Integral formulae for codimension-one foliated Finsler spaces

Mon, 18/01/2016 - 14:00

Recent decades brought increasing interest in Finsler spaces $(M,F)$,
especially, in extrinsic geometry of their hypersurfaces.
Randers metrics (i.e., $F=\alpha+\beta$, $\alpha$ being the norm of a Riemannian structure
and $\beta$ a 1-form of $\alpha$-norm smaller than $1$ on~$M$),
appeared in Zermelo's control problem, are of special interest.

After a short survey of above, we will discuss
Integral formulae, which provide obstructions for existence of foliations
(or compact leaves of them) with given geometric properties.
The first known Integral formula (by G.\,Reeb) for codimension-1 foliated closed manifolds tells us that
the total mean curvature $H$ of the leaves is zero (thus, either $H\equiv0$ or $H(x)H(y)<0$  for some $x,y\in M$).

Using a unit normal to the leaves of a codimension-one foliated $(M,F)$,
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 a foliation in terms of $F$;
then express them in terms of invariants of $\alpha$ and~$\beta$.
Using our results \cite{rw2} for Riemannian case, we present new Integral formulae
for codimension-one foliated $(M, F)$ and $(M, \alpha+\beta)$.
Some of them generalize Reeb's formula.