Integral formulae for codimension-one foliated Randers spaces

TL;DR

This paper develops new integral formulae for codimension-one foliated Randers spaces, generalizing classical results like Reeb's formula, and explores their implications for the geometry and curvature of foliations.

Contribution

It introduces novel integral formulae for Randers spaces, extending classical curvature results and providing tools for analyzing foliations with specific geometric properties.

Findings

Generalized Reeb's formula for Randers spaces

Total mean curvature of leaves is zero under certain conditions

Total second mean curvature relates to Ricci curvature in the normal direction

Abstract

Integral formulae for foliated Riemannian manifolds provide obstructions for existence of foliations or compact leaves of them with given geometric properties. This paper continues our recent study and presents new integral formulae and their applications for codimension-one foliated Randers spaces. The goal is a generalization of Reeb's formula (that the total mean curvature of the leaves is zero) and its companion (that twice total second mean curvature of the leaves equals to the total Ricci curvature in the normal direction). We also extend results by Brito, Langevin and Rosenberg (that total mean curvatures of arbitrary order for a codimension-one foliated Riemannian manifold of constant curvature don't depend on a foliation). All of that is done by a comparison of extrinsic and intrinsic curvatures of the two Riemannian structures which arise in a natural way from a given Randers structure.