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
,
:
![{\displaystyle \left({\frac {\partial }{\partial \phi _{j}}}\right)_{p}:{\mathcal {C}}^{n}(M)\to \mathbb {R} ,\left({\frac {\partial }{\partial \phi _{j}}}\right)_{p}(\varphi )={\begin{cases}\left(\partial _{x_{j}}(\varphi \circ \phi ^{-1})\right)(\phi (p))&p\in O\\0&p\notin O\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fda6faedca2d42705b963435ed7b585ad78971f0)
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
:
![{\displaystyle \left({\frac {\partial }{\partial \phi _{j}}}\right)_{p}\in T_{p}M}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b5dd8b6b07c303b981570a9ca75746d84a05d552)
i. e. the function
is contained in the tangent space
.
Proof:
Let
.
1. We show linearity.
![{\displaystyle {\begin{aligned}\left({\frac {\partial }{\partial \phi _{j}}}\right)_{p}(\varphi +c\vartheta )&=\left(\partial _{x_{j}}((\varphi +c\vartheta )\circ \phi ^{-1})\right)(\phi (p))\\&=\left(\partial _{x_{j}}(\varphi \circ \phi ^{-1}+c\vartheta \circ \phi ^{-1})\right)(\phi (p))\\&=\left(\partial _{x_{j}}(\varphi \circ \phi ^{-1})+c\partial _{x_{j}}(\vartheta \circ \phi ^{-1}))\right)(\phi (p))\\&=\left({\frac {\partial }{\partial \phi _{j}}}\right)_{p}(\varphi )+c\left({\frac {\partial }{\partial \phi _{j}}}\right)_{p}(\vartheta )\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0bd1699be27a0b5f5813e2f4113775c80ce0f5ee)
From the second to the third line, we used the linearity of the derivative.
2. We show the product rule.
![{\displaystyle {\begin{aligned}\left({\frac {\partial }{\partial \phi _{j}}}\right)_{p}(\varphi \vartheta )&=\left(\partial _{x_{j}}((\varphi \vartheta )\circ \phi ^{-1})\right)(\phi (p))\\&=\left(\partial _{x_{j}}((\varphi \circ \phi ^{-1})(\vartheta \circ \phi ^{-1}))\right)(\phi (p))\\&=(\varphi \circ \phi ^{-1})(\phi (p))\left(\partial _{x_{j}}(\vartheta \circ \phi ^{-1})\right)(\phi (p))+(\vartheta \circ \phi ^{-1})(\phi (p))\left(\partial _{x_{j}}(\varphi \circ \phi ^{-1})\right)(\phi (p))\\&=\varphi (p)\left({\frac {\partial }{\partial \phi _{j}}}\right)_{p}(\vartheta )+\vartheta (p)\left({\frac {\partial }{\partial \phi _{j}}}\right)_{p}(\varphi )\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ef79c98eb87398de6a6c1abe73b186402751aab3)
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
![{\displaystyle \phi |_{U\cap O}\circ \theta |_{O\cap U}^{-1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/68846cf0189caedffc00fcef34471c1b2e7f4a03)
is of class
. But if we denote by
the function
![{\displaystyle \pi _{k}:\mathbb {R} ^{d}\to \mathbb {R} ,\pi _{k}(x_{1},\ldots ,x_{d})=x_{k}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3edfe3b9065f660fa0660e0464e60249c6ebb38d)
, which is also called the projection to the
-th component, then we have:
![{\displaystyle \phi _{k}|_{U\cap O}\circ \theta |_{O\cap U}^{-1}=\pi _{k}\circ \phi |_{U\cap O}\circ \theta |_{O\cap U}^{-1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ac680fe93e87f52c50756f55a38ab0b710a297dd)
It is not difficult to show that
is contained in
, and therefore the function
![{\displaystyle \pi _{k}\circ \phi |_{U\cap O}\circ \theta |_{O\cap U}^{-1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ff3183783361acb923c0badc4544e59fc305b29e)
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:
![{\displaystyle \left({\frac {\partial }{\partial \phi _{j}}}\right)_{p}(\phi _{k})={\begin{cases}1&j=k\\0&j\neq k\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d393167a9e1440bd1904a11eb2a3315179575b9c)
Note that due to lemma 2.3,
for all
, which is why the above expression makes sense.
Proof:
We have:
![{\displaystyle {\begin{aligned}\left({\frac {\partial }{\partial \phi _{j}}}\right)_{p}(\phi _{k})&=\left(\partial _{x_{j}}(\phi _{k}\circ \phi ^{-1})\right)(\phi (p))\\&=\lim _{y\to 0}{\frac {(\phi _{k}\circ \phi ^{-1})(x_{1},\ldots ,x_{j-1},x_{j}+y,x_{j+1},\ldots ,x_{d})-(\phi _{k}\circ \phi ^{-1})(x_{1},\ldots ,x_{d})}{y}}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2afba79f840003d01b2c14e4b761b6ff8f62a389)
Further,
![{\displaystyle (\phi _{k}\circ \phi ^{-1})(x_{1},\ldots ,x_{d})=x_{k}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/941eaed6c59ced6ea89f4e19b374cde1e653947d)
and
![{\displaystyle (\phi _{k}\circ \phi ^{-1})(x_{1},\ldots ,x_{j-1},x_{j}+y,x_{j+1},\ldots ,x_{d})={\begin{cases}x_{k}+y&k=j\\x_{k}&k\neq j\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/17d84e756381bfba27f991935ce72ca8ac3b07f8)
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
![{\displaystyle \left({\frac {\partial }{\partial \phi _{j}}}\right)_{p}\in T_{p}M,j\in \{1,\ldots ,d\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9241ad72cbbfec76b603ba89317ce3631bfa556e)
are linearly independent.
Proof:
We write again
.
Let
. Then we have for all
:
![{\displaystyle 0=0_{p}(\phi _{k})=\sum _{j=1}^{d}a_{j}\left({\frac {\partial }{\partial \phi _{j}}}\right)_{p}(\phi _{k})=a_{k}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f3a11505a5d2d24f36f830a6c8f2438fb1963ce8)
![{\displaystyle \Box }](https://wikimedia.org/api/rest_v1/media/math/render/svg/029b77f09ebeaf7528fc831fe57848be51f2240b)
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:
![{\displaystyle \varphi \circ \theta |_{U\cap V}(x)=c}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3af17930a8269f99b699b437718b5b5976b8c417)
This is contained in
.
2. We show that
.
We define
. Using the two rules linearity and product rule for tangent vectors, we obtain:
![{\displaystyle \mathbf {V} _{p}(\varphi )=\mathbf {V} _{p}(\vartheta \varphi )=1\mathbf {V} _{p}(\varphi )+\varphi (p)\mathbf {V} _{p}(\vartheta )=\mathbf {V} _{p}(\varphi )+\mathbf {V} _{p}(\vartheta \varphi (p))=\mathbf {V} _{p}(\varphi )+\mathbf {V} _{p}(\varphi )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/86ecdad409f036da8375f0d2c175e60831348d1a)
Substracting
, 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
![{\displaystyle \mu _{x}(\xi ):=F(\xi x+(1-\xi )\phi (p))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/96ddd3192c55700340a25d266c66f1b20933d6fa)
From the fundamental theorem of calculus, the multi-dimensional chain rule and the linearity of the integral follows for each
, that
![{\displaystyle {\begin{aligned}F(x)&=\mu _{x}(1)\\&=\mu _{x}(0)+\int _{0}^{1}\mu _{x}'(\xi )d\xi \\&=F(\phi (p))+\sum _{j=1}^{d}(x_{j}-\phi (p)_{j})\int _{0}^{1}\partial _{x_{j}}F(\xi \phi (p)+(1-\xi )x)d\xi \end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3e6dd457dd8c4ad41eaabdeda658b565b9b21255)
If one sets
for
, one obtains, inserting the definition of
:
![{\displaystyle \varphi (q)=\varphi (p)+\sum _{j=1}^{d}(\phi (q)_{j}-\phi (p)_{j})\int _{0}^{1}\partial _{x_{j}}(\varphi \circ \phi ^{-1})(\xi \phi (p)+(1-\xi )\phi (q))d\xi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b109bc4f5f8fe0156d00d23dcd6d6301920d54e)
Now we define the functions
![{\displaystyle f_{j}:W\to \mathbb {R} ,f_{j}(q):=\int _{0}^{1}\partial _{x_{j}}(\varphi \circ \phi ^{-1})(\xi \phi (p)+(1-\xi )\phi (q))d\xi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/0cf164f3fa8fdd518d33b912599d789f54c5a622)
These are contained in
since they are defined on
which is open, and further, if
, then
![{\displaystyle f_{j}|_{V\cap W}\circ \theta |_{V\cap W}^{-1}=\int _{0}^{1}\partial _{x_{j}}(\varphi \circ \phi ^{-1})(\xi \phi |_{V\cap W}\circ \theta |_{V\cap W}^{-1}+(1-\xi )\phi |_{V\cap W}\circ \theta |_{V\cap W}^{-1})d\xi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/a54b2a1b22503226fbc58225d9e9f868f9061edc)
, 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:
![{\displaystyle {\begin{aligned}\mathbf {V} _{p}(\varphi |_{W})&=\mathbf {V} _{p}\left(\varphi (p)+\sum _{j=1}^{d}(\phi _{j}-\phi (p)_{j})f_{j}\right)\\&=\sum _{j=1}^{d}\left(\phi _{j}(p)\mathbf {V} _{p}(f_{j})+f_{j}(p)\mathbf {V} _{p}(\phi _{j})-\phi (p)_{j}\mathbf {V} _{p}(f_{j})\right)\\&=\sum _{j=1}^{d}f_{j}(p)\mathbf {V} _{p}(\phi _{j})\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/454cdc0ee84a53d34fa199b134c5556271b0914f)
, since due to our notation it's clear that
.
But
![{\displaystyle {\begin{aligned}f_{j}(p)&=\int _{0}^{1}\partial _{x_{j}}(\varphi \circ \phi ^{-1})(\xi \phi (p)+(1-\xi )\phi (p))d\xi \\&=\int _{0}^{1}\partial _{x_{j}}(\varphi \circ \phi ^{-1})(\phi (p))d\xi \\&=\partial _{x_{j}}(\varphi \circ \phi ^{-1})(\phi (p))\\&=\left({\frac {\partial }{\partial \phi _{j}}}\right)_{p}(\varphi )\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/64703b559d48bd719aca63e2fde4dd4e88cffa2c)
Thus we have successfully shown
![{\displaystyle \mathbf {V} _{p}(\varphi |_{W})=\sum _{j=1}^{d}\mathbf {V} _{p}(\phi _{j})\left({\frac {\partial }{\partial \phi _{j}}}\right)_{p}(\varphi )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4a6590c9f1b3649e6a97e12e2016355770b53173)
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:
![{\displaystyle \mathbf {V} _{p}(\varphi |_{W}-\varphi )=\mathbf {V} _{p}(0)=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/090ab6ca194cc5467553ed649a91ec3430f0e340)
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
![{\displaystyle \left\{\left({\frac {\partial }{\partial \phi _{j}}}\right)_{p}{\Bigg |}j\in \{1,\ldots ,d\}\right\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ac3594783d1afe5b6154e2380ead27301023b1f7)
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
:
![{\displaystyle (d\phi _{j})_{p}:T_{p}M\to \mathbb {R} ,(d\phi _{j})_{p}(\mathbf {V} _{p}):=\mathbf {V} _{p}(\phi _{j})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f307ec02dad9e63cb766f7d205775dc3c4f71546)
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:
![{\displaystyle {\begin{aligned}(d\phi _{j})_{p}(\mathbf {V} _{p}+b\mathbf {W} _{p})&=(\mathbf {V} _{p}+b\mathbf {W} _{p})(\phi _{k})\\&=\mathbf {V} _{p}(\phi _{k})+b\mathbf {W} _{p}(\phi _{k})\\&=(d\phi _{j})_{p}(\mathbf {V} _{p})+b(d\phi _{j})_{p}(\mathbf {W} _{p})\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e67c03bac045a2bdbcc46e7c98c42ef95fa3149b)
![{\displaystyle \Box }](https://wikimedia.org/api/rest_v1/media/math/render/svg/029b77f09ebeaf7528fc831fe57848be51f2240b)
Lemma 2.10: Let
be a
-dimensional manifold of class
and atlas
, let
and let
. For
, the following equation holds:
![{\displaystyle (d\phi _{j})_{p}\left(\left({\frac {\partial }{\partial \phi _{k}}}\right)_{p}\right)={\begin{cases}1&k=j\\0&k\neq j\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/79702e969fcf1f9a09c7aff3fc4a62cecfd3bb3c)
Proof:
We have:
![{\displaystyle (d\phi _{j})_{p}\left(\left({\frac {\partial }{\partial \phi _{k}}}\right)_{p}\right)=\left({\frac {\partial }{\partial \phi _{k}}}\right)_{p}(\phi _{j}){\overset {\text{lemma 2.4}}{=}}{\begin{cases}1&k=j\\0&k\neq j\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7611cde68f10d73423d505d7ba47670c7926c340)
![{\displaystyle \Box }](https://wikimedia.org/api/rest_v1/media/math/render/svg/029b77f09ebeaf7528fc831fe57848be51f2240b)
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
:
![{\displaystyle 0=\sum _{j=1}^{d}a_{j}(d\phi _{j})_{p}\left(\left({\frac {\partial }{\partial \phi _{k}}}\right)_{p}\right){\overset {\text{lemma 2.10}}{=}}a_{k}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/55ec03794f68e25ca44216762d0dc0a0f1997ffc)
![{\displaystyle \Box }](https://wikimedia.org/api/rest_v1/media/math/render/svg/029b77f09ebeaf7528fc831fe57848be51f2240b)
Proof:
Let
and
. Due to theorem 2.7, we have
![{\displaystyle \mathbf {V} _{p}=\sum _{j=1}^{d}\mathbf {V} _{p}(\phi _{j})\left({\frac {\partial }{\partial \phi _{j}}}\right)_{p}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/268633ed97bfc7157783fa082e425a17d973e819)
Therefore, and due to the linearity of
(because
was the space of linear functions to
):
![{\displaystyle {\begin{aligned}\alpha _{p}(\mathbf {V} _{p})&=\sum _{j=1}^{d}\mathbf {V} _{p}(\phi _{j})\alpha _{p}\left(\left({\frac {\partial }{\partial \phi _{j}}}\right)_{p}\right)\\&=\sum _{j=1}^{d}\alpha _{p}\left(\left({\frac {\partial }{\partial \phi _{j}}}\right)_{p}\right)(d\phi _{j})_{p}(\mathbf {V} _{p})\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8a5a09366cfcfb8d7ec01e0ae0c3162e10fd4f65)
Since
was arbitrary, the theorem is proven.
From theorems 2.11 and 2.12 follows, as in the last subsection, that
![{\displaystyle \left\{(d\phi _{j})_{p}{\big |}j\in \{1,\ldots ,d\}\right\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e3c12d2be2877186a774c2fde04767718b1c550a)
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
[edit | edit source]
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
:
![{\displaystyle \left({\frac {\partial }{\partial \phi _{j}}}\right)_{p}=\sum _{k=1}^{d}\left({\frac {\partial }{\partial \phi _{j}}}\right)_{p}(\theta _{k})\left({\frac {\partial }{\partial \theta _{k}}}\right)_{p}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ce0fd317981efc953abed84c3dc29b1094e38add)
From this follows:
![{\displaystyle {\begin{aligned}\mathbf {V} _{p}&=\sum _{j=1}^{d}a_{j}\left({\frac {\partial }{\partial \phi _{j}}}\right)_{p}\\&=\sum _{j=1}^{d}a_{j}\sum _{k=1}^{d}\left({\frac {\partial }{\partial \phi _{j}}}\right)_{p}(\theta _{k})\left({\frac {\partial }{\partial \theta _{k}}}\right)_{p}\\&=\sum _{k=1}^{d}\sum _{j=1}^{d}a_{j}\left({\frac {\partial }{\partial \phi _{j}}}\right)_{p}(\theta _{k})\left({\frac {\partial }{\partial \theta _{k}}}\right)_{p}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f797b58ce49b9c0254241588586764032bb52187)
![{\displaystyle \Box }](https://wikimedia.org/api/rest_v1/media/math/render/svg/029b77f09ebeaf7528fc831fe57848be51f2240b)
Proof:
Due to theorem 2.12, we have for
:
![{\displaystyle (d\phi _{j})_{p}=\sum _{k=1}^{d}(d\phi _{j})_{p}\left(\left({\frac {\partial }{\partial \phi _{k}}}\right)_{p}\right)(d\theta _{k})_{p}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1b34f0bfc2d9ff6c1350b29e142aa888a80bcb40)
Thus we obtain:
![{\displaystyle {\begin{aligned}\alpha _{p}&=\sum _{j=1}^{d}a_{j}(d\phi _{j})_{p}\\&=\sum _{j=1}^{d}a_{j}\sum _{k=1}^{d}(d\phi _{j})_{p}\left(\left({\frac {\partial }{\partial \phi _{k}}}\right)_{p}\right)(d\theta _{k})_{p}\\&=\sum _{k=1}^{d}\sum _{j=1}^{d}a_{j}(d\phi _{j})_{p}\left(\left({\frac {\partial }{\partial \phi _{k}}}\right)_{p}\right)(d\theta _{k})_{p}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3c7edbbb22384ad44cf53a468cc2ba752eb0a8e0)
![{\displaystyle \Box }](https://wikimedia.org/api/rest_v1/media/math/render/svg/029b77f09ebeaf7528fc831fe57848be51f2240b)
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
![{\displaystyle \mathbb {R} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc)
- 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
![{\displaystyle \theta \circ \psi \circ \phi |_{\phi (O\cap \psi ^{-1}(U))}^{-1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2f5071458438b05fa17dcd2bc7592fc1eb12c941)
is contained in
, and therefore continuous. But
and
are charts and therefore homeomorphisms, and thus the function
![{\displaystyle \psi |_{O\cap \psi ^{-1}(U)}:O\cap \psi ^{-1}(U)\to N,\psi =\theta ^{-1}\circ \theta \circ \psi \circ \phi |_{O\cap \psi ^{-1}(U)}^{-1}\circ \phi |_{O\cap \psi ^{-1}(U)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/96dd67f198219ac5dcbc6caad9b2c1dd64397365)
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
![{\displaystyle (\varphi \circ \psi |_{\psi ^{-1}(U)}\circ \phi |_{\psi ^{-1}(U)\cap O}^{-1})|_{\phi (\psi ^{-1}(U\cap V)\cap O)}=\varphi |_{\psi (\psi ^{-1}(U\cap V)\cap O)}\circ \theta |_{\psi (\psi ^{-1}(U\cap V)\cap O)}^{-1}\circ \theta |_{\psi (\psi ^{-1}(U\cap V)\cap O)}\circ \psi |_{\psi ^{-1}(U\cap V)\cap O}\circ \phi |_{\phi (\psi ^{-1}(U\cap V)\cap O)}^{-1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f443e1174ea3d577e3d36b196c3e8490991cfb80)
is
-times continuously differentiable (or continuous if
) at
as the composition of two
times continuously differentiable (or continuous if
) functions. Thus, the function
![{\displaystyle \varphi \circ \psi |_{\psi ^{-1}(U)}\circ \phi |_{\psi ^{-1}(U)\cap O}^{-1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/54f5d2447180c56a210a07cc5d394186a2756852)
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.
![{\displaystyle {\begin{aligned}(\mathbf {V} _{p}\circ \psi ^{*})(\varphi +c\vartheta )&=\mathbf {V} _{p}(\psi ^{*}(\varphi +c\vartheta ))\\&=\mathbf {V} _{p}((\varphi +c\vartheta )\circ \psi |_{\psi ^{-1}(O\cap U)})\\&=\mathbf {V} _{p}(\varphi |_{O\cap U}\circ \psi |_{\psi ^{-1}(O\cap U)}+c\vartheta |_{O\cap U}\circ \psi |_{\psi ^{-1}(O\cap U)})\\&=\mathbf {V} _{p}(\varphi |_{O\cap U}\circ \psi |_{\psi ^{-1}(O\cap U)})+c\mathbf {V} _{p}(\vartheta |_{O\cap U}\circ \psi |_{\psi ^{-1}(O\cap U)})\\&=\mathbf {V} _{p}(\psi ^{*}(\varphi ))+c\mathbf {V} _{p}(\psi ^{*}(\vartheta ))\\&=(\mathbf {V} _{p}\circ \psi ^{*})(\varphi )+c(\mathbf {V} _{p}\circ \psi ^{*})(\vartheta )\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bcee308fb85c079a690ecd9db2ab1dbd9d3e611e)
2. We prove the product rule.
![{\displaystyle {\begin{aligned}(\mathbf {V} _{p}\circ \psi ^{*})(\varphi \vartheta )&=\mathbf {V} _{p}(\psi ^{*}(\varphi \vartheta ))\\&=\mathbf {V} _{p}((\varphi |_{O\cap U}\circ \psi |_{\psi ^{-1}(O\cap U)})(\vartheta |_{O\cap U}\circ \psi |_{\psi ^{-1}(O\cap U)}))\\&=(\varphi |_{O\cap U}\circ \psi |_{\psi ^{-1}(O\cap U)})(p)\mathbf {V} _{p}(\vartheta |_{O\cap U}\circ \psi |_{\psi ^{-1}(O\cap U)})+(\vartheta |_{O\cap U}\circ \psi |_{\psi ^{-1}(O\cap U)})(p)\mathbf {V} _{p}(\varphi |_{O\cap U}\circ \psi |_{\psi ^{-1}(O\cap U)})\\&=\varphi (\psi (p))\mathbf {V} _{p}(\psi ^{*}\vartheta )+\vartheta (\psi (p))\mathbf {V} _{p}(\psi ^{*}\varphi )\\&=\varphi (\psi (p))(\mathbf {V} _{p}\circ \psi ^{*})(\vartheta )+\vartheta (\psi (p))(\mathbf {V} _{p}\circ \psi ^{*})(\varphi )\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/405518e0508e781e6559906dfca9b5e26b9a653f)
![{\displaystyle \Box }](https://wikimedia.org/api/rest_v1/media/math/render/svg/029b77f09ebeaf7528fc831fe57848be51f2240b)
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
![{\displaystyle (\varphi \circ \gamma )|_{\gamma ^{-1}(O\cap U)\cap I}=\varphi \circ \phi ^{-1}\circ \phi \circ \gamma |_{\gamma ^{-1}(O\cap U)\cap I}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bc78d11f845bc2471a666ee1ab6f0d2ff5b641fd)
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:
![{\displaystyle {\begin{aligned}\gamma '_{y}(\varphi +c\vartheta )&=((\varphi +c\vartheta )\circ \gamma )'(y)\\&=(\varphi \circ \gamma +c\vartheta \circ \gamma )'(y)\\&=(\varphi \circ \gamma )'(y)+c(\vartheta \circ \gamma )'(y)\\&=\gamma '_{y}(\varphi )+c\gamma '_{y}(\vartheta )\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7c114e262755f7d26cfc9746e828c6df927c8ee5)
2.2 We prove the product rule.
![{\displaystyle {\begin{aligned}\gamma '_{y}(\varphi \vartheta )&=((\varphi \vartheta )\circ \gamma )'(y)\\&=((\varphi \circ \gamma )(\vartheta \circ \gamma ))'(y)\\&=(\varphi \circ \gamma )(y)(\vartheta \circ \gamma )'(y)+(\vartheta \circ \gamma )(y)(\varphi \circ \gamma )'(y)\\&=\varphi (\gamma (y))\gamma '_{y}(\vartheta )+\vartheta (\gamma (y))\gamma '_{y}(\varphi )\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/326bb2c3ef3d4fc961914cfe984895085882d343)
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
[edit | edit source]
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
![{\displaystyle (\varphi +c\vartheta )|_{O\cap U}\circ \phi |_{O\cap U}^{-1}=\varphi |_{O\cap U}\circ \phi |_{O\cap U}^{-1}+c\vartheta |_{O\cap U}\circ \phi |_{O\cap U}^{-1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/193a5b96ffe03fac9a26be00c3116ae2495dc242)
is contained in
as the linear combination of two
functions.
2. We show that
.
For all
and
, we have:
![{\displaystyle d(\varphi +c\vartheta )_{p}(\mathbf {V} _{p})=\mathbf {V} _{p}(\varphi +c\vartheta )=\mathbf {V} _{p}(\varphi )+c\mathbf {V} _{p}(\vartheta )=d\varphi _{p}(\mathbf {V} _{p})+cd\vartheta _{p}(\mathbf {V} _{p})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2c003f3ffaeadb68951b00f2203b9018b54b7b86)
![{\displaystyle \Box }](https://wikimedia.org/api/rest_v1/media/math/render/svg/029b77f09ebeaf7528fc831fe57848be51f2240b)
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
![{\displaystyle (\varphi \vartheta )|_{O\cap U}\circ \phi |_{O\cap U}^{-1}=\varphi |_{O\cap U}\circ \phi |_{O\cap U}^{-1}\vartheta |_{O\cap U}\circ \phi |_{O\cap U}^{-1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9656905717e47e3031d0439b818ea2a10e277c8a)
is contained in
as the product of two
functions.
2. We show that
.
For all
and
, we have:
![{\displaystyle d(\varphi \vartheta )_{p}(\mathbf {V} _{p})=\mathbf {V} _{p}(\varphi \vartheta )=\varphi (p)\mathbf {V} _{p}(\vartheta )+\vartheta (p)\mathbf {V} _{p}(\varphi )=\varphi (p)d\vartheta _{p}+\vartheta (p)d\varphi _{p}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dfa8c2cd21ad8d60731d41e67e204aa911d9e0d7)
![{\displaystyle \Box }](https://wikimedia.org/api/rest_v1/media/math/render/svg/029b77f09ebeaf7528fc831fe57848be51f2240b)
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
![{\displaystyle {\frac {\varphi }{\vartheta }}{\big |}_{O\cap U}\circ \phi |_{O\cap U}^{-1}={\frac {\varphi |_{O\cap U}\circ \phi |_{O\cap U}^{-1}}{\vartheta |_{O\cap U}\circ \phi |_{O\cap U}^{-1}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2ded2a5f8dad45bf7b9bde25f27f39f5099b131a)
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:
![{\displaystyle 0=d\left(\vartheta {\frac {1}{\vartheta }}\right)=\vartheta d\left({\frac {1}{\vartheta }}\right)+{\frac {1}{\vartheta }}d\vartheta }](https://wikimedia.org/api/rest_v1/media/math/render/svg/074e3843dd18119358df4045d1e2e565bab4c163)
which, through equivalent transformations, can be transformed to
![{\displaystyle d\left({\frac {1}{\vartheta }}\right)=-{\frac {d\vartheta }{\vartheta ^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/377d54ff301199a4c21d3f8d32dd80eb3f1c65d4)
From this and from the product rule we obtain:
![{\displaystyle d\left(\varphi {\frac {1}{\vartheta }}\right)={\frac {1}{\vartheta }}d\varphi -{\frac {\varphi d\vartheta }{\vartheta ^{2}}}={\frac {\vartheta d\varphi -\varphi d\vartheta }{\vartheta ^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/544b63bc246bd3ac9e324bbd5a260ac4c837e237)
![{\displaystyle \Box }](https://wikimedia.org/api/rest_v1/media/math/render/svg/029b77f09ebeaf7528fc831fe57848be51f2240b)
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:
![{\displaystyle {\begin{aligned}(d\varphi _{\psi (p)}\circ d\psi _{p})(\mathbf {V} _{p})&=d\varphi _{\psi (p)}(d\psi _{p}(\mathbf {V} _{p}))\\&=d\varphi _{\psi (p)}(\mathbf {V} _{p}\circ \psi )\\&=(\mathbf {V} _{p}\circ \psi ^{*})(\varphi )\\&=\mathbf {V} _{p}(\psi ^{*}(\varphi ))\\&=\mathbf {V} _{p}(\varphi \circ \psi )\\&=d(\varphi \circ \psi )_{p}(\mathbf {V} _{p})\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5507e150b84e50cc6c1bc48ee9480e0c7e5c36f5)
![{\displaystyle \Box }](https://wikimedia.org/api/rest_v1/media/math/render/svg/029b77f09ebeaf7528fc831fe57848be51f2240b)
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
![{\displaystyle (\chi \circ \psi )^{*}=\psi ^{*}\circ \chi ^{*}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/585a28fb24f70b591cf0e7cf9b46231b11ce6dda)
Proof:
Let
. Then we have:
![{\displaystyle {\begin{aligned}(\chi \circ \psi )^{*}(\varphi )&=\varphi \circ (\chi \circ \psi )\\&=(\varphi \circ \chi )\circ \psi \\&=\chi ^{*}(\varphi )\circ \psi \\&=\psi ^{*}(\chi ^{*}(\varphi ))\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ad04d8b086740b4a3f8183226f240c32407d244b)
![{\displaystyle \Box }](https://wikimedia.org/api/rest_v1/media/math/render/svg/029b77f09ebeaf7528fc831fe57848be51f2240b)
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
![{\displaystyle \theta ^{-1}\circ (\chi \circ \psi )\circ \phi |_{W}^{-1}=\theta ^{-1}\circ \chi \circ \eta ^{-1}\circ \eta \circ \psi |_{W}\circ \phi |_{W}^{-1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9e9d71a81505f5125938d3eb840c50b5f2554583)
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:
![{\displaystyle {\begin{aligned}(d\chi _{\psi (p)}\circ d\psi _{p})(\mathbf {V} _{p})&=d\chi _{\psi (p)}(d\psi _{p}(\mathbf {V} _{p}))\\&=d\chi _{\psi (p)}(\mathbf {V} _{p}\circ \psi ^{*})\\&=\mathbf {V} _{p}\circ \psi ^{*}\circ \chi ^{*}\\&{\overset {\text{theorem 2.26}}{=}}\mathbf {V} _{p}\circ (\chi \circ \psi )^{*}\\&=d(\chi \circ \psi )_{p}(\mathbf {V} _{p})\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d850ec3757b24c108a178c39ebb8d20c2ed88e9b)
![{\displaystyle \Box }](https://wikimedia.org/api/rest_v1/media/math/render/svg/029b77f09ebeaf7528fc831fe57848be51f2240b)
Proof:
1. Among another thing, theorem 2.22 states that
is contained in
.
2. We show that
:
![{\displaystyle {\begin{aligned}d\varphi _{\gamma (y)}(\gamma '_{y})&=\gamma '_{y}(\varphi )\\&=(\varphi \circ \gamma )'(y)\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/36773c4ff103087aa996d49aebee88134b2c6022)
![{\displaystyle \Box }](https://wikimedia.org/api/rest_v1/media/math/render/svg/029b77f09ebeaf7528fc831fe57848be51f2240b)
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.