Nash embedding theorem n-dimensional Riemannian manifold M, assume that there exists an embedding ε : M → V where V is an mdimensional Hilbert space (n ≤ m).

http://stat.fsu.edu/~anuj/pdf/papers/Y2009/TuragaChapterVideoManifolds.pdf

4 Statistical Inference on Manifolds What are the challenges in performing a statistical analysis if the underlying state space is non-Euclidean? Take the case of the simplest statistic, the sample mean, for a sample set (x1, x2, . . . , xn) on R n: x¯k = 1 k X k i=1 xi , xi ∈ R n . (16) Since ¯xk is a widely used and studied statistic, one already knows the pros and cons of using ¯xk as an estimate of the population mean. For example, we know that ¯xk is an unbiased and efficient estimator, but it is susceptible to the outliers. Now what if the underlying space is not R n but a non-Euclidean manifold instead? To answer this question we consider an n-dimensional Riemannian manifold M. Let d(p, q) denote the length of the shortest geodesic between arbitrary points p, q ∈ M. To facilitate a general discussion, we will assume that there exists an embedding ε : M → V where V is an mdimensional Hilbert space (n ≤ m). We have chosen V to be a vector space so that we can perform a statistical analysis in V using standard techniques from multivariate calculus. The distance between any two elements p, q ∈ M is the geodesic distance d(p, q) when the geodesic is restricted to be in M and it is kε(p) − ε(q)k, with the norm of V , when the geodesic is allowed to be in V . The latter distance, of course, depends on the choice of the embedding ε. We start the analysis by assuming that we are given a probability density function f on M. This function, by definition, satisfies the properties that f : M → R≥0 and R M f(p)dp = 1, where dp denotes the reference measure on M with respect to which the density f is defined. We can extend f to the larger set V by simply setting: ˜f(x) = f(p) if x = ε(p), p ∈ M 0 if x 6∈ ε(M) . (17) Naturally, ˜f is a probability density function on V . There are two possibilities for computing statistics on M – intrinsic and extrinsic. We describe them next.

Nash embedding theorem

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The Nash embedding theorems (or imbedding theorems), named after John Forbes Nash, state that every Riemannian manifold can be isometrically embeddedinto some Euclidean space. Isometric means preserving the length of every path. For instance, bending without stretching or tearing a page of paper gives anisometric embedding of the page into Euclidean space because curves drawn on the page retain the same arclength however the page is bent.

The first theorem is for continuously differentiable (C¹) embeddings and the second for analytic embeddings or embeddings that are smooth of class C^k, 3 ≤ k ≤ ∞. These two theorems are very different from each other; the first one has a very simple proof and leads to some very counterintuitive conclusions, while the proof of the second one is very technical but the result is not that surprising.

The C¹ theorem was published in 1954, the C^k-theorem in 1956. The real analytic theorem was first treated by Nash in 1966; his argument was simplified considerably by Greene & Jacobowitz (1971). (A local version of this result was proved by Élie Cartan and Maurice Janet in the 1920s.) In the real analytic case, the smoothing operators (see below) in the Nash inverse function argument can be replaced by Cauchy estimates. Nash's proof of the C^k- case was later extrapolated into the h-principle and Nash–Moser implicit function theorem. A simplified proof of the second Nash embedding theorem was obtained by Günther (1989) who reduced the set of nonlinear partial differential equations to an elliptic system, to which the contraction mapping theorem could be applied.