Curvature-dimension inequalities on sub-Riemannian manifolds obtained from Riemannian foliations, Part I

TL;DR

This paper introduces a generalized curvature-dimension inequality for sub-Riemannian manifolds derived from Riemannian foliations, linking geometry with sub-Laplacian properties and enabling eigenvalue bounds.

Contribution

It establishes a new curvature-dimension inequality applicable to a broad class of sub-Riemannian manifolds from Riemannian foliations, with geometric interpretation and eigenvalue estimates.

Findings

Validates the inequality on a large class of manifolds

Provides geometric interpretation of invariants

Derives lower bounds for sub-Laplacian eigenvalues

Abstract

We give a generalized curvature-dimension inequality connecting the geometry of sub-Riemannian manifolds with the properties of its sub-Laplacian. This inequality is valid on a large class of sub-Riemannian manifolds obtained from Riemannian foliations. We give a geometric interpretation of the invariants involved in the inequality. Using this inequality, we obtain a lower bound for the eigenvalues of the sub-Laplacian. This inequality also lays the foundation for proving several powerful results in Part~II.