On the geometry of tangent bundle of Finsler manifold with Cheeger-Gromoll metric

Let (M, F) be a Finsler manifold and G be the Cheeger-Gromoll metric on TM inducedby F. We show that the curvature tensor field of the Levi-Civita connection on (TM,G) isdetermined by the curvature tensor field of Vrãnceanu connection and some adapted tensorfields on TM. Then we prove that (TM,G) is locally symmetric if and only if (M, F) is locallyEuclidean. Also, we express the flag curvature of the Finsler manifold (M, F).

Keywords:

Finsler manifold , Cheeger-Gromoll metric , Levi-Civita connection , Curvature tensor , Locally symmetric manifold