On Geodesics of Warped Sasaki Metric

Abstract

In this paper we establish a necessary and sufficient conditions under which a curve be a geodesic respect to the warped Sasaki metric.

Some results on proper biharmonic vector field

“…First we introduce a new metric called Mus-Sasaki metric on the tangent bundle T M . This new natural metric will lead us to interesting results (see [16] and [17]). Afterward we establish necessary and sufficient conditions under which a vector field be biharmonic ( Theorem 2, Theorem 3, Theorem 4, Theorem 5 and Theorem 6).…”

“…Note that, if f = 1 then g f is the Sasaki metric [15]. For more detail on geometry of Mus-Sasaki metric see [11], [16].…”

Some Notes on Geodesics of Vertical Rescaled Berger Deformation Metric in Tangent Bundle

“…Proof. Let (U, x i ) be a local chart on M in x ∈ M and (π −1 (U), x i , y j ) be the induced chart on T M, if 14]). Let (M m , g) be a Riemannian manifold and ∇ denotes the Levi-Civita connection of (M m , g).…”

Mus-Sasaki Metric and Harmonicity

“…First we introduce a new metric called Mus-Sasaki metric on the tangent bundle T M . This new natural metric will lead us to interesting results [18]. Afterward we establish necessary and sufficient conditions under which a vector field is harmonic with respect to the Mus-Sasaki metric (Theorem 3.2 and Theorem 3.3).…”