Isometries of the Ebin metric

TL;DR

This paper investigates the symmetries of the space of Riemannian metrics under the Ebin metric, establishing conditions under which two such metric spaces are isometric based on the diffeomorphism of their underlying manifolds.

Contribution

It characterizes the self-isometries of the space of Riemannian metrics with the Ebin metric and proves an isometry criterion based on manifold diffeomorphisms.

Findings

Self-isometries of the space are characterized.

Two metric spaces are isometric iff their manifolds are diffeomorphic.

Provides a classification of symmetries in the space of Riemannian metrics.

Abstract

We study the space of Riemannian metrics over a compact manifold equipped with the Ebin metric. We characterize its self-isometries and prove that two such spaces are isometric if and only if their underlying manifolds are diffeomorphic.