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Geodesics of the cheeger gromoll metric

In Cheeger and Gromoll study complete manifolds of nonnegative curvature and suggest a construction of Riemannian metrics useful in that contex. The main purpose of the paper is to investigate geodesics on the tangent bundle with respect to the Cheeger-Gromoll metric. | Turk J Math 33 (2009) , 99 – 105. ¨ ITAK ˙ c TUB doi:10.3906/mat-0804-24 Geodesics of the Cheeger-Gromoll Metric A. A. Salimov, S. Kazimova Abstract The main purpose of the paper is to investigate geodesics on the tangent bundle with respect to the Cheeger-Gromoll metric. Key Words: Geodesics, Cheeger-Gromoll metric, Horizontal and vertical lift. 1. Introduction In [1] Cheeger and Gromoll study complete manifolds of nonnegative curvature and suggest a construction of Riemannian metrics useful in that contex. Inspired by a paper of Cheeger and Gromoll, in [4] Musso and Tricerri defined a new Riemannian metric CG g on tangent bundle of Riemannian manifold which they called the Cheeger-Gromoll metric. The Levi-Civita connection of CG g and its Riemannian curvature tensor are calculated by Sekizawa in [5] (for more details see [2],[3]). The main purpose of this paper is to investigate geodesics of the Cheeger-Gromoll metrics on tangent bundle. Let Mn be a Riemannian manifold with metric g . We denote by pq (Mn ) the set of all tensor fields of type (p, q) on Mn . Manifolds, tensor field and connections are always assumed to be differentiable and of class C∞ . Let T (Mn ) be a tangent bundle of Mn , and π the projection π : T (Mn ) → Mn . Let the manifold Mn be covered by system of coordinate neighbourhoods (U, xi ), where (xi ), i = 1, ., n is a local coordinate system defined in the neighbourhood U . Let (yi ) be the Cartesian coordinates in each tangent spaces Tp (Mn ) ∂ at P ∈ Mn with respect to the natural base ∂x , P being an arbitrary point in U whose coordinates are i xi . Then we can introduce local coordinates (xi , yi ) in open set π −1 (U ) ⊂ T (Mn ) . We call them coordinates induced in π −1 (U ) from (U, xi ). The projection π is represented by (xi , yi ) → (xi ). We use the notations xI = (xi , x¯ı ) and x¯ı = yi . The indices I, J, . run from 1 to 2n, the indices ¯ı, ¯j, . run from n+1 to 2n . ∂ 1 i Let X ∈ 0 (Mn ), which locally