Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

Article

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Bull. Korean Math. Soc. 2012; 49(5): 939-947

Printed September 30, 2012

https://doi.org/10.4134/BKMS.2012.49.5.939

Copyright © The Korean Mathematical Society.

Embedding Riemannian manifolds via their eigenfunctions and their heat kernel

Hiba Abdallah

Laboratoire de Math\'ematiques associ\'e au CNRS

Abstract

In this paper, we give a generalization of the embeddings of Riemannian manifolds via their heat kernel and via a finite number of eigenfunctions. More precisely, we embed a family of Riemannian manifolds endowed with a time-dependent metric analytic in time into a Hilbert space via a finite number of eigenfunctions of the corresponding Laplacian. If furthermore the volume form on the manifold is constant with time, then we can construct an embedding with a complete eigenfunctions basis.

Keywords: Riemannian manifold, Laplacian, eigenvalues/eigenfunctions, heat equation, embedding

MSC numbers: 47B40, 53B21, 54A20, 58A32, 58C25, 58C40