Manifolds (unknown book chapter).pdf

Manifolds

Intuitively, a manifold is a generalization of curves and surfaces to arbitrary dimen-

sion. While there are many different kinds of manifolds—topological manifolds,

C k -manifolds, analytic manifolds, and complex manifolds, in this book we are con-

cerned mainly with smooth manifolds.

5.1 Topological Manifolds

We ﬁrst recall a few deﬁnitions from point-set topology. For more details, see Ap-

pendix A. A topological space is second countable if it has a countable basis. A

neighborhood of a point p in a topological space M is any open set containing p .An

open cover of M is a collection

{

U α } α ∈ A of open sets in M whose union α ∈ A U α

is M .

n ) a chart , U a coordinate

neighborhood or a coordinate open set , and φ a coordinate map or a coordinate

system on U . We say that a chart (U, φ) is centered at p

n . We call the pair (U, φ

−→ R

∈

U if φ(p)

0. A chart

(U, φ) about p simply means that (U, φ) is a chart and p

∈

U .

Deﬁnition 5.2. A topological manifold of dimension n is a Hausdorff, second count-

able, locally Euclidean space of dimension n .

For the dimension to be well deﬁned, we need to know that for n

m an open

m . This is indeed true, but is

not easy to prove (see [4] for a discussion and further references). We will not pursue

this point as we are mainly interested in smooth manifolds, for which the analogous

result is easy to prove (Corollary 8.8). Of course, if a topological manifold has

several connected components, it is possible for each component to have a different

dimension.

n is not homeomorphic to an open subset of

Deﬁnition 5.1. A topological space M is locally Euclidean of dimension n if every

point p in M has a neighborhood U such that there is a homeomorphism φ from U

onto an open subset of

subset of

5 Manifolds

Example 5.3 . The Euclidean space

is covered by a single chart (

n , 1

n ) , where

n is the identity map. It is the prime example of a topological manifold.

Every open subset of

: R

−→ R

n is also a topological manifold, with chart (U, 1 U ) .

Recall that the Hausdorff condition and second countability are “hereditary prop-

erties’’; that is, they are inherited by subspaces: a subspace of a Hausdorff space is

Hausdorff (Proposition A.23) and a subspace of a second countable space is second

countable (Proposition A.19). So any subspace of

n is automatically Hausdorff and

second countable.

Example 5.4 ( The cusp ) . The graph of y

x 2 / 3

is a topological manifold

2 , it is Hausdorff and second

countable. It is locally Euclidean, because it is homeomorphic to

via (x, x 2 / 3 )

→

x .

(a) Cusp

(b) Cross

Fig. 5.1.

2 in Figure 5.1 with the subspace

topology is not locally Euclidean at p , and so cannot be a topological manifold.

Solution. If a space is locally Euclidean of dimension n at p , then p has a neighbor-

hood U homeomorphic to an open ball B

B( 0 ,)

⊂ R

n with p mapping to 0.

The homeomorphism

: U −→ B restricts to a homeomorphism

: U −{ p }→ B −{

}

Now B −{

}

is either connected if n ≥

2 or has two connected components if n =

Since U −{ p }

has four connected components, there can be no homeomorphism from

U −{ p }

to B −{

}

. This contradiction proves that the cross is not locally Euclidean

at p .

5.2 Compatible Charts

Deﬁnition 5.6. Two charts (U, φ

−→ R

n ) , (V , ψ

−→ R

n ) of a topological

manifold are C ∞ -compatible if the two maps

φ ◦ ψ − 1

are C ∞ (Figure 5.2). These two maps are called the transition functions between the

charts. If U

ψ(U

∩

−→

φ(U

∩

V), ψ ◦ φ − 1

φ(U

∩

−→

ψ(U

∩

V is empty, then the two charts are automatically C ∞ -compatible.

To simplify the notation, we will sometimes write U αβ for U α ∩

∩

U β and U αβγ for

U α ∩

U β ∩

U γ .

1 R

(Figure 5.1(a)). By virtue of being a subspace of

Example 5.5 ( The cross ) . Show that the cross in

5.2 Compatible Charts

φ(U ∩ V)

Fig. 5.2. The transition function ψ ◦ φ − 1 is deﬁned on φ(U ∩ V) .

α U α .

Although the C ∞ compatibility of charts is clearly reﬂexive and symmetric, it

is not transitive. The reason is as follows. Suppose (U 1 ,φ 1 ) is C ∞ -compatible

with (U 2 ,φ 2 ) , and (U 2 ,φ 2 ) is C ∞ -compatible with (U 3 ,φ 3 ) . Note that the three

coordinate functions are simultaneously deﬁned only on the triple intersection U 123 .

Thus, the composite

(U α ,φ α )

}

of C ∞ -compatible charts that cover M , i.e., such that M

1 )

is C ∞ but only on φ 1 (U 123 ) , not necessarily on φ 1 (U 13 ) (Figure 5.3). A priori we

know nothing about φ 3 ◦ φ − 1

φ 3 ◦ φ − 1

(φ 3 ◦ φ − 1

) ◦ (φ 2 ◦ φ − 1

on φ 1 (U 13 −

U 123 ) and so we cannot conclude that

(U 1 ,φ 1 ) and (U 3 ,φ 3 ) are C ∞ -compatible.

φ 1

U 1

U 2

φ 2

φ 1 (U 123 )

U 3

φ 3

Fig. 5.3. φ 3 ◦ φ − 1

is C ∞ on φ 1 (U 123 ) .

We say that a chart (V , ψ) is compatible with an atlas

{

(U α ,φ α )

}

if it is compatible

with all the charts (U α ,φ α ) of the atlas.

Lemma 5.8. Let

be an atlas on a locally Euclidean space. If two charts

(V , ψ) and (W, σ ) are both compatible with the atlas

{

(U α ,φ α )

}

{

(U α , φ α )

, then they are

compatible with each other.

{

Since we are interested only in C ∞ -compatible charts, we often omit to mention

C ∞ and speak simply of compatible charts.

Deﬁnition 5.7. A C ∞ atlas or simply an atlas on a locally Euclidean space M is a

collection

}

5 Manifolds

Proof. (See Figure 5.4.) Let p

∈

∩

W . We need to show that σ ◦ ψ − 1 is C ∞ at

ψ(p) . Since

{

(U α ,φ α )

}

is an atlas for M , p

∈

U α for some α . Then p is in the triple

intersection V

∩

U α .

U α

φ α

ψ(p)

φ α ◦ ψ − 1

φ α (p)

σ ◦ φ − 1

σ (p)

Fig. 5.4. Two charts (V , ψ) , (W, σ ) compatible with an atlas.

By the remark above, σ ◦ ψ − 1

(σ ◦ φ − 1

) ◦ (φ α ◦ ψ − 1 ) is C ∞ on ψ(V

∩

W , this proves that σ ◦ ψ − 1

U α ) , hence at ψ(p) . Since p is an arbitrary point of V

∩

is C ∞ on ψ(V

∩

W) . Similarly, ψ ◦ σ − 1 is C ∞ on σ(V

∩

W) .

5.3 Smooth Manifolds

An atlas A on a locally Euclidean space is said to be maximal if it is not contained in

a larger atlas; in other words, if M is any other atlas containing A, then M = A.

Deﬁnition 5.9. A smooth or C ∞ manifold is a topological manifold M together with

a maximal atlas. The maximal atlas is also called a differentiable structure on M .

A manifold is said to have dimension n if all of its connected components have

dimension n . Amanifold of dimension n is also called an n -manifold .

In Corollary 8.8 we will prove that if an open set U

⊂ R

n is diffeomorphic to an

open set V

⊂ R

m , then n

m . As a consequence, the dimension of a manifold at a

point is well deﬁned.

In practice, to check that a topological manifold M is a smooth manifold, it is

not necessary to exhibit a maximal atlas. The existence of any atlas on M will do,

because of the following proposition.

Proposition 5.10. Any atlas A ={

(U α ,φ α )

}

on a locally Euclidean space is con-

tained in a unique maximal atlas.

5.4 Examples of Smooth Manifolds

Proof. Adjoin to the atlas A all charts (V i ,ψ i ) that are compatible with A.By

Proposition 5.8 the charts (V i ,ψ i ) are compatible with one another. So the enlarged

collection of charts is an atlas. Any chart compatible with the new atlas must be

compatible with the original atlas A and so by construction belongs to the new atlas.

This proves that the new atlas is maximal.

Let M be the maximal atlas containing A that we have just constructed. If M is

another maximal atlas containing A, then all the charts in M are compatible with A

and so by construction must belong to M. This proves that M ⊂ M. Since both are

maximal, M = M. Therefore, the maximal atlas containing A is unique.

In summary, to show that a topological space M is a C ∞ manifold, it sufﬁces to

check:

(i) M is Hausdorff and second countable,

(ii) M has a C ∞ atlas (not necessarily maximal).

From now on by a manifold we will mean a C ∞ manifold. We use the words

smooth and C ∞ interchangeably.

5.4 Examples of Smooth Manifolds

Example 5.11 . The Euclidean space

is a smooth manifold with a single chart

n ,r 1 ,...,r n ) , where r 1 ,...,r n are the standard coordinates on

n .

Example 5.12 . Any open subset V of a manifold M is also a manifold. If

{

(U α ,φ α )

}

is an atlas for M , then

{

(U α ∩

V,φ α | U α ∩ V }

is an atlas for V , where φ α | U α ∩ V :

U α ∩

−→ R

n denotes the restriction of φ α to the subset U α ∩

V .

Example 5.13 ( The graph of a smooth function ) . For U an open subset of

n and

−→ R

m a C ∞ function, the graph of f is deﬁned to be the subspace

(f )

(x, f (x))

∈

× R

}

The two maps

(f )

−→

(x, f (x))

→

and

−→

(f ),

→

(x, f (x))

m has an atlas with a single chart ((f ), φ) , and is

therefore a C ∞ manifold. This shows that many of the familiar surfaces of calculus,

for example an elliptic paraboloid or a hyperbolic paraboloid, are manifolds.

−→ R

Example 5.14 . For any two positive integers m and n let

n be the vector space

of all m

n matrices. Since

is isomorphic to

mn , we give it the topology of

mn . The general linear group GL (n,

) is by deﬁnition

(

are continuous and inverse to each other, and so are homeomorphisms. The graph

(f ) of a C ∞ function f

Plik z chomika:

Inne pliki z tego folderu:

Inne foldery tego chomika: