Non-negative curvature on certain product manifolds

TL;DR

This paper proves that certain product manifolds, homotopy equivalent to specific homogeneous spaces, admit metrics with non-negative sectional curvature, expanding the class of manifolds known to support such geometric structures.

Contribution

It establishes conditions under which manifolds homotopy equivalent to homogeneous spaces admit non-negative curvature metrics when producted with spheres, generalizing previous results.

Findings

Product manifolds with homotopy type of certain homogeneous spaces admit non-negative curvature.

Many homogeneous manifolds, including compact rank-one symmetric spaces, satisfy the key assumption.

Existence of non-negative curvature metrics on these product manifolds is proven.

Abstract

Let $G/H$ be a closed, simply connected homogeneous manifold. Suppose every stable class of real vector bundles over $G/H$ contains a homogeneous bundle. Then, for any closed, simply connected smooth manifold $M$ homotopy equivalent to $G/H$, there exists $n>\mathrm{dim}(M)$ such that the product manifold $M\times S^{n}$ admits a metric with non-negative sectional curvature. Many homogeneous manifolds satisfy this assumption, including simply connected compact rank-one symmetric spaces, and among others.