New structures on the tangent bundles and tangent sphere bundles
نویسنده
چکیده
In this paper we study a Riemanian metric on the tangent bundle T (M) of a Riemannian manifold M which generalizes Sasaki metric and Cheeger Gromoll metric and a compatible almost complex structure which together with the metric confers to T (M) a structure of locally conformal almost Kählerian manifold. This is the natural generalization of the well known almost Kählerian structure on T (M). We compute the Levi Civita connection, the curvature tensor, the sectional curvature and the scalar curvature of this metric and we found conditions under which T (M) is almost Kählerian or Kählerian or when T (M) has constant sectional curvature or constant scalar curvature. Then we will restrict to the unit tangent bundle and we find an isometry with the tangent sphere bundle with the Sasaki metric. Moreover, we found that this map preserves also the natural almost contact structures obtained from the almost Hermitian ambient structures on the unit tangent bundle and the tangent sphere bundle, respectively. Mathematics Subject Classifications (2000): 53B35, 53C07, 53C25, 53C55.
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