Non-canonical variations of Riemannian submersions with totally geodesic fibers

TL;DR

This paper explores how variations in Riemannian metrics affect sectional curvatures and structures of submersions with totally geodesic fibers, providing conditions for positive curvature and new geometric structures.

Contribution

It introduces new conditions and methods for modifying Riemannian submersions to achieve desired curvature properties and geometric structures.

Findings

Conditions for positive sectional curvature on product manifolds.

Existence of fat Riemannian submersions with non-constant vertizontal curvatures.

Variations preserving isometric group actions while altering horizontal distributions.

Abstract

Using variations of Riemannian metric that preserve a given Riemannian submersion, keep its fibers totally geodesic and the metric restricted to the fibers fixed, but change the horizontal distribution, we examine changes of sectional curvatures in horizontal and vertical directions. We obtain conditions, in terms of a 1-form defining a variation, to locally make all sectional curvatures positive on the product of a manifold with positive curvature and a circle, while preserving the Riemannian submersion with geodesic fibers defined by the projection from the product. We examine conditions for obtaining weak contact metric structures from K-contact structures. We demonstrate existence of fat Riemannian submersions with totally geodesic fibers and vertizontal (i.e., spanned by a horizontal and a vertical vector) curvatures non-constant along a fiber. For a Riemannian submersion defined by an isometric group action, with totally geodesic fibers of dimension higher than one, we find variations that preserve the isometric action, while changing the horizontal distribution and its curvatures.