In this article, we construct a new metric G = R ∇ + m i,j=1 a ji δp j δp i in the cotangent bundle, where R ∇ is the Riemannian extension and a ji is a symmetric (2,0)-tensor field on a differentiable manifold.

The main aim of this paper is to study paraholomorpic Sasakian metric and Killing vector field with respect to the Sasakian metric in the cotangent bundle.

No abstract

Let $(M,\nabla)$ be an $n$-dimensional differentiable manifold with a torsion-free linear connection and $T^{*}M$ its cotangent bundle. In this context we study some properties of the natural Riemann extension (M. Sekizawa (1987), O. Kowalski and M. Sekizawa (2011)) on the cotangent bundle $T^{*}M$. First, we give an alternative definition of the natural Riemann extension with respect to horizontal and vertical lifts. Secondly, we investigate metric connection for the natural Riemann extension. Finally, we present geodesics on the cotangent bundle $T^{*}M$ endowed with the natural Riemann extension.

By [5] it is known that a geodesic γ in an abstract reflection space X (in the sense of Loos, without any assumption of differential structure) canonically admits an action of a 1-parameter subgroup of the group of transvections of X. In this article, we modify these arguments in order to prove an analog of this result stating that, if X contains an embedded hyperbolic plane H ⊂ X, then this yields a canonical action of a subgroup of the transvection group of X isomorphic to a perfect central extension of PSL2(R). This result can be further extended to arbitrary Riemannian symmetric spaces of non-compact type Y lying in X and can be used to prove that a Riemannian symmetric space and, more generally, the Kac-Moody symmetric space G/K for an algebraically simply connected two-spherical Kac-Moody group G, as defined in [5], satisfies a universal property similar to the universal property that the group G satisfies itself.