Splitting, parallel gradient and Bakry-Emery Ricci curvature

TL;DR

This paper proves a splitting theorem for Riemannian manifolds with a symmetric diffusion operator, showing under certain curvature and gradient conditions that the manifold splits as a product space.

Contribution

It establishes a new splitting theorem involving the Bakry-Emery Ricci curvature and the behavior of the diffusion operator, extending classical results to weighted manifolds.

Findings

Manifold splits as a Riemannian product under given conditions.

Conditions involve non-negative Ricci curvature and maximum of gradient magnitude.

The proof leverages properties of functions with parallel gradient vector fields.

Abstract

In this paper we obtain a splitting theorem for the symmetric diffusion operator $\Delta_\phi=\Delta-\left<\nabla\phi,\nabla \right>$ and a non-constant $C^3$ function $f$ in a complete Riemannian manifold $M$, under the assumptions that the Ricci curvature associated with $\Delta_\phi$ satisfies ${\rm Ric}_\phi(\nabla f,\nabla f)\ge 0$, that $|\nabla f|$ attains a maximum at $M$ and that $\Delta_\phi$ is non-decreasing along the orbits of $\nabla f$. The proof uses the general fact that a complete manifold $M$ with a non-constant smooth function $f$ with parallel gradient vector field must be a Riemannian product $M=N\times \mathbb{R}$, where $N$ is any level set of $f$.