Extension of metrics

11 SUMMARY In this thesis, same metrics in the tangent bundle of a Riemannian manifold were investigated. Christoffel symbols FCB of the metric II were obtained with respect to ~ A ~ the induced coordinates in T{Mn) and the components KDCB of curvature tensor K were found. I + II metric was defined and it was shown that the Levi-Civita connection of the metric II and the metric I + II coincide. The metric connection V -A - - is defined and the components Rjjbc of curvature tensor R of the V were calculated. In addition, the metric I + III in the Tangent Bundle of a Riemannian manifold was defined and its components were investigated with respect to the induced coordinates in T(Mn). The concept of the adapted frame was defined and the metric I + III was considered with respect to the adapted frame. Finally, we considered the almost Kahler structure and the differential equations of the geodezics of the tangent bundle T(Mn) with respect to the metric / + III.