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The Embedding Theorems of Whitney and Nash

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The Embedding Theorems of Whitney and Nash GENERAL ARTICLE The Embedding Theorems of Whitney and Nash Harish Seshadri and Kaushal Verma We begin by briefly motivating the idea of a manifold and then discuss the embedding the- orems of Whitney and Nash that allow us to view these objects inside appropriately large Eu- clidean spaces. 1. Introduction (left) Harish Seshadri is with the Department of Let us begin by motivating the concept of a manifold. Mathematics, Indian Start with the unit circle C in the plane given by Institute of x2 + y2 =1. Science, Bengaluru. His research interest is in This is not a graph of a single function y = f(x). How- differential geometry. ever, apart√ from (±1, 0), C is the union of the graphs y ± − x2 (right) Kaushal Verma is of = 1 . Each graph is in one-to-one (bijec- with the Department of tive) correspondence with its domain, namely the inter- Mathematics, Indian val (−1, 1) on the x-axis – apart from the end points Institute of ±1. To take care of points on C near (±1, 0), we see Science, Bengaluru. His that apart from (0, ±1), C is the union of the graphs of research interest is in complex analysis. x = ± 1 − y2. Again, each graph is in bijective cor- respondence with its domain, namely the interval from (0, −1) to (0, 1) on the y-axis. Thus, to accommodate the two exceptional points (±1, 0) we had to change our coordinate axes and hence the functions that identify the pieces of C. Doing all this allows us to describe all points on C in a bijective manner by associating them with points either on the x-axis or the y-axis. It is nat- ural to ask whether such a bijection can be given for all points of the circle in a single stroke without having to resort to a two-step approach given above. The answer is no. Let us look at another example. We consider the sphere Keywords S R3 Manifold, embedding, immer- in given by sion, Nash, Whitney, smooth x2 + y2 + z2 =1. map, chart, Riemannian metric. RESONANCE ⎜ September 2016 815 GENERAL ARTICLE The theorems of As before, it is possible to solve for one of the vari- Whitney and Nash ables in terms of the other two. For example, writing concern general z = ± 1 − x2 − y2 exhibits two hemispheres of S as spaces called graphs over the unit disc in the xy-plane. The analog of ‘compact the exceptional points in the case of C above is a much 2 2 manifolds’ like the bigger set, namely the ‘equator’ x + y = 1. To accom- circle and the modate it, we write y (or x) in terms of the other two sphere. variables and proceed as before. Again, the conclusion of doing all this is that we are able to create a bijective correspondence between all points on S and discs that lie in the xy, yz or the zx planes. If we ask whether it is possible to build such a correspondence at one stroke instead of so many steps, the answer again is no. Both C and S furnish examples of what are called manifolds. These are spaces that ‘locally’ look Euclidean – that is, locally look like an open interval or an open rectangular region. Indeed, on C, consider the point (0, 1) and note that the entire upper arc is a neighbourhood containing 1 A topological space is second it, that is in bijective correspondence with an interval countable, if there exists some − , x countable collection of open sub- ( 1 1) on the -axis. The theorems of Whitney and sets such that any open subset Nash concern general spaces called ‘compact manifolds’ of the space can be written as a like the circle and the sphere; in a neighbourhood of union of elements of some sub- each of its points, a manifold of dimension n looks like family of this countable collec- the n-space Rn. The theorems address the question of tion. The property of being sec- ond countable restricts the num- whether these abstract spaces can be thought of as sit- N ber of open sets that a space ting inside R for some N. Of course, in doing so, there can have. For the usual Euclid- must be no distortions – the image must be an exact ean space, the set of all open replica of the given manifold. balls with rational radii and whose centers have rational coordi- The purpose of this note is to provide an overview of nates, is countable and forms a some of the main ideas behind the embedding theo- basis as above. rems of Whitney and Nash. To describe them briefly, 2 A Hausdorff space is a topo- we start with the definition of what we formally mean logical space in which any two by a smooth real n-dimensional manifold M. The man- different points admit disjoint ifolds we define are firstly required to be well-behaved open neighbourhoods. This property allows one to deduce sets in the sense of topology; they are topological spaces 1 2 uniqueness of limits of se- that are second countable and Hausdorff .Ahome- quences. Hausdorff was one of omorphism between topological spaces is a one-to-one the founders of topology. 816 RESONANCE ⎜ September 2016 GENERAL ARTICLE continuous map whose inverse is also continuous. The notion of diffeomorphism is defined analogously when the continuity assumption is replaced by differentiabil- ity. A topological space M which is second countable and has the Hausdorff property, can be equipped with a so-called smooth structure which enables us to do differ- ential calculus near each point. More precisely, a smooth structure on M consists of pairs (Uα,φα)whereUα is open in M and n φα : Uα → φα(Uα) ⊂ R is a homeomorphism such that M is covered by the union of the Uα’s, and for all α, β with Uα ∩ Uβ = ∅,the homeomorphism −1 φαβ = φα ◦ φβ : φβ(Uα ∩ Uβ) → φα(Uα ∩ Uβ)(1) between open sets in Rn is infinitely differentiable. Thus, M is made by gluing together open pieces (usually called ‘charts’) that are equivalent to open subsets of Rn.Fur- thermore, the collection {(Uα,φα)} is called an atlas; it is maximal in the sense that it contains all possible pairs (Uγ ,φγ) that satisfy the compatibility condition (1). The pair (Uα,φα) can be thought of as giving coor- dinates since points in Uα can be assigned coordinates by n identifying them with their images φα(p) ∈ R .There is, of course, an ambiguity in this process since the same point on M can belong to several such coordinate charts. This is accounted for by viewing the transition maps φαβ as a change of coordinates. The manifold M is said to have dimension n. Loosely speaking, Whitney’s theorem – there are, in fact, several such – says that every smooth manifold can be viewed as sitting inside RN for some large N, i.e., it can be embedded in RN . The word ‘embedding’ is used in a technically precise sense which will be ex- plained shortly. It is of interest to determine the least N that works for all manifolds with a given dimension. RESONANCE ⎜ September 2016 817 GENERAL ARTICLE ‘Riemannian Taking this one step further, we may consider a notion metric’ provides a of distance on a manifold M; this is a so-called ‘Rieman- notion of length of nian metric’ g on M. It provides a notion of length of a a tangent vector tangent vector and hence a way to measure the distance and hence a way between two points. The pair (M, g) is called a ‘Rie- to measure the mannian manifold’. More precisely, fix p ∈ M and let 1 2 n distance between x =(x ,x ,...,x ) be local coordinates near p.Fora 1 2 n two points. tangent vector v =(v ,v ,...,v )inTpM, the tangent space to M at p, the square of the length of v is given by n i j g(v,v)= gij(x)v v , i,j=1 where the matrix (gij(x))1≤i,j≤n, each of whose entries is a smooth function of x, is symmetric and positive definite. Now, there is a natural notion of a submanifold of a manifold, and every submanifold S of Rn is clearly Riemannian since the standard inner product on Rn may be restricted to S. Nash’s theorem says that not only can we embed M in RN (for some N) but we can do this in such a way that the intrinsic notion of length of a tangent vector on M is inherited from RN .We discuss these theorems in more detail in Sections 2 and 3 wherein we adopt the convention that ‘smooth’ will always mean infinitely differentiable (denoted by C∞), either for manifolds or maps between them. 2. The Whitney Embedding Theorem(s) Let us begin by recalling two definitions. Let M, N be smooth manifolds of dimensions m, n respectively and f : M → N a smooth map. The tangent space at a point p ∈ M is a linear space Rn;itisthespaceofall directions in which one can pass through p tangentially. The above informal description is dependent on think- ing of the manifold as sitting inside a Euclidean space already. However, there is an intrinsic definition of tan- gent vectors and tangent space at each point which we do not recall here. The key property of tangent spaces 818 RESONANCE ⎜ September 2016 GENERAL ARTICLE is that a smooth map f : M → N gives a natural linear The map f is said to be map an immersion if the df (p):TpM → Tf(p)N.
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