Fiberwise volume decreasing diffeomorphisms on product manifolds

TL;DR

This paper investigates fiberwise volume decreasing diffeomorphisms on product manifolds, proving they must preserve fibers under certain conditions, and explores implications for isometries, group actions, and extensions of classical theorems.

Contribution

It establishes conditions under which fiberwise volume decreasing diffeomorphisms are fiber volume preserving and demonstrates splitting of isometries on product manifolds.

Findings

Fiberwise volume decreasing diffeomorphisms are fiber volume preserving under cohomological conditions.

Isometries of product manifolds split into factors.

Generalization of Bieberbach theorem and extension of Talelli's conjecture for group actions.

Abstract

Given a closed connected Riemannian manifold M and a connected Riemannian manifold N, we study fiberwise volume decreasing diffeomorphisms on the product M x N. Our main theorem shows that in the presence of certain cohomological condition on M and N such diffeomorphisms must map a fiber diffeomorphically onto another fiber and are therefore fiber volume preserving. As a first corollary, we show that the isometries of M x N split. We also study properly discontinuous actions of a discrete group on M x N. In this case, we generalize the first Bieberbach theorem and prove a special case of an extension of Talelli's conjecture.