Chapter 5 - Submanifolds

Question 1

Define: embedded submanifold, codimension, ambient manifold, embedded hypersurface, properly embedded

Answer 1

Suppose M is a smooth manifold with or without boundary. An EMBEDDED SUBMANIFOLD of M is a subset S < M that is a manifold (without boundary) in the subspace topology, endowed with a smooth structure with respect to which the inclusion map S –> M is a smooth embedding.

The difference dim M - dim S is called the CODIMENSION OF S IN M. M is the AMBIENT MANIFOLD for S. An EMBEDDED HYPERSURFACE is an embedded submanifold of codimension 1.

An embedded submanifold S < M is PROPERLY EMBEDDED if the inclusion S –> M is a proper map (ie inverse images of compact sets are compact - infinite does not go to finite region. Send infinity to infinity)

Question 2

What are the embedded submanifolds of codimension 0 in M?

Answer 2

They are exactly the open submanifolds.

Pf. Show if U < M is an open submanifold, then U is embedded submanifold with codim 0 and conbversely. pg 99

Question 3

What is relationship between smooth embeddings and embedded submanifolds? Proof?

Answer 3

Embedded submanifolds = images of smooth embeddings.

The definition of embedded submanifold includes condition that inclusion S –> M is a smooth embedding.

Let F: N –> M be a smooth embedding and S = F(N). Give S the subspace topology. The assumption that F is an embedding means F is a homeomorphism from N –> S. Give S smooth structure (F(U), phi o F^-1), where (U, phi) is any smooth chart for N. With this structure, F is diffeomorphism onto its image. Inclusion S –> M is F o F^-1|S a composition of a diffeo followed by smooth embedding hence a smooth embedding

Question 4

Conditions for an embedded submanifold to be properly embedded?

Answer 4

1. S is properly embedded <=> S is a closed subset of M
2. Every compact embedded submanifold is properly embedded
3. Global graphs are properly embedded. Say f: M –> N is smooth, Gamma(f) is properly embedded in M x N.

Question 5

Define: k-slice, local k-slice condition, slice chart, slice coordinates

Answer 5

If U is an open subset of R^n, a K-DIMENSIONAL SLICE OF U is any subset of the form S = { (x^1, … , x^k, x^k+1, … , x^n) in U : x^k+1 = c^k+1, …. , x^n = c^n} for some constants c^k+1, … , c^n. (in particular think of the c = 0}

Now let M be a smooth n-manifold, and let (U, phi) be a smooth chart on M. If S is a subset of U such that phi(S) is a k-slice of phi(U), then we say that S is a K-SLICE OF U.

Given a subset S < M and a nonnegative integer k, we say that S satisfies the LOCAL K-SLICE CONDITION if each point of S is contained in the domain of a smooth chart (U, phi) for M such that S intersect U is a single k-slice in U. Any such chart is called a SLICE CHART FOR S IN M and the corresponding coordinates (x^1, … , x^n) are called SLICE COORDINATES.

Question 6

What is the relationship between embedded submanifolds and k-slices? proof?

Answer 6

A subset S < M satisfies the local k -slice condition <=> S is an embedded k-dimensional submanifold.

(<=) Apply rank theorem to inclusion map which is a smooth embedding

(=>) If (x^1, … , x^n) are slice coordinate for S, we can use (x^1, … , x^k) as local coordinates for S.

pg 101 - 103

So at this point we have S < M satisfies local k-slice condition <=> S is an embedded k-dimensional submanifold <=> S is the image of a smooth embedding of a k-dimensional manifold

Question 7

Prove: If M is a smooth n-manifold with boundary, then with the subspace topology, boundary M is a topological (n-1)-dimensional manifold without boundary and has a smooth structure such that it is a properly embedded submanifold of M

Answer 7

pg 104

Question 8

How do submanifolds most often arise in practice?

Answer 8

In practice, submanifolds are most often presented asd solution sets of systems of equations. Of phi : M –> N is any map and c is any point of N, we call the set phi^-1(c) a LEVEL SET OF PHI. (if into R^k, often speak of ZERO SET phi^-1(0)).

Question 9

What is the Constant-Rank Level Set Theorem?

Proof?

Answer 9

Thm. Let M and N be smooth manifolds and phi : M –> N be a smooth map with constant rank r. Each level set of phi is a properly embedded submanifold of codimension r in M.

Pf. Let S = phi^-1(c). For each p in S, the Rank Theorem furnishes coordinates in which S is a k-slice. So S satisfies local k-slice condition => S is an embedded submanifold. It is closed in M by continuity, so properly embedded.

Cor. If phi a submersion, then each level set is properly embedded submanifold whose codimension equals the dimension of N.
pg 105

Question 10

Define: regular point/value, critical point/value, regular level set

What is the Regular Level Set Theorem? Proof?

Answer 10

REGULAR POINT if dphi_p is surjective, CRITICAL POINT otherwise. REGULAR VALUE if every point in the level set phi^-1(c) is a regular point, CRITICAL VALUE otherwise. A level set phi^-1(c) is called a REGULAR LEVEL SET if c is a regular value of phi.

Thm. Every regular level set of a smooth map between smooth manifolds is a properly embedded submanifold whose codimension is equal to the dimension of the codomain.

Pf. Could do directly as in constant-rank level set thm, but a slicker approach in book. Let phi : M –> N be smooth and c in N be regular value. The set of p in M where rank dphi_p = dim N is open and contains phi^-1(c) because of assumption that c is regular value. It follows that phi|U : U –> N is a smooth submersion hence phi^-1(c) is an embedded submanifold of U by c-r level set thm. Since the composition of smooth embedding phi^-1(C) –> U –> M is again a smooth embedding, it follows that phi^-1(c) is an embedded submanifold of M, and it is closed => proper by continuity.

Question 11

Define: defining map/function, local defining map

Is every embedded submanifold a level set of a smooth submersion? If not, what?

Answer 11

No. However the following theorem holds.

If S < M is an embedded submanifold, a smooth map phi : M –> N s.t S is a regular level set of phi is called a DEFINING MAP FOR S. If N = R^m-k, usually called DEFINING FUNCTION. More generally, if U is an open subset of M and phi: U –> N is a smooth map such that S int U is a regular level set of phi, then phi is called a LOCAL DEFINING MAP for S.

Thm. Let S be a subset of a smooth m-manifold M. Then S is an embedded k-submanifold of M <=> every point of S has a neighborhood U in M which admits a local defining function for S.

Pf. (=>) Use slice coordinates for S to define submersion.
(<=) Use suvmersion level set thm to show S int U is an embedded submanifold so satisfies k-slice condition.
pg 106 -107

Question 12

Define: immersed submanifold, codimension

Answer 12

Let M be a smooth manifold with or without boundary. An IMMERSED SUBMANIFOLD OF M is a subset S < M endowed with a topology (not necessarily the subspace topology) with respect to which it is a topological manifold (without boundary) and a smooth structure with respect to which the inclusion map S –> M is a smooth immersion. Codim again is just dim M - dim S.

Note: Every embedded submanifold is also an immersed submanifold

Question 13

What is the relationship between immersed submanifolds and immersions?

Answer 13

S < M is an immersed submanifold <=> S is the image of an injective smooth immersion F: N –> M.

(=>) Definition of immersed requires inclusion to be a smooth immersion.

(<=) Similar to embedded case. First, need to define topology on S. Let U in S be open <=> F^-1(U) is open in N. Smooth structure charts (F(U), phi o F^-1), where (U, phi) is any smooth chart for N. (We have a set bijection between N and S so we just transfer topology and smooth structure from N to S) With this structure, F is clearly a diffeomorphism on to S. The inclusion S –> M can be written S – F^-1 –> N – F –> M, where F^-1 is a diffeo and F is a smooth immersion, so composition is smooth immersion.

Thus, at this point we have:

1. Embedded submanifold = Image of a smooth embedding
2. Immersed submanifold = Image of injective smooth immersion.

Question 14

In what sense is the local structure of an immersed submanifold the same as that of an embedded one?

Answer 14

Thm. If M is a smooth manifold with or without boundary, and S < M is an immersed submanifold, then for each p in S there exists a neighborhood U of p in S that is an embedded submanifold of M.

Pf. Recall the Local Embedding Theorem (Ch 4) which says F : M –> N is a smooth immersion <=> Every point in M has a neighborhood U < M s.t. F|U : U –> N is a smooth embedding

Be careful, we are taking a neighborhood U of p in S which embedds into M. It may not be possible to find a neighborhood V of p in M s.t. V int. S is embedded. pg 111

Question 15

Define: (smooth) local parameterization of S, global parameterization, graph parameterization

Answer 15

Suppose S < M is an immersed k-dim submanifold. A LOCAL PARAMETERIZATION of S is a continuous map X : U –> M whose domain is an open subset U < R^k, whose image is an open subset of S, and which, considered as a map into S, is a homeomorphism onto its image. It is called a SMOOTH LOCAL PARAMETERIZATION if it is a diffeomorphism onto its image.

If the image of X is all of S, it is called a GLOBAL PARAMETRIZATION

Notice, we can just flip a chart around and compose with inclusion to get a local parameterization.

f: U –> R^k, then (u , f(u)) is GRAPH PARAMETERIZATION of lambda(f)

Question 16

Discuss restricting maps to submanifolds

Weakly embedded submanifolds?

Uniqueness of smooth structures on submanifolds?

Answer 16

Restricting domain of smooth manifold to immersed or embedded submanifold always preserves smoothness. Indeed, if S < M, inclusion i: S –> M is smooth by def of immersed submanifold. Since F|S = F o i, the result follows.

When the codomain is restricted, the resulting map may not be smooth. Any time the map fails to be smooth, it actually fails to even be continuous.

Thm. Suppose M is a smooth manifold, S < M is an immersed submanifold, and F : N –> M is a smooth map whose image is contained in S. If F is continuous as a map from N to S, then F: N –> S is smooth.

Everything works well for embedded submanifolds

Corollary. If S < M is embedded, then every smooth map F: N –> M whose image is contained in S is also smooth as a map from N to S

If M is a smooth manifold and S < M is an immersed submanifold, then S is said to be WEAKLY EMBEDDED IN M if every smooth map F : N –> M whose image lies in S is smooth as a map from N to S.

Now concerning uniqueness of smooth structures:

Thm. Suppose M is a smooth manifold and S < M is an embedded submanifold. The subspace topology on S and the smooth structure described above are the only topology and smooth structure with respect to which S is an embedded or immersed submanifold.

Question 17

Discuss extending smooth functions from submanifolds

Answer 17

With S < M, when we talk about f in C^inf(S) we always mean that f is smooth on the smooth manifold S (not that it is smooth in the sense of subsets).

Then the extension problem is nontrivial. We have the following

Suppose M is a smooth manifold, S < M is a smooth submanifold, and f in C^inf(S).

(a) If S is embedded, then there exist a neighborhood U of S in M and a smooth function F in C^inf(U) s.t. F|S = f.
(b) If S is properly embedded, then the neighborhood U in (a) can be taken to be all of M

Question 18

How can we view the tangent space to a submanifold as a subspace of the tangent space of the ambient manifold?

Answer 18

Let M be a smooth manifold wwb, S < M be immersed or embedded submanifold. Since the inclusion i : S –> M is a smooth immersion, at each p in S we have an injective linear map di_p : T_pS –> T_pM.

v tilda f = di_p(v)f = v(f o i) = v(f|S).

Adopt convention of identifying T_pS with its image under this map, thinking of T_pS as a linear subspace of T_pM.

Question 19

Discuss alternative ways of characterizing T_pS as a subspace of T_pM

Answer 19

1. v in T_pM is in T_pS <=> there is a smooth curve gamma: J –> M with gamma(0) = p whose image is contained in S, and which is smooth as a map into S, s.t. gamma’(0) = v.
2. If S is an embedded submanifold, then
   T_pS = { v in T_pM : vf = 0 whenever f in C^inf(M) and f|S = 0}
3. If phi is any local defining map for S, then T_pS = ker dphi_p : T_pM –> T_phi(p) N for each p in S int. U.

R^k. A vector v in T_pM is tangent to S <=> vphi^1 = … = vphi^k = 0. Pg116-117

Question 20

Discuss how to partition the tangent space of a boundary point

Answer 20

partition into inward facing, outward facing, and tangent

If p in boundary M, a vector c in T_pM \ T_p del M is INWARD POINTING if for some epsilon > 0 there exists a smooth curve g : [0, epsilon) –> M s.t, g(0) = p and g’(0) = v and is OUTWARD POINTING if there exists a curve whose domain is (-epsilon, 0].

In coordinates (x^1, … , x^n), v in T_pM inward pointing if x_n > 0, outward pointing if x_n < 0, tangent to boundary if x_n = 0.

Question 21

What is a boundary defining function? Does every smooth manifold with boundary have one?

Answer 21

A BOUNDARY DEFINING FUNCTION for M is a smooth function f: M –> [0, inf) s.t. f^-1(0) = del M and df_p != 0 for all p in del M

Yes every smooth manifold admits a boundary defining function.

Question 22

Discuss submanifolds with boundary

Answer 22

Important type: REGULAR DOMAIN in M is a properly embedded codimension-0 submanifold with boundary.

Arise as sublevel sets f^-1((-inf, b]) where b is a regular value.

Most results about manifolds without boundary have analogues to manifolds with boundary.

Generalize the k-slice condition:

If (U, xi) is a chart for M a K-DIMENSIONAL HALF-SLICE OF U is a subset {(x^1, … , x^n) in U : x^k+1 = c^k+1 … x^n = c^n, and x^k >= 0}.

Define LOCAL K-SLICE CONDITION FOR SUBMANIFOLDS WITH BOUNDARY