Integral formula for codimension-one foliated $(\alpha,\beta)$-spaces

TL;DR

This paper extends integral formulae for codimension-one foliated spaces to a broader class of $(eta,eta)$-spaces, providing tools to analyze geometric properties and obstructions for such foliations.

Contribution

It introduces a new metric framework for $(eta,eta)$-spaces and derives a Reeb type integral formula applicable to various Finsler spaces.

Findings

Derived a set of perturbed metrics for $(eta,eta)$-spaces.

Calculated the Weingarten operator for these metrics.

Established a Reeb type integral formula for $(eta,eta)$-spaces.

Abstract

Integral formulae for foliated Riemannian manifolds provide obstructions for existence of foliations or compact leaves of them with given geometric properties. Recently, we associated a new Riemannian metric to a codimension-one foliated Finsler space and proved integral formulae for general and for Randers spaces. In the paper, we study this metric for a wider class of codimension-one foliated $(\alpha,\beta)$-spaces and embody it in a set of metrics that can be viewed as a perturbed metric associated with $\alpha$. For such metrics we calculate the Weingarten operator of the leaves and deduce the Reeb type integral formula, which can be used for $(\alpha,\beta)$-spaces, e.g. Randers and Kropina spaces.