Decompositions of the space of Riemannian metrics on a compact manifold with boundary

TL;DR

This paper establishes new decomposition and slice theorems for the space of Riemannian metrics on a compact manifold with boundary, providing insights into the structure of metrics with fixed boundary conformal class and characterizing relative Einstein metrics.

Contribution

It introduces Koiso-type decomposition and Ebin-type slice theorems for metrics with boundary conditions, advancing understanding of metric space structure.

Findings

Decomposition of metric space with boundary conditions

Slice theorem for metrics with fixed boundary conformal class

Characterization of relative Einstein metrics

Abstract

In this paper, for a compact manifold $M$ with non-empty boundary, we give a Koiso-type decomposition theorem, as well as an Ebin-type slice theorem, for the space of all Riemannian metrics on $M$ endowed with a fixed conformal class on the boundary. As a corollary, we give a characterization of relative Einstein metrics.