New structures on the tangent bundles and tangent sphere bundles

Abstract

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 found conditions under which $T(M)$ is almost Kählerian, locally conformal 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 (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.

20 pages, LaTeX2e 4.17

Cited by in corpus (7)

• On vector bundle manifolds with spherically symmetric metrics
• Weighted metrics on tangent sphere bundles
• Product and anti-Hermitian structures on the tangent space
• Conformality of a differential with respect to Cheeger-Gromoll type metrics
• A note about scalar curvature on the total space of a vector bundle
• On the geometry of the normal bundle with a metric of Cheeger-Gromoll type
• Tensor sphere bundle of Cheeger-Gromoll type