Geodesics of Sasakian Metrics on Tensor Bundles

Abstract

On the basis of the phase completion the notion of vertical and horizontal lifts of vector fields is defined in the tensor bundles over a Riemannian manifold. Such a tensor bundle is made into a manifold with a Riemannian structure of special type by endowing it with Sasakian metric. The components of the Levi-Civita and other metric connections with respect to Sasakian metrics on tensor bundles with respect to the adapted frame are presented. This having been done, it is shown that it is possible to study geodesics of Sasakian metrics dealing with geodesics of the base manifolds. Mathematics Subject Classification (2000). Primary 53C22; Secondary 53C25.

On the geometry of the (1, 1)-tensor bundle with Sasaki type metric

“…For tensor bundles of type (p, q), see [18]. We now consider local 1-forms ω α in π −1 (U ) defined by…”

“…The Sasaki metric on the cotangent bundle was studied by several authors, including Mok [14], Salimov and Agca [21]. In [3,11,17,18], the Sasaki metric was studied on the tensor bundles of different types (p, q) over differentiable manifolds. In [13], Mok studied the Sasaki metric on frame bundles and calculated its curvature tensor (for details, see [4]).…”

Some notes concerning Riemannian metrics of Cheeger Gromoll type

“…Then the Lie algebra of Killing vectors on (M, g) and the Lie algebra of Killing vectors on (TM, g a,b ) are isomorphic, via the correspondence X → C X . (TM, g a,b ) An important geometric problem is to find the geodesics on the smooth manifolds with respect to the Riemannian metrics (see [21,[40][41][42][43][44]). In [21], Yano and Ishihara proved that the curves on the tangent bundles of Riemannian manifolds are geodesics with respect to certain lifts of the metric from the base manifold, if and only if the curves are obtained as certain types of lifts of the geodesics from the base manifold.…”

Geometry of the cotangent bundle with Sasakian metrics and its applications

“…Sasakian metrics (diagonal lifts of metrics) on tangent bundles were also studied in [4,8,16]. In a more general case of tensor bundles of type (1, q), (0, q) and (p, q), Sasakian metrics and their geodesics are considered in [3,10,11]. Curvature properties for the Sasakian metric of the tangent bundle are given in [1,4,6,9,16].…”