A general splitting principle on RCD spaces and applications to spaces with positive spectrum

TL;DR

This paper introduces a broad analytic splitting principle for RCD spaces, demonstrating that certain functions with specific Laplacian and Hessian properties imply the space is a warped product, extending known results to non-smooth contexts.

Contribution

It develops a general splitting principle applicable to RCD spaces, unifying and extending existing splitting results, and applies it to non-smooth spaces to generalize properties of Riemannian manifolds.

Findings

Spaces with suitable functions are isomorphic to warped products

The splitting principle covers most existing RCD splitting results

Extended structural properties of Riemannian manifolds to non-smooth spaces

Abstract

In this paper we develop a general `analytic' splitting principle for RCD spaces: we show that if there is a function with suitable Laplacian and Hessian, then the space is (isomorphic to) a warped product. Our result covers most of the splitting-like results currently available in the literature about RCD spaces. We then apply it to extend to the non-smooth category some structural property of Riemannian manifolds obtained by Li and Wang.