The weighted mixed curvature of a foliated manifold

TL;DR

This paper introduces a new concept of weighted mixed curvature for foliated Riemannian manifolds, providing tools to estimate diameters, prove splitting theorems, and explore generalizations of Toponogov's conjecture.

Contribution

It defines weighted mixed curvature and related conditions, enabling new geometric estimates and splitting theorems for foliated manifolds with weighted curvatures.

Findings

Established diameter estimates for foliated manifolds with weighted mixed curvature.

Proved new splitting theorems under nonnegative weighted mixed scalar curvature.

Explored generalizations of Toponogov's conjecture for positively curved foliations.

Abstract

In this paper, we introduce the weighted mixed (sectional, Ricci and scalar) curvature of a foliated (and almost-product) Riemannian manifold $(M,g)$ equipped with a vector field $X$. We define several functions ($q$th Ricci type curvatures), which "interpolate" between the weighed sectional and Ricci curvatures. The novel concepts of the "mixed curvature-dimension" condition and "synthetic dimension of a distribution" allow us to update the estimate of the diameter of a compact Riemannian foliation and to prove new splitting theorems for almost-product manifolds of nonnegative/nonpositive weighted mixed scalar curvature. In the case of positive (and nonnegative) weighted mixed sectional curvature we explore the weighted generalization of Toponogov's conjecture on totally geodesic foliations.