Tangent bundle and indicatrix bundle of a Finsler manifold

Let F-m = (M, F) be a Finsler manifold and G be the Sasaki-Finsler metric on TM degrees. We show that the curvature tensor field of the Levi-Civita connection on (TM degrees, G) is completely determined by the curvature tensor field of Vranceanu connection and some adapted tensor fields on TM degrees. Then we prove that (TM degrees, 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 degrees, G). Finally, for any c not equal 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 degrees, G).