A decomposition theorem for immersions of product manifolds

TL;DR

This paper introduces polar metrics on product manifolds, characterizes their local decompositions, and describes their isometric immersions into space forms, generalizing classical decomposition theorems.

Contribution

It presents a new class of metrics called polar metrics, a de Rham-type theorem for their local decomposition, and a comprehensive description of their isometric immersions into space forms.

Findings

Polar metrics unify product and warped product metrics.

A de Rham-type theorem characterizes manifolds with polar metrics.

Complete description of isometric immersions with adapted second fundamental forms.

Abstract

We introduce polar metrics on a product manifold, which have product and warped product metrics as special cases. We prove a de Rham-type theorem characterizing Riemannian manifolds that can be locally decomposed as a product manifold endowed with a polar metric. For a product manifold endowed with a polar metric, our main result gives a complete description of all its isometric immersions into a space form whose second fundamental forms are adapetd to its product structure, in the sense that the tangent spaces to each factor are preserved by all shape operators. This is a far-reaching generalization of a basic decomposition theorem for isometric immersions of Riemannian products due to Moore as well as its extension by N\"olker to isometric immersions of warped products.