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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.

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    $\begingroup$ In fact, those three authors do not impose any condition on a closed manifold in order to embed it in $\ell ^2$ (because compactness itself guarantees the boundedness of its Ricci curvature and of its diameter). What they do is to show that under supplementary conditions placed on the diameter and the Ricci curvature, all the closed manifolds satisfying them may be uniformly (in a certain way) embedded into the same Hilbert space. Furthermore, their embedding is not an isometry (but it is continuous)! $\endgroup$
    – Alex M.
    Commented Jan 13, 2023 at 8:42

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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.

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    $\begingroup$ 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 $g$. $\endgroup$
    – Ho Man-Ho
    Commented Sep 9, 2018 at 15:45
  • $\begingroup$ $\Xi _\varepsilon$ is not an isometry, though, so strictly speaking this looks like half an answer to the OP's question. I believe that the OP was asking about the existence of a Nash isometric embedding theorem but under the weaker requirement of the target space being infinite-dimensional (which, hopefully, would make the proof easier and shorter). $\endgroup$
    – Alex M.
    Commented Jan 13, 2023 at 8:45

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