Integral formulae for codimension-one foliated Finsler manifolds

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 ${\cal F}$.