Conformal deformations of the Ebin metric and a generalized Calabi metric 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 Kähler 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.

arXiv comment28 pages

References in corpus (2)

• An overview of the Riemannian metrics on spaces of curves using the Hamiltonian approach
• The Riemannian L^2 topology on the manifold of Riemannian metrics

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• Smooth perturbations of the functional calculus and applications to Riemannian geometry on spaces of metrics
• Geodesics of positive Lagrangians in Milnor fibers
• Weak topology on CAT(0) spaces
• Conformal transformations of the pseudo-Riemannian metric of a homogeneous pair