Tangent bundle and indicatrix bundle of a Finsler manifold

Abstract

Let F^m = (M, F) be a Finsler manifold and G be the Sasaki-Finsler metric on TM°. We show that the curvature tensor field of the Levi-Civita connection on (TM°, G) is completely determined by the curvature tensor field of Vrănceanu connection and some adapted tensor fields on TM°. Then we prove that (TM°, G) is locally symmetric if and only if F^m is locally Euclidean. Also, we show that the flag curvature of the Finsler manifold F^m is determined by some sectional curvatures of the Riemannian manifold (TM°, G). Finally, for any c ≠ 0 we introduce the c-indicatrix bundle IM (c) and obtain new and simple characterizations of F^m of constant flag curvature c by means of geometric objects on both IM (c) and (TM°, G).