Semi-Riemannian manifolds with a doubly warped structure

TL;DR

This paper studies the structure of semi-Riemannian manifolds formed as quotients of doubly warped products, providing conditions for their decomposition into simpler geometric components and analyzing their uniqueness.

Contribution

It establishes when such quotient manifolds are essentially doubly warped products and explores conditions for their unique decomposition, extending understanding of their geometric structure.

Findings

Manifolds are covered by products of leaves under certain conditions.

Necessary and sufficient conditions for manifold decomposition are provided.

Uniqueness of product decomposition is analyzed in non-simply connected cases.

Abstract

We investigate manifolds obtained as a quotient of a doubly warped product. We show that they are always covered by the product of two suitable leaves. This allows us to prove, under regularity hypothesis, that these manifolds are a doubly warped product up to a zero measure subset formed by an union of leaves. We also obtain a necessary and sufficient condition which ensures the decomposition of the whole manifold and use it to give sufficient conditions of geometrical nature. Finally, we study the uniqueness of direct product decomposition in the non simply connected case.