CHAPTER 1

Riemannian manifolds

1.1 Introduction

A Riemannian metric is a family of smoothly varying inner products on the tangent spaces of
a smooth manifold. Riemannian metrics are thus infinitesimal objects, but they can be used to
measure distances on the manifold. They were introduced by Riemann in his seminal work [Rie53]
in 1854. At that time, the concept of a manifold was extremely vague and, except for some known
global examples, most of the work of the geometers focused on local considerations, so the modern
concept of a Riemannian manifold took quite some time to evolve to its present form. We point
out the seemingly obvious fact that a given smooth manifold can be equipped with many different
Riemannian metrics. This is really one of the great insights of Riemann, namely, the separation
between the concepts of space and metric.
This chapter is mainly concerned with examples.

1.2 Riemannian metrics

Let M be a smooth manifold. A Riemannian metric g on M is a smoothly varying family of inner

products on the tangent spaces of M . Namely, g associates to each p ∈ M a positive definite
symmetric bilinear form on Tp M ,

gp : Tp M × Tp M → R,

and the smoothness condition on g refers to the fact that the function

p ∈ M 7→ gp (Xp , Yp ) ∈ R

must be smooth for every locally defined smooth vector fields X, Y in M . A Riemannian manifold
is a pair (M, g) where M is a differentiable manifold and g is a Riemannian metric on M . Later
on (but not in this chapter), we will often simplify the notation and refer to M as a Riemannian
manifold where the Riemannian metric is implicit.
Let (M, g) be a Riemannian manifold. If (U, ϕ = (x1 , . . . , xn )) is a chart of M , a local ex-
pression for g can be given as follows. Let { ∂x∂ 1 , . . . , ∂x∂n } be the coordinate vector fields, and let
{dx1 , . . . , dxn } be the dual 1-forms. For p ∈ U and u, v ∈ Tp M , we write
X ∂ X ∂
u= ui and v= vj .
∂xi p ∂xj p
i j

c Claudio Gorodski 2012

Then, by bilinearity,
 
X
i j ∂ ∂
gp (u, v) = u v gp i
, j
∂x ∂x
i,j
X
= gij (p)ui v j ,
i,j

where we have set  

∂ ∂
gij (p) = gp i
, j .
∂x ∂x
Note that gij = gji . Hence we can write
X X
(1.2.1) g= gij dxi ⊗ dxj = g̃ij dxi dxj ,
i,j i≤j

where g̃ii = gii , g̃ij = 2gij if i < j, and dxi dxj = 12 (dxi ⊗ dxj + dxj ⊗ dxi ).
Next, let (U ′ , ϕ′ = (x′ 1 , . . . , x′ n )) be another chart of M such that U ∩ U ′ 6= ∅. Then
∂ X ∂xk ∂
= ,
∂x′i ∂x′i ∂xk
k

so the relation between the local expressions of g with respect to (U, ϕ) and (U ′ , ϕ′ ) is given by
  X k
′ ∂ ∂ ∂x ∂xl
gij = g , = gkl .
∂x′i ∂x′j ∂x′i ∂x′j
k,l

1.2.2 Examples (a) The canonical Euclidean metric is expressed in Cartesian coordinates by
g = dx2 + dy 2 . Changing to polar coordinates x = r cos θ, y = r sin θ yields that
dx = cos θdr − r sin θdθ and dy = sin θdr + r cos θdθ,
so
g = dx2 + dy 2
= (cos2 θdr2 + r2 sin2 θdθ2 − 2r sin θ cos θdrdθ)
+(sin2 θdr2 + r2 cos2 θdθ2 + 2r sin θ cos θdrdθ)
= dr2 + r2 dθ2 .
(b) A classical example is the surface of revolution parametrized by
x(r, θ) = (a(r) cos θ, a(r) sin θ, b(r)),
where a > 0, b are smooth functions defined on some interval and the generatrix γ(r) = (a(r), 0, b(r))
has ||γ ′ ||2 = (a′ )2 + (b′ )2 = 1, equipped with the metric g induced from R3 . Namely, the tangent
spaces to the surface are subspaces of R3 , so we can endow them with inner products just by taking
the restrictions of the Euclidean dot product in R3 . The tangent spaces are spanned by the partial
∂y ∂z ∂x ∂y ∂z
derivatives xr = ( ∂x 2
∂r , ∂r , ∂r ), xθ = ( ∂θ , ∂θ , ∂θ ), and then g = (xr · xr ) dr + 2(xr · xθ ) drdθ + (xθ ·
2
xθ ) dθ . Equivalently, from
dx = a′ (r) cos θ dr − a(r) sin θ dθ
dy = a′ (r) sin θ dr + a(r) cos θ dθ
dz = b′ (r) dr

26
we obtain

g = dx2 + dy 2 + dz 2
= dr2 + a(r)2 dθ2 .

The functions gij are smooth on U and, for each p ∈ U , the matrix (gij (p)) is symmetric and
positive-definite. Conversely, a Riemannian metric in U can be obviously specified by these data.

1.2.3 Proposition Every smooth manifold can be endowed with a Riemannian metric.

Proof. Let M = ∪α Uα be a covering of M by domains of charts {(Uα , ϕα )}. For each α, consider
the Riemannian metric gα in Uα whose local expression ((gα )ij ) is the identity matrix. Let {ρα }
be a smooth partition of unity of M subordinate to the covering {Uα }, and define
X
g= ρα g α .
α

Since the family of supports of the ρα is locally finite, the above sum is locally finite, and hence g
is well
P defined and smooth, and it is bilinear and symmetric at each point. Since ρα ≥ 0 for all α
and α ρα = 1, it also follows that g is positive definite, and thus is a Riemannian metric in M . 
The proof of the preceding proposition suggests the fact that there exists a vast array of Rie-
mannian metrics on a given smooth manifold. Even taking into account equivalence classes of
Riemannian manifolds, the fact is that there many uninteresting examples of Riemannian mani-
folds, so an important part of the work of the differential geometer is to sort out relevant families
of examples.
Let (M, g) and (M ′ , g ′ ) be Riemannian manifolds. A isometry between (M, g) and (M ′ , g ′ ) is
diffeomorphism f : M → M ′ whose differential is a linear isometry between the corresponding
tangent spaces, namely,
gp (u, v) = gf′ (p) (dfp (u), dfp (v)),
for every p ∈ M and u, v ∈ Tp M . We say that (M, g) and (M ′ , g ′ ) are isometric Riemannian
manifolds if there exists an isometry between them. This completes the definition of the category
of Riemannian manifolds and isometric maps. Note that the set of all isometries of a Riemannian
manifold (M, g) forms a group, called the isometry group of (M, g), with respect to the operation
of composition of mappings, which we will denote by Isom(M, g). Here we quote without proof the
following important theorem [MS39].

1.2.4 Theorem (Myers-Steenrod) The isometry group Isom(M, g) of a Riemannian manifold

(M, g) has the structure of a Lie group with respect to the compact-open topology. Its isotropy
subgroup at an arbitrary fixed point is compact. Moreover, Isom(M, g) is compact if M is compact.

The isometry group is a Riemannian-geometric invariant in the sense that if f : (M, g) → (M ′ , g)

is an isometry between Riemannian manifolds, then α 7→ f ◦ α ◦ f −1 defines an isomorphism
Isom(M, g) → Isom(M ′ , g ′ ).
A local isometry from (M, g) into (M ′ , g ′ ) is a smooth map f : M → M ′ satisfying the condition
that every point p ∈ M admits a neighborhood U such that the restriction of f to U is an isometry
onto its image. In particular, f is a local diffeomorphism. Note that a local isometry which is
bijective is an isometry.

27
1.3 Examples

The Euclidean space

The Euclidean space is Rn equipped with its standard scalar product. The essential feature of Rn
as a smooth manifold is that, since it is the model space for finite dimensional smooth manifolds, it
admits a global chart given by the identity map. Of course, the identity map establishes canonical
isomorphisms of the tangent spaces of Rn at each of its points with Rn itself. Therefore an
arbitrary Riemannian metric in Rn can be viewed as a smooth family of inner products in Rn . In
particular, by taking the constant family given by the standard scalar product, we get the canonical
Riemannian structure in Rn . In this book, unless explicitly stated, we will always use its canonical
metric when referring to Rn as a Riemannian manifold.
If (x1 , . . . , xn ) denote the standard coordinates on Rn , then it is readily seen that the local
expression of the canonical metric is

(1.3.1) dx21 + · · · + dx2n .

More generally, if a Riemannian manifold (M, g) admits local coordinates such that the local
expression of g is as in (1.3.1), then (M, g) is called flat and g is called a flat metric on M . Note
that, if g is a flat metric on M , then the coordinates used to express g as in (1.3.1) immediately
define a local isometry between (M, g) and Euclidean space Rn .

Riemannian submanifolds and isometric immersions

Let (M, g) be a Riemannian manifold and consider an immersed submanifold ι : N → M . This

means that N is a smooth manifold and ι is an injective immersion. Then the Riemannian metric
g induces a Riemannian metric gN in N as follows. Let p ∈ N . The tangent space Tp N can be
viewed as a subspace of Tp M via the injective map dιp : Tp N → Tι(p) M . We define (gN )p to be
simply the restriction of g to this subspace, namely,

(gN )p (u, v) = gι(p) (dιp (u), dιp (v)),

where u, v ∈ Tp N . It is clear that gN is a Riemannian metric. We call gN the induced Riemannian

metric in N , and we call (N, gN ) a Riemannian submanifold of (M, g).
Note that the definition of gN makes sense even if ι is an immersion that is not necessarily
injective. In general, we call gN the pulled-back metric, write gN = ι∗ g, and say that ι : (N, gN ) →
(M, g) is an isometric immersion (of course, any immersion must be locally injective). On another
note, an isometry f : (M, g) → (M ′ , g ′ ) is a diffeomorphism satisfying f ∗ (g ′ ) = g.
A very important particular case is that of Riemannian submanifolds of Euclidean space (com-
pare example 1.2.2(b)) Historically speaking, the study of Riemannian manifolds was preceded by
the theory of curves and surfaces in R3 . In the classical theory, one uses parametrizations instead
of local charts, and these objects are called parametrized curves and parametrized surfaces since
they usually already come with the parametrization. In the most general case, the parametrization
is only assumed to be smooth. One talks about a regular curve or a regular surface if one wants
the parametrization to be an immersion. Of course, in this case it follows that the parametrization
is locally an embedding. This is good enough for the classical theory, since it is really concerned
with local computations.

28
The sphere S n
The canonical Riemannian metric in the sphere S n is the Riemannian metric induced by its embed-
ding in Rn+1 as the sphere of unit radius. When one refers to S n as a Riemannian manifold with its
canonical Riemannian metric, sometimes one speaks of “the unit sphere”, or “the metric sphere”,
or the “Euclidean sphere”, or “the round sphere”. One also uses the notation S n (R) to specify
a sphere of radius R embedded in Rn+1 with the induced metric. In this book, unless explicitly
stated, we will always use the canonical metric when referring to S n as a Riemannian manifold.

Product Riemannian manifolds

Let (Mi , gi ), where i = 1, 2, denote two Riemannian manifolds. Then the product smooth manifold
M = M1 × M2 admits a canonical Riemannian metric g, called the product Riemannian metric,
given as follows. The tangent space of M at a point p = (p1 , p2 ) ∈ M1 × M2 splits as Tp M =
Tp1 M1 ⊕ Tp2 M2 . Given u, v ∈ Tp M , write accordingly u = u1 + u2 and v = v1 + v2 , and define

gp (u, v) = gp1 (u1 , v1 ) + gp2 (u2 , v2 ).

It is clear that g is a Riemannian metric. Note that it follows from this definition that Tp1 M1 ⊕ {0}
is orthogonal to {0} ⊕ Tp2 M2 . We will sometimes write that (M, g) = (M1 , g1 ) × (M2 , g2 ), or that
g = g1 + g2 .
It is immediate to see that Euclidean space Rn is the Riemannian product of n copies of R.

Conformal Riemannian metrics

Let (M, g) be a Riemannian manifold. If f is a nowhere zero smooth function on M , then f 2 g
defined by
(f 2 g)p (u, v) = f 2 (p)gp (u, v),
where p ∈ M , u, v ∈ Tp M , is a new Riemannian metric on M which is said to be conformal to g.
The idea behind this definition is that g and f 2 g define the same angles between pairs of tangent
vectors. We say that (M, g) is conformally flat if M can be covered by open sets on each of which
g is conformal to a flat metric.
A particular case happens if f is a nonzero constant in which f 2 g is said to be homothetic to g.

The real hyperbolic space RH n

To begin with, consider the Lorentzian inner product in Rn+1 given by

hx, yi = −x0 y0 + x1 y1 + · · · + xn yn ,

where x = (x0 , . . . , xn ), y = (y0 , . . . , yn ) ∈ Rn+1 . We will write R1,n to denote Rn+1 with such
a Lorentzian inner product. Note that if p ∈ R1,n is such that hp, pi < 0, then the restriction of
h, i to hpi⊥ (the the orthogonal complement to p with regard to h, i) is positive-definite (compare
Exercise 15). Note also that the equation hx, xi = −1 defines a two-sheeted hyperboloid in R1,n .
Now we can define the real hyperbolic space as the following submanifold of R1,n ,

RH n = { x ∈ R1,n | hx, xi = −1 and x0 > 0 },

equipped with a Riemannian metric g given by the restriction of h, i to the tangent spaces at its
points. Since the tangent space of the hyperboloid at a point p is given by hpi⊥ , the Riemannian

29
metric g turns out to be well defined. Actually, this submanifold is sometimes called the hyperboloid
model of RH n (compare Exercises 3 and 4). This model brings about the duality between S n and
RH n in the sense that one can think of the hyperboloid as the sphere of unit imaginary radius in
R1,n . Of course, as a smooth manifold, RH n is diffeomorphic to Rn .

Flat tori
A lattice Γ in Rn (or, more generally, in a real vector space) is the additive subgroup of Rn
consisting of integral linear combinations of thePvectors in a fixed basis. Namely, if {v1 , . . . , vn } is
a basis of Rn , then it defines the lattice Γ = { nj=1 mj vj | m1 , . . . , mn ∈ Z }. For a given lattice Γ
we consider the quotient group Rn /Γ in which two elements p, q ∈ Rn are identified if q − p ∈ Γ.
We will show that M = Rn /Γ has the structure of a compact smooth manifold of dimension n
diffeomorphic to a product of n copies of S 1 , which we denote by T n . Moreover there is a naturally
defined flat metric gΓ on M ; the resulting Riemannian manifold is called a flat torus. We also
denote it by (T n , gΓ ).
Relevant for the topology of M will be the discreteness of Γ as an additive subgroup P of Rn ,
n
namely: any bounded subset of Rn meets Γ in finitely many points only. In fact, if p = j=1 mj vj
is a lattice point viewed as a column vector, then
 
m1
p = M  ... 
 

mn

where M is the (invertible) matrix having the v1 , . . . , vn as columns. We obtain

 1/2
Xn
|mi | ≤  m2j  ≤ ||M −1 || ||p||
j=1

for all i = 1, . . . , n, where || · || denotes the Euclidean norm. Therefore if we require p to lie in a
given bounded subset of Rn , then there are only finitely many possibilities for the integers mj , and
thus only finitely many such lattice points. Note that discreteness of Γ implies that Γ, and thus
any equivalence class p + Γ, is a closed subset of Rn .
Equip M with the quotient topology induced by the canonical projection π : Rn → M that maps
each p ∈ Rn to its equivalence class [p]P = p + Γ. Then π is continuous. It follows that M is compact
since it coincides with the image of { nj=1 xj vj | 0 ≤ xj ≤ 1 } under the projection π. Moreover,
π is an open map, as for an open subset W of Rn we have that π −1 (π(W )) = ∪γ∈Γ (W + γ) is a
union of open sets and thus open. It follows that the projection of a countable basis of open sets
of Rn is a countable basis of open sets of M . We also see that the quotient topology is Hausdorff.
In fact, given [p], [q] ∈ Rn /Γ, [p] 6= [q], the minimal distance rpq from p to a point in the closed
r
subset q + Γ is positive. Let Wp , Wq be the balls of radius 2pq centered at p, q, respectively. A
point x ∈ Wp ∩ (Wq + Γ) satisfies d(x, p) < 2r and d(x, q + γ) < 2r for some γ ∈ Γ, and therefore
d(p, q + γ) ≤ d(p, x) + d(x, q + γ) < r leading to a contradiction. It follows that Wp ∩ (Wq + Γ) = ∅
and hence π(Wp ), π(Wq ) are disjoint open neighborhoods of [p], [q], respectively.
We next check that π : Rn → M is a covering. In fact, discreteness of Γ implies that the
minimal distance s from a non-zero lattice point to the origin is positive. Note that s is also the
minimal distance from any given point p ∈ Rn to another point in p + Γ. Let V be the ball of
radius 2s centered at p. Then V ∩ (V + γ) = ∅ 0for all γ ∈ Γ r {0}. Note also that π : V → π(V ) is
continuous, open and injective, thus a homemorphism. Now π −1 (π(V )) = ∪γ∈Γ (V + γ) is a disjoint

30
union of open sets on each of which π is a homeomorphism onto π(V ), proving that π(V ) is an
evenly covered neighborhood and hence π is a covering map. Since Rn is simply-connected, this is
the universal covering and the fundamental group of M is isomorphic to Γ.
Now we have natural local charts for M defined on any evenly covered neighborhood U = π(V )
as above. Indeed, write π −1 U = ∪γ∈Γ (V + γ) and take as chart ϕV = (π|V )−1 : U → V . If
U ′ = π(V ′ ) is another evenly covered neighborhood as above with U ∩ U ′ 6= ∅, consider a connected
component W of U ∩ U ′ , take p ∈ V such that [p] ∈ W and note that there is a unique γ ∈ Γ such
that p + γ ∈ V ′ . Now τγ ◦ ϕV |W and ϕV ′ |W , where τγ denotes the translation by γ, are both lifts
of the identity map of π(W ) and coincide on [p], hence τγ ◦ ϕV |W = ϕV ′ |W (Theorem 0.2.12). This
proves that the transition map ϕV ′ ◦ ϕ−1V coincides with τγ on W and is thus smooth. In this way
we have defined a smooth atlas for M . The covering map π : Rn → M is smooth and in fact a
local diffeomorphism because π|V composed with ϕV on the left yields as local representation the
identity, so we indeed have a smooth covering. The smooth structure on M is the unique one that
makes π : Rn → M into a smooth covering (this is more than a covering whose covering map is
smooth, compare page 8!).
The transition maps of the above atlas are restrictions of translations of Rn and thus isometries.
In account of this, M acquires a natural quotient Riemannian metric gΓ , which is the unique one
making the covering map π into a local isometry. In fact this requirement implies uniqueness
of gΓ , as it imposes that on an evenly covered neighborhood U = π(V ) as above, the local chart
ϕV = (π|V )−1 must be a local isometry and so gΓ = ϕ∗V g on U , where g denotes the canonical metric
in Rn . To have existence of gΓ , we need to check that it is well defined, namely, for another evenly
covered neighborhood U ′ = π(V ′ ) as above with U ∩ U ′ 6= ∅ it holds that ϕ∗V g = ϕ∗V ′ g on U ∩ U ′ .

∗
However, this follows from ϕ∗V ′ g = (ϕV ′ ϕ−1 V )ϕV g = ϕ∗V (ϕV ′ ϕ−1 ∗ ∗ −1 ∗
V ) g = ϕV g as (ϕV ′ ϕV ) g = g.
Note that gΓ is a flat metric.
As a smooth manifold, M is diffeomorphic to the n-torus T n . In fact, define a map f : Rn → T n
by setting
Xn 
f xj vj = (e2πix1 , . . . , e2πixn ),
j=1

where we view S1
as the set of unit complex numbers. Then f is constant on Γ, so it induces a
¯
bijection f : M → T n . Suitable restrictions of
(e2πix1 , . . . , e2πixn ) 7→ (x1 , . . . , xn )
define local chartsP of T n whose domains cover it. Now f = f¯ ◦ π composed on the left with such
charts of T give nj=1 xj vj 7→ (x1 , . . . , xn ), the restriction of an invertible linear map. It follows
n

that f¯ is a local diffeomorphism and hence a diffeomorphism.

We remark that different lattices may give rise to nonisometric flat tori, although they will
always be locally isometric one to the other since they are all isometrically covered by Euclidean
space; in other words, for two given lattices Γ, Γ′ , suitable restrictions of the identity map id :
Rn → Rn induce locally defined isometries Rn /Γ → Rn /Γ′ .
One way to globally distinguish the isometry classes of tori obtained from different lattices is
to show that they have different isometry groups. To fix ideas, let n = 2, and√consider in R2 the
lattices Γ, Γ′ respectively generated by the bases {(1, 0), (0, 1)} and {(1, 0), ( 21 , 23 )}. Then R2 /Γ is
called a square flat torus and R2 /Γ′ is called an hexagonal flat torus. The isotropy subgroup of the
square torus at an arbitrary point is isomorphic to the dihedral group D4 (of order 8) whereas the
isotropy subgroup of the hexagonal torus at an arbitrary point is isomorphic to the dihedral group
D3 . Hence R2 /Γ and R2 /Γ′ are not isometric. See exercise 9 for a characterization of isometric
flat tori.

31
 We finish the discussion of this example by noting that we could have introduced the smooth
structure on M and the smooth covering π : Rn → M by invoking Theorem 0.2.13, which we
have avoided only for pedagogical reasons. In fact, the elements of Γ can be identified with the
translations of Rn that they define and, in this way, Γ becomes a discrete group acting on Rn .
Plainly, the action is free. It is also proper, as this follows from the existence of r > 0 such that
d(p, q +Γ) ≥ r if p 6= q and d(p, p+Γr{0}) ≥ r, which was shown above. In the next subsection, we
follow and extend this alternative approach to incorporate the construction of the quotient metric.

Riemannian coverings
A Riemannian covering between two Riemannian manifolds is a smooth covering that is also a
local isometry. For instance, for a lattice Γ in Rn the projection π : Rn → Rn /Γ is a Riemannian
covering.
If M̃ is a smooth manifold and Γ is a discrete group acting freely and properly by diffeomor-
phisms on M̃ , then the quotient space M = Γ\M̃ endowed with the quotient topology admits a
unique structure of smooth manifold such that the projection π : M̃ → M is a smooth covering,
owing to Theorem 0.2.13. If we assume, in addition, that M̃ is equipped with a Riemannian metric
g̃ and Γ acts on M̃ by isometries, then we can show that there is a unique Riemannian metric
g on M , called the quotient metric, so that π : (M̃ , g̃) → (M, g) becomes a Riemannian cover-
ing, as follows. Around any point p ∈ M , there is an evenly covered neighborhood U such that
π −1 U = ∪i∈I Ũi . If π is to be a local isometry, we must have
 ∗
g = (π|Ũi )−1 g̃

on U , for any i ∈ I. In more pedestrian terms, we are forced to have

(1.3.2) gq (u, v) = g̃q̃i ((dπq̃i )−1 (u), (dπq̃i )−1 (v)),

for all q ∈ U , u, v ∈ Tq M , i ∈ I, where q̃i = (π|Ũi )−1 (q) is the unique point in the fiber π −1 (q) that
lies in Ũi . We claim that this definition of gq does not depend on the choice of point in π −1 (q). In
fact, if q̃j is another point in π −1 (q), there is a unique γ ∈ Γ such that γ(q̃i ) = q̃j . Since π ◦ γ = π,
the chain rule gives that dπq̃j ◦ dγq̃i = dπq̃i , so

g̃q̃i ((dπq̃i )−1 (u), (dπq̃i )−1 (v)) = g̃q̃i ((dγq̃i )−1 (dπq̃j )−1 (u), (dγq̃i )−1 (dπq̃j )−1 (v))
= g̃q̃j ((dπq̃j )−1 (u), (dπq̃j )−1 (v)),

since dγq̃i : Tq̃i M̃ → Tq̃j M̃ is a linear isometry, checking the claim. Note that g is smooth since it
is locally given as a pull-back metric.
On the other hand, if we start with a Riemannian manifold (M, g) and a smooth covering
π : M̃ → M , then π is in particular an immersion, so we can endow M̃ with the pulled-back
metric g̃ and π : (M̃ , g̃) → (M, g) becomes a Riemannian covering. Let Γ denote the group of
deck transformations of π : M̃ → M . An element γ ∈ Γ satisfies π ◦ γ = π. Since π is a local
isometry, we have that γ is a local isometry, and being a bijection, it must be a global isometry.
Hence the group Γ consists of isometries of M̃ . If we assume, in addition, that π : M̃ → M is a
regular covering (meaning that Γ acts transitively on each fiber of π; this is true, for instance, if
π : M̃ → M is the universal covering), then M is diffeomorphic to the orbit space Γ\M̃ , and since
we already know that π : (M̃ , g̃) → (M, g) is a Riemannian covering, it follows from the uniqueness
result of the previous paragraph that g must be the quotient metric of g̃.

32
The real projective space RP n
As a set, RP n is the set of all lines through the origin in Rn+1 . It can also be naturally viewed as
a quotient space in two ways. In the first one, we define an equivalence relation among points in
Rn+1 r {0} by declaring x and y to be equivalent if they lie in the same line, namely, if there exists
λ ∈ R r {0} such that y = λx. In the second one, we simply note that every line meets the unit
sphere in Rn+1 in two antipodal points, so we can also view RP n as a quotient space of S n and,
in this case, x, y ∈ S n are equivalent if and only if y = ±x. Of course, in both cases RP n acquires
the same quotient topology.
Next, we reformulate our point of view slightly by introducing the group Γ consisting of two
isometries of S n , namely the identity map and the antipodal map. Then Γ obviously acts freely and
properly (it is a finite group!) on S n , and the resulting quotient smooth structure makes RP n into
a smooth manifold. Furthermore, as the action of Γ is also isometric, RP n immediately acquires a
Riemannian metric such that π : S n → RP n is a Riemannian covering.

The Klein bottle

Let M̃ = R2 , let {v1 , v2 } be a basis of R2 , and let Γ be the discrete group of transformations of
R2 generated by the affine linear maps
 
1
γ1 (x1 v1 + x2 v2 ) = x1 + v1 − x2 v2 and γ2 (x1 v1 + x2 v2 ) = x1 v1 + (x2 + 1)v2 .
2

It is easy to see that Γ acts freely and properly on R2 , so we get a quotient manifold R2 /Γ which is
called the Klein bottle K 2 . It is a compact non-orientable manifold, since γ1 reverses the orientation
of R2 . It follows that K 2 cannot be embedded in R3 by the Jordan-Brouwer separation theorem;
however, it is easy to see that it can be immersed there.
Consider R2 equipped with its canonical metric. Note that γ2 is always an isometry of R2 , but
so is γ1 if and only if the basis {v1 , v2 } is orthogonal. In this case, Γ acts by isometries on R2 and
K 2 inherits a flat metric so that the projection R2 → K 2 is a Riemannian covering.

Riemannian submersions
Let π : M → N be a smooth submersion between two smooth manifolds. Then Vp = ker dπp
for p ∈ M defines a smooth distribution on M which is called the vertical distribution. Clearly,
V can also be given by the tangent spaces of the fibers of π. In general, there is no canonical
choice of a complementary distribution of V in T M , but in the case in which M comes equipped
with a Riemannian metric, one can naturally construct such a complement H by setting Hp to be
the orthogonal complement of Vp in Tp M . Then H is a smooth distribution which is called the
horizontal distribution. Note that dπp induces an isomorphism between Hp and Tπ(p) N for every
p ∈ M.
Having these preliminary remarks at hand, we can now define a smooth submersion π : (M, g) →
(N, h) between two Riemannian manifolds to be a Riemannian submersion if dπp induces an isom-
etry between Hp and Tπ(p) N for every p ∈ M . Note that Riemannian coverings are particular cases
of Riemannian submersions.
Let (M, g) and (N, h) be Riemannian manifolds. A quite trivial example of a Riemannian
submersion is the projection (M × N, g + h) → (M, g) (or (M × N, g + h) → (N, h)). More
generally, if f is a nowhere zero smooth function on N , the projection from (M × N, f 2 g + h) onto
(N, h) is a Riemannian submersion. In this case, the fibers of the submersion are homothetic but

33
not necessarily isometric one to the other. A Riemannian manifold of the form (M × N, f 2 g + h)
is called a warped product.
Recall that if M̃ is a smooth manifold and G is a Lie group acting freely and properly on
M̃ , then the quotient space M = G\M̃ endowed with the quotient topology admits a unique
structure of smooth manifold such that the projection π : M̃ → M is a (surjective) submersion
(Theorem 0.4.16). If in addition we assume that M̃ is equipped with a Riemannian metric g̃ and
G acts on M̃ by isometries, then we can show that there is a unique Riemannian metric g on M ,
called the quotient metric, so that π : (M̃ , g̃) → (M, g) becomes a Riemannian submersion. Indeed,
given a point p ∈ M and tangent vectors u, v ∈ Tp M , we set

(1.3.3) gp (u, v) = g̃p̃ (ũ, ṽ),

where p̃ is any point in the fiber π −1 (p) and ũ, ṽ are the unique vectors in Hp̃ satisfying dπp̃ (ũ) = u
and dπp̃ (ṽ) = v. The proof that g̃ is well defined is similar to the proof that the quotient metric is
well defined in the case of a Riemannian covering, namely, choosing a different point p̃′ ∈ π −1 (p),
one has unique vectors ũ′ , ṽ ′ ∈ Hp̃′ that project to u, v, but g̃p̃′ (ũ′ , ṽ ′ ) gives the same result as
above because p̃′ = Φ(g)p̃ for some g ∈ G, d(Φ(g))p̃ : Hp̃ → Hp̃′ is an isometry and maps ũ, ṽ
to ũ′ , ṽ ′ respectively. The proof that g̃ is smooth is also similar, but needs an extra ingredient.
Let Pp̃ : Tp̃ M̃ → Hp̃ denote the orthogonal projection. It is known that π : M̃ → M admits
local sections, so let s : U → M̃ be a local section defined on an open set U of M . Now we can
rewrite (1.3.3) as
gq (u, v) = g̃s(q) (Ps(q) dsq (u), Ps(q) dsq (v)),

where q ∈ U . Since V as a distribution is locally defined by smooth vector fields, it is easy to check
that P takes locally defined smooth vector fields on T M to locally defined smooth vector fields on
T M . It follows that g is smooth. Finally, the requirement that π be a Riemannian submersion
forces g to be given by formula (1.3.3), and this shows the uniqueness of g.

The complex projective space CP n

The definition of CP n is similar to that of RP n in that we replace real numbers by complex

numbers. Namely, as a set, CP n is the set of all complex lines through the origin in Cn+1 , so it can
be viewed as the quotient of Cn+1 r {0} by the multiplicative group C r {0} as well as the quotient
of the unit sphere S 2n+1 of Cn+1 (via its canonical identification with R2n+2 ) by the multiplicative
group of unit complex numbers S 1 . Here the action of S 1 on S 2n+1 is given by multiplication of
the coordinates (since C is commutative, it is unimportant whether S 1 multiplies on the left or
on the right). This action is clearly free and it is also proper since S 1 is compact. Further, the
multiplication Lz : S 2n+1 → S 2n+1 by a unit complex number z ∈ S 1 is an isometry. In fact, S 2n+1
has the induced metric from R2n+2 , the Euclidean scalar product is the real part of the Hermitian
inner product (·, ·) of Cn+1 and (Lz x, Lz y) = (zx, zy) = ||z||2 (x, y) = (x, y) for all x, y ∈ Cn+1 . It
follows that CP n = S 2n+1 /S 1 has the structure of a compact smooth manifold of dimension 2n.
Moreover there is a natural Riemannian metric which makes the projection π : S 2n+1 → CP n
into a Riemannian submersion. This quotient metric is classically called the Fubini-Study metric
on CP n .
We want to explicitly construct the smooth structure on CP n and prove that π : S 2n+1 → CP n
is a submersion in order to better familiarize ourselves with such an important example. For each
p ∈ CP n , we construct a local chart around p. View p as a one-dimensional subspace of Cn+1 and
denote its Hermitian orthogonal complement by p⊥ . The subset of all lines which are not parallel

34
to p⊥ is an open subset of CP n , which we denote by CP n r p⊥ . Fix a unit vector p̃ lying in the
line p. The local chart is
1
ϕp : CP n r p⊥ → p⊥ , q 7→ q̃ − p̃,
(q̃, p̃)

where q̃ is any nonzero vector lying in q. In other words, q meets the affine hyperplane p̃ + p⊥
1
at a unique point (q̃,p̃) q̃ which we orthogonally project to p⊥ to get ϕp (q). (Note that p⊥ can
be identified with R2n simply by choosing a basis.) The inverse of ϕp is the map that takes
v ∈ p⊥ to the line through p̃ + v. Therefore, for p′ ∈ CP n , we see that the transition map
ϕp ◦ (ϕp )−1 : { v ∈ p⊥ | v + p̃ 6∈ p′ ⊥ } → { v ′ ∈ p′ ⊥ | v ′ + p̃′ 6∈ p⊥ } is given by
′

1
(1.3.4) v 7→ (v + p̃) − p̃′ ,
(v + p̃, p̃′ )
and hence smooth.
Next we prove that the projection π : S 2n+1 → CP n is a smooth submersion. Let p̃ ∈ S 2n+1 .
Since the fibers of π are just the S 1 -orbits, the vertical space Vp̃ = R(ip̃). It follows that the
horizontal space Hp̃ ⊂ Tp̃ S 2n+1 is the Euclidean orthogonal complement of R{p̃, ip̃} = Cp̃ in
C2n+1 , namely, p⊥ where p = π(p̃). It suffices to check that dπp̃ is an isomorphism from Hp̃
onto Tp CP n , or, d(ϕp ◦ π)p̃ is an isomorphism from p⊥ to itself. Let v be a unit vector in p⊥ .
Then t 7→ cos t p̃ + sin t v is a curve in S 2n+1 with initial point p̃ and initial speed v, so using that
(cos t p̃ + sin t v, p̃) = cos t we have
d
d(ϕp ◦ π)p̃ (v) = (ϕp ◦ π)(cos t p̃ + sin t v)
dt t=0
d 1
= (cos t p̃ + sin t v) − p̃
dt t=0 cos t
= v,

completing the check.

One-dimensional Riemannian manifolds

Let (M, g) be a Riemannian manifold and let γ : [a, b] → M be a piecewise C 1 curve. Then the
length of γ is defined to be
Z b
(1.3.5) L(γ) = gγ(t) (γ ′ (t), γ ′ (t))1/2 dt.
a

It is easily seen that the length of a curve does not change under re-parametrization. Moreover,
every regular curve (i.e. satisfying γ ′ (t) 6= 0 for all t) admits a natural parametrization given by
arc-length. Namely, let Z t
s(t) = gγ(τ ) (γ ′ (τ ), γ ′ (τ ))1/2 dτ.
a
ds
Then = gγ(t) (γ ′ (t), γ ′ (t))1/2 (t) > 0, so s can be taken as a new parameter, and then
dt
L(γ|[0,s] ) = s

and

(1.3.6) (γ ∗ g)t = gγ(t) (γ ′ (t), γ ′ (t))dt2 = ds2 .

35
 Suppose now that (M, g) is a one-dimensional Riemannian manifold. Then any connected
component of M is diffeomorphic either to R or to S 1 . In any case, a neighborhood of any point
p ∈ M can be viewed as a regular smooth curve in M and, in a parametrization by arc-length,
the local expression of the metric g is the same, namely, given by (1.3.6). It follows that all the
one-dimensional Riemannian manifolds are locally isometric among themselves.

Lie groups ⋆
The natural class of Riemannian metrics to be considered in Lie groups is the class of Riemannian
metrics that possess some kind of invariance, be it left, right or both. Let G be a Lie group.
A left-invariant Riemannian metric on G is a Riemannian metric with respect to which the left
translations of G are isometries. Similarly, a right-invariant Riemannian metric is defined. A
Riemannian metric on G that is both left- and right-invariant is called a bi-invariant Riemannian
metric.
Left-invariant Riemannian metrics (henceforth, left-invariant metrics) are easy to construct on
any given Lie group G. In fact, given any inner product h, i in its Lie algebra g, which we identify
with the tangent space at the identity T1 G, one sets g1 = h, i and uses the left translations to pull
back g1 to the other tangent spaces, namely one sets


gx (u, v) = g1 d(Lx−1 )x (u) , d(Lx−1 )x (v) ,
where x ∈ G and u, v ∈ Tx G. This defines a smooth Riemannian metric, since g(X, Y ) is constant
(and hence smooth) for any pair (X, Y ) of left-invariant vector fields, and any smooth vector field
on G is a linear combination of left-invariant vector fields with smooth functions as coefficients. By
the very construction of g, the d(Lx )1 for x ∈ G are linear isometries, so the composition of linear
isometries d(Lx )y = d(Lxy )1 ◦ d(Ly )−1
1 is also a linear isometry for x, y ∈ G. This checks that all
the left-translations are isometries and hence that g is left-invariant. (Equivalently, one can define
g by choosing a global frame of left-invariant vector fields on G and declaring it to be orthonormal
at every point of G.) It follows that the set of left-invariant metrics in G is in bijection with the
set of inner products on g. Of course, similar remarks apply to right-invariant metrics.
Bi-invariant metrics are more difficult to come up with. Starting with a fixed left-invariant
metric g on G, we want to find conditions for g to be also right-invariant. Reasoning similarly as
in the previous paragraph, we see that it is necessary and sufficient that the d(Rx )1 for x ∈ G be
linear isometries. Further, by differentiating the obvious identity Rx = Lx ◦ Inn(x−1 ) at 1, we get
that
d(Rx )1 = d(Lx )1 ◦ Ad(x−1 )
for x ∈ G. From this identity, we get that g is right-invariant if and only if the Ad(x) : g → g for
x ∈ G are linear isometries with respect to h, i = g1 . In this case, h, i is called an Ad-invariant
inner product on g.
In view of the previous discussion, applying the following proposition to the adjoint repre-
sentation of a compact Lie group on its Lie algebra yields that any compact Lie group admits a
bi-invariant Riemannian metric.

1.3.7 Proposition Let ρ : G → GL(V ) be a representation of a Lie group on a real vector space
V such that the closure ρ(G) is relatively compact in GL(V ). Then there exists an inner product
h, i on V with respect to which the ρ(x) for x ∈ G are orthogonal transformations.
Proof. Let G̃ denote the closure of ρ(G) in GL(V ). Then ρ factors through the inclusion
ρ̃ : G̃ → GL(V ) and it suffices to prove the result for ρ̃ instead of ρ. By assumption, G̃ is compact,
so without loss of generality we may assume in the following that G is compact.

36
 Let h, i0 be any inner product on V and fix a right-invariant Haar measure dx on G. Set
Z
hu, vi = hρ(x)u, ρ(x)vi0 dx,
G

where u, v ∈ V . It is easy to see that this defines a positive-definite bilinear symmetric form h, i
on V . Moreover, if y ∈ G, then
Z
hρ(y)u, ρ(y)vi = hρ(x)ρ(y)u, ρ(x)ρ(y)vi0 dx
ZG
= hρ(xy)u, ρ(xy)vi0 dx
G
= hu, vi,

where in the last equality we have used that dx is right-invariant. Note that we have used the
compactness of G only to guarantee that the above integrands have compact support. 
In later chapters, we will explain the special properties that bi-invariant metrics on Lie groups
have.

Homogeneous spaces ⋆
It is apparent that for a generic Riemannian manifold (M, g), the isometry group Isom(M, g) is
trivial. Indeed, Riemannian manifolds with large isometry groups have a good deal of symmetries.
In particular, in the case in which Isom(M, g) is transitive on M , (M, g) is called a Riemannian
homogeneous space or a homogeneous Riemannian manifold . Explicitly, this means that given any
two points of M there exists an isometry of M that maps one point to the other. In this case, of
course it may happen that a subgroup of Isom(M, g) is already transitive on M .
Let (M, g) be a homogeneous Riemannian manifold, and let G be a subgroup of Isom(M, g)
acting transitively on M . Then the isotropy subgroup H at an arbitrary fixed point p ∈ M is
compact and M is diffeomorphic to the quotient space G/H. In this case, we also say that the
Riemannian metric g on M is G-invariant.
Recall that if G is a Lie group and H is a closed subgroup of G, then there exists a unique
structure of smooth manifold on the quotient G/H such that the projection G → G/H is a sub-
mersion and the action of G on G/H by left translations is smooth. (Theorem 0.4.18). A manifold
of the form G/H is called a homogeneous space. In some cases, one can also start with a homoge-
neous space G/H and construct G-invariant metrics on G/H. For instance, if G is equipped with a
left-invariant metric that is also right-invariant with respect to H, then it follows that the quotient
G/H inherits a quotient Riemannian metric such that the projection G → G/H is a Riemannian
submersion and the action of G on G/H by left translations is isometric. In this way, G/H becomes
a Riemannian homogeneous space. A particular, important case of this construction is when the
Riemannian metric on G that we start with is bi-invariant; in this case, G/H is called a normal
homogeneous space. In general, a homogeneous space G/H for arbitrary G, H may admit several
distinct G-invariant Riemannian metrics, or may admit no such metrics at all.
Let M = G/H be a homogeneous space, where H is the isotropy subgroup at p ∈ M . Then the
isotropy representation at p is the homomorphism

H → O(Tp M ), h 7→ dhp .

1.3.8 Lemma The isotropy representation of G/H at p is equivalent to the adjoint representation
of H on g/h.

37
1.3.9 Proposition a. There exists a G-invariant Riemannian metric on G/H if and only if
the image of the adjoint representation of H on g/h is relatively compact in GL(g/k).
b. In case the condition in (a) is true, the G-invariant metrics on G/H are in bijective corre-
spondence with the AdG (H)-invariant inner products on g/h.

1.4 Exercises

1 Show that the Riemannian product of (0, +∞) and S n−1 is isometric to the cylinder

C = { (x0 , . . . , xn ) ∈ Rn+1 | x21 + · · · + x2n = 1 and x0 > 0 }.

2 The catenoid is the surface of revolution in R3 with the z-axis as axis of revolution and the
catenary x = cosh z in the xz-plane as generating curve. The helicoid is the ruled surface in
R3 consisting of all the lines parallel to the xy plane that intersect the z-axis and the helix t 7→
(cos t, sin t, t).
a. Write natural parametrizations for the catenoid and the helicoid.
b. Consider the catenoid and the helicoid with the metrics induced from R3 , and find the local
expressions of these metrics with respect to the parametrizations in item (a).
c. Show that the local expressions in item (b) coincide, possibly up to a change of coordinates,
and deduce that the catenoid and the helicoid are locally isometric.
d. Show that the catenoid and the helicoid cannot be isometric because of their topology.

3 Consider the real hyperbolic space (RH n, g) as defined in section 1.3. Let Dn be the open unit
disk of Rn embedded in Rn+1 as

Dn = { (x0 , . . . , xn ) ∈ Rn+1 | x0 = 0 and x21 + · · · + x2n < 1 }.

Define a map f : RH n → Dn by setting f (x) to be the unique point of Dn lying in the line joining
x ∈ RH n and the point (−1, 0, . . . , 0) ∈ Rn+1 . Prove that f is a diffeomorphism and, setting
g1 = (f −1 )∗ g, we have that

4

g1 |x = 2
dx21 + · · · + dx2n ,
(1 − hx, xi)

where x = (0, x1 , . . . , xn ) ∈ Dn . Deduce that RH n is conformally flat.

(Dn , g1 ) is called the Poincaré disk model of RH n .

4 Consider the open unit disk Dn = { (x1 , . . . , xn ) ∈ Rn | x21 + · · · x2n < 1 } equipped with the
metric g1 as√in Exercise 3. Prove that the inversion of Rn on the sphere of center (−1, 0, . . . , 0)
and radius 2 defines a diffeomorphism f1 from Dn onto the upper half-space

Rn+ = { (x1 , . . . , xn ) ∈ Rn | x1 > 0 },

and that the metric g2 = (f1−1 )∗ g1 is given by

1

g2 |x = 2 dx21 + · · · + dx2n ,
x1

where x = (x1 , . . . , xn ) ∈ Rn+ .

(Rn+ , g2 ) is called the Poincaré upper half-space model of RH n .

38
5 Consider the Poincaré

upper half-plane model R2+ = { (x, y) ∈ R2 | y > 0 } with the metric
g2 = y2 dx + dy (case n = 2 in Exercise 4). Check that the following transformations of R2+
1 2 2

into itself are isometries:

a. τa (x, y) = (x + a, y) for a ∈ R;
b. hr (x, y) = (rx, ry) for r > 0; 
x y
c. R(x, y) = , .
x2 + y 2 x2 + y 2
Deduce from (a) and (b) that R+ 2 is homogeneous.

6 Use stereographic projection to prove that S n is conformally flat.

7 Consider the parametrized curve


x = t − tanh t
1
y = cosh t

The surface of revolution in R3 constructed by revolving it around the x-axis is called the pseudo-
sphere. Note that the pseudo-sphere is singular along the circle obtained by revolving the point
(0, 1).
a. Prove that the pseudo-sphere with the singular circle taken away is locally isometric to the
upper half plane model of RH 2 .
b. Show that the Gaussian curvature of the pseudo-sphere is −1.

8 Let Γ be the lattice in Rn defined by the basis {v1 , . . . , vn }, and denote by gΓ the Riemannian
metric that it defines on T n . Show that in some product chart of T n = S 1 × · · · × S 1 the local
expression X
gΓ = hvi , vj i dxi ⊗ dxj
i,j

holds, where h, i denotes the standard scalar product in Rn .

9 Let Γ and Γ′ be two lattices in Rn , and denote by gΓ , gΓ′ the Riemannian metrics that they
define on T n , respectively.
a. Prove that (T n , gΓ ) is isometric to (T n , gΓ′ ) if and only if there exists an isometry f : Rn → Rn
such that f (Γ) = Γ′ . (Hint: You may use the result of exercise 2 of chapter 3.)
b. Use part (a) to see that (T n , gΓ ) is isometric to the Riemannian product of n copies of S 1 if
and only if Γ is the lattice associated to an orthonormal basis of Rn .

10 Let Γ be the lattice of R2 spanned by an orthogonal basis {v1 , v2 } and consider the associated
rectangular flat torus T 2 .
a. Prove that the map γ of R2 defined by γ(x1 v1 +x2 v2 ) = (x1 + 21 )v1 −x2 v2 induces an isometry
of T 2 of order two.
b. Prove that T 2 double covers a Klein bottle K 2 .

11 Prove that Rn r {0} is isometric to the warped product ((0, +∞) × S n−1 , dr2 + r2 g), where r
denotes the coordinate on (0, +∞) and g denotes the standard Riemannian metric on S n−1 .

12 Let G be a Lie group equal to one of O(n), U(n) or SU(n), and denote its Lie algebra by g.
Prove that for any c > 0
hX, Y i = −c trace (XY ),
where X, Y ∈ g, defines an Ad-invariant inner product on g.

39
13 Consider the special unitary group SU(2) equipped with a bi-invariant metric induced from
an Ad-invariant inner product on su(2) as in the previous exercise with c = 21 . Show that the map
   
α −β̄ α
7→
β ᾱ β

where α, β ∈ C and |α|2 + |β|2 = 1, defines an isometry from SU(2) onto S 3 . Here C2 is identified
with R4 and S 3 is viewed as the unit sphere in R4 .

14 Show that RP 1 equipped with the quotient metric from S 1 (1) is isometric to S 1 ( 21 ). Show
that CP 1 equipped with the Fubini-Study metric is isometric to S 2 ( 12 ).

15 (Sylvester’s law of inertia) Let B : V × V → R be a symmetric bilinear form on a finite-

dimensional real vector space V . For each basis E = (e1 , . . . , en ) of V , we associate a symmetric
matrix BE = (B(ei , ej )).
a. Check that B(u, v) = vE t B u for all u, v ∈ V , where u (resp. v ) denotes the column
E E E E
vector representing the vector u (resp. v) in the basis E.
b. Suppose F = (f1 , . . . , fn ) is another basis of V such that
   
e1 f1
 ..   . 
 .  = A  ..  .
en fn

for a real matrix A of order n. Show that BE = ABF At .

c. Prove that there exists a basis E of V such that BE has the form
 
In−i−k 0 0
 0 −Ii 0  ,
0 0 0k

where Im denotes an identity block of order m, and 0m denotes a null block of order m.
d. Prove that there is a B-orthogonal decomposition

V = V+ ⊕ V− ⊕ V0

where B is positive definite on V+ and negative definite on V− , V0 is the kernel of B (the

set of vectors B-orthogonal to V ), i = dim V− and k = dim V0 . Prove also that i is the
maximal dimension of a subspace of V on which B is negative definite. Deduce that i and k
are invariants of B. They are respectively called the index and nullity of B. Of course, B is
nondegenerate if and only if k = 0, and it is positive definite if and only if k = i = 0.
e. Check that the Lorentzian metric of R1,n restricts to a positive definite symmetric bilinear
form on the tangent spaces to the hyperboloid modeling RH n .

1.5 Additional notes

§1 Riemannian manifolds were defined as abstract smooth manifolds equipped with Riemannian
metrics. One class of examples of Riemannian manifolds is of course furnished by the Riemannian
submanifolds of Euclidean space. On the other hand, a very deep theorem of Nash [Nas56] states
that every abstract Riemannian manifold admits an isometric embedding into Euclidean space, so
that it can be viewed as an embedded Riemannian submanifold of Euclidean space. In view of this,

40
one might be tempted to ask why bother to consider abstract Riemannian manifolds in the first
place. The reason is that Nash’s theorem is an existence result: for a given Riemannian manifold,
it does not supply an explicit embedding of it into Euclidean space. Even if an isometric embedding
is known, there may be more than one (up to congruence) or there may be no canonical one. Also,
an explicit embedding may be too complicated to describe. Finally, a particular embedding is
sometimes distracting because it highlights some specific features of the manifold at the expense of
some other features, which may be undesirable.
§2 From the point of view of foundations of the theory of smooth manifolds, the following
assertions are equivalent for a smooth manifold M whose underlying topological space is assumed
to be Hausdorff but not necessarily second-countable:
a. The topology of M is paracompact.
b. M admits smooth partitions of unity.
c. M admits Riemannian metrics.
In fact, as is standard in the theory of smooth manifolds, second-countability of the topology of
M (together with the Hausdorff property) implies its paracompactness and this is used to prove
the existence of smooth partitions of unity [War83, chapter 1]. Next, Riemannian metrics are
constructed on M by using partitions of unity as we did in Proposition 1.2.3. Finally, the underlying
topology of a Riemannian manifold is metrizable according to Proposition 3.2.3, and every metric
space is paracompact.
§3 The pseudo-sphere constructed in Exercise 7 was introduced by Beltrami [Bel68] in 1868 as a
local model for the Lobachevskyan geometry. This means that the geodesic lines and their segments
on the pseudo-sphere play the role of straight lines and their segments on the Lobachevsky plane.
In 1900, Hilbert posed the question of whether there exists a surface in three-dimensional Euclidean
space whose intrinsic geometry coincides completely with the geometry of the Lobachevsky plane.
Using a simple reasoning, it follows that if such a surface does exist, it must have constant negative
curvature and be complete (see chapter 3 for the notion of completeness).
As early as 1901, Hilbert solved this problem [Hil01] (see also [Hop89, chapter IX]), and in
the negative sense, so that no complete surface of constant negative curvature exists in three-
dimensional Euclidean space. This theorem has attracted the attention of geometers over a number
of decades, and continues to do so today. The reason for this is that a number of interesting
questions are related to it and to its proof. For instance, the occurrence of a singular circle on the
pseudo-sphere is not coincidental, but is in line with Hilbert’s theorem.

41