Properties of some five dimensional Einstein metrics

TL;DR

This paper investigates the geometric, spectral, and dynamical properties of a new family of five-dimensional Einstein metrics on $S^3$ bundles over $S^2$, revealing integrability and spectral characteristics relevant to theoretical physics.

Contribution

It introduces and analyzes a new class of five-dimensional Einstein metrics, demonstrating their integrability, spectral properties, and implications for stability in compactifications and black hole solutions.

Findings

Geodesic flow is completely integrable.

Laplace equation separates in these metrics.

Computed the zeta function for large p in $T^{p,1}$.

Abstract

The volumes, spectra and geodesics of a recently constructed infinite family of five-dimensional inhomogeneous Einstein metrics on the two $S^3$ bundles over $S^2$ are examined. The metrics are in general of cohomogeneity one but they contain the infinite family of homogeneous metrics $T^{p,1}$. 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 $T^{p,1}$ 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 generalised black holes.