Conformal deformations of the Ebin metric and a generalized Calabi metric on the space of Riemannian metrics

TL;DR

This paper explores conformal deformations of the Ebin metric on the space of Riemannian metrics, introduces a generalized Calabi metric, and analyzes its geometry, geodesics, and completion, revealing new geometric properties.

Contribution

It characterizes a new conformally related metric generalizing Calabi's metric to Riemannian metrics and explicitly solves its geodesic equation, providing insights into its geometric structure.

Findings

Explicit geodesic solutions using a constant of motion

Identification of the metric's completion as strictly smaller than Ebin's

First example of a metric with a smaller completion on the space of Riemannian metrics

Abstract

We consider geometries on the space of Riemannian metrics conformally equivalent to the widely studied Ebin L^2 metric. Among these we characterize a distinguished metric that can be regarded as a generalization of Calabi's metric on the space of K\"ahler metrics to the space of Riemannian metrics, and we study its geometry in detail. Unlike the Ebin metric, the geodesic equation involves non-local terms, and we solve it explicitly by using a constant of the motion. We then determine its completion, which gives the first example of a metric on the space of Riemannian metrics whose completion is strictly smaller than that of the Ebin metric.