New structures on the tangent bundles and tangent sphere bundles

Details New structures on the tangent bundles and tangent sphere bundles New structures on the tangent bundles and tangent sphere bundles

2005-11-15

arxiv.org/abs/math/0511377v2

Mediterranean J. Mathematics, 5 (2008) 1, 43 - 60 (part of this article)

Mathematics

Differential Geometry

20 pages, LaTeX2e 4.17

Scientific paper

10.1007/s00009-008-0135-4

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 K\"ahlerian manifold. This is the natural generalization of the well known almost K\"ahlerian structure on $T(M)$. We found conditions under which $T(M)$ is almost K\"ahlerian, locally conformal K\"ahlerian or K\"ahlerian 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 (not necessary unitary) endowed with the restriction of the Sasaki metric from $T(M)$. 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.

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