Tangent Spaces

Luigi T. Sousa
July 2020

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Contents
1 Why tangent spaces? 3

2 A quick addendum on maps between manifolds 4

3 The main dish 5

3.1 Tangent Vectors and Spaces . . . . . . . . . . . . . . . . . . . . . 5
3.2 Cotangent Vectors and Spaces . . . . . . . . . . . . . . . . . . . . 7
3.3 Tensors on differentiable manifolds . . . . . . . . . . . . . . . . . 8

4 References 12

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1 Why tangent spaces?
The reason tangent spaces are important is so we can approximate our man-
ifolds by linear spaces and as such, use the usual rules of calculus and linear
(and multilinear) algebra to work with them, at least at a local scale, since this
spaces can sometimes be too general to build a consistent algebraic structure,
for instance, how do one define a ”origin” in a sphere?
So, we must then define our tangent spaces, and we may want to look back
to how we dealt with tangent planes in multivariable calculus.

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2 A quick addendum on maps between mani-
folds
Before we talk about our main topic, we may wanna talk about another type
of map, which is the main type of map used when dealing with differentiable
manifolds: diffeomorphisms.
-Definition(2): let M and N be two differentiable manifolds and let

f :M →N

be a homeomorphism, if we have charts φ and ψ on M and N , respectively, then

if the composite map
φ−1 ◦ f ◦ ψ : Rm → Rn
and it’s inverse
ψ −1 ◦ f −1 ◦ φ : Rn → Rm

f −1
M N
f

φ φ−1 ψ ψ −1

ψ −1 ◦ f −1 ◦ φ
Rm Rn
φ−1 ◦ f ◦ ψ

are both C ∞ , then we say f is a diffeomorphism1 and that M is diffeomor-

phic to N , denoted as M ≡ N .
*Note 1: every diffeomorphism is a homeomorphism, but not every √ home-
omorphism is a diffeomorphism, for instance, take the function f (x) = 3 x, it
is a homeomorphism, but it fails to be a diffeomorphism since it’s derivative
2
f 0 (x) = 13 x− 3 isn’t continuous at x = 0.
*Note 2: since every diffeomorphism is also a homeomorphism, if M ≡ N
then it follows from the Invariance of Domain theorem that dim(M ) = dim(N ).

1 Actually, this is a smooth diffeomorphism, where a r-diffeomorphism is the same but

where our composite maps are C r , 0 < r < ∞, and we denote M ' N

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3 The main dish
To talk about tangent spaces, it would be nice to recap some concepts from
multivariable calculus: curves and (multivariable) functions. In calculus, we
learn that a (open) curve is a function

c : (a, b) ⊂ R → Rn ,

usually with a < 0 < b for convenience, while a closed curve is parametrized as

c : S 1 → Rn ,

and a scalar function in Rn is just

F : Rn → R.

Generalizing these concepts to a arbitrary m-manifold is quite straight for-

ward actually: a curve in M will just be a function

c : (a, b) ⊂ R → M

for a open curve, and

c : S1 → M
for an closed curve, while our scalar function will just be

F : M → R.

If a chart (U, φ) is given, then c and F have, respectively, coordinates represen-

tation
φ ◦ c : (a, b) ⊂ R → Rm
or
φ ◦ c : S 1 → Rm
and
F ◦ φ−1 : Rm → R,
which are just a regular curve and multivariable function in Rm . We also denote
the set of smooth functions on M by F(M ). It is important to note that, just as
our manifold exists independent of any coordinate system, so does curves and
functions defined on manifolds, and coordinates must be used just for conve-
nience when needed.

3.1 Tangent Vectors and Spaces

With the concepts of curves and functions on manifolds, we are now able to
define a important geometrical object: vectors. This will be done in a similar
manner to how it was done in multivariable calculus, using directional deriva-
tives.

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 -Definitions(3.1): let M be a m-manifold and c and F be, respectively, a
curve and a scalar function defined

on M , where t = 0 ∈ (a, b) and F is defined
along the curve c, i.e, c (a, b) ⊂ Dom{F }(the domain
of F ) and such that
c(0) = p. Then, if we take the derivative of F c(t) at t = 0, we have (given a
chart (U, x))



dF c(t)  ∂F dxµ c(t) 
= , (1)
dt 
t=0 ∂xµ dt 
t=0
where we made use of the chain rule to express the derivative in terms of the local
coordinates xµ , giving us the directional derivative of F in the direction given
µ
by the curve c. If we rearrange things on equation (1) and write dx dt |t=0 = V
µ

we get


∂F dxµ c(t)  ∂F ∂
= V µ µ = V µ µ (F ) =: V [F ], (2)
∂xµ dt 
t=0 ∂x ∂x
where the last equality defines the operator V = V µ ∂x∂ µ , which is what is
def ined as the tangent vector to M at the point p.
We can define V µ ∂x∂ µ as a vector since differential operators are also linear
operators. Being a bit more mathematical, given two curves c1 (t) and c2 (t), if
they satisfy

(i) c1 (0) = c2 (0) = p and


dxµ c1 t  dxµ c2 t 

(ii) = ,
dt 
t=0 dt 
t=0
then we make an equivalence relation between c1 (t) and c2 (t), since both of
them will yield the same differential operator on F at p, so we identify our
tangent vector with the equivalence class of curves
dxµ (c̃(t))  dxµ (c(t)) 
    

[c(t)] = c̃(t)c̃(0) = c(0) and
 = .
dt 
t=0 dt 
t=0

The set of all the equivalence classes of curves at p ∈ M form the tangent
space of M at p, denoted Tp M ,

Tp M := [c]p c : R → M , c ∈ C ∞ and c(0) = p ,

 

which, since we can identify every equivalence class of curves as a differential

operator (vector), we may as well view Tp M as the space formed by all possible
such differential operators. A natural choice for a basis on Tp M would be
eµ = ∂x∂ µ (1 ≤ µ ≤ m), which is called the coordinate basis, and we have
that dim(Tp M ) = dim(M ). Since curves exist independently of any coordinate
systems, so does tangent vectors to M , and hence we can find a transformation
property for said vectors between any two given coordinate systems at p: given
two charts (Ui , x) and (Uj , y) with p ∈ Ui ∩ Uj we give rise to two distinct
expressions for the operator V :

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 ∂ ∂
V =Vµ µ
= Ve µ µ , (3)
∂x ∂y
which suggests that V µ and Ve µ are related by
µ
∂y
Ve µ = V ν ν , (4)
∂x
since, if we substitute (4) into (3) we get
∂ ∂y µ ∂ ∂ ∂
Ve µ µ = V ν ν µ = V ν ν = V µ µ ,
∂y ∂x ∂y ∂x ∂x
where we made use of the chain rule again and of the fact that summation indices
are dummy µ
indices, so we can change them at will without problem. The coef-
ficients ∂y∂x are the components of the Jacobian matrix of the transformations
ν

between the two coordinate systems.

It’s also good to realise that the basis for Tp M need not to be the coordinate
basis {eµ } and one may want to instead use a basis ẽν = Aνµ eµ , where A =
(Aνµ ) ∈ GL(m, R), which is called a non − coordinate basis.

3.2 Cotangent Vectors and Spaces

Since we can construct a vector space Tp M from our manifold M , it would
be nice to also talk about it’s dual space:
-Definitions(3.2): the cotangent space to M at p is defined as the
dual vector space of Tp M , denoted as Tp∗ M , i.e, the space of all linear function-
als ω : Tp M → R, and an element ω ∈ Tp∗ M is called a cotangent vector, dual
vector or, when talking about differential forms, a one-form. The simplest
example of a one-form is the differential dF of a function F ∈ F(M ), where the
action of dF ∈ Tp∗ M on V ∈ Tp M is defined as
∂F
hdF, V i := V [F ] = V µ ∈ R. (5)
∂xµ
If we are given a chart (U, x) for the neighbourhood of p, then we can expand
dF in terms of the local coordinates xµ (p) as the total differential of F ,
∂F
dF = dxµ ,
∂xµ
giving rise to a natural candidate to a basis on Tp∗ M : {dxµ }. Because of our
definition of the action of dF on V , it also agrees with the intuition of it being
the dual basis:
∂xµ
 
µ ∂
dx , ν = = δνµ .
∂x ∂xν
If two different charts (Ui , x) and (Uj , y) are given, if we have p ∈ Ui ∩ Uj ,
them we can find the relation between the representation of a element ω ∈ Tp∗ M
on both coordinate systems:
ω = ωµ dxµ = ω̃µ dy µ =⇒

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 ∂xν
=⇒ ω̃µ = ων , (6)
∂y µ
since if we substitute just like when dealing with vectors, we will get
∂xν µ
ω̃µ dy µ = ων dy = ων dxν = ωµ dxµ ,
∂y µ
where once again summation indices are dummy indices, but unlike the case of
Tp M , instead of the chain rule we just made use of the identity of differentials
of functions
∂xν µ
dxν = dy .
∂y µ

3.3 Tensors on differentiable manifolds

Since we are able to assign a tangent vector and covector to each point of our
manifold M , we can also talk about another object with geometrical importance:
tensors.
Note that in equation (4) we had the coefficients of the transformation being
∂y µ
∂xν , which is the Jacobian matrix of the transformation from coordinates xν
ν
∂x
into coordinates y µ , but in (6) we have ∂y µ , i.e, the Jacobian matrix of the

inverse transformation going from coordinates y µ into coordinates xν , which

are themselves inverse matrices to each other, since
∂y µ ∂xα ∂y µ
α ν
= = δνµ , or
∂x ∂y ∂y ν
∂xµ ∂y α ∂xµ
= = δνµ ,
∂y α ∂xν ∂xν
by the chain rule once more. It’s precisely these transformation rules that distin-
guishes regular (or contra-variant) vectors from dual (or co-variant) covectors,
and is also a important notion to characterize what are tensors.
-Definition(3.3): a tensor of type (q, r) at a point p of M is a multi-linear
map which takes q elements from Tp∗ M and r elements from Tp M and sends
them to a real number T (ω(1) , ..., ω(q) ; V(1) , ..., V(r) ). The set of tensors of type
q
(q, r) at p ∈ M is the tensor space of type (q, r) at p ∈ M , denoted by Tr,p (M ).
q
An element T ∈ Tr,p (M ) can be expanded in terms of local coordinates using
the bases discussed earlier:
∂ ∂
T = T µ1 ...µqν1 ...νr ⊗ ... ⊗ ⊗ dxν1 ⊗ ... ⊗ dxνr .
∂xµ1 ∂xµq
µi ∂ νj
Given r vectors V(i) = V(i) ∂xµi and q covectors ω(j) = ω(j)νj dx , then the
action of T on all V(i) and ω(j) is (index mayhem alert):

T (ω(1) , ..., ω(q) ; V(1) , ..., V(r) ) =

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 ∂ ∂
= T µ1 ...µqν1 ...νr µ
⊗ ... ⊗ µq ⊗ dxν1 ⊗ ... ⊗ dxνr (ω(1) , ..., ω(q) ; V(1) , ..., V(r) ) =
∂x 1 ∂x
∂
= T µ1 ...µqν1 ...νr µ1 (ω(1)ν1 dxν1 )... 2
∂x
   
∂ νq ν1 µ1 ∂ νr µr ∂
... µq (ω(q)νq dx )dx V(1) µ1 ...dx V(r) µr =
∂x ∂x ∂x
∂
= T µ1 ...µqν1 ...νr ω(1)ν1 µ1 (dxν1 )...
∂x
   
∂ µ1 ∂ µr ∂
...ω(q)νq µq (dxνq )V(1) dxν1 ...V(r) dx νr
=
∂x ∂xµ1 ∂xµr
µ1 µr ν1
= T µ1 ...µqν1 ...νr ω(1)ν1 ...ω(q)νq V(1) ...V(r) δµ1 ...δµνqq δµν11 ...δµνrr =
ν1 νr
= T µ1 ...µqν1 ...νr ω(1)µ1 ...ω(q)µq V(1) ...V(r) ,
thus arriving at the identity
ν1 νr
T (ω(1) , ..., ω(q) ; V(1) , ..., V(r) ) = T µ1 ...µqν1 ...νr ω(1)µ1 ...ω(q)µq V(1) ...V(r) ,

which holds for any tensor constructed from Tp M and Tp∗ M .

Recall from earlier that a tensor exists independently of any coordinate sys-
tem, since it’s just a direct product of q vectors with r covectors, which them-
selves do not depend on the choice of coordinates and have very specific trans-
formation rules. Let’s use this transformation rules to get the transformation
q
rules for general tensors: given a element T ∈ Tr,p (M ) and two charts (Ui , x)
and (Uj , y), with p ∈ Ui ∩ Uj , it’s representation in the local coordinate basis is

∂ ∂
T = T µ1 ...µqν1 ...νr ⊗ ... ⊗ ⊗ dxν1 ⊗ ... ⊗ dxνr =
∂xµ1 ∂xµq
 α1   αq 
µ1 ...µq ∂y ∂ ∂y ∂
=T ν1 ...νr ⊗ ... ⊗ ⊗
∂xµ1 ∂y α1 ∂xµq ∂y αq
 ν1   νr 
∂x β1 ∂x βr
⊗ dy ⊗ ... ⊗ dy = (7)
∂y β1 ∂y βr
∂y α1 ∂y αq ∂xν1 ∂xνr ∂ ∂
= T µ1 ...µqν1 ...νr µ
... µq β1 ... βr α1 ⊗ ... ⊗ αq ⊗ dy β1 ⊗ ... ⊗ dy βr ,
∂x 1 ∂x ∂y ∂y ∂y ∂y
which, by construction, we know must be equal to the representation of T in
the y coordinates:

α ...α ∂ ∂
Te 1 qβ1 ...βr α1 ⊗ ... ⊗ αq ⊗ dy β1 ⊗ ... ⊗ dy βr =
∂y ∂y
∂y α1 ∂y αq ∂xν1 ∂xνr ∂ ∂
= T µ1 ...µqν1 ...νr µ
... µq β1 ... βr α1 ⊗ ... ⊗ αq ⊗ dy β1 ⊗ ... ⊗ dy βr ,
∂x 1 ∂x ∂y ∂y ∂y ∂y
2 The notation here is a bit wonky, here we’re NOT differentiating the one-forms ωj but
rather taking the inner product of ∂x∂µj and ωj

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and since T is a linear quantity, both sides of the equation will be equal iff the
components multiplying the bases are equal, and since the bases on both sides
are exactly the same, we can focus our attention on just the components, giving
us the transformation rule for a general tensor T :

α ...α ∂y α1 ∂y αq ∂xν1 ∂xνr

Te 1 qβ1 ...βr = T µ1 ...µqν1 ...νr µ1 ... µq β1 ... βr . (8)
∂x ∂x ∂y ∂y

-Examples: a tensor of type (0, 1) is just a dual vector: it maps a single

element of Tp M to a scalar and has coordinate representation

ωµ dxµ .

Similarly, a type (1, 0) tensor is just a vector mapping a covector to a scalar,

which in component form looks like
∂
Vµ .
∂xµ
A linear transformation is a example of a tensor of type (1, 1), since when
applied to a single vector it maps it to another vector and same for covectors,
maps one covector to another, but may act on one vector and one dual vector
to give a scalar:
Aµν V ν ≡ Ve µ
ωµ Aµν ≡ ω̃ν
ωµ Aµν V ν = ω̃ν V ν = ωµ Ve µ .
To get a better understanding of why A is a (1, 1)-tensor and not a (2, 0) or
(0, 2)-tensor, recall from linear algebra that a property all linear operators have
is that when switching basis, they transform via the relation
e = U−1 AU,
A

where U is the change of basis matrix, which in our context is just the Jacobian
∂y µ
∂xν , so we have:
∂y α µ ∂xν α
µ ∂y ∂x
ν
A ν = A ν ,
∂xµ ∂y β ∂xµ ∂y β
which is precisely the transformation law for (1, 1)-tensors.
Last but not least, a example of a (0, 2)-tensor that is quite important for
differential geometry and for physics is the metric tensor, which maps two
vectors from Tp M to a real number, namely the inner product between the two
vectors, with coordinate representation

gµν dxµ ⊗ dxν

V µ gµν U ν = Vν U ν = V µ Uµ ,

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which, when applied to a difference of two vectors, gives the squared distance
between both vectors

(V µ − U µ )gµν (V ν − U ν ) ≡ hV − U, V − U i ≡ d(V, U ).

One can also think of g acting on just one vector, in which case it is a iso-
morphism between Tp M and Tp∗ M , sending a element V ∈ Tp M to it’s unique
counterpart in Tp∗ M , turning V into a dual vector, which in turn maps one more
vector to a scalar.
The tensor g is called the metric tensor because it can be used to construct
a sensible notion of a metric on a manifold, when applied to infinitesimally
close vectors, being in a way the derivative of the distance function on M , and
through integration being able to define arc lengths along curves in M .

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4 References
1. Geometry, Topology and Physics, Second Edition, by Mikio Nakahara
2. INTRODUCTION TO DIFFERENTIAL GEOMETRY, by Joel W. Rob-
bin and Dietmar A. Salamon
3. https://en.wikipedia.org/wiki/Diffeomorphism

4. https://en.wikipedia.org/wiki/Jacobian_matrix_and_determinant
5. https://en.wikipedia.org/wiki/Tensor

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