An orbit space of a nonlinear involution of $S^2\times S^2$ with nonnegative sectional curvature

TL;DR

This paper constructs new nonnegative sectional curvature metrics on certain nonorientable 4-manifolds and 2-sphere bundles, expanding the known examples of such geometries.

Contribution

It introduces a novel method to produce nonnegative curvature metrics on specific nonorientable 4-manifolds and 2-sphere bundles, including previously unknown homotopy classes.

Findings

New metrics on nonorientable 4-manifolds with fundamental group of order two.

Nonnegative curvature metrics on 2-sphere bundles over S^2 and RP^2.

Realization of new homotopy classes with nonnegative sectional curvature.

Abstract

We describe a construction of Riemannian metrics of nonnegative sectional curvature on a closed smooth nonorientable 4-manifold with fundamental group of order two that realizes a homotopy class that was not previously known to contain nonnegatively curved manifolds. The procedure yields new metrics of nonnegative sectional curvature on any 2-sphere bundle with base space the 2-sphere or the real projective plane.