Some Aspects on the Geometry of the Tangent Bundles and Tangent Sphere Bundles of a Riemannian Manifold

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.Mathematics Subject Classifications (2000): 53B35, 53C07, 53C25, 53C55.

On the tangent bundle of a hypersurface in a Riemannian manifold

“…Lemma 3.4 (cf. [8]) Suppose that (N n+1 (c), G) is a space form of the constant sectional curvature c. Then the Riemannian curvature tensor R of (T N n+1 (c), G s ) is given by…”

“…Munteanu [8] computed the Riemannian curvature tensor of T N endowed with the general metric G a,b . For (T N n+1 (c), G s ), we have the following lemma.…”

Conformality of a differential with respect to Cheeger-Gromoll type metrics

“…For any p, q, α, the Riemannian metric h p,q,α is a special case of a metric considered in [7]. Notice that if p, q, α are constants and α = 1 then h p,q,α becomes a metric from [1].…”

Some notes concerning Riemannian metrics of Cheeger Gromoll type

“…A compatible almost complex structure is also introduced and the tangent bundle TM becomes a locally conformal almost Kählerian manifold. In [11], Munteanu studied another Riemannian metric on the tangent bundle TM of a Riemannian manifold M which generalizes the Sasaki metric and Cheeger Gromoll metric and a compatible almost complex structure which confers a structure of locally conformal almost K ählerian manifold to TM together with the metric. He found conditions under which the tangent bundle TM is almost Kählerian, locally conformal Kählerian or Kählerian when the tangent bundle TM has constant sectional curvature or constant scalar curvature.…”