An intrinsic curvature condition for submersions over Riemannian manifolds

Abstract

Let $\pi :(M,\mathcal{H})\to (B,b)$ be a submersion equipped with a horizontal connection $\mathcal{H}$ over a Riemannian manifold $(B,b)$. We present an intrinsic curvature condition that only depends on the pair $(\mathcal{H},b)$. By studying a set of relative flat planes, we prove that a certain class of pairs $(\mathcal{H},b)$ admits a compatible metric with positive sectional curvature only if they are \textit{fat}, verifying Wilhelm's Conjecture in this class.

5 pages