ! Anteā enim trīstēs istās noctēs
sōlitārius in ōtiō et tenebrīs dēgerat. Jam vērō cum sociō ad
lampadem et focum sedēns, in aliquā rē gerendā occupātus,
cōnfābulātur, neque umquam gravī dēsidiae pondere opprimitur.
Atque ex sociō didicit variās artēs quibus barbarī nōnnūllās sibi

commoditātēs comparant ; Rōbinson quoque multa illum docuit,
quae barbarīs latent. Ita in diēs perītiōrēs factī multa jūnctā operā
cōnfēcēre, quae neuter sōlus suscipere potuisset. Tum ambō
intellēxēre quanta ex hominum amīcā cōnspīrātiōne oriantur
commoda, quibus illī carērent, sī bēstiārum mōre singulī vagārentur.
In prīmīs Vendredi artem callēbat, quā tenuēs dēnsāsque librō
strāgulās necteret corporī vestiendō aptissimās. Quod cum
Rōbinson ex eō didicisset, tum illī certātim tam multās ejusmodī
texuērunt, [165]ut idōneum utrīque vestīmentum suppeteret. Ō quam
jūcundum Rōbinsōnī fuit abjicere amiculum istud molestum, ē
pellibus rigidīs nec subāctīs cōnfectum, quō hūc ūsque corpus ipsī
tegendum fuerat !
Vendredi etiam ē fibrīs nucum cocossae variīsque herbīs līnī

nātūram referentibus fīla ēdūcendī artem tenēbat, quae fūniculōs ā
Rōbinsōne hūc ūsque cōnfectōs longē superābant. Ē fīlīs rētia
piscātōria propriō ac singulārī artificiō nectēbat. Quibus in operibus
fabricandīs breviōrēs factae sunt vesperae, quae, omnibus hīs
dēficientibus, multum ipsīs taediī attulissent.
Intereā dum sedent, Rōbinson amīcī ingenium rude sēnsim

excolere, ejusque mentem certā vērāque nōtitiā Deī imbuere.
Quantīs vērō errōribus mēns illīus hāc in rē labōrāverit, ex sequentī
cōnfābulātiōne facile erit intellegere.
Rōb. Dīc mihi, quaesō, ō bone, nōstīne quis mare, quis terram,
quis animālia, tē ipsum dēnique creāverit ?
[166]
Vendr. Profectō ! Tupan ista creāvit.
Rōb. Quisnam est Tupan ille ?
Vendr. Is quī tonat.

Rōb. Quisnam vērō ille est quī tonat ?
Vendr. Senex est aetāte prōvectissimus, quī omnibus cēterīs

rēbus superstes est, quīque tonitrū efficit. Aetāte sōlem, lūnam
stēllāsque longē superat, omnēsque animantēs eum adōrant, Ō !
dīcentēs.
Rōb. Quemnam in locum post mortem commigrant tuī

populārēs ?
Vendr. Ad Tupan revertuntur.
Rōb. Ubinam vērō ille habitat ?
Vendr. In excelsīs montibus.
Rōb. Num aliquis eum ibi vīdit ?
Vendr. Nēminī fās est eum adīre, nisi Ovocacēīs (id est
sacerdōtibus). Illī Ō dīcentēs eum interrogant, ac deinde nōbīs
referunt ejus respōnsa.
Rōb. Quī autem post mortem ad eum migrant, num fēlīcitāte

aliquā fruuntur ?
Vendr. Sānē illī quidem, sī magnam [167]hostium cōpiam

mactāverint atque comēderint.
Quō audītō, Rōbinson cohorruit, statimque illum meliōra dē Deō

vītāque futūrā docēre coepit ; Deum nempe esse omnipotentem,
sapientissimum benignissimumque ; quī omnia creāverit, omnia
regat, omnibus cōnsulat ; ipsum autem numquam orīginem
habuisse, ubīque adesse, illam intellegere quaecumque nōs
cōgitāmus, audīre quaecumque loquimur, vidēre quaecumque
agimus, quamvīs ipse ā nēmine vidērī queat ; habēre probōrum
atque improbōrum ratiōnem ; eamque ob causam cum in hāc, tum in
futūrā vītā nēminem beātum reddere, nisi illum quī ex animō virtūtī
studuerit.
Quae praeclāra plēnaque sōlāciī praecepta Vendredi magnā

cum reverentiā audiit, audītaque altē in animō īnfīxit. Cum magister
nōn minōrī docendī, quam discipulus discendī studiō flagrāret, hic
brevī praecipuam dē Deō ac religiōne doctrīnam te [168]nuit, quantum
huic ille explānāre poterat. Ex eō tempore Vendredi fēlīcissimum sē
exīstimāvit quod in hanc īnsulam dēferrī sibi contigisset.
Posteā Rōbinson precēs suās semper praesente sociō fundere

solēbat ; jūcundissimumque adspectū fuisset, quantō gaudiō, quantā
pietāte hic verba dominī ad verbum sequerētur. Tunc vērō tantā
ambō fēlīcitāte fruēbantur, quantam assequī possunt hominēs ab
hūmānā societāte sējūnctī. Sīc ēlāpsum est pluviārum tempus ūllā
sine molestiā. Caelō tandem serēna faciēs redierat ; cessābant
ventī, nimbīque aufūgerant. Rōbinson cum fīdō sociō pūram
tepidamque auram vēris spīrat. Ambō novīs vīribus auctōs sē
sentiunt, atque alacrī animō ad arduum opus susceptum sē
accingunt.
Vendredi admōtō igne truncum excavāvit, atque duōs intrā

mēnsēs id absolvit ; quod aegrē multōrum annōrum spatiō Rōbinson
sōlus cōnfēcisset. Jam praeter vēla [169]rēmōsque nihil dēfuit. Hōs
quidem Rōbinson, illa vērō Vendredi parātūrum sē spondet.
Ambō simul opus susceptum absolvēre ; sed īnstrūctā nāvī, nihil

superfuit nisi ut haec ā lītore in mare dēmitterētur. Quoniam vērō
locus, in quō nāvem fabricāverant, longē ā marī distābat, nōn satis
patēbat quā ratiōne, ut erat gravissima, ad mare aut dēdūcerētur, aut
dēportārētur, aut traherētur, aut dēnique prōvolverētur. Et nunc ex
illīs difficultātibus quōmodo sē expedient ?
Neque vērō Rōbinson oblītus erat ūtilitātis quam longa pertica

ipsī praestiterat. Quam ob rem nunc quoque illam adhibuit. Sed
nāvis tam lentē prōvolūta est, ut facile intellegerent sē integrum
mēnsem in hāc operā cōnsūmptūrōs esse.
Tandem opportūnē recordātus est alīus īnstrūmentī aequē

simplicis parābilisque, quō fabrī lignāriī aliīque in Eurōpā ūtī
[170]solent ad magna pondera prōmovenda, cylindrī nīmīrum.
Quod quidem vix Rōbinson expertus magnā cum laetitiā vīdit

quam facile nāvis prōmovērētur. Post bīduum marī quoque committī
potuit, magnōque cum voluptātis sēnsū uterque comperit illam ad
nāvigandum prōrsus esse idōneam.
Jam nihil superfuit nisi ut quae ad proficīscendum essent

necessāria parārentur, scīlicet ut nāvis tot onerārētur commeātibus
quot vehendīs sufficeret. Quōnam vērō illī tendent ? Vendredi
optābat ut in patriam redīret ; Rōbinson autem, ut ad continentem
Americae vēla darentur, ubi sē Hispānōs, aliōsque ab Eurōpā
profectōs inventūrum spērābat. Vendredi vērō patria nōnnisi quattuor
circiter mīliāribus aberat, terra continēns duodecim aut quīndecim.
Quod sī prius ad īnsulam tenderent, tunc ā continentī terrā plūribus
adhūc mīliāribus recēdēbant, atque hōc ipsō perīculum itineris
augēbātur.
[171]
Vendredi nihil ex iīs quae pertinent ad nātūram maris nōverat
nisi quod necesse esset ut ad īnsulam suam pervenīret. Ista autem
magis etiam Rōbinson ignōrābat, quoniam in hōc marī numquam
nāvigāverat.
Tandem quā Rōbinson ārdēbat hominēs hūmānitāte excultōs

revīsendī cupiditās omnem dubitātiōnem vīcit. Quamvīs Vendredi
īnstāret, multaque et varia objiceret, statūtum est proximō diē iter
parāre, et ubi prīmum ventus adspīrāret, vēla dare, Deōque favente
illūc tendere ubi Vendredi spērābat sē proximam continentis ōram
inventūrum.
[172]
Caput vīcēsimum prīmum.
Rōbinson et Vendredi, īnsulā relictā, marī sē committunt. —

Summa perīcula in quibus versantur.
R ōbinson, arce relictā, in tumulō imminente restitit, sēcum

paulisper meditātūrus, sociumque praeīre jussit. Tum vītae
sōlitāriae hīc āctae vicissitūdinēs mente repetit ; ac recordātus
quanta accēpisset ā suprēmō nūmine beneficia, lacrimās grātī animī
indicēs effundit, manibusque expānsīs, ex intimō pectore summā
cum pietāte Deō grātiās agit.
Tum regiōnem illam, eō sibi cāriōrem, quod eam mox relictūrus

erat, oculīs perlūstrāvit. Hominis īnstar quī patriam linquit nūllā cum
spē illīus umquam revīsendae, [173]oculī trīstēs madentēsque in
arbore quāvīs cujus umbrā ōlim recreātus fuerat, in opere quōlibet
quod propriīs manibus multōque sūdōre cōnfēcerat, dēfīxī haerent.
Ab amīcīs disjungī sibi vidētur. Cum vērō tandem lamās ad īmum
montem pāscentēs cōnspexisset, faciem āvertit, nē cārissimōrum
sibi animantium adspectū ipse ā prōpositō cōnsiliō āvocārētur.
Tandem vīcit cāritātem animī cōnstantia. Ad fortitūdinem sē ipse

exacuit, ulnīsque ad regiōnem tōtam, velutī eam amplexūrus,
expānsīs, clārā vōce exclāmāvit : « Valēte, ō calamitātum meārum
testēs ! valēte ! » Atque hōc ultimō valē inter singultūs ēmissō, in
viam quae ad lītus dūcēbat, sē contulit.
Eundō fīdissimum sibi psittacum per arborēs volitandō

sequentem animadvertit. Tum vērō victus voluntāte ejus sēcum
abdūcendī, vocat Pol, Pol. Ille vērō celerrimē dēsilit, atque ē dominī
manū in umerum prōvolat. Intereā Vendredi morae [174]impatiēns in
lītore exspectābat ; cumque Rōbinson ad eum pervēnisset, ambō
nāvem cōnscendunt.
Trīcēsimō diē novembris, hōrā octāvā mātūtīnā, annō post

Rōbinsōnis in hanc īnsulam adventum nōnō, caelō serēnissimō,
ventōque māximē secundō profectī sunt. Vix autem circiter duo mīlia
passuum prōgressī, ad continuam scopulōrum seriem pervēnēre
longē in mare prōcurrentem. Uterque perīculōsum putāvit saxa ista
superāre : itaque vēlō in aliam partem dīrēctō, circumīre nītuntur.
Cum autem illī extrēmam partem scopulōrum vix attigissent,
scapham summā cum vēlōcitāte abripī animadvertunt : exterritī
ambō vēlum colligere, exīstimantēs quippe repentīnum esse ventī
impetum. Frūstrā vērō. Scapha enim in praeceps prōna rapitur,
mediōque in flūmine marīnō versārī sē intellegunt.
Tum ambō vīribus conjūnctīs rēmīs pertinācius mare verberāre,

sī possint sca [175]pham ēripere. Illa vērō fūgit, īnstar sagittae, tantā
celeritāte, ut brevī ōra īnsulae oculīs subtraherētur. Jam in discrīmen
vītae sē pervēnisse sentiunt ; nec multō post summa quoque
montium cacūmina ex adspectū recessēre. Quod sī tunc etiam
impetus paululum remīsisset, āctum tamen dē illīs erat ; quippe quī,
cum carērent pyxide nauticā, viam invenīre nōn possent, quā ad
īnsulam reverterentur. Quid autem tētrius excōgitārī potest quam
mediō in ōceanō scaphā exiguā fragilīque vectōs jactārī, et sōlō
aliquot diērum vīctū īnstrūctōs !
Quantō autem praesidiō sincēra pietās integraque animī

cōnscientia iīs sit quī in calamitātibus versantur, nunc documentō est
Rōbinsōnis condiciō. Quibus sī noster tunc caruisset, quōmodo
malōrum vim dēnuō urgentem sustinuisset ? In hāc rērum
dēspērātiōne manūs sibi intulisset violentās, nē morte omnium
saevissimā, famē scīlicet, necārētur.
Socius vērō pietāte nōndum satis cōnfir [176]mātā, nec diuturnīs

calamitātibus spectātā, dē salūte prōrsus dēspērābat. Corporis enim
et animī vīribus frāctīs, rēmum abjicit ; vēcorsque et āmēns dominum
intuētur interrogāns, annōn satius sit ē nāve sē praecipitēs in
profundum dare, an morte celerrimā īnstantibus malīs sē ēripiant.
Initiō Rōbinson blandīs verbīs eum ērigere firmāreque tentat ; tum
castīgātor dēspērātiōnis lēnī vōce exprobrat, quod tam parum Deō
benignissimō cōnfīdat, eīque revocat in mentem quae ipsum dē
dīvīnā prōvidentiā docuerat. Vendredi, Rōbinsōnis cohortātiōnibus
permōtus, ērubuit ; rēmīsque receptīs ambō flūmen subigere, nec
bracchia remittere, etsī nūlla spēs salūtis affulgēbat. « Dēbitō, inquit
Rōbinson, officiō fungimur. Quamdiū spīritus nōn dēficit, nostrum est
ad vītam servandam omnibus quae supersunt nervīs ēnītī. Tunc, sī
ad ultimum moriendum est, morientibus erit sōlāciō, quod Deus sīc
voluerit. »
Intereā flūmen eādem cum violentiā ruit. [177]Scapha aestū

abripitur, omnisque spēs salūtis recuperandae ēvānēscit.
Vērum enim vērō cum jam mortālēs animīs dēficiunt, cum sunt
ab omnī spē dēstitūtī, tunc alma nūminis prōvidentia iīs praesentius

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Lectures on the Geometry of Manifolds

Third Edition Liviu I Nicolaescu

A Graduate Course in Probability 1st Edition Liviu I.

Lectures on Poisson Geometry Crainic

Lectures on the Philosophy of Religion The Lectures of

Analysis and Geometry on Graphs and Manifolds London

Manifolds, Vector Fields, and Differential Forms: An

Foundations of Hyperbolic Manifolds Third Edition 3rd

Differential Geometry of Curves and Surfaces: Third

Lectures on the Mechanical Foundations of

Introduction to Real Analysis

Library of Congress Cataloging-in-Publication Data

British Library Cataloguing-in-Publication Data

Copyright © 2021 by World Scientific Publishing Co. Pte. Ltd.

For any available supplementary material, please visit

Desk Editor: Liu Yumeng

lin the current edition, this is Chapter 10.

application of these results we consider a Kazhdan-Warner type equation which recently

mathematics. In particular, he convinced me that behind each formalism there must be a

Ann Arbor, 1996.

Preface to the second edition

Preface to the third edition

2 • Natural Constructions on Manifolds 23

4. Riemannian Geometry 139

5. Elements of the Calculus of Variations 201

5.2.1 Variational formulae 211

6. The Fundamental Group and Covering Spaces 227

8. Characteristic Classes 329

9. Classical Integral Geometry 373

9. 1.3 Affine Grassmannians . 395

10. Elliptic Equations on Manifolds 451

10.2.1 Sobolev spaces in RN . . . . . . . . . . . . . 464

11. Spectral Geometry 523

12. Dirac Operators 611

12.1.7 The complex spin group 633

Bibliog raphy 669

1.1.1 Space and Coordinatization

Euclidian Space -> R3 7 P I

In general, any coordinatization replaces geometry by algebra and we get a two-way

Study of Space < > Study of Equations.

Fig. 1.1 Polar coordinates.

P >(1~(p),9(p))@(0,oo) X ( or, fl-

at rosH y=rsin9. (1.1.1)

1.1.2 The implicit function theorem

Example 1.1.2. Consider F : U C R" -> Run. Using Cartesian coordinates

in Run, we can think of F as a collection of m functions on U

Theorem 1.1.4 (Inverse function theorem). Let X , Y be two Banach spaces, U C X

More formally, if we set T DuoF, then

Theorem 1.1.5 (Implicit function theorem). Let X, Y , Z be Banach spaces, W C X ,

{(a:,y)€UxV; F(w,y)=20}={(w,G(w)l G U x V ; a2€U}.

Proof. Consider the map

1.2 Smooth manifolds

1.2.1 Basic definitions

Definition 1.2.1. A smooth manifold of dimension m is a locally compact, second count-

(a) An open cover {U'i}i@I of M,

\Pi u -> \p(u-> C Run; i 6 I}

\IJjo 'Pi 1 \P¢(U¢ M Url C RM \Ifjlu¢ O Url C RM

Fig. 1.2 Transition maps.

when referring to this function.

l]l L -I-SS)'1 (HL + SS)'1S*

l]l L -I-SSl-1 (aL -|- SS)'1S*