The Scalar Curvature of the Tangent Bundle of a Finsler Manifold

Let Fm = (M,F ) be a Finsler manifold and G be the Sasaki– Finsler metric on the slit tangent bundle TM0 = TM {0} of M . We express the scalar curvature ρ̃ of the Riemannian manifold (TM0, G) in terms of some geometrical objects of the Finsler manifold Fm. Then, we find necessary and sufficient conditions for ρ̃ to be a positively homogenenous function of degree zero with respect to the fiber coordinates of TM0. Finally, we obtain characterizations of Landsberg manifolds, Berwald manifolds and Riemannian manifolds whose ρ̃ satisfies the above condition. Introduction The geometry of the tangent bundle TM of a Riemannian manifold (M, g) goes back to Sasaki [10], who constructed on TM a Riemannian metric G which in our days is called the Sasaki metric. Then, several papers on the interrelations between the geometries of (M, g) and (TM,G) have been published (see Gudmundsson and Kappos [6] for results and references). The extension of the study from Riemannian manifolds to Finsler manifolds is not an easy task. This is because a Finsler manifold F = (M,F ) does not admit a canonical linear connection on M , that plays the role of the Levi–Civita connection on a Riemannian manifold. Recently, the first author (cf. [3]) has initiated a study of the interrelations between the geometries of both the tangent bundle and indicatrix bundle of a Finsler manifold on one side, and the geometry of the manifold itself, on the other side. The main tool in the study was the Vrănceanu connection induced by the Levi–Civita connection on (TM0, G), where G is the Sasaki–Finsler metric on TM0. We study the geometry of a Finsler manifold F = (M,F ) under the assumption that the scalar curvature ρ̃ of (TM0, G) is a positively homogeneous function of degree zero with respect to the fiber coordinates (y) of TM0. In the first part 2010 Mathematics Subject Classification: Primary 53C60, 53C15.