Properties of some five-dimensional Einstein metrics

Abstract

The volumes, spectra and geodesics of a recently constructed infinite family of five-dimensional inhomogeneous Einstein metrics on the two S3 bundles over S2 are examined. The metrics are in general of cohomogeneity one but they contain the infinite family of homogeneous metrics Tp,1. The geodesic flow is shown to be completely integrable, in fact both the Hamilton–Jacobi and the Laplace equation separate. As an application of these results, we compute the zeta function of the Laplace operator on Tp,1 for large p. We discuss the spectrum of the Lichnerowicz operator on symmetric transverse tracefree second rank tensor fields, with application to the stability of Freund–Rubin compactifications and generalized black holes.