Differentiable Manifolds/Maximal atlases, second-countable spaces and partitions of unity

Differentiable Manifolds
 ← Bases of tangent and cotangent spaces and the differentials Maximal atlases, second-countable spaces and partitions of unity Submanifolds → 

Maximal atlases

edit

Definition 3.1:

Let   be a  -dimensional manifold of class   and let   be it's atlas. We call the set

 

the maximal atlas of  .

Lemma 3.2: We have  .

Proof: This is because if  , then by definition of an atlas it is compatible with all the elements of   and hence, by definition of  , contained in  . 

Theorem 3.3: The maximal atlas really is an atlas; i. e. for every point   there exists   such that  , and every two charts in it are compatible.

Proof:

1.

We first show that for every point   there exists   such that  :

From lemma 3.2 we know that the atlas of   is contained in  .

Let now  . Due to the definition of an atlas, we find an   such that  . Since  , we obtain  .

2.

We prove that every two charts   such that  , are compatible.

So let   such that   be 'arbitrary' (of course we still require  ).

If we have  , this directly implies compatibility (recall that we defined compatibility so that if   for two charts  , then the two are by definition automatically compatible).

So in this case, we are finished. Now we shall prove the other case, which namely is  .

Due to the definition of compatibility of class  , we have to prove that the function

 

is contained in   and

 

is contained in  .

Let  . Since   is the atlas of  , we find a chart   such that  . Due to the definition of  ,   and   are compatible and   and   are compatible. Hence, the functions

 

and

 

are  -times differentiable (or, if  , continuous), in particular at  ,   respectively. Since   was arbitrary, since

 

and

 

(which you can show by direct calculation!) and since   are bijective, this shows the theorem. 

Theorem 3.4:

Let   be a  -dimensional manifold with atlas  , and let   be it's maximal atlas. There does not exist an atlas   such that   (this notation shall mean that   is contained in  , but   is 'strictly larger than  ' (by this obscure saying we shall mean that there exists at least one element in   which is not contained in  )).

This is, in fact, the reason why the word maximal atlas for   does not completely miss the point.

Proof: We show that there does not exist an atlas   of   such that  .

Assume by contradiction that there exists such an atlas. Then we find an element  . But since   is an atlas,   is compatible to all other charts   for which  . This means, due to lemma 3.2, that it is compatible to every  . Hence, due to the definition of  ,  . This is a contradiction! 

Second-countable spaces

edit

Definition 3.5:

Let   be a topological space and let   be a set of open sets. We call   a basis of the topology of   iff every open set   can be written as the union of elements of  , i. e.

 

where  .

Definition 3.6:

Let   be a topological space. We call   second-countable iff  's topology has a countable basis.

Locally finite refinements and partitions of unity

edit

Definition 3.7:

Let   be a topological space. An open cover of   is a set   of open subsets of   such that

 

Example 3.8:

The set   is an open cover of the real numbers.

Definition 3.9:

Let   be a manifold of class  . We say that   admits partitions of unity iff for every open cover   there exist functions   such that:

  1. For all  , there exists a   such that  ,
  2. for all   and  ,   AND
  3. for all  ,  .

Definition 3.10:

Let   be a topological space and let   be an open cover of  . A locally finite refinement of   is defined to be another open cover of  , say  , such that:

  • for each  , there exists a   such that  , AND
  • for each  , the set   is finite.

We will now prove a few lemmas, which will help us to prove that every manifold whose topology has a countable basis admits partition of unity. Then, we will prove that every manifold whose topology has a countable basis admits partition of unity :-)

Lemma 3.11:

Let   be a manifold with a countable basis. Then   has a countable basis   such that for each  ,   is compact.

Proof:

Let   be a countable basis of  . For each  , we choose a chart   such that  . Then we choose  . Since in  , sets are compact if and only if bounded and closed,   is compact. There is a theorem from topology, which states that the image of a compact set under a homeomorphism is again compact. Hence,   is a compact subset of  .

Further, if   is an cover of   by open subsets of  , then the set   is a cover of   by open subsets of  . Since   is compact in  , we may pick out of the latter a finite subcover  . Then, since

 

, the set   is a finite subcover of  . Thus,   is also a compact subset of  .

As   is a homeomorphism,   is open in  , and from  , it follows  . Thus, also

 

since the closure of   is, by definition (with the definition of some lectures), equal to

 

Further, another theorem from topology states that closed subsets of compact sets are compact. Hence,   is compact.

Since   was a basis, each of the   can be written as the union of elements of  . We choose now our new basis as consisting of the union over   of the elements of   with smallest index  , such that   and  . Now the closures of the   are compact: From   follows that  , and since  ,   is compact as the closed subset of a compact set.

Since our new basis is a subset of a countable set, it is itself countable (we include finite sets in the category 'countable' here). Thus, we have obtained a countable basis the elements of which have compact closure. 

Lemma 3.12:

Let   be a manifold with a countable base (i. e. a second-countable manifold). Then for every cover of   there is a locally finite refinement.

Proof:

Let   be a cover of  . Due to lemma 3.11, we may choose a countable basis   of   such that each   is compact. We now define a sequence of compact sets   inductively as follows: We set  . Once we defined  , we define

 

, where   is smallest such that we have:

 

This is compact, since a theorem from topology states that the finite union of compact sets is compact. Since, as mentioned before, there is a theorem from topology stating that closed subsets of compact sets are compact, the sets defined by   and

 

for   (intuitively the closed annulus) are compact. Further, the sets  , defined by  ,   and

 

for   (intuitively the next bigger open annulus) are open, and we have for all  :

 

Now since   is covered by  , so is each of the sets  . Now we compose our locally finite refinement as follows: We include all the sets, which are the intersection of   and the (by compactness existing) sets of the finite subcovers of   out of  . This is a locally finite refinement. 

Lemma 3.13:

Let   be a  -dimensional manifold of class   with atlas  , let  , let   be open in   (with respect to the subspace topology and let   and let   be such that  . If we define

 

, and

 ,

then we have  .

Proof:

Let  . Then we have for  :

 

This function is   times differentiable (or continuous if  ) as the composition of   times differentiable (or continuous if  ) functions. 

Theorem 3.14:

Let   be a manifold of class   with a countable basis, i. e. a second-countable manifold. Then   admits partitions of unity.

Proof:

Let   be an open cover of  .

We choose for each point   an atlas   such that  . Further, we choose an arbitrary   in the open cover such that  . By definition of the subspace topology we have that   is open in  . Therefore, due to lemma 3.13, we may choose   such that  . Since   is continuous, all the   are open; this is because they are preimages of the open set  . Further, since there is a   for every  , and always  , the   form a cover of  . Due to lemma 3.12 we may choose a locally finite refinement. This open cover, this set of open sets we shall denote by  .

We now define the function

 

This function is of class   as a finite sum (because for each   there are only finitely many   such that  , because   was a locally finite subcover) of   functions (that finite sums of   functions are again   follows from theorem 2.22 and induction) and does not vanish anywhere (since for every   there is a   such that   is in it; remember that a finite refinement is an open cover), and therefore follows from theorem 2.26, that all the functions   are contained in  . It is not difficult to show that these functions are non-negative and that they sum up to   at every point. Further due to the construction, each of their supports is contained in one  . Thus they form the desired partition of unity. 

Sources

edit
  • Lang, Serge (2002). Introduction to Differentiable Manifolds. New York: Springer. ISBN 0-387-95477-5.
Differentiable Manifolds
 ← Bases of tangent and cotangent spaces and the differentials Maximal atlases, second-countable spaces and partitions of unity Submanifolds →