Abstract
PurposeLet (M, g) be a n-dimensional smooth Riemannian manifold. In the present paper, the authors introduce a new class of natural metrics denoted by gf and called gradient Sasaki metric on the tangent bundle TM. The authors calculate its Levi-Civita connection and Riemannian curvature tensor. The authors study the geometry of (TM, gf) and several important results are obtained on curvature, scalar and sectional curvatures.Design/methodology/approachIn this paper the authors introduce a new class of natural metrics called gradient Sasaki metric on tangent bundle.FindingsThe authors calculate its Levi-Civita connection and Riemannian curvature tensor. The authors study the geometry of (TM, gf) and several important results are obtained on curvature scalar and sectional curvatures.Originality/valueThe authors calculate its Levi-Civita connection and Riemannian curvature tensor. The authors study the geometry of (TM, gf) and several important results are obtained on curvature scalar and sectional curvatures.
Highlights
We recall some basic facts about the geometry of the tangent bundle
We introduce the gradient Sasaki metric on the tangent bundle TM as a new natural metric non-rigid on TM
First we investigate the geometry of the gradient Sasaki metric and we characterize the sectional curvature (Proposition 2.1) and the scalar curvature (Proposition 2.2)
Summary
We recall some basic facts about the geometry of the tangent bundle. In the present paper, we denote by ΓðTMÞ the space of all vector fields of a Riemannian manifold ðM ; gÞ: Let ðM ; gÞ be an n-dimensional Riemannian manifold and ðTM; π; M Þ be its tangent bundle. The geometry of a tangent bundle has been studied by using a new metric gs, which is called Sasaki metric, with the aid of a Riemannian metric g on a differential manifold M in 1958 by Sasaki [1]. The explicit formulas of another natural metric gCG, which is called Cheeger-Gromoll metric, on the tangent bundle TM of a Riemannian manifold ðM ; gÞ. The geometry of the tangent bundles with Cheeger-Gromoll metric has been studied by many mathematicians First we investigate the geometry of the gradient Sasaki metric and we characterize the sectional curvature (Proposition 2.1) and the scalar curvature (Proposition 2.2)
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