# Embedding Riemannian manifolds into some infinite dimensional manifolds?

First of all I am new to the field of embedding one manifold into another other.

I have recently come across with the paper "Embedding Riemannian manifolds by their heat kernel" by P. BERARD, G. BESSON, S. GALLOT (published in Geometric and Functional Analysis in 1984), who prove that one can embed a closed Riemannian manifold with certain assumptions on its Ricci curvature and its diameter into $\ell^2$.

My question is if there is other results on isometric embedding of a closed Riemannian manifold without any assumption on its Ricci curvature and its diameter into some infinite dimensional manifold (Banach, Hilbert, or Frechet manifolds)? Or into some $L^2$ space? The point is the ambient space must be an infinite dimensional manifold, not finite dimension. Or if not, what other "weaker" assumptions have to be imposed on the manifold?

Thank you.

There is a more general result. Fix an even Schwartz function $\newcommand{\bR}{\mathbb{R}}$ $w:\bR\to[0,\infty)$.

Let $\Delta$ be the Laplacian of the compact connected Riemann manifold $(M,g)$, $\dim M=m$. Its eigenvalues are

$$0=\lambda_0< \lambda_1\leq \lambda_2\leq \cdots$$

where each eigenvalue appears as many times as its multiplicity. Fix an orthonormal eigen-basis $(\Psi_k)_{k\geq 0}$ of $L^2(M,g)$,

$$\Delta\Psi_k=\lambda_k\Psi_k.$$

For each $\newcommand{\ve}{\varepsilon}$ $\ve >0$ define $\Xi_\ve: M\to L^2(M,g)$ by setting

$$\Xi_\ve(p)= \left(\frac{\ve^{m+2}}{d_m}\right)^{\frac{1}{2}}\sum_{k\geq 0}w\bigl(\,\ve \sqrt{\lambda_k}\,\bigr)^{\frac{1}{2}}\Psi_k(p)\Psi_k,$$

where

$$d_m:=\frac{2\pi^{\frac{m}{2}}}{m \Gamma(\frac{m}{2})}\int_0^\infty w(r) r^{m+1} dr.$$

Then for $\ve>0$ sufficiently small the map $\Xi_\ve$ is an embedding. Moreover, as $\ve\to 0$ the induced metric converges to the original metric. No assumption on the metric $g$ is required. Note that when $w$ is compactly supported the above sum consists of finitely many terms so $\Xi_\ve$ is actually an embedding into a finite dimensional space.

The result of Berard-Besson-Gallot corresponds to $w(r)=e^{-r^2}$. For details see this paper.

• Thank you Prof. Nicolaescu. To me this is a very amazing result in the sense that no conditions have to be imposed on the metric $g$. – Ho Man Ho Sep 9 '18 at 15:45