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embedding of differentiable manifolds

Contents

Idea

Embeddings of smooth manifolds are submanifold inclusions.

Definition

Definition

(embedding of smooth manifolds)

An embedding of smooth manifolds is a smooth function f:XYf : X \hookrightarrow Y between smooth manifolds XX and YY such that

  1. ff is an immersion;

  2. the underlying continuous function is an embedding of topological spaces.

A closed embedding is an embedding such that the image f(X)Yf(X) \subset Y is a closed subset.

Examples

Nonexample

(immersions that are not embeddings)

Consider an immersion f:(a,b) 2f \;\colon\; (a,b) \to \mathbb{R}^2 of an open interval into the Euclidean plane (or the 2-sphere) as shown on the right. This is not a embedding of smooth manifolds: around the points where the image crosses itself, the function is not even injective, but even a#t the points where it just touches itself, the pre-images under ff of open subsets of 2\mathbb{R}^2 do not exhaust the open subsets of (a,b)(a,b), hence do not yield the subspace topology.

As a concrete examples, consider the function (sin(2),sin()):(π,π) 2(sin(2-), sin(-)) \;\colon\; (-\pi, \pi) \longrightarrow \mathbb{R}^2. While this is an immersion and injective, it fails to be an embedding due to the points at t=±πt = \pm \pi “touching” the point at t=0t = 0.

graphics grabbed from Lee

Properties

Basic properties

Proposition

(proper injective immersions are equivalently the closed embeddings)

Let XX and YY be smooth manifolds, and let f:XYf \colon X \to Y be a smooth function. Then the following are equivalent

  1. ff is a proper injective immersion;

  2. ff is a closed embedding (def. 1).

Embedding into Euclidean space

Every smooth manifold has a embedding of smooth manifolds into a Euclidean space k\mathbb{R}^k of some dimension kk.

For compact smooth manifolds this is easy to see (prop. 2 below), while the generalization to non-compact smooth manifolds requires a tad more work (theorem 1 below). What is non-trivial is to find the minimum dimension of the ambient Euclidean space for which embedding in general still exist. The Whitney embedding theorem (prop. 3 below) says that this is twice the dimension of the manifolds to be embedded.

Proposition

For every compact smooth manifold XX (of finite dimension), there exists some kk \in \mathbb{N} such that XX has an embedding (def. 1) into the Euclidean space of dimension kk:

Xembd k X \overset{\text{embd}}{\hookrightarrow} \mathbb{R}^k
Proof

Let

{ nϕ iU iX} iI \{\mathbb{R}^n \underoverset{\simeq}{\phi_i}{\longrightarrow} U_i \subset X\}_{i \in I}

be an atlas exhibiting the smooth structure of XX. In particular this is an open cover, and hence by compactness there exists a finite subset JIJ \subset I such that

{ nϕ iU iX} iJI \{\mathbb{R}^n \underoverset{\simeq}{\phi_i}{\to} U_i \subset X\}_{i \in J \subset I}

is still an open cover.

Since XX is a smooth manifold, there exists a partition of unity {f iC (X,)} iJ\{f_i \in C^\infty(X,\mathbb{R})\}_{i \in J } subordinate to this cover with smooth functions f if_i (by this prop.).

This we may use to extend the inverse chart identifications

XU iψ i n X \supset \;\; U_i \underoverset{\simeq}{\psi_i}{\longrightarrow} \mathbb{R}^n

to smooth functions

ψ^ i:X n \hat \psi_i \;\colon\; X \to \mathbb{R}^{n}

by setting

ϕ^ i:x{f i(x)ψ i(x) | xU iX 0 | otherwise. \hat \phi_i \;\colon\; x \mapsto \left\{ \array{ f_i(x) \cdot \psi_i(x) &\vert& x \in U_i \subset X \\ 0 &\vert& \text{otherwise} } \right. \,.

The idea now is to combine all these functions to obtain an injective function

(ψ^ i) iJ:X( n) |J| n|J|. (\hat \psi_i)_{i \in J} \;\colon\; X \longrightarrow (\mathbb{R}^n)^{\vert J\vert } \simeq \mathbb{R}^{n \cdot {\vert J \vert }} \,.

But while this is injective, it need not be an immersion, since the derivatives of the product functions f iψ if_i \cdot \psi_i may vanish, even though the derivatives of the two factors do not vanish separately. However this is readily fixed by adding yet more ambient coordinates and considering the function

(ψ^ i,f i) iI:X( n+1)) |J| (n+1)|J|. (\hat \psi_i, f_i)_{i \in I} \;\colon\; X \longrightarrow \left(\mathbb{R}^{n+1})\right)^{\vert J \vert} \simeq \mathbb{R}^{(n+1)\cdot {\vert J \vert}} \,.

This is an immersion. Hence it remains to see that it is also an embedding of topological spaces.

By this prop it is sufficient to see that the injective continuous function is a closed map. But this follows generally since XX is a compact topological space by assumption, and since YY is a Hausdorff topological space by definition of manifolds, and since maps from compact spaces to Hausdorff spaces are closed and proper.

Proposition

For every smooth manifold XX of dimension nn (Hausdorff, sigma-compact), there exists some kk \in \mathbb{N} such that XX has an embedding into the Euclidean space of dimension kk.

What is harder to prove is that kk may be chosen to be as small as 2n2n

Proposition

(Whitney's strong embedding theorem)

For every smooth manifold XX of dimension nn (Hausdorff, sigma-compact), there is an embedding into the Euclidean space of dimension 2n2n.

In fact this bound is minimal, there are smooth manifolds of dimension nn which have no embdding into k\mathbb{R}^k for k<2nk \lt 2n.

Space of Embeddings

This section follows from a discussion on the Algebraic Topology mailing list as to making concrete the contractibility of the space of embeddings of a smooth manifold in ()\mathbb{R} ^{(\infty )}, the union of the Euclidean spaces. There are two parts to showing that this space is contractible. The first is showing that it is not empty. The second is to exhibit a contraction.

Showing that it is not empty is a partition of unity argument. As the space that we are embedding in is infinite dimensional, we do not need to resort to either compactness or Whitney’s embedding theorem to obtain an embedding. One of these would be required if we wanted to show that our embedding was actually in some finite Euclidean space.

The contraction depends on certain properties of ()\mathbb{R} ^{(\infty )} which are quite general and seemingly unrelated to the question of embeddings. The key property is the existence of a shift map. The same idea is used in showing that many other related spaces are contractible, including the infinite sphere.

Existence of Embeddings

Theorem

Let MM be a smooth manifold. Let VV be an infinite dimensional locally convex topological vector space. Then there is an embedding (def. 1) MVM \hookrightarrow V.

Proof

We shall start with the case V= ()V = \mathbb{R} ^{(\infty )}. The more general case will follow from that.

As MM is a smooth manifold, we can choose a partition of unity {ρ i:iI}\{ \rho _{i} : i \in I\} on MM with the property that for each ρ i\rho _{i}, its support is contained in the domain of some chart for MM and the family of those domains is locally finite. Let us suppose that these charts are ϕ i:U i n\phi _{i} \colon U_{i} \to \mathbb{R} ^{n}, where n=dimMn = \dim M. As MM is second countable, this partition of unity will be countable.

Using the partition of unity, we extend ϕ i:U i n\phi _{i} \colon U_{i} \to \mathbb{R} ^{n} to a smooth map ψ i:M n+1\psi _{i} \colon M \to \mathbb{R} ^{n+1} by setting, for xU ix \in U_{i}, ψ i(x)=(ρ i(x)ϕ i(x),ρ i(x))\psi _{i}(x) = (\rho _{i}(x) \phi _{i}(x),\rho _{i}(x)) and, for xU ix \notin U_{i}, ψ i(x)=0\psi _{i}(x) = 0. Using the ψ i\psi _{i} we define ψ:M ()\psi \colon M \to \mathbb{R} ^{(\infty )} by ψ(x)=(ψ i(x))\psi (x) = (\psi _{i}(x)). This is well-defined as, for any xMx \in M, there are only finitely many of the ρ i\rho _{i} for which ρ i(x)0\rho _{i}(x) \ne 0 and thus only finitely many of the ψ i\psi _{i} for which ψ i(x)0\psi _{i}(x) \ne 0. We define ψ:MV\psi \colon M \to V by composing this with the inclusion ()V\mathbb{R} ^{(\infty )} \to V.

We claim that this is an embedding.

The first thing to show is that it is continuous. Let xMx \in M. Then there is some ii such that xU ix \in U_{i}. As the family U iU_{i} is locally finite, there are only a finite number of other U jU_{j} with non-trivial intersection with U iU_{i}. Thus there are only a finite number of ρ j\rho _{j} which take non-zero values on U iU_{i}. Hence the image of U iU_{i} under ψ\psi lies in some k\mathbb{R} ^{k} and the induced map U i kU_{i} \to \mathbb{R} ^{k} is built up from a finite number of the ϕ j\phi _{j} and ρ j\rho _{j}, whence is continuous. Hence ψ\psi is continuous.

The next thing to show is that it is injective. Suppose that x,yMx,y \in M are such that ψ(x)=ψ(y)\psi (x) = \psi (y). As the partition of unity functions can be recovered from ψ\psi , we must therefore have ρ i(x)=ρ i(y)\rho _{i}(x) = \rho _{i}(y) for all ii. Since {ρ i}\{ \rho _{i}\} is a partition of unity, there must be some ii such that ρ i(x)0\rho _{i}(x) \ne 0 (whence also ρ i(y)0\rho _{i}(y) \ne 0). This means that xx and yy lie in the domain of the chart ϕ i\phi _{i}. Moreover, as ψ(x)=ψ(y)\psi (x) = \psi (y), we must have ψ i(x)=ψ i(y)\psi _{i}(x) = \psi _{i}(y). Lastly, as ρ i(x)=ρ i(y)\rho _{i}(x) = \rho _{i}(y) and this is non-zero, we have ϕ i(x)=ϕ i(y)\phi _{i}(x) = \phi _{i}(y). Therefore x=yx = y.

We want to show that ψ\psi is a topological embedding. To do this, we consider first what happens to the sets ρ i 1(0,)\rho _{i}^{-1}(0,\infty ). Pick some iIi \in I. Let p i: () n+1p_{i} \colon \mathbb{R} ^{(\infty )} \to \mathbb{R} ^{n+1} be the projection onto the n+1n + 1 coordinates corresponding to the position of ψ i\psi _{i} in ψ\psi , so that p iψ=ψ ip_{i} \psi = \psi _{i}. Let W n+1W \subseteq \mathbb{R} ^{n+1} be the open subset where the last coordinate is non-zero. Then p i 1(W)p_{i}^{-1}(W) is open in ()\mathbb{R} ^{(\infty )}. Now xψ(M)p i 1(W)x \in \psi (M) \cap p_{i}^{-1}(W) if and only if ρ i(x)0\rho _{i}(x) \ne 0, thus:

ψ(M)p i 1(W)=ψ(ρ i 1(0,)). \psi (M) \cap p_{i}^{-1}(W) = \psi (\rho _{i}^{-1}(0,\infty )).

We define a map W nW \to \mathbb{R} ^{n} by (x,t)t 1x(x,t) \mapsto t^{-1} x. The composition

ρ i 1(0,)ψp i 1(W)p iW n \rho _{i}^{-1}(0,\infty ) \xrightarrow{\psi } p_{i}^{-1}(W) \xrightarrow{p_{i}} W \to \mathbb{R} ^{n}

is then seen to be the same as the restriction of ϕ i\phi _{i} to ρ i 1(0,)\rho _{i}^{-1}(0,\infty ). As ϕ i\phi _{i} is a homeomorphism and ρ i\rho _{i} is continuous, the above map ρ i 1(0,) n\rho _{i}^{-1}(0,\infty ) \to \mathbb{R} ^{n} is a homeomorphism onto its image which is an open subset of n\mathbb{R} ^{n}.

We therefore see that if Vρ i 1(0,)V \subseteq \rho _{i}^{-1}(0,\infty ) is open then ψ(V)\psi (V) is equal to the inverse image of ϕ i(V)\phi _{i}(V) under the composition p i 1(W)W np_{i}^{-1}(W) \to W \to \mathbb{R} ^{n} and is thus open in ψ(M)\psi (M).

Now suppose that VMV \subseteq M is an arbitrary open subset. Then for each iIi \in I, Vρ i 1(0,)V \cap \rho _{i}^{-1}(0,\infty ) is open and the union of these is VV. So ψ(V)\psi (V) is the union of ψ(Vρ i 1(0,))\psi (V \cap \rho _{i}^{-1}(0,\infty )) and thus is open in ψ(M)\psi (M). Hence ψ\psi is an embedding.

To complete the proof for ()\mathbb{R} ^{(\infty )}, we need to show that ψ\psi is an immersion. This is a local condition and follows from the fact that we can recover ϕ i\phi _{i} from ψ i\psi _{i}.

The case for an arbitrary VV is a simple adaptation of the above. As VV is infinite dimensional, it admits linear injections ()V\mathbb{R} ^{(\infty )} \to V and composing one of those with ψ\psi gives an injective map MVM \to V. To ensure that it is an embedding, we need suitable projections V n+1V \to \mathbb{R} ^{n+1}. To know that these exist, we simply choose our map ()V\mathbb{R} ^{(\infty )} \to V carefully as follows. Pick a non-zero vector v 1Vv_{1} \in V. By the Hahn-Banach theorem, there is some f 1V *f_{1} \in V^{*} with f 1(v 1)=1f_{1}(v_{1}) = 1. Now choose a non-zero v 2kerf 1v_{2} \in \ker f_{1}. Again, we have f 2V *f_{2} \in V^{*} such that f 2(v 2)=1f_{2}(v_{2}) = 1 and f 2(v 1)=0f_{2}(v_{1}) = 0. By construction, f 1(v 1)=0f_{1}(v_{1}) = 0 also. We continue this recursively, choosing at each stage a non-zero v kkerf jv_{k} \in \bigcap \ker f_{j} and f kV *f_{k} \in V^{*} with f k(v k)=1f_{k}(v_{k}) = 1 and f k(v j)=0f_{k}(v_{j}) = 0 for j<kj \lt k. The result is an injection ()V\mathbb{R} ^{(\infty )} \to V together with functions f jV *f_{j} \in V^{*} such that the composition ()Vf j\mathbb{R} ^{(\infty )} \to V \xrightarrow{f_{j}} \mathbb{R} is the jjth coordinate function.

With an assumption on VV and a careful choice of partition of unity, we can ensure that our embedding so constructed has closed image.

Theorem

Let MM be a smooth manifold. Let VV be a locally convex topological vector space and assume that there exists a family of continuous linear functionals {f j:j}V *\{f_j : j \in \mathbb{N}\} \subseteq V^* with the properties:

  1. For jj \in \mathbb{N}, f jf_j is non-trivial on kjkerf k\bigcap_{k \ne j} \ker f_k, and
  2. The family of functionals { jFf j:|F|<}\{\sum_{j \in F} f_j : {|F|} \lt \infty\} is equicontinuous

then there is an embedding ψ:MV\psi \colon M \to V with closed image.

Proof

From the first property, we can find v j kjkerf kv_j \in \bigcap_{k \ne j} \ker f_k such that f j(v j)=1f_j(v_j) = 1. We use the family {v j}\{v_j\} to define a linear injection ()V\mathbb{R}^{(\infty)} \to V (continuous by necessity). The functionals f jf_j provide the necessary coordinate projections so that the proof of Theorem 1 works.

In addition, we shall assume that our partition of unity used in the proof of Theorem 1 is such that the functions therein have compact support.

Let (x λ)(x_\lambda) be a net in MM for which (ψ(x λ))(\psi(x_\lambda)) converges in VV, say to xx. The assumption on the functionals is there to show that x0x \ne 0. Let us show why this is the case. For a finite set FF \subseteq \mathbb{N}, let f F= jFf jf_F = \sum_{j \in F} f_j. The assumption of equicontinuity means that Ff F 1(1,1)\bigcap_F f_F^{-1}(-1,1) is an open neighbourhood of 00 in VV. Now for each yMy \in M, there is some finite FNF \subseteq \N for which f j(y)=1\sum f_j(y) = 1. This is because we can choose FF to correspond to those terms in the partition of unity which are non-zero on yy. Hence ψ(M)\psi(M) has empty intersection with the above neighbourhood of 00, and so no net in ψ(M)\psi(M) can converge to 00.

Thus x0x \ne 0 and, moreover, there is some finite FF \subseteq \mathbb{N} for which f F(x)0f_F(x) \ne 0 and thus there is some jj \in \mathbb{N} for which f j(x)0f_j(x) \ne 0. Since (ψ(x λ))x(\psi(x_\lambda)) \to x, there is therefore a cofinal subset on which f jf_j is never zero. Let us pass to that subnet. Now on MM, the functions f iψf_i \circ \psi are either the functions in the partition of unity or are dominated by them. Hence there is some ii \in \mathbb{N} with the property that for all λ\lambda, ρ i(x λ)0\rho_i(x_\lambda) \ne 0 (now that we’ve passed to the subnet). This means that (x λ)(x_\lambda) lies in the support of ρ i\rho_i, which is (by the first assumption) a compact subset of MM. It therefore has a convergent subnet with limit, say xx'. But by continuity of ψ\psi and uniqueness of limits in VV, it must be the case that ψ(x)=x\psi(x') = x.

Hence ψ(M)\psi(M) is closed in VV.

There are various situations in which it is straightforward to find a suitable family of functionals. If VV is barrelled, then finding a family for which f j\sum f_j weakly converges will do since then the family of partial sums is simply bounded and hence, as VV is barrelled, equicontinuous.

Thus the coordinate projections will do for ()\mathbb{R}^{(\infty)} and for 1\ell^1. For 2\ell^2 we can take weighted projections, say f j=1jp jf_j = \frac{1}{j} p_j. In this case, our dual vectors will be {je j}\{j e_j\}. For \ell^\infty we can take weighted projections corresponding to a sequence in 1\ell^1.

It is interesting to note what is going on in these latter cases where we take weighted projections. If we embedded our manifold using the standard basis vectors as our “anchor points” then the structure of the ambient space is insufficient to guarantee that our manifold does not contain a convergent net with limit point outside. More exactly, there could be a net in our manifold such that in the image it converges to zero. The partition of unity is not enough to keep our points away from zero, since as we get further along the family in our partition we could need more and more terms. That is, we could have the situation where we have a sequence (x n)(x_n) in MM where x nx_n is “seen” by nn of the functions in the partition of unity. Then we could have ψ(x n)\psi(x_n) to be something like (0,,0,1/n,1/n,,1/n,0,)(0,\dots,0,1/n,1/n,\dots,1/n,0,\dots). In, say, 2\ell^2 then this sequence converges to 00. Essentially, ψ(x n)\psi(x_n) gets more and more diffuse. Introducing the weights, or rather their inverses, counters this by ensuring that terms “far down” the sequence are given more consideration.

Contractibility of Embeddings

Once we know that the space of embeddings of a compact smooth manifold in ()\mathbb{R} ^{(\infty )} is not empty, its contractibility follows from rather general properties. It follows from the existence of a split map.

Theorem

Let MM be a smooth manifold. Let VV be a locally convex topological vector space which admits a split map. Then the space of embeddings, Emb(M,V)\Emb(M,V), is contractible.

Proof

Recall that to have a split map on VV means that we have a decomposition VVVV \cong V \oplus V with a non-degeneracy property. This implies that one of the induced maps VVV \to V, called the splitting map, has no eigenvectors.

Let S +:VVS^{+} \colon V \to V be a splitting map and let S S^{-} be the remainder. Then the map H:[0,1]:VVH \colon [0,1] \colon V \to V given by H(t,v)=H t(v)=(1t)v+tS +vH(t,v) = H_{t}(v) = (1 - t) v + t S^{+} v is a homotopy between the identity on VV and S +S^{+}.

At t=0t = 0, H t=IH_{t} = I which is an inclusion. For t0t \ne 0 then if H t(v)=0H_{t}(v) = 0 we have that S +v=t 1(1t)vS^{+} v = t^{-1}(1 - t) v whence vv is an eigenvector of S +S^{+}. But the assumption that S +S^{+} is a splitting map includes the property that it has no eigenvectors, and thus H tH_{t} is an inclusion for all tt. Moreover, H tH_{t} is an embedding for all tt since I=H t+tS vI = H_{t} + t S^{-} v.

Hence we can apply H tH_{t} to Emb(M,V)\Emb(M,V) to obtain a homotopy between the identity on Emb(M,V)\Emb(M,V) and the inclusion of Emb(M,V)\Emb(M,V) in itself as the subspace of all embeddings that lie in the even subspace.

Now we pick some embedding ψ 0:MV\psi _{0} \colon M \to V which, by applying S S^{-} if necessary, we assume to exist in the image of S S^{-}. We define a homotopy G tG_{t} on the embedding space by G t(ψ)=tψ 0+(1t)ψG_{t}(\psi ) = t \psi _{0} + (1 - t) \psi , using the additive structure in VV. That this passes through the space of embeddings is clear since for t1t \ne 1 we can recover the embedding ψ 0\psi _{0} by composing with S S^{-}. Combining H tH_{t} and G tG_{t} as homotopies leads to the desired contraction of the embedding space.

References

Lecture notes include

Revised on May 24, 2017 11:55:08 by Urs Schreiber (131.220.184.222)