In this chapter, we will show what submanifolds are, and how we can obtain, under a condition, a submanifold out of some
functions.
Definition 4.1:
Let
be a
-dimensional manifold of class
, and let
be it's maximal atlas. If
, we call a subset
a submanifold of dimension
iff
is the largest number in
such that for each
there exists
such that
and
![{\displaystyle \phi (O\cap N)\subseteq \{(x_{1},\ldots ,x_{d})\in \mathbb {R} ^{d}|x_{d-m+1}=x_{d-m+2}=\ldots =x_{d}=0\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0cfbe746889bda778e9ea093e1a58e76c869a414)
How to obtain a submanifold out of a set of certain functions
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Lemma 4.2: Let
be a
-dimensional manifold of class
with atlas
, let
be it's maximal atlas, let
, and let
be an open subset of
. Then
.
Proof:
1. We show that
is a chart.
It is a homeomorphism since the restriction of a homeomorphism is a homeomorphism, and if
is open, then
is open in
since
is a homeomorphism, and further, due to the definition of the subspace topology and since
is open in
, we have
for an open set
, and hence
is open in
as the intersection of two open sets.
2. We show that
is compatible with all
.
Let
.
We have:
![{\displaystyle \theta |_{V\cap U}\circ (\phi |_{V})|_{V\cap U}^{-1}=(\theta |_{O\cap U}\circ \phi |_{O\cap U}^{-1})|_{\phi (V\cap U)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b0ac9e3c18f4d39f4a40bd5eb2d8e7c01e5b84fe)
and
![{\displaystyle (\phi |_{V})|_{V\cap U}\circ \theta ^{-1}|_{V\cap U}=(\phi |_{O\cap U}\circ \theta |_{O\cap U}^{-1})|_{\theta (V\cap U)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/42ae2e10c12dd6c5c653c1ed7543426b35621a73)
, which can be verified by direct calculation. But these are
-times differentiable (or continuous if
), since they are restrictions of
-times differentiable (or continuous if
) functions; this is since
and
are compatible. Due to the definitions of
and
respectively, the lemma is proved.
Lemma 4.3: Let
be a
-dimensional manifold of class
with atlas
, let
be it's maximal atlas, let
, and let
be a diffeomorphism of class
. Then we have:
.
Proof:
1. We show that
is a chart.
By invariance of domain, and since
is open in
since
is a chart,
is open in
. Furthermore,
and
are homeomorphisms (
is a homeomorphism because every diffeomorphism is a homeomorphism), and therefore
is a homeomorphism as well. Thus,
is a chart.
2. We show that
is compatible with all
.
Let
.
We have:
![{\displaystyle \theta |_{U\cap O}\circ (\Phi \circ \phi )|_{U\cap O}^{-1}=\theta |_{U\cap O}\circ \phi |_{U\cap O}^{-1}\circ \Phi |_{\phi (O\cap U)}^{-1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9560e316852c0c9d69a4ae51eabb73630c5a34c4)
And also:
![{\displaystyle (\Phi \circ \phi )|_{U\cap O}\circ \theta |_{U\cap O}^{-1}=\Phi |_{\phi (O\cap U)}\circ \phi |_{O\cup U}\circ \theta |_{U\cap O}^{-1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b87a2e480aed7114cdacda37ea04fe99691592db)
These functions are
-times differentiable (or continuous if
), because they are compositions of functions, which are
-times differentiable (or continuous if
); this is since
and
are compatible. By definition of
and
respectively, we are finished with the proof of this lemma.
Theorem 4.4:
Let
be a
-dimensional manifold of class
, where
must be
here, with maximal atlas
, let
and let
(remember def. 1.5). If for each
there exists
such that
and the matrix
![{\displaystyle {\begin{pmatrix}\left(\partial _{x_{1}}(f_{1}\circ \phi ^{-1})\right)(\phi (p))&\cdots &\left(\partial _{x_{d}}(f_{1}\circ \phi ^{-1})\right)(\phi (p))\\\vdots &\ddots &\vdots \\\left(\partial _{x_{1}}(f_{m}\circ \phi ^{-1})\right)(\phi (p))&\cdots &\left(\partial _{x_{d}}(f_{m}\circ \phi ^{-1})\right)(\phi (p))\end{pmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e40f8bb10db7b5cc4b8ff9bff52f550b152e78fc)
has rank
, then the set
is a submanifold of dimension
of
.
Proof:
Since the matrix
![{\displaystyle {\begin{pmatrix}\left(\partial _{x_{1}}(f_{1}\circ \phi ^{-1})\right)(\phi (p))&\cdots &\left(\partial _{x_{d}}(f_{1}\circ \phi ^{-1})\right)(\phi (p))\\\vdots &\ddots &\vdots \\\left(\partial _{x_{1}}(f_{m}\circ \phi ^{-1})\right)(\phi (p))&\cdots &\left(\partial _{x_{d}}(f_{m}\circ \phi ^{-1})\right)(\phi (p))\end{pmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e40f8bb10db7b5cc4b8ff9bff52f550b152e78fc)
has rank
, it has
linearly independent columns (this is a theorem from linear algebra). Therefore there exists a permutation
such that the last
columns of thee matrix
![{\displaystyle {\begin{pmatrix}\left(\partial _{x_{\sigma (1)}}(f_{1}\circ \phi ^{-1})\right)(\phi (p))&\cdots &\left(\partial _{x_{\sigma (d)}}(f_{1}\circ \phi ^{-1})\right)(\phi (p))\\\vdots &\ddots &\vdots \\\left(\partial _{x_{\sigma (1)}}(f_{m}\circ \phi ^{-1})\right)(\phi (p))&\cdots &\left(\partial _{x_{\sigma (d)}}(f_{m}\circ \phi ^{-1})\right)(\phi (p))\end{pmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/20c7b2e3a5695d1541cba34b11d40c05ae306a93)
Hence, the
matrix
![{\displaystyle {\begin{pmatrix}1&0&\cdots &0&0&0&\cdots &0\\0&1&\cdots &0&0&&&\\\vdots &\ddots &\ddots &\ddots &\vdots &\vdots &\ddots &\vdots \\0&0&\cdots &1&0&&&\\0&0&\cdots &0&1&0&\cdots &0\\\left(\partial _{x_{\sigma (1)}}(f_{1}\circ \phi ^{-1})\right)(\phi (p))&&\cdots &&&&\cdots &\left(\partial _{x_{\sigma (d)}}(f_{1}\circ \phi ^{-1})\right)(\phi (p))\\\vdots &&&&\ddots &&&\vdots \\\left(\partial _{x_{\sigma (1)}}(f_{m}\circ \phi ^{-1})\right)(\phi (p))&&\cdots &&&&\cdots &\left(\partial _{x_{\sigma (d)}}(f_{m}\circ \phi ^{-1})\right)(\phi (p))\end{pmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e542240c75a449e8e6f1505e1dad47afbd193f18)
is invertible (one can prove the invertibility of the transpose by induction and Laplace's formula). But the latter matrix is the Jacobian matrix of the function
given by
![{\displaystyle \Phi (x_{1},\ldots ,x_{d})={\begin{pmatrix}x_{1}\\\vdots \\x_{d-m}\\(f_{1}\circ \phi ^{-1})(x_{\sigma (1)},\ldots ,x_{\sigma (d)}\\\vdots \\(f_{m}\circ \phi ^{-1})(x_{\sigma (1)},\ldots ,x_{\sigma (d)})\end{pmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c6634176e4c8f40d8e17bf708866b551de525345)
at
. By the inverse function theorem, there exists an open set
such that
and
is a diffeomorphism.
Since
is a homeomorphism, and in particular is continuous,
is an open subset of
. Due to lemma 4.2,
. Due to lemma 4.3,
. But it also holds that for
such that
:
![{\displaystyle \Phi \circ \phi |_{\phi ^{-1}(V)}(q)=(\phi _{1}(q),\ldots ,\phi _{d-m}(q),f_{1}(q),\ldots ,f_{m}(q))=(\phi _{1}(q),\ldots ,\phi _{d-m}(q),0,\ldots ,0)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/72cd6c2d3f8f27881fd8e931e8b747e8e2c8166f)
Hence,
is a submanifold of dimension
.
- Torres del Castillo, Gerardo (2012). Differentiable Manifolds. Boston: Birkhäuser. ISBN 978-0-8176-8271-2.