Differentiable Manifolds/Submanifolds

Differentiable Manifolds
 ← Maximal atlases, second-countable spaces and partitions of unity Submanifolds Vector fields, covector fields, the tensor algebra and tensor fields → 

In this chapter, we will show what submanifolds are, and how we can obtain, under a condition, a submanifold out of some functions.

Definition of a submanifold

edit

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

 

How to obtain a submanifold out of a set of certain functions

edit

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:

 

and

 

, 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:

 

And also:

 

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

 

has rank  , then the set   is a submanifold of dimension   of  .

Proof:

Since the matrix

 

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

 

Hence, the   matrix

 

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

 

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  :

 

Hence,   is a submanifold of dimension  . 

Sources

edit
  • Torres del Castillo, Gerardo (2012). Differentiable Manifolds. Boston: Birkhäuser. ISBN 978-0-8176-8271-2.
Differentiable Manifolds
 ← Maximal atlases, second-countable spaces and partitions of unity Submanifolds Vector fields, covector fields, the tensor algebra and tensor fields →