Metrics of constant scalar curvature on sphere bundles

Abstract

Let $G/H$ be a Riemannian homogeneous space. For an orthogonal representation $ϕ$ of $H$ on the Euclidean space $\mathbb{R}^{k+1}$, there corresponds the vector bundle $E=G\times_ϕ\mathbb{R}^{k+1} \to G/H$ with fiberwise inner product. Provided that $ϕ$ 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.