In this section we shall
- give one base for the tangent and cotangent space for each chart at a point of a manifold,
- show how to convert representations in one base into another,
- define the differentials of functions from a manifold to the real line, from an interval to a manifold and from a manifold to another manifold,
- and prove the chain, product and quotient rules for those differentials.
Definition 2.1:
Let
be a
-dimensional manifold of class
with
and atlas
, let
and let
. We define for every
and
,
:

In the following, we will show that these functionals are a basis of the tangent space.
Theorem 2.2: Let
be a
-dimensional manifold of class
with
and atlas
, let
and let
. For all
:

i. e. the function
is contained in the tangent space
.
Proof:
Let
.
1. We show linearity.

From the second to the third line, we used the linearity of the derivative.
2. We show the product rule.

From the second to the third line, we used the product rule of the derivative.
3. It follows from the definition of
, that
if
is not defined at
.
Lemma 2.3: Let
be a
-dimensional manifold of class
with atlas
, and let
. If we write
, then we have for each
, that
.
Proof:
Let
. Since
is an atlas,
and
are compatible. From this follows that the function

is of class
. But if we denote by
the function

, which is also called the projection to the
-th component, then we have:

It is not difficult to show that
is contained in
, and therefore the function

is contained in
as a composition of
-times continuously differentiable functions (or continuous functions if
).
Lemma 2.4: Let
be a
-dimensional manifold of class
with
and atlas
, let
and let
. If we write
we have:

Note that due to lemma 2.3,
for all
, which is why the above expression makes sense.
Proof:
We have:

Further,

and

Inserting this in the above limit gives the lemma.
Theorem 2.5: Let
be a
-dimensional manifold of class
with
and atlas
, let
and let
. The tangent vectors

are linearly independent.
Proof:
We write again
.
Let
. Then we have for all
:


Lemma 2.6:
Let
be a manifold with atlas
,
,
be open, let
and
for a
; i. e.
is a constant function. Then
and
.
Proof:
1. We show
.
By assumption,
is open. This means the first part of the definition of a
is fulfilled.
Further, for each
and
, we have:

This is contained in
.
2. We show that
.
We define
. Using the two rules linearity and product rule for tangent vectors, we obtain:

Subtracting
, we obtain
.
Proof:
Let
be open, and let
be contained in
.
Case 1:
.
In this case,
and
, since
is not defined at
and both
and
are tangent vectors. From this follows the formula.
Case 2:
.
In this case, we obtain that the set
is open in
as follows: Since
is a homeomorphism by definition of charts, the set
is open in
. By definition of the subspace topology, we have
for a
open in
. But
is open in
as the intersection of two open sets; recall that
was required to be open in the definition of a chart.
Furthermore, from
and
it follows that
, and therefore
. Since
is open, we find an
such that the open ball
is contained in
. We define
. Since
is bijective,
, and since
is a homeomorphism, in particular continuous,
is open in
with respect to the subspace topology of
. From this also follows
open in
, because if
is open in
, then by definition of the subspace topology it is of the form
for an open set
, and hence it is open as the intersection of two open sets.
We have that
, is contained in
:
is an open subset of
, and if
, then
,
(check this by direct calculation!), which is contained in
as the restriction of an arbitrarily often continuously differentiable function.
We now define the function
,
, and further for each
, we define

From the fundamental theorem of calculus, the multi-dimensional chain rule and the linearity of the integral follows for each
, that

If one sets
for
, one obtains, inserting the definition of
:

Now we define the functions

These are contained in
since they are defined on
which is open, and further, if
, then

, which is arbitrarily often differentiable by the Leibniz integral rule as the integral of a composition of arbitrarily often differentiable functions on a compact set.
Further, again denoting
, the functions
,
are contained in
due to lemma 2.3.
Since
,
is defined. We apply the rules (linearity and product rule) for tangent vectors and lemma 2.6 (we are allowed to do so because all the relevant functions are contained in
), and obtain:

, since due to our notation it's clear that
.
But

Thus we have successfully shown

But due to the definition of subtraction on
, due to lemma 2.6, and due to the fact that the constant zero function is a constant function:

Due to linearity of
follows
, i. e.
. Now, inserting in the above equation gives the theorem.
Together with theorem 2.5, this theorem shows that

is a basis of
, because a basis is a linearly independent generating set. And since the dimension of a vector space was defined to be the number of elements in a basis, this implies that the dimension of
is equal to
.
Definition 2.8:
Let
be a
-dimensional manifold of class
and atlas
, let
and let
. We write
. Then we define for
:

Note that
is well-defined because of lemma 2.3.
Theorem 2.9: Let
be a
-dimensional manifold of class
and atlas
, let
and let
. For all
,
is contained in
.
Proof:
By definition,
maps from
to
. Thus, linearity is the only thing left to show. Indeed, for
and
, we have, since addition and scalar multiplication in
are defined pointwise:


Lemma 2.10: Let
be a
-dimensional manifold of class
and atlas
, let
and let
. For
, the following equation holds:

Proof:
We have:


Theorem 2.11: Let
be a
-dimensional manifold of class
and atlas
, let
and let
. The cotangent vectors
are linearly independent.
Proof:
Let
, where by
we mean the zero of
. Then we have for all
:


Proof:
Let
and
. Due to theorem 2.7, we have

Therefore, and due to the linearity of
(because
was the space of linear functions to
):

Since
was arbitrary, the theorem is proven.
From theorems 2.11 and 2.12 follows, as in the last subsection, that

is a basis for
, and that the dimension of
is equal to
, like the dimension of
.
Expressing elements of the tangent and cotangent spaces in different bases
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If
is a manifold,
and
are two charts in
's atlas such that
and
. Then follows from the last two subsections, that
and
are bases for
, and
and
are bases for
.
One could now ask the questions:
If we have an element
in
given by
, then how can we represent
as linear combination of the basis
?
Or if we have an element
in
given by
, then how can we represent
as linear combination of the basis
?
The following two theorems answer these questions:
Proof:
Due to theorem 2.7, we have for
:

From this follows:


Proof:
Due to theorem 2.12, we have for
:

Thus we obtain:


In this subsection, we will define the pullback and the differential. For the differential, we need three definitions, one for each of the following types of functions:
- functions from a manifold to another manifold
- functions from a manifold to

- functions from an interval
to a manifold (i. e. curves)
For the first of these, the differential of functions from a manifold to another manifold, we need to define what the pullback is:
Lemma 2.16: Let
be a
-dimensional and
be a
-dimensional manifold, let
and let
be differentiable of class
. Then
is continuous.
Proof:
We show that for an arbitrary
,
is continuous on an open neighbourhood of
. There is a theorem in topology which states that from this follows continuity.
We choose
in the atlas of
such that
, and
in the atlas of
such that
. Due to the differentiability of
, the function

is contained in
, and therefore continuous. But
and
are charts and therefore homeomorphisms, and thus the function

is continuous as the composition of continuous functions.
Lemma 2.17: Let
be two manifolds, let
be differentiable of class
, and let
be defined on the open set
. In this case, the function
is contained in
; i. e. the pullback with respect to
really maps to
.
Proof:
Since
is continuous due to lemma 2.16,
is open in
. Thus
is defined on an open set.
Let
be an arbitrary element of the atlas of
and let
be arbitrary. We choose
in the atlas of
such that
. The function

is
-times continuously differentiable (or continuous if
) at
as the composition of two
times continuously differentiable (or continuous if
) functions. Thus, the function

is
-times continuously differentiable (or continuous if
) at every point, and therefore contained in
.
Theorem 2.19:
Let
be two manifolds of class
, let
be differentiable of class
and let
. We have
; i. e. the differential of
at
really maps to
.
Proof:
Let
be open,
and
be arbitrary. In the proof of the following, we will use that for all open subsets
,
(which follows from the linearity of
).
1. We prove linearity.

2. We prove the product rule.


Theorem 2.22: Let
be a manifold of class
,
, let
be an interval, let
and let
be a differentiable curve of class
. Then
is contained in
for every
and
is a tangent vector of
at
.
Proof:
1. We show
Let
be arbitrary, and let
be the set where
is defined (
is open by the definition of
functions. We choose
in the atlas of
such that
. Then the function

is contained in
as the composition of two
times continuously differentiable (or continuous if
) functions.
Thus,
is
times continuously differentiable (or continuous if
) at every point, and hence
times continuously differentiable (or continuous if
).
2. We show that
in three steps:
Let
and
.
2.1 We show linearity.
We have:

2.2 We prove the product rule.

2.3 It follows from the definition of
that
is equal to zero if
is not defined at
.
Linearity of the differential for Ck(M), product, quotient and chain rules
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In this subsection, we will first prove linearity and product rule for functions from a manifold to
.
Proof:
1. We show that
.
Let
be the (open as intersection of two open sets) set on which
is defined, and let
be contained in the atlas of
. The function

is contained in
as the linear combination of two
functions.
2. We show that
.
For all
and
, we have:


Remark 2.24: This also shows that for all
,
.
Proof:
1. We show that
.
Let
be the (open as intersection of two open sets) set on which
is defined, and let
be contained in the atlas of
. The function

is contained in
as the product of two
functions.
2. We show that
.
For all
and
, we have:


Proof:
1. We show that
:
Let
be the (open as the intersection of two open set) set on which
is defined, and let
be in the atlas of
such that
. The function

is contained in
as the quotient of two
from which the function in the denominator vanishes nowhere.
2. We show that
:
Choosing
as the constant one function, we obtain from 1. that the function
is in
. Hence follows from the product rule:

which, through equivalent transformations, can be transformed to

From this and from the product rule we obtain:


Proof:
1. We already know that
is differentiable of class
; this is what lemma 2.17 says.
2. We prove that
.
Let
. Then we have:


Now, let's go on to proving the chain rule for functions from manifolds to manifolds. But to do so, we first need another theorem about the pullback.
Theorem 2.28:
Let
be three manifolds, and let
and
be two functions differentiable of class
. Then

Proof:
Let
. Then we have:


Proof:
1. We prove that
is differentiable of class
.
Let
be contained in the atlas of
and let
be contained in the atlas of
such that
, and let
be arbitrary. We choose
in the atlas of
such that
.
We have
; indeed,
due to the choice of
and
because
. Further, we choose
. Then the function

is contained in
as the composition of two
functions.
Thus,
is
times continuously differentiable (or continuous if
) at every point, and thus
times continuously differentiable (or continuous if
).
2. We prove that
.
For all
and
, we have:


Proof:
1. Among another thing, theorem 2.22 states that
is contained in
.
2. We show that
:


In this section, we want to prove that what we defined as the tangent space is isomorphic to a space whose elements are in analogy to tangent vectors to, say, tangent vectors of a function
.
We start by proving the following lemma from linear algebra:
Proof:
We only prove that
is a vector space isomorphism; that
and
are also vector space isomorphisms will follow in exactly the same way.
From
and
follows that
is the inverse function of
.
- Torres del Castillo, Gerardo (2012). Differentiable Manifolds. Boston: Birkhäuser. ISBN 978-0-8176-8271-2.