Metrics of constant scalar curvature on sphere bundles

TL;DR

This paper constructs constant scalar curvature metrics on sphere bundles over homogeneous spaces, especially spheres, and analyzes their conformal classes, advancing understanding of geometric structures on such bundles.

Contribution

It introduces new methods to build constant scalar curvature metrics on sphere bundles over homogeneous spaces with specific representation conditions.

Findings

Constructed metrics of constant scalar curvature on sphere bundles

Analyzed the number of such metrics in conformal classes over spheres

Extended the understanding of scalar curvature on fiber bundles

Abstract

Let $G/H$ be a Riemannian homogeneous space. For an orthogonal representation $\phi$ of $H$ on the Euclidean space $\mathbb{R}^{k+1}$, there corresponds the vector bundle $E=G\times_{\phi}\mathbb{R}^{k+1} \to G/H$ with fiberwise inner product. Provided that $\phi$ is the direct sum of at most two representations which are either trivial or irreducible, we construct metrics of constant scalar curvature on the unit sphere bundle $UE$ of $E$. When $G/H$ is the round sphere, we study the number of constant scalar curvature metrics in the conformal classes of these metrics.