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The Nash Embedding Theorem

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The Nash Embedding Theorem The Nash embedding theorem Khang Manh Huynh March 13, 2018 Abstract This is an attempt to present an elementary exposition of the Nash embedding theorem for the graduate student who at least knows what a vector field is. We mainly rely on [Tao16] and [How99]. 1 Preliminary definitions Definition 1. A Riemannian manifold (M, g) is a smooth manifold M equipped with a smooth Rieman- nian metric g. In other words, for any vector fields X, Y (i.e. X, Y 2 XM), g(X, Y) is a smooth function on M and for any p in M, there is a positive-definite inner product gp : Tp M × Tp M ! R such that ¥ g(X, Y)(p) = gp(Xp, Yp). As a consequence, for any f 2 C (M) : g( f X, Y) = f g(X, Y). An isometric embedding f from (M1, g1) to (M2, g2) (both Riemannian manifolds) is a smooth embed- ding f : M1 ! M2 that preserves the metric, i.e. 8X, Y 2 XM1, 8p 2 M1 : (g1)p (Xp, Yp) = (g2)f(p) (df · Xp, df · Yp) We write e = h·, ·i for the Euclidean metric on Rn, formed by the usual Euclidean dot product. Unless indicated otherwise, Rn is always equipped with e and we write Rn for (Rn, e). Throughout this note, everything we work with is assumed to be smooth unless indicated otherwise. Every metric is a Riemannian metric unless indicated otherwise. Now we can state the main theorem: Theorem 2 (Nash embedding). Any compact Riemannian manifold (M, g) without boundary can be isometrically embedded into Rn for some n. 2 Sketch of proof We will first give a sketch of the proof, leaving the technical details as lemmas to be proven later. The first tool we require is Whitney embedding. For any u 2 C¥(M, Rn) and v 2 C¥(M, Rk), we can define u ⊕ v : p 7! (u(p), v(p)). Glueing chart functions (properly cut off) leads to Whitney. Lemma 3 (Baby Whitney). Any compact smooth manifold can be smoothly embedded into Rn for some n. Because of this, and because M is compact, we can embed M into a torus, which will simplify our calculations greatly. Lemma 4 (Torus embedding). WLOG, we can assume M = Tm = (R/Z)m, i.e. the m-dimensional torus, equipped with an arbitrary Riemannian metric g. 1 Then, we introduce some definitions: Definition 5. Let Sym denote the set of symmetric tensors on Tm. In other words, h 2 Sym when h = m (hab)1≤a,b≤m is a smooth function from T into Symm(R) (the set of symmetric m × m matrices). Any m metric g of T can be considered a symmetric tensor and gab = g(¶a, ¶b) where (¶a) are the standard coordinate vector fields on Tm. Conversely, if h 2 Sym and h > 0 (i.e. h(p) is a positive matrix 8p), h is a metric. ¥ m n m n Let Map = [n2NC (T , R ) be the set of all maps from T into some Euclidean space R . We define the function Q : Map ! Sym such that for any u 2 Map: Q(u)ab = ¶au, ¶bu where h·, ·i is the usual Euclidean dot product. We note that Q(u) ≥ 0. A metric g on Tm is called good when we can find u 2 Map such that Q(u) = g (and such u would have to be immersions, though not necessarily embeddings). We write Good for the set of good metrics. Because Q(u ⊕ v) = Q(u) + Q(v), Good is closed under addition. Let Emb ⊂ Map be the set of maps that are embeddings. Nash’s theorem says every metric is in Q(Emb). However, if every metric is good, Nash’s theorem is proven. Indeed, let g be a metric and W 2 Emb be a Whitney embedding. By rescaling W, WLOG Q(W) < g. Then g = Q(W) + g0 where g0 is a metric, therefore good and g0 = Q(v) for some v 2 Map. Then W ⊕ v 2 Emb and Q(W ⊕ v) = g. We’re done. Before we can prove every metric is good, we can prove it is approximately good by finding a very clever symmetric tensor. Lemma 6 (Approximation). For any metric g, there is h 2 Sym such that g + #2h 2 Good 8# > 0. This reduces Nash’s theorem into a local perturbation problem, where Nash made his fundamental contribution. We need another definition: m n Definition 7. (Gromov-Rohklin) An injective smooth map f : T ! R is called free when f¶af(p)ga [ m f¶a¶bf(p)ga≤b is linearly independent for any p 2 T . Note that injective immersions on a compact manifold are automatically embeddings. Let Free ⊂ Emb be the set of embeddings that are free. We can also define free maps on open submanifolds of Tm and Rm. C¥ Ck Now we can state the perturbation lemma. Recall that h −! 0 means h −! 0 8k. Lemma 8 (Perturbation). Let u 2 Free. Then for any h 2 Sym small enough (in the C¥ topology), Q(u) + h 2 Good. We use the flexibility of free maps make the symmetric tensor good. Now let us prove Nash’s theorem. Proof. Let g be any metric on Tm. We first find a free map. m m m ¥ Because T is just like R locally, we can find a finite open cover (Ui) of T with cutoffs yi 2 Cc (Ui), 0 ≤ m yi ≤ 1 such that Vi = intfyi = 1g also form an open cover of T , and there are diffeomorphisms fi : Ui ! BRm (0, 2) such that fi(Vi) = BRm (0, 1) and dfi · ¶a = ¶a 8a. m L Obviously yi and yifi can be defined on T by zero extension. Then we define F = i (yi ⊕ yifi). So F is an injective immersion, therefore embedding from Tm into Rk for some k. k(k+1) k k+ 2 a k a a b Then define the Veronese embedding ik : R ! R , (x )a=1 7! (x , x x )1≤a≤b≤k and let u = ◦ j ik F. By looking at u Vi , it’s easy to see u is a free embedding. m By rescaling u, and by the compactness of T , we can WLOG assume Q(u)(p) < gp 8p (as positive matrices). Then g = g0 + Q(u) where g0 is a metric. 2 By approximation, there is h 2 Sym such that g0 + #2h 2 Good 8# > 0. By perturbation, for # small enough, Q(u) − #2h 2 Good. So g = (g0 + #2h) + (Q(u) − #2h) 2 Good. So every metric is good and Nash’s theorem is proven. So we now only need to prove our lemmas. 3 Embedding into the torus Proof of baby Whitney. Let M be any compact manifold. We can find a finite open cover (Ui) of M with ¥ cutoffs yi 2 Cc (Ui), 0 ≤ yi ≤ 1 such that Vi = intfyi = 1g also form an open cover of M, and there are diffeomorphisms fi : Ui ! BRm (0, 2) such that fi(Vi) = BRm (0, 1). Obviously yi and yifi can be defined L on M by zero extension. Then we define W = i (yi ⊕ yifi). So W is an injective immersion, therefore embedding from M into Rk for some k. Proof of torus embedding. Let (M, g) be any compact Riemannian manifold. By Whitney, there is an em- bedding W : M ! Rm for some m. Because M is compact, by translation and rescaling, WLOG assume W(M) ⊂ (0, 1)m. So WLOG, M ⊂ Tm. We want to extend g from M to Tm. We first do this locally. Let p 2 M. As M is a regular submanifold of Tm, there is a neighborhood U m containing p, open in T with a diffeomorphism F : U ! BRm (0, 1) such that for U = U \ M, F(U) = m l m−l BRl (0, 1) × f0g where l = dim M. Parametrize R = f(y, z) : y 2 R , z 2 R g, then WLOG, via the diffeomorphism, U = f(y, z) : jyj2 + jzj2 < 1g and U = f(y, 0) : jyj2 < 1g where U is equipped with a metric g. Then simply define g on U: g(y,z)((a1, b1), (a2, b2)) = gy(a1, b1) + hb1, b2i where h·, ·i is the usual Euclidean dot product. Then going back via the diffeomorphism we get g on the original U. Finally we use partition of unity to extend g globally. We can find a finite collection of Riemannian manifolds (U , g )N such that U are open in Tm and cover M, while g are extensions of gj . Let i i i=1 i i M\Ui m U0 = T nM. Then U0 is open and (U0, g0) is a Riemannian manifold where g0 is the usual Euclidean ( )N N = N metric. Then we can find a partition of unity yi i=0 subordinate to Ui i=0 and we define g ∑i=0 yi gi. So we can find an isometric embedding (M, g) ! (Tm, g). So we just need to prove Nash’s theorem for the torus. 4 Approximating the metric Firstly, by the Gram-Schmidt process, for any p, we can find an orthonormal basis for the inner product m m m a m space TpT , gp . Identify TpT with R . Let v = (v ) 2 R be any vector, then define the rank-1 tensor T a b m v ⊗ v = vv = (v v )ab 2 Symm(R). So there are vectors (vi)i=1 such that gp = ∑i vi ⊗ vi.
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