On the geometry of the tangent bundle with gradient Sasaki metric

Purpose – Let (M, g) be a n-dimensional smooth Riemannian manifold. In the present paper, the authors introduce a new class of natural metrics denoted by g 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, g) and several important results are obtained on curvature, scalar and sectional curvatures. Design/methodology/approach – In this paper the authors introduce a new class of natural metrics called gradient Sasaki metric on tangent bundle. Findings – The authors calculate its Levi-Civita connection and Riemannian curvature tensor. The authors study the geometry of ðTM; g Þ and several important results are obtained on curvature scalar and sectional curvatures. Originality/value – The authors calculate its Levi-Civita connection and Riemannian curvature tensor. The authors study the geometry of ðTM; g Þ and several important results are obtained on curvature scalar and sectional curvatures.


Introduction
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.
A local chart ðU ; x i Þ i¼1...n on M induces a local chart ðπ −1 ðU Þ; x i ; y i Þ i¼1...n on TM. Denote by Γ k ij the Christoffel symbols of g and by ∇ the Levi-Civita connection of g.
where ðx; uÞ ∈ TM, such that T ðx;uÞ TM ¼ H ðx;uÞ ⊕ V ðx;uÞ . Let X ¼ X i v vx i be a local vector field on M. The vertical and the horizontal lifts of X are defined by is a local adapted frame in TTM.
The geometry of tangent bundle of a Riemannian manifold ðM ; gÞ is very important in many areas of mathematics and physics. In recent years, a lot of studies about their local or global geometric properties have been published in the literature. When the authors studied this topic, they used different metrics which are called natural metrics on the tangent bundle. First, the geometry of a tangent bundle has been studied by using a new metric g s , which is called Sasaki metric, with the aid of a Riemannian metric g on a differential manifold M in 1958 by Sasaki [1]. It is uniquely determined by for all vector fields X and Y on M. More intuitively, the metric g s is constructed in such a way that the vertical and horizontal subbundles are orthogonal and the bundle map π : ðTM; g s Þ → ðM ; gÞ is a Riemannian submersion.
After that, the tangent bundle could be split to its horizontal and vertical subbundles with the aid of Levi-Civita connection ∇ on ðM ; gÞ. Later, the Lie bracket of the tangent bundle TM, the Levi-Civita connection ∇ s on TM and its Riemannian curvature tensor R s have been obtained in Refs. [2,3]. Furthermore, the explicit formulas of another natural metric g CG , which is called Cheeger-Gromoll metric, on the tangent bundle TM of a Riemannian manifold ðM ; gÞ. It is uniquely determined by where X ; Y ∈ ΓðTMÞ, ðx; uÞ ∈ TM, α ¼ 1 þ g x ðu; uÞ. This metric has been given by Musso and Tricerri in Ref. [4], using Cheeger and Gromoll's study [5]. The Levi-Civita connection ∇ CG and the Riemannian curvature tensor R CG of ðTM; g CG Þ have been obtained in Refs. [6,7], respectively. The sectional curvatures and the scalar curvature of this metric have been obtained in Refs. [8][9][10][11][12][13][14][15][16]. These results are completed in 2002 by S. Gudmundson and E. Kappos in Ref. [6]. They have also shown that the scalar curvature of the Cheeger-Gromoll metric is never constant if the metric on the base manifold has constant sectional curvature. Furthermore, in Ref. [17] M.T.K. Abbassi, M. Sarih have proved that TM with the Cheeger-Gromoll metric is never a space of constant sectional curvature. A more general metric is given by M. Anastasiei in Ref. [18] which generalizes both of the two metrics mentioned above in the following sense: it preserves the orthogonality of the two distributions, on the horizontal distribution it is the same as on the base manifold, and finally the Sasaki and the Cheeger-Gromoll metric can be obtained as particular cases of this metric. A compatible almost complex structure is also introduced and hence TM becomes a locally conformal almost K€ aherian manifold. V.Oproiu and his collaborators constructed a family of Riemannian metrics on the tangent bundles of Riemannian manifolds which possess interesting geometric properties (see Refs. [19,20]). In particular, the scalar curvature of TM can be constant also for a non-flat base manifold with constant sectional curvature. Then M.T.K. Abbassi and M. Sarih proved in Ref. [21] that the considered metrics by Oproiu form a particular subclass of the so-called g-natural metrics on the tangent bundle. Recently, the geometry of the tangent bundles with Cheeger-Gromoll metric has been studied by many mathematicians (see Refs. [17,22,23] and etc). Zayatuev in [24] introduced a Riemannian metric on TM given by In this paper, 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).

Gradient Sasaki metric
Definition 2.1. Let ðM ; gÞ be a Riemannian manifold and f : M → ½0; þ ∞. then the gradient Sasaki metric g f on the tangent bundle TM of M is given by for all vector fields X ; Y ∈ ΓðTMÞ, ðx; uÞ ∈ TM.
(1) If f is constant, then g f is the Sasaki metric. ( 2.1 Levi-Civita connection of g f Lemma 2.1. Let ðM ; gÞ be a Riemannian manifold and ∇ (resp ∇ f ) denote the Levi-Civita connection of ðM ; gÞ ðrespðTM; g f ÞÞ, then we have: The geometry of the tangent bundle for all p ¼ ðx; uÞ ∈ TM and X ; Y ; Z ∈ ΓðTMÞ.

Sectional curvature of the gradient Sasaki metric
Let V and W be two orthonormal tangent vectors V ; W ∈ T ðx;uÞ TM. The sectional curvatures of the tangent bundle ðTM; g f Þ is given by gÞ be a Riemannian manifold and ðTM; g f Þ its tangent bundle equipped with the gradient Sasaki metric, then for any orthonormal vectors fields X ; Y ∈ ΓðTMÞ, we have gÞ be a Riemannian manifold and ðTM; g f Þ its tangent bundle equipped with the gradient Sasaki metric. If K, (resp K f ) denotes the sectional curvature of ðM ; gÞ ðresp:; ðTM; g f ÞÞ, then for any orthonormal vectors fields X ; Y ∈ ΓðTMÞ, we have gÞ be a Riemannian manifold and ðTM; g f Þ its tangent bundle equipped with the gradient Sasaki metric. If ðE 1 ; . . . ; E m Þ (resp ðF 1 ; . . . ; F 2m Þ) are local orthonormal on M (resp., TM), then for all i; j ¼ 1; m et k; l ¼ 2; m, we have