Some Examples of Randers Spaces

A Riemannian almost product structure on a manifold induces on a submanifold of codimension 1 a structure generalizing the paracontact structures and containing a Riemannain metric and an one form . We show that the pair consisting of this Riemannian metric and one form defines a strongly convex Randers metric on submanifold. We establish some properties of this Randers metric and we provide some examples. 1. Randers metrics provided by induced structures Let (M̃, g̃, P̃ ) be a Riemannian almost product manifold. This means that P̃ is an almost product structure on M̃ i.e. P̃ 2 = I (identity) and g̃ is a Riemannian structure on M̃ which is compatible with P̃ , i.e. g̃(P̃X, P̃Y ) = g̃(X, Y ) for any vector fields X, Y on M̃ . Let M be a submanifold of codimension 1 in M̃ . We denote by g the Riemannian metric induced by g̃ on M and by N a field of unitary vectors that are normal to M . Then for any vector field X tangent to M , the vector field P̃X decomposes in a tangent and a normal component: (1.1) P̃X = PX + b(X)N, X ∈ X (M), (X (M) denotes the Lie algebra of vector fields on M). It is clear that b is an 1-form on M . Also, the vector field P̃N decomposes in the form (1.2) P̃N = ξ + aN, where ξ is a vector field tangent to M and a is a function on M . 2000 Mathematics Subject Classification. 53C60, 53C25.