Abstract
We consider a Finsler vector bundle, i. e. a vector bundle ξ : (E,p,M) endowed with a smooth function F:E →IR; (x,y) → F(x,y), that is positively homogeneous of degree 1 with respect to the variables y in fibres of ξ. Then F(x; y) = 1 with a fixed x defines the indicatrix of the given Finsler bundle in the fibre E<sub>x</sub>and F(x; y) =1 for every x and y is its indicatrix bundle. We show in Section 2 that the indicatrix is a totally umbilical submanifold in E<sub>x</sub> of constant mean curvature (-1). The indicatrix bundle is a submanifold of E \ 0 . Assuming that ξ is endowed with a nonlinear connection compatible with F and the base M is a Riemannian manifold, we define a Riemannian metric on E \ 0 and determine the normal to the indicatrix bundle.