Variations of metric that preserve a Riemannian submersion and geometry of its fibers

TL;DR

This paper studies how specific metric variations on a Riemannian submersion affect fiber geometry, deriving conditions for preserving fiber properties and analyzing curvature behavior under these variations.

Contribution

It provides formulas and conditions for metric variations that preserve fiber geometry and examines the impact on curvature and critical points of geometric functionals.

Findings

Derived a formula for the variation of the second fundamental form.

Established conditions for preserving fiber properties like being totally geodesic or minimal.

Analyzed how variations affect sectional curvatures and identified non-constant vertizontal curvatures.

Abstract

On the domain of a Riemannian submersion, we consider variations (i.e., smooth one-parameter families) of Riemannian metrics, for which the submersion is Riemannian and which all keep the metric induced on its fibers fixed. We obtain a formula for the variation of the second fundamental form of the fibers with respect to such changes of metric. We find a choice of parameters defining the variations, that allows to easily formulate the necessary and sufficient conditions for preserving particular geometry of the fibers, i.e., keeping them totally geodesic, totally umbilical, or minimal. These conditions are related to the existence of Killing, conformal Killing and divergence-free vector fields on the fibers. We find conditions for metric to be a critical point of integrated squared norms of the mean curvature and the second fundamental form of the fibers, with respect to the considered variations, and prove that at all critical points of these functionals the second variation is non-negative. We also examine variations of sectional curvatures of planes defined by the horizontal lifts of vectors from the image of the submersion. In particular, we find that some variations preserving Riemannian submersions with totally geodesic fibers can make vertizontal curvatures not constant on the fibers.