LECTURE 6: SMOOTH SUBMANIFOLDS

1. Smooth submanifolds as images Let M be a smooth manifold of dimension n, and k < n. Definition 1.1. A subset S ⊂ M is a k-dimensional (embedded/regular) smooth sub- manifold of M if for every p ∈ S, there is a chart (ϕ, U, V ) around p in M such that k k+1 n ϕ(U ∩ S) = V ∩ (R × {0}) = {x ∈ ϕ(U) | x = ··· = x = 0}. We will call codim(S) = n − k the codimension of S.

Example. The sphere Sn is a smooth submanifold of Rn+1. Can you construct a local chart of Rn+1 near every point of Sn which satisfies the condition in the definition 1.1? Example. For any smooth map f : M → N, the graph

Γf = {(p, q) | q = f(p)} is a smooth submanifold of M × N.

Recall that an embedding of a smooth manifold M into a smooth manifold N is an immersion f : M → N (meaning that dfp is injective for all p ∈ M) so that M is homeomorphic to f(M) ⊂ N. It is not suprising that a smooth submanifold (with the subspace topology) is always a smooth manifold by itself, and the inclusion map from the submanifold to the ambient manifold is always an embedding: Theorem 1.2. Let S be a k-dimensional submanifold of M. Then with the subspace topology, S admits a unique smooth structure so that (1) S is a smooth manifold of dimension k. (2) The inclusion map ι : S,→ M is a smooth embedding.

Proof. With the subspace topology, S satisfies the Hausdorff and second-countable conditions. To construct local charts on S, we denote n k 1 n 1 k π : R → R , (x , ··· , x ) 7→ (x , ··· , x ) k n 1 k 1 k  : R ,→ R , (x , ··· , x ) 7→ (x , ··· , x , 0, ··· , 0). Fix any chart (ϕ, U, V ) of M satisfying definition 1.1. Let X = U ∩ S, Y = π ◦ ϕ(X) −1 −1 and ψ = π ◦ ϕ. Then ψ|X = ϕ ◦ . So (ψ, X, Y ) is a chart on S. Moreover, charts of this type are compatible, since the transition map −1 −1 ψβ ◦ ψα = π ◦ ϕβ ◦ ϕα ◦  = π ◦ ϕα,β ◦  is smooth.

1 2 LECTURE 6: SMOOTH SUBMANIFOLDS

We need to check that the inclusion map ι : S,→ M is an embedding. Obviously if we endow ι(S) with the subspace topology, the map ι : S → ι(S) is a homeomorphism. It is an immersion because in each pair of charts as constructed above, ι = ϕ−1 ◦  ◦ ψ. We refer to theorem 5.31 in Lee’s book (page 114) for the uniqueness of the topolo- gy/smooth structures on S that makes ι an embedding.  It turns out that smooth submanifolds are exactly the image of embeddings: Theorem 1.3. Let f : M → N be an embedding. Then the image f(M) is a smooth submanifold of N. Proof. Let p ∈ M and q = f(p). Since f is an immersion, the canonical immersion theorem implies that there exists charts (ϕ1,U1,V1) near p and (ψ1,X1,Y1) near q such −1 m n that on V1, ψ1 ◦ f ◦ ϕ1 is the canonical embedding  : R → R restricted to V1, i.e.

ψ1 ◦ f =  ◦ ϕ1 on U1. Since f is a homeomorphism onto its image, f(U1) is open in f(M) ⊂ N. In other words, there exists an open set X ⊂ N such that f(U1) = f(M) ∩ X. Replace X1 by X1 ∩ X, and Y1 by ψ1(X1 ∩ X). Then for this new chart (ψ1,X1,Y1), m ψ1(X1 ∩ f(M)) = Y1 ∩ ψ1(f(U1)) = Y1 ∩ (ϕ1(U1)) = Y1 ∩ (R × {0}).  Now let S ⊂ M be a submanifold, and p ∈ S. Since ι : S,→ M is an embedding, dιp : TpS → TpM is injective. We might identify TpS as the vector subspace dιp(TpS) of TpM for every p ∈ S. In other words, we can identify any vector Xp ∈ TpS with the ∞ vector Xep = dιp(Xp) in TpM so that for any f ∈ C (M),

Xep(f) = (dιp(Xp))f = Xp(f ◦ ι) = Xp(f|S).

A natural question is: which vectors in TpM can be regarded as vectors in TpS? Theorem 1.4. Suppose S ⊂ M is a submanifold, and p ∈ S. Then ∞ TpS = {Xp ∈ TpM | Xp(f) = 0 for all f ∈ C (M) with f|S = 0}.

∞ Proof. Obviously if Xp ∈ TpS, then for f ∈ C (M) with f|S = 0, Xep(f) = 0.

Conversely, if Xp ∈ TpM satisfies Xp(f) = 0 for all f that vanishes on S, we need to show Xp ∈ TpS. Take a coordinate chart (ϕ, U, V ) on M such that near p, S is k+1 n given by x = ··· = x = 0. Then TpM is the span of ∂1, ··· , ∂n, while TpS is the P i subspace spanned by ∂1, ··· , ∂k. In other words, a vector Xp = X ∂i lies in TpS if and only if Xi = 0 for i > k. Now let h be a smooth bump function supported in U that equals 1 in a neighbor- j hood of p. For any j > k, consider the function fj(x) = h(x)x (ϕ(x)), extended to be zero on M \ U. Then fj|S = 0. So X ∂(h(ϕ−1(x))xj) 0 = X (f ) = Xi (ϕ(p)) = Xj p j ∂xi LECTURE 6: SMOOTH SUBMANIFOLDS 3 for any j > k. It follows that Xp ∈ TpS.  Remark. Note the difference between an immersion and an embedding: • If f : M → N is an immersion, then by the canonical immersion theorem, any point p ∈ M has a neighborhood in M whose image is nice in N. • If f : M → N is an embedding, then by theorem 1.3, any point q ∈ f(M) has a neighborhood in f(M) that is nice in N. We will call the image of an injective immersion an immersed submanifold. Unlike embedded submanifolds, the two topologies of an immersed submanifold f(M), one from the topology of M via the map f and the other from the subspace topology of N, might be different, as we have seen from the examples we constructed last time. Remark. More generally, we have

Theorem 1.5. If f : M → N is a smooth map with constant rank k (i.e. dfp is of constant rank k at any point p ∈ M), then the image of f is an immersed submanifold with tangent space the image of the tangent map dfp.

2. Smooth submanifolds as level sets Recall that a smooth map f : M → N is a submersion at p ∈ M if the differential dfp : TpM → Tf(p)M is surjective. Definition 2.1. Suppose f : M → N is a smooth map between smooth manifolds. A point q ∈ N is called regular value if f is a submersion at each p ∈ f −1(q). One can regard the next theorem as a generalization of the implicit function theo- rem we mentioned in lecture 2. Theorem 2.2. If q is a regular value of a smooth map f : M → N, then S = f −1(q) is a submanifold of M of dimension dim M − dim N. Moreover, for every p ∈ S, TpS is the kernel of the map dfp : TpM → TqN. Proof. Let p ∈ S = f −1(q). Then by the canonical submersion theorem, there are charts (ϕ1,U1,V1) centered at p and (ψ1,X1,Y1) centered at q such that f(U1) ⊂ X1, −1 −1 −1 −1 and π = ψ1 ◦ f ◦ ϕ1 . It follows that ϕ1 maps U1 ∩ f (q) onto V1 ∩ π (0). So f (q) is a submanifold of M. Now denote the inclusion by ι : S → M. Then for any p ∈ S, f ◦ ι(p) = q. In other words, f ◦ ι is a constant map on S. So dfp ◦ dιp = 0, i.e. dfp = 0 on the image of dιp : TpS,→ TpM, or in other words, TpS ⊂ ker(dfp). By dimension counting we conclude that TpS coincides with the kernel of dfp.  Theorem 2.2 gives us a very powerful tool to construct smooth manifolds. Example. Consider the map n+1 1 n+1 1 2 n+1 2 f : R → R, (x , ··· , x ) 7→ (x ) + ··· + (x ) . Then Sn = f −1(1). Since the Jacobian Jf(x) = 2(x1, ··· , xn+1) 6= 0 4 LECTURE 6: SMOOTH SUBMANIFOLDS on Sn, we conclude that Sn is an n-dimensional submanifold of Rn+1. Moreover, for any a = (a1, ··· , an+1) ∈ Sn, n n+1 TaS = {v ∈ R | v · a = 0}. Remark. The theorem above can be generalized to smooth maps of constant ranks. Theorem 2.3 (Constant rank level set theorem). Let M,N be smooth mani- fold, and f : M → N be a smooth map with constant rank k. Then each level set of f is a closed submanifold of codimension k in M. The proof is left as an exercise.