Integral formulae for codimension-one foliated Finsler manifolds

TL;DR

This paper derives integral formulae relating geometric invariants of codimension-one foliations on Finsler manifolds, generalizing classical results and providing new tools for understanding extrinsic geometry in Finsler spaces.

Contribution

It introduces a new approach to express geometric invariants of foliations on Finsler manifolds, leading to generalized integral formulae that extend previous results.

Findings

Derived integral formulae for Finsler foliations

Expressed invariants in terms of Randers data

Generalized classical curvature integral relations

Abstract

We study extrinsic geometry of a codimension-one foliation ${\cal F}$ of a closed Finsler space $(M,F)$, in particular, of a Randers space $(M,\alpha+\beta)$. Using a unit vector field $\nu$ orthogonal (in the Finsler sense) to the leaves of ${\cal F}$ we define a new Riemannian metric $g$ on $M$, which for Randers case depends nicely on $(\alpha,\beta)$. For that $g$ we derive several geometric invariants of ${\cal F}$ (e.g. the Riemann curvature and the shape operator) in terms of $F$, then under natural assumptions on $\beta$ which simplify derivations, we express them in terms of corresponding invariants arising from $\alpha$ and $\beta$. Using our approach (2012), we produce the integral formulae for ${\cal F}$ on $(M, F)$ and $(M, \alpha+\beta)$, which relate integrals of mean curvatures with those involving algebraic invariants obtained from the shape operator of a foliation, and the Riemann curvature in the direction $\nu$. They generalize the formulae by Brito, Langevin and Rosenberg, which show that total mean curvatures (of arbitrary order $k$) for codimension-one foliations on a closed $(m+1)$-dimensional manifold of constant curvature $K$ don't depend on a choice of ${\cal F}$.