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

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Differentiable Manifolds
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Maximal atlases

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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

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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

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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

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  • 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 →