Isometric immersions of RCD$(K,N)$ spaces via heat kernels

Abstract

For an RCD(K, N ) space (X, d, m), one can use its heat kernel ρ to embed it into L 2 (m) by a locally Lipschitz map Φ t (x) := ρ(x, •, t). In particular, an RCD(K, N ) space is said to be an isometric heat kernel immersing space, if its associated Φ t is an isometric immersion multiplied by a constant depending on t for any t > 0. We prove that any compact isometric heat kernel immersing RCD(K, N ) space is isomorphic to an unweighted closed smooth Riemannian manifold. More generally, it is proved that any noncollapsed RCD(K, N ) space with an isometrically immersing eigenmap is also isomorphic to an unweighted closed smooth Riemannian manifold. As an application, we give some diffeomorphic finiteness theorems for a certain class of Riemannian manifolds with a curvature-dimension-diameter bound and an almost isometrically immersing eigenmap.