MIHAI ANASTASIEI ^{1}, MANUELA GÃŽRÅ¢U ^{2} 1. University "Al. I. Cuza" IaÅŸi, Faculty of Mathematics, 700506, IaÅŸi, Romania, and Mathematics Institute "O.Mayer", Romanian Academy IaÅŸi Branch, 700506, IaÅŸi, Romania, e-mail: anastas@uaic.ro
2. "Vasile Alecsandri" University of BacÄƒu, Faculty of Sciences, Department of Mathematics and Informatics, Calea MÄƒrÄƒÅŸeÅŸti 157, BacÄƒu 600115, Romania, e-mail: girtum@yahoo.com.

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_{x}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_{x} 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.