Properties of Modified Riemannian Extensions

Abstract

Let M be an n−dimensional differentiable manifold with a symmetric connection ∇ and T * M be its cotangent bundle. In this paper, we study some properties of the modified Riemannian extension g ∇,c on T * M defined by means of a symmetric (0, 2)-tensor field c on M. We get the conditions under which T * M endowed with the horizontal lift H J of an almost complex structure J and with the metric g ∇,c is a Kähler-Norden manifold. Also curvature properties of the Levi-Civita connection and another metric connection of the metric g ∇,c are presented.2000 Mathematics Subject Classification. Primary 53C07, 53C55; Secondary 53C22.

Ricci Flow on Modified Riemann Extensions

“…Here in this paper we discuss some interesting properties satisfied by curvature tensors under the influence of Ricci flow on modified Riemann extensions. We give a brief introduction to modified Riemann Extensions [3] in section 2. In Section 3 we find the rate of change of concircular, conharmonic and conformal curvature tensors under Ricci flow.…”

SOME NOTES ON THE MODIFIED RIEMANNIAN EXTENSION g ̃_(∇,c) ON COTANGENT BUNDLE

“…For a given symmetric connection on an dimensional manifold , the cotangent bundle can be equipped with a pseudo-Riemannian metric ̃ of signature the Riemannian extension of [3], given by . The modified Riemannian extension ̃ has the following properties [1]:…”

Notes About a New Metric on the Cotangent Bundle

“…Calviño-Louzao et.al [5] introduced the modified Riemannian extension using a symmetric tensor field of type (0, 2) and studied some geometric applications. Gezer and his collaborators studied the curvature properties and the Kähler-Norden structure with respect to the modified Riemannian extension [8]. Aslanci and Cakan [1] discussed the curvature properties of the deformed Riemannian extension in the cotangent bundle by means of musical isomorphism between tangent and cotangent bundle.…”