On the geometry of the tangent bundle with gradient Sasaki metric

Belarbi, Lakehal; Elhendi, Hichem

doi:10.1108/AJMS-11-2020-0125

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^f 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^f) 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 $(TM, g^{f})$ 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 $(TM, g^{f})$ and several important results are obtained on curvature scalar and sectional curvatures.

1. Introduction

We recall some basic facts about the geometry of the tangent bundle. In the present paper, we denote by $Γ (TM)$ the space of all vector fields of a Riemannian manifold $(M, g) .$ Let $(M, g)$ be an n-dimensional Riemannian manifold and $(TM, π, M)$ be its tangent bundle.

A local chart ${(U, x^{i})}_{i = 1 \dots n}$ on M induces a local chart ${(π^{- 1} (U), x^{i}, y^{i})}_{i = 1 \dots n}$ on TM. Denote by $Γ_{i j}^{k}$ the Christoffel symbols of g and by $\nabla$ the Levi-Civita connection of g.

We have two complementary distributions on $TM$ ⁠, the vertical distribution $V$ and the horizontal distribution $H$ ⁠, defined by

V_{(x, u)} = \ker (d π_{(x, u)}) = {a^{i} \frac{\partial}{\partial y^{i}} | (x, u); a^{i} \in ℝ}

H_{(x, u)} = {a^{i} {\frac{\partial}{\partial x^{i}}}_{| (x, u)} - a^{i} y^{j} Γ_{i j}^{k} {\frac{\partial}{\partial y^{k}}}_{| (x, u)}; a^{i} \in ℝ}

where $(x, u) \in TM$ ⁠, such that $T_{(x, u)} TM = H_{(x, u)} \oplus V_{(x, u)}$ ⁠.

Let $X = X^{i} \frac{\partial}{\partial x^{i}}$ be a local vector field on M. The vertical and the horizontal lifts of X are defined by

X^{V} = X^{i} \frac{\partial}{\partial y^{i}}

X^{H} = X^{i} \frac{δ}{δ x^{i}} = X^{i} {\frac{\partial}{\partial x^{i}} - y^{j} Γ_{i j}^{k} \frac{\partial}{\partial y^{k}}}

(1.1)

For consequences, we have ${(\frac{\partial}{\partial x^{i}})}^{H} = \frac{δ}{δ x^{i}}$ and ${(\frac{\partial}{\partial x^{i}})}^{V} = \frac{\partial}{\partial y^{i}}$ ⁠, then ${(\frac{δ}{δ x^{i}}, \frac{\partial}{\partial y^{i}})}_{i = 1.. n}$ is a local adapted frame in $TTM$ ⁠.

The geometry of tangent bundle of a Riemannian manifold $(M, g)$ 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

\begin{array}{l} g^{s} (X^{H}, Y^{H}) = g (X, Y) \circ π \\ g^{s} (X^{H}, Y^{V}) = 0 \\ g^{s} (X^{V}, Y^{V}) = g (X, Y) \circ π \end{array}

(1.2)

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 $π : (TM, g^{s}) \to (M, g)$ 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 $\nabla$ on $(M, g)$ ⁠. Later, the Lie bracket of the tangent bundle $TM$ ⁠, the Levi-Civita connection $\nabla^{s}$ 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 $(M, g)$ ⁠. It is uniquely determined by

\begin{array}{l} g_{CG} (X^{H}, Y^{H}) = g (X, Y) \circ π \\ g_{CG} (X^{H}, Y^{V}) = 0 \\ g_{CG} (X^{V}, Y^{V}) = \frac{1}{α} {g (X, Y) + g (X, u) g (Y, u)} \circ π \end{array}

(1.3)

where $X, Y \in Γ (TM)$ ⁠, $(x, u) \in TM$ ⁠, $α = 1 + g_{x} (u, u)$ ⁠. This metric has been given by Musso and Tricerri in Ref. [4], using Cheeger and Gromoll's study [5]. The Levi-Civita connection $\nabla^{CG}$ and the Riemannian curvature tensor $R^{CG}$ of $(TM, g_{CG})$ have been obtained in Refs. [6, 7], respectively. The sectional curvatures and the scalar curvature of this metric have been obtained in Refs. [8–16]. 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 Käherian 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

\begin{array}{l} g_{f}^{s} (X^{H}, Y^{H}) = f (p) g_{p} (X, Y) \\ g_{f}^{s} (X^{H}, Y^{V}) = 0 \\ g_{f}^{s} (X^{V}, Y^{V}) = g_{p} (X, Y) \end{array}

(1.4)

for all vector fields X and Y on $(M, g)$ ⁠, where f is strictly positive smooth function on $(M, g)$ ⁠. In Ref. [25] J. Wang, Y. Wang called $g_{f}^{s}$ the rescaled Sasaki metric and studied the geometry of $TM$ endowed with $g_{f}^{s}$ ⁠.

H. M. Dida, F. Hathout in Ref. [26], we define a new class of naturally metric on $TM$ given by

\begin{array}{l} G_{(p, u)}^{f} (X^{H}, Y^{H}) = g_{p} (X, Y) \\ G_{(p, u)}^{f} (X^{H}, Y^{V}) = 0 \\ G_{(p, u)}^{f} (X^{V}, Y^{V}) = f (p) g_{p} (X, Y) \end{array}

(1.5)

for some strictly positive smooth function f in $(M, g)$ and any vector fields X and Y on M. We call $G^{f}$ vertical rescaled metric.

L. Belarbi, H. El Hendi in Ref. [27], we define a new class of naturally metric on $T M$ given by

\begin{array}{l} G_{(p, u)}^{f, h} (X^{H}, Y^{H}) = f (p) g_{p} (X, Y) \\ G_{(p, u)}^{f, h} (X^{V}, Y^{H}) = 0 \\ G_{(p, u)}^{f, h} (X^{V}, Y^{V}) = h (p) g_{p} (X, Y) \end{array}

(1.6)

where $f, h$ be strictly positive smooth functions on M and any vector fields X and Y on M. For $h = 1$ the metric $G^{f, h}$ is exactly the rescaled Sasaki metric. If $f = 1$ _, the metric $G^{f, h}$ is exactly the vertical rescaled metric. We call $G^{f, h}$ the twisted Sasaki metric.

Motivated by the above studies, we define a new class of naturally metric on $TM$ given by

\begin{array}{l} g^{f} {(X^{H}, Y^{H})}_{(x, u)} = g_{x} (X, Y) \\ g^{f} {(X^{V}, Y^{H})}_{(x, u)} = 0 \\ g^{f} {(X^{V}, Y^{V})}_{(x, u)} = g_{x} (X, Y) + X_{x} (f) Y_{x} (f) \end{array}

(1.7)

where f be strictly positive smooth functions on M and any vector fields X and Y on M. If f is constant the metric $g^{f}$ is exactly the Sasaki metric.

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).

2. Gradient Sasaki metric

Definition 2.1.

Let $(M, g)$ be a Riemannian manifold and $f : M \to [0, + \infty]$ ⁠. then the gradient Sasaki metric $g^{f}$ on the tangent bundle $TM$ of M is given by

g^{f} {(X^{H}, Y^{H})}_{(x, u)} = g_{x} (X, Y)

g^{f} {(X^{V}, Y^{H})}_{(x, u)} = 0

g^{f} {(X^{V}, Y^{V})}_{(x, u)} = g_{x} (X, Y) + X_{x} (f) Y_{x} (f)

for all vector fields

X, Y \in Γ (TM)

⁠,

(x, u) \in TM

⁠.

Remark 2.1.

If f is constant, then $g^{f}$ is the Sasaki metric.
$g^{f} (X^{H}, {(gradf)}^{H}) = g (X, gradf) = X (f)$
$g^{f} (X^{V}, {(grad f)}^{V}) = (1 + {‖ grad f ‖}^{2}) X (f) = α X (f)$ ⁠, where $α = 1 + {‖ grad f ‖}^{2}$ ⁠.
$g^{f} (X^{V}, Y^{V}) - g^{f} (X^{H}, Y^{H}) = X (f) Y (f)$ ⁠, where $X, Y \in Γ (TM)$ ⁠.

2.1 Levi-Civita connection of $g^{f}$

Lemma 2.1.

Let $(M, g)$ be a Riemannian manifold and $\nabla$ (resp $\nabla^{f}$ ⁠) denote the Levi-Civita connection of $(M, g)$ $(resp (TM, g^{f}))$ ⁠, then we have:

$g^{f} (\nabla_{X^{H}}^{f} Y^{H}, Z^{H}) = g^{f} ({(\nabla_{X} Y)}^{H}, Z^{H})$
$g^{f} (\nabla_{X^{H}}^{f} Y^{H}, Z^{V}) = - \frac{1}{2} g^{f} ({(R (X, Y) u)}^{V}, Z^{V})$
$g^{f} (\nabla_{X^{H}}^{f} Y^{V}, Z^{H}) = \frac{1}{2} g^{f} ({(R (u, Y) X)}^{H} + Y (f) {(R (u, grad f) X)}^{H}, Z^{H})$
$g^{f} (\nabla_{X^{H}}^{f} Y^{V}, Z^{V}) = g^{f} ({(\nabla_{X} Y)}^{V}, Z^{V}) + \frac{1}{2} Y (f) g^{f} ({(\nabla_{X} grad f)}^{V}, Z^{V}) + \frac{1}{2 α} [g (Y, \nabla_{X} grad f) - \frac{1}{2} X (α) Y (f)] g^{f} ({(grad f)}^{V}, Z^{V})$
$g^{f} (\nabla_{X^{V}}^{f} Y^{H}, Z^{H}) = \frac{1}{2} g^{f} ({(R (u, X) Y)}^{H} + X (f) {(R (u, grad f) Y)}^{H}, Z^{H})$
$g^{f} (\nabla_{X^{V}}^{f} Y^{H}, Z^{V}) = \frac{1}{2} X (f) g^{f} ({(\nabla_{Y} grad f)}^{V}, Z^{V}) + \frac{1}{2 α} [g (X, \nabla_{Y} grad f) - \frac{1}{2} Y (α) X (f)] g^{f} ({(grad f)}^{V}, Z^{V})$
$g^{f} (\nabla_{X^{V}}^{f} Y^{V}, Z^{H}) = - \frac{1}{2} g^{f} (X (f) {(\nabla_{Y} grad f)}^{H} + Y (f) {(\nabla_{X} grad f)}^{H}, Z^{H})$
$g^{f} (\nabla_{X^{V}}^{f} Y^{V}, Z^{V}) = 0$

Using Lemma 2.1, we have the theorem

Theorem 2.1.

Let $(M, g)$ be a Riemannian manifold and $\nabla^{f}$ be the Levi-Civita connection of the tangent bundle $(TM, g^{f})$ ⁠. Then, we have

{(\nabla_{X^{H}}^{f} Y^{H})}_{p} = {(\nabla_{X} Y)}_{p}^{H} - \frac{1}{2} {(R (X, Y) u)}_{p}^{V}

{(\nabla_{X^{H}}^{f} Y^{V})}_{p} = \frac{1}{2} {(R (u, Y) X)}_{p}^{h} + \frac{1}{2} Y (f) {(R (u, grad f) X)}^{H} + \frac{1}{2} Y (f) {(\nabla_{X} grad f)}^{V} + {(\nabla_{X} Y)}^{V} + \frac{1}{2 α} [g (Y, \nabla_{X} grad f) - \frac{1}{2} X (α) Y (f)] {(grad f)}^{V}

{(\nabla_{X^{V}}^{f} Y^{H})}_{p} = \frac{1}{2} {(R (u, X) Y)}_{p}^{h} + \frac{1}{2} X (f) {(R (u, grad f) Y)}^{H} + \frac{1}{2} X (f) {(\nabla_{Y} grad f)}^{V} + \frac{1}{2 α} [g (X, \nabla_{Y} grad f) - \frac{1}{2} Y (α) X (f)] {(grad f)}^{V}

{(\nabla_{X^{V}}^{f} Y^{V})}_{p} = - \frac{1}{2} X (f) {(\nabla_{Y} grad f)}^{H} - \frac{1}{2} Y (f) {(\nabla_{X} grad f)}^{H}

for all vector fields

X, Y \in Γ (TM)

⁠,

p = (x, u) \in TM

⁠.

2.2 Curvature tensor of gradient Sasaki metric

Using Theorem 2.1 and the formula of curvature, we have

Theorem 2.2.

Let $(M, g)$ be a Riemannian manifold and $(TM, g^{f})$ its tangent bundle equipped with the gradient Sasaki metric. If R (resp $R^{f}$ ⁠) denote the Riemann curvature tensor of M (resp $TM$ ⁠), then we have the following formulas

(1)
$\begin{array}{l} R_{p}^{f} (X^{H}, Y^{H}) Z^{H} = (R (X, Y) Z_{p}^{H} + \frac{1}{2} {(R (u, R (X, Y) u) Z)}_{p}^{H} \\ + \frac{1}{4} (R (u, R (X, Z) u) Y_{p}^{H} - \frac{1}{4} {(R (u, R (Y, Z) u) X)}_{p}^{H} \\ - \frac{1}{4} g_{x} (R (Y, Z) u, grad f) {(R (u, grad f) X)}_{p}^{H} \\ + \frac{1}{4} g_{x} (R (X, Z) u, grad f) {(R (u, grad f) Y)}_{p}^{H} \\ + \frac{1}{2} g_{x} (R (X, Y) u, grad f) {(R (u, grad f) Z)}_{p}^{H} \\ + \frac{1}{2} {((\nabla_{Z} R) (X, Y) u)}_{p}^{V} + \frac{1}{2} g_{x} (R (X, Y) u, grad f) {(\nabla_{Z} grad f)}_{p}^{V} \\ + \frac{1}{4} g_{x} (R (X, Z) u, grad f) {(\nabla_{Y} grad f)}_{p}^{V} \\ - \frac{1}{4} g_{x} (R (Y, Z) u, grad f) {(\nabla_{X} grad f)}_{p}^{V} \\ + \frac{1}{4 α} [g_{x} (R (X, Z) u, \nabla_{Y} grad f) - g_{x} (R (Y, Z) u, \nabla_{X} grad f) \\ + 2 g_{x} (R (X, Y) u, \nabla_{Z} grad f)] {(grad f)}_{p}^{V} \\ + \frac{1}{8 α} [X_{x} (α) g_{x} (R (Y, Z) u, grad f) - Y_{x} (α) g_{x} (R (X, Z) u, grad f) \\ - 2 Z_{x} (α) g_{x} (R (X, Y) u, grad f)] {(grad f)}_{p}^{V} \end{array}$
(2)
$\begin{array}{l} R_{p}^{f} (X^{H}, Y^{V}) Z^{V} = - \frac{1}{2} Y_{x} (f) {(\nabla_{X} \nabla_{Z} grad f)}_{p}^{H} - \frac{1}{4} Y_{x} (f) {(R (u, grad f) R (u, Z) X)}_{p}^{H} \\ - \frac{1}{2} Z_{x} (f) {(\nabla_{X} \nabla_{Y} grad f)}_{p}^{H} - \frac{1}{4} {(R (u, Y) R (u, Z) X)}_{p}^{H} \\ - \frac{1}{2} Z_{x} (f) {(R (Y, grad f) X)}_{p}^{H} - \frac{1}{4} Z_{x} (f) {(R (u, Y) R (u, grad f) X)}_{p}^{H} \\ - \frac{1}{4} Y_{x} (f) Z_{x} (f) (R (u, grad f) {(R (u, grad f) X)}_{p}^{H} - \frac{1}{2} {(R (Y, Z) X)}_{p}^{H} \\ + \frac{1}{4} Y_{x} (f) Z_{x} (f) {(\nabla_{(\nabla_{X} grad f)} grad f)}_{p}^{H} + \frac{1}{2} Z_{x} (f) {(\nabla_{(\nabla_{X} Y)} grad f)}_{p}^{H} \\ - \frac{1}{2} g_{x} (Y, \nabla_{X} grad f) {(\nabla_{Z} grad f)}_{p}^{H} + \frac{1}{2} Y_{x} (f) {(\nabla_{(\nabla_{X} Z)} grad f)}_{p}^{H} \\ + Y_{x} (f) [\frac{1}{8 α} g_{x} (Z, \nabla_{X} grad f) - \frac{1}{16 α} X_{x} (α) Z_{x} (f)] {(grad α)}_{p}^{H} \\ + [\frac{1}{8 α} X_{x} (α) Z_{x} (f) - \frac{1 + α}{4 α} g_{x} (Z, \nabla_{X} grad f)] {(\nabla_{Y} grad f)}_{p}^{H} \\ + \frac{1}{4} Y_{x} (f) {(R (X, \nabla_{Z} grad f) u)}_{p}^{V} + \frac{1}{4} Z_{x} (f) {(R (X, \nabla_{Y} grad f) u)}_{p}^{V} \\ - \frac{1}{4} Y_{x} (f) {(\nabla_{(R (u, Z) X)} grad f)}_{p}^{V} - \frac{1}{4} Y_{x} (f) Z_{x} (f) {(\nabla_{(R (u, grad f) X)} grad f)}_{p}^{V} \\ + [\frac{1}{8 α} Y_{x} (f) g_{x} (R (u, Z) X, grad α) - \frac{1}{4 α} g_{x} (R (u, Z) X, \nabla_{Y} grad f) \\ + \frac{1}{8 α} Y_{x} (f) Z_{x} (f) g_{x} (R (u, grad f) X, grad α) - \frac{1}{4 α} Z_{x} (f) g_{x} (R (u, grad f) X, \nabla_{Y} grad f)] {(grad f)}_{p}^{V} \end{array}$
(3)
$\begin{array}{l} R_{p}^{f} (X^{V}, Y^{V}) Z^{H} = \frac{1}{4} Y_{x} (f) {(R (u, X) R (u, grad f) Z)}_{p}^{H} + \frac{1}{2} Y_{x} (f) {(R (X, grad f) Z)}_{p}^{H} \\ - \frac{1}{4} X_{x} (f) {(R (u, Y) R (u, grad f) Z)}_{p}^{H} - \frac{1}{2} X_{x} (f) {(R (Y, grad f) Z)}_{p}^{H} \\ + \frac{1}{4} {(R (u, X) R (u, Y) Z)}_{p}^{H} - \frac{1}{4} {(R (u, Y) R (u, X) Z)}_{p}^{H} \\ + {(R (X, Y) Z)}_{p}^{H} - \frac{1}{4} X_{x} (f) (R (u, grad f) {(R (u, Y) Z)}_{p}^{H} \\ - \frac{1}{4} Y_{x} (f) (R (u, grad f) {(R (u, X) Z)}_{p}^{H} \\ - [\frac{1}{8 α} Y_{x} (f) Z_{x} (α) + \frac{α - 1}{4 α} g_{x} (Y, \nabla_{Z} grad f)] {(\nabla_{X} grad f)}_{p}^{H} \\ + [\frac{1}{8 α} X_{x} (f) Z_{x} (α) + \frac{α - 1}{4 α} g_{x} (X, \nabla_{Z} grad f)] {(\nabla_{Y} grad f)}_{p}^{H} \\ + \frac{1}{8 α} [Y_{x} (f) g_{x} (X, \nabla_{Z} grad f) - X_{x} (f) g_{x} (Y, \nabla_{Z} grad f)] {(grad α)}_{p}^{H} \\ + \frac{1}{4} X_{x} (f) {(\nabla_{(R (u, Y) Z)} grad f)}_{p}^{V} - \frac{1}{4} Y_{x} (f) {(\nabla_{(R (u, X) Z)} grad f)}_{p}^{V} \\ + [\frac{1}{8 α} Y_{x} (f) g_{x} (R (u, X) Z, grad α) - \frac{1}{8 α} X_{x} (f) g_{x} (R (u, Y) Z, grad α) \\ + \frac{1}{4 α} g_{x} (R (u, Y) Z, \nabla_{X} grad f) - \frac{1}{4 α} g_{x} (R (u, X) Z, \nabla_{Y} grad f) \\ + \frac{1}{4 α} Y_{x} (f) g_{x} (R (u, grad f) Z, \nabla_{X} grad f) - \frac{1}{4 α} X_{x} (f) g_{x} (R (u, grad f) Z, \nabla_{Y} grad f)] {(grad f)}_{p}^{V} \end{array}$
(4)
$\begin{array}{l} R_{p}^{f} (X^{H}, Y^{V}) Z^{H} = \frac{1}{2} {((\nabla_{X} R) (u, Y) Z)}_{p}^{H} + \frac{1}{2} Y_{x} (f) ((\nabla_{X} R) (u, grad f) Z_{p}^{H} \\ + \frac{1}{2} Y_{x} (f) {(R (u, \nabla_{X} grad f) Z)}_{p}^{H} + \frac{1}{2} g_{x} (Y, \nabla_{X} grad f) {(R (u, grad f) Z)}_{p}^{H} \\ + \frac{1}{4} Y_{x} (f) {(R (u, \nabla_{Z} grad f) X)}_{p}^{H} + \frac{1}{4} g_{x} (Y, \nabla_{Z} grad f) {(R (u, grad f) X)}_{p}^{H} \\ - \frac{1}{4} Y_{x} (f) {(\nabla_{(R (X, Z) u)} grad f)}_{p}^{H} - \frac{1}{4} g_{x} (R (X, Z) u, grad f) {(\nabla_{Y} grad f)}_{p}^{H} \\ + \frac{1}{2} {(R (X, Z) Y)}_{p}^{V} + \frac{1}{2} g_{x} (Y, \nabla_{X} grad f) {(\nabla_{Z} grad f)}_{p}^{V} \\ + \frac{1}{2} Y_{x} (f) {(\nabla_{X} \nabla_{Z} grad f)}_{p}^{V} - \frac{1}{2} Y_{x} (f) {(\nabla_{(\nabla_{X} Z)} grad f)}_{p}^{V} \\ - \frac{1}{4} {(R (X, R (u, Y) Z) u)}_{p}^{V} - \frac{1}{4} Y_{x} (f) {(R (X, R (u, grad f) Z) u)}_{p}^{V} \\ + [\frac{α + 1}{4 α} g_{x} (Y, \nabla_{Z} grad f) - \frac{1}{8 α} Y_{x} (f) Z_{x} (α)] {(\nabla_{X} grad f)}_{p}^{V} \\ + [\frac{1}{4 α} Y_{x} (f) g_{x} (\nabla_{Z} grad f, \nabla_{X} grad f) - \frac{1}{2 α} g_{x} (\nabla_{X} Z, \nabla_{Y} grad f) \\ + \frac{1}{2 α} g_{x} (\nabla_{X} \nabla_{Z} grad f, Y) - \frac{1}{4 α} Y_{x} (f) g_{x} (Z, \nabla_{X} grad α) \\ - \frac{1}{8 α^{2}} X_{x} (α) Y_{x} (f) Z_{x} (α) - \frac{1}{4 α} Z_{x} (α) g_{x} (Y, \nabla_{X} grad f) \\ - \frac{α + 2}{8 α^{2}} X_{x} (α) g_{x} (Y, \nabla_{Z} grad f)] {(grad f)}_{p}^{V} \end{array}$
(5)
$\begin{array}{l} R_{p}^{f} (X^{H}, Y^{H}) Z^{V} = \frac{1}{2} {((\nabla_{X} R) (u, Z) Y)}_{p}^{H} + \frac{1}{4} g_{x} (Z, \nabla_{X} grad f) {(R (u, grad f) Y)}_{p}^{H} \\ - \frac{1}{2} {((\nabla_{Y} R) (u, Z) X)}_{p}^{H} - \frac{1}{4} g_{x} (Z, \nabla_{Y} grad f) {(R (u, grad f) X)}_{p}^{H} \\ + \frac{1}{2} Z_{x} (f) {((\nabla_{X} R) (u, grad f) Y)}_{p}^{H} - \frac{1}{2} Z_{x} (f) {((\nabla_{Y} R) (u, grad f) X)}_{p}^{H} \\ + \frac{1}{4} Z_{x} (f) {(R (u, \nabla_{X} grad f) Y)}_{p}^{H} - \frac{1}{4} Z_{x} (f) {(R (u, \nabla_{Y} grad f) X)}_{p}^{H} \\ - \frac{1}{2} g_{x} (R (X, Y) u, grad f) {(\nabla_{Z} grad f)}_{p}^{H} - \frac{1}{2} Z_{x} (f) {(\nabla_{(R (X, Y) u)} grad f)}_{p}^{H} \\ + {(R (X, Y) Z)}_{p}^{V} + \frac{1}{2} Z_{x} (f) {(R (X, Y) grad f)}_{p}^{V} \\ - \frac{1}{4} {(R (X, R (u, Z) Y) u)}_{p}^{V} - \frac{1}{4} Z_{x} (f) {(R (X, R (u, grad f) Y) u)}_{p}^{V} \\ + \frac{1}{4} {(R (Y, R (u, Z) X) u)}_{p}^{V} + \frac{1}{4} Z_{x} (f) {(R (Y, R (u, grad f) X) u)}_{p}^{V} \\ - [\frac{1}{8 α} Z_{x} (f) Y_{x} (α) + \frac{α - 1}{4 α} g_{x} (Z, \nabla_{Y} grad f)] {(\nabla_{X} grad f)}_{p}^{V} \\ + [\frac{1}{8 α} Z_{x} (f) X_{x} (α) + \frac{α - 1}{4 α} g_{x} (Z, \nabla_{X} grad f)] {(\nabla_{Y} grad f)}_{p}^{V} \\ + [\frac{α - 2}{8 α^{2}} X_{x} (α) g_{x} (\nabla_{Y} grad f, Z) + \frac{1}{2 α} g_{x} (R (X, Y) grad f, Z) - \frac{α - 2}{8 α^{2}} Y_{x} (α) g_{x} (\nabla_{X} grad f, Z)] {(grad f)}_{p}^{V} \end{array}$
(6)
$\begin{array}{l} R_{p}^{f} (X^{V}, Y^{V}) Z^{V} = \frac{1}{4} X_{x} (f) {(R (u, Y) (\nabla_{Z} grad f))}_{p}^{H} - \frac{1}{4} Y_{x} (f) {(R (u, X) (\nabla_{Z} grad f))}_{p}^{H} \\ + \frac{1}{4} Z_{x} (f) {(R (u, Y) (\nabla_{X} grad f))}_{p}^{H} - \frac{1}{4} Z_{x} (f) {(R (u, X) (\nabla_{Y} grad f))}_{p}^{H} \\ + \frac{1}{4} Y_{x} (f) Z_{x} (f) {(R (u, grad f) (\nabla_{X} grad f))}_{p}^{H} \\ - \frac{1}{4} X_{x} (f) Z_{x} (f) {(R (u, grad f) (\nabla_{Y} grad f))}_{p}^{H} \\ - \frac{1}{8 α} Y_{x} (f) Z_{x} (f) g_{x} (\nabla_{X} grad f, grad α) {(grad f)}_{p}^{V} \\ + \frac{1}{8 α} X_{x} (f) Z_{x} (f) g_{x} (\nabla_{Y} grad f, grad α) {(grad f)}_{p}^{V} \\ - \frac{1}{4 α} Y_{x} (f) g_{x} (\nabla_{X} grad f, \nabla_{Z} grad f) {(grad f)}_{p}^{V} \\ + \frac{1}{4 α} X_{x} (f) g_{x} (\nabla_{Y} grad f, \nabla_{Z} grad f) {(grad f)}_{p}^{V} \\ + \frac{1}{4} Y_{x} (f) Z_{x} (f) {(\nabla_{(\nabla_{X} grad f)} grad f)}_{p}^{V} \\ - \frac{1}{4} X_{x} (f) Z_{x} (f) {(\nabla_{(\nabla_{Y} grad f)} grad f)}_{p}^{V} \end{array}$

for all

p = (x, u) \in TM

and

X, Y, Z \in Γ (TM)

⁠.

2.3 Sectional curvature of the gradient Sasaki metric

Let V and W be two orthonormal tangent vectors $V, W \in T_{(x, u)} TM$ ⁠. The sectional curvatures of the tangent bundle $(TM, g^{f})$ is given by

K^{f} (V, W) = \frac{G^{f} (V, W)}{Q^{f} (V, W)}

(2.1)

where

Q^{f} (V, W) = g^{f} (V, V) g^{f} (W, W) - {| g^{f} (V, W) |}^{2} and G^{f} (V, W) = g^{f} (R^{f} (V, W) W, V)

Lemma 2.2.

Let $(M, g)$ be a Riemannian manifold and $(TM, g^{f})$ its tangent bundle equipped with the gradient Sasaki metric, then for any orthonormal vectors fields $X, Y \in Γ (TM)$ ⁠, we have

(1) $Q^{f} (X^{H}, Y^{H}) = 1$
(2) $Q^{f} (X^{H}, Y^{V}) = 1 + {| Y (f) |}^{2}$
(3) $Q^{f} (X^{V}, Y^{V}) = 1 + {| X (f) |}^{2} + {| Y (f) |}^{2}$
(4) $G^{f} (X^{H}, Y^{H}) = g (R (X, Y) Y, X) - \frac{3}{4} {‖ R (X, Y) u ‖}^{2} - \frac{3}{4} {| g (R (X, Y) u), grad f |}^{2}$
(5)
$\begin{matrix} G^{f} (X^{H}, Y^{V}) = Y (f) [g (\nabla_{X} grad f, \nabla_{X} Y) - g (\nabla_{X} \nabla_{Y} grad f, X) \\ + \frac{1}{4 α} X (α) g (\nabla_{X} grad f, Y) + \frac{1}{2} g (R (u, Y) X, R (u, grad f) X)] \\ + {| Y (f) |}^{2} [\frac{1}{4} ‖ R (u, grad f) {X ‖}^{2} + \frac{1}{4} {‖ \nabla_{X} grad f ‖}^{2} - \frac{1}{16 α} {| X (α) |}^{2} \\ \frac{1}{4} {‖ R (u, Y) X ‖}^{2} - \frac{3 α + 1}{4 α} {| g (\nabla_{X} grad f, Y) |}^{2} \end{matrix}$
(6)
$G^{f} (X^{V}, Y^{V}) = \frac{1}{4} {| Y (f) |}^{2} {‖ \nabla_{X} grad f ‖}^{2} + \frac{1}{4} {| X (f) |}^{2} \nabla_{Y} grad f^{2} - \frac{1}{2} X (f) Y (f) g (\nabla_{X} grad f, \nabla_{Y} grad f)$

Proposition 2.1.

Let $(M, g)$ be a Riemannian manifold and $(TM, g^{f})$ its tangent bundle equipped with the gradient Sasaki metric. If K, (resp $K^{f}$ ⁠) denotes the sectional curvature of $(M, g)$ $(resp ., (TM, g^{f}))$ ⁠, then for any orthonormal vectors fields $X, Y \in Γ (TM)$ ⁠, we have

(1) $K^{f} (X^{H}, Y^{H}) = K (X, Y) - \frac{3}{4} {‖ R (X, Y) u ‖}^{2} - \frac{3}{4} {| g (R (X, Y) u, grad f |}^{2}$
(2)
$\begin{matrix} K^{f} (X^{H}, Y^{V}) = \frac{Y (f)}{1 + {| Y (f) |}^{2}} [g (\nabla_{X} grad f, \nabla_{X} Y) - g (\nabla_{X} \nabla_{Y} grad f, X) \\ + \frac{1}{4 α} X (α) g (\nabla_{X} grad f, Y) + \frac{1}{2} g (R (u, Y) X, R (u, grad f) X)] \\ + \frac{{| Y (f) |}^{2}}{1 + {| Y (f) |}^{2}} [\frac{1}{4} {‖ R (u, grad f) X ‖}^{2} + \frac{1}{4} {| | \nabla_{X} grad f | |}^{2} - \frac{1}{16 α} {| X (α) |}^{2}] \\ + \frac{1}{1 + {| Y (f) |}^{2}} [\frac{1}{4} {‖ R (u, Y) X ‖}^{2} - \frac{3 α + 1}{4 α} {| g (\nabla_{X} grad f, Y) |}^{2}] \end{matrix}$
(3)
$K^{f} (X^{V}, Y^{V}) = \frac{1}{1 + {| X (f) |}^{2} + {| Y (f) |}^{2}} [- \frac{1}{2} X (f) Y (f) g (\nabla_{X} grad f, \nabla_{Y} grad f) + \frac{1}{4} {| Y (f) |}^{2} {‖ \nabla_{X} grad f ‖}^{2} + \frac{1}{4} {| X (f) |}^{2} {‖ \nabla_{Y} grad f ‖}^{2}]$

Proof. The proof of Proposition 2.1 is deduced from equation (2.1) and Lemma 2.2.

Lemma 2.3.

Let $(M, g)$ be a Riemannian manifold and $(TM, g^{f})$ its tangent bundle equipped with the gradient Sasaki metric. If $(E_{1}, \dots, E_{m})$ be a local orthonormal frame on M such that $E_{1} = \frac{grad f}{grad f}$ ⁠. Then $(F_{1}, \dots, F_{2 m})$ is a local orthonormal on $(TM, g^{f})$ ⁠.

Where $F_{i} = E_{i}^{H}$ ⁠, $F_{m + 1} = \frac{1}{\sqrt{α}} E_{1}^{V}$ and $F_{m + j} = E_{j}^{V}$ ⁠, $i = \overline{1, m}, j = \overline{2, m}$ ⁠.

Lemma 2.4.

Let $(M, g)$ be a Riemannian manifold and $(TM, g^{f})$ its tangent bundle equipped with the gradient Sasaki metric. If $(E_{1}, \dots, E_{m})$ (resp $(F_{1}, \dots, F_{2 m})$ ⁠) are local orthonormal on M (resp., $TM$ ⁠), then for all $i, j = \overline{1, m}$ et $k, l = \overline{2, m}$ ⁠, we have

(1) $K^{f} (F_{i}, F_{j}) = K (E_{i}, E_{j}) - \frac{3}{4} {‖ R (E_{i}, E_{j}) u ‖}^{2} - \frac{3}{4} ‖ g {(R (E_{i}, E_{j}) u, grad f) ‖}^{2}$
(2)
$K^{f} (F_{i}, F_{m + 1}) = \frac{α + 3}{4 α} {‖ \nabla_{E_{i}} grad f ‖}^{2} - \frac{α^{2} - α + 4}{16 α^{2} (α - 1)} {| E_{i} (α) |}^{2} + \frac{α}{4 (α - 1)} {| | R (u, grad f) E_{i} | |}^{2} - \frac{1}{2 α} g (\nabla_{E_{i}} grad α, E_{i})$
(3) $K^{f} (F_{i}, F_{m + l}) = \frac{1}{4} ‖ R (u, E_{l}) E_{i} ‖^{2} - \frac{3 α + 1}{4 α} {| g (\nabla_{E_{i}} grad f, E_{l}) |}^{2}$
(4) $K^{f} (F_{m + k}, F_{m + 1}) = \frac{α - 1}{4 α} {‖ \nabla_{E_{k}} grad f ‖}^{2}$
(5) $K^{f} (F_{m + k}, F_{m + l}) = 0$

Proof. Using proposition 2.1, we have

(1)direct application
(2)
$\begin{array}{l} K^{f} (F_{i}, F_{m + 1}) = G^{f} (E_{i}^{H}, \frac{1}{\sqrt{α (α - 1)}} {(grad f)}^{V}) \\ = \frac{1}{α (α - 1)} G^{f} (E_{i}^{H}, {(grad f)}^{V}) \\ = \frac{1}{α (α - 1)} [(α - 1) [g (\nabla_{E_{i}} grad f, \nabla_{E_{i}} grad f) - g (\nabla_{E_{i}} \nabla_{grad f} grad f, E_{i}) \\ = + \frac{1}{4 α} E_{i} (α) g (\nabla_{E_{i}} grad f, grad f) + \frac{1}{2} g (R (u, grad f) E_{i}, R (u, grad f) E_{i})] \\ + {(α - 1)}^{2} [\frac{1}{4} {‖ R (u, grad f) E_{i} ‖}^{2} + \frac{1}{4} ‖ \nabla_{E_{i}} grad {f ‖}^{2} - \frac{1}{16 α} {| E_{i} (α) |}^{2} \\ + \frac{1}{4} ‖ R (u, grad f) E_{i}^{2} \frac{3 α + 1}{4 α} g (\nabla_{E_{i}} grad f, grad f)]] \\ = \frac{1}{α} ‖ \nabla_{E_{i}} grad f^{2} - \frac{1}{2 α} g (\nabla_{E_{i}} grad α, E_{i}) + \frac{1}{8 α^{2}} | E_{i} (α) |^{2} \\ + \frac{1}{2 α} {‖ R (u, grad f) E_{i} ‖}^{2} + \frac{α - 1}{4 α} R (u, grad f) E_{i} ‖^{2} + \frac{α - 1}{4 α} \nabla_{E_{i}} grad f^{2} \end{array}$
$\begin{array}{l} - \frac{α - 1}{16 α^{2}} {| E_{i} (α) |}^{2} + \frac{1}{4 α (α - 1)} {‖ R (u, grad f) E_{i} ‖}^{2} - \frac{3 α + 1}{16 α^{2} (α - 1)} {| E_{i} (α) |}^{2} \\ = \frac{3 + α}{4 α} \nabla_{E_{i}} grad f ‖^{2} - \frac{α^{2} - α + 4}{16 α^{2} (α - 1)} {| E_{i} (α) |}^{2} \\ + \frac{α}{4 (α - 1)} {‖ R (u, grad f) E_{i} ‖}^{2} - \frac{1}{2 α} g (\nabla_{E_{i}} grad α, E_{i}) \end{array}$
(3)
$K^{f} (F_{i}, F_{m + l}) = K^{f} (E_{i}^{H}, E_{l}^{V}) = \frac{1}{4} ‖ R (u, E_{l}) E_{i} ‖^{2} - \frac{3 α + 1}{4 α} {| g (\nabla_{E_{i}} grad f, E_{l}) |}^{2}$
(4)
$\begin{array}{l} K^{f} (F_{m + k}, F_{m + 1}) = G^{f} (E_{k}^{V}, \frac{1}{\sqrt{α - 1}} {(grad f)}^{V}) \\ = \frac{1}{α - 1} G^{f} (E_{k}^{H}, {(grad f)}^{V}) \\ = \frac{1}{α (α - 1)} \frac{{(α - 1)}^{2}}{4} ‖ \nabla_{E_{k}} grad f ‖^{2} \\ = \frac{α - 1}{4 α} ‖ \nabla_{E_{k}} grad f ‖^{2} \end{array}$
(5)direct application□

Lemma 2.5.

[18] Let $(E_{1}, \dots, E_{m}$ be local orthonormal frame on M, then for all $i, j = \overline{1, m}$ ⁠, we have

\sum_{i, j = 1}^{m} ‖ R (u, E_{i}) E_{j} ‖^{2} = \sum_{i, j = 1}^{m} {‖ R (E_{i}, E_{j}) u ‖}^{2}

Proposition 2.2.

Let $(M, g)$ be a Riemannian manifold and $(TM, g^{f})$ its tangent bundle equipped with the metric of the gradient Sasaki metric. If σ (resp., $σ^{f}$ denote the scalar curvature of $(M, g)$ (resp, $(TM, g^{f})$ ⁠), then for any local orthonormal frame $(E_{1}, \dots, E_{m})$ on M, we have

\begin{matrix} σ^{f} = σ - \frac{1}{4} \overset{m}{\sum_{i, j = 1}} {‖ R (E_{i}, E_{j}) u ‖}^{2} - \frac{3}{4} \overset{m}{\sum_{i, j = 1}} {| g (R (E_{i}, E_{j}) u, grad f) |}^{2} \\ - \frac{3 α + 1}{2 α} \overset{m}{\sum_{i, j = 1}} {| g (\nabla_{E_{i}} grad f, E_{j} |}^{2} - \frac{α^{2} - α + 4}{8 α^{2} (α - 1)} {‖ grad α ‖}^{2} \\ + \frac{α + 1}{α} {‖ \nabla grad f ‖}^{2} - \frac{1}{2} {trace}_{g} ({(R (u, grad f) *)}^{2} - \frac{1}{α} \nabla_{*} grad α) \end{matrix}

Proof. Using Lemma 2.3, we have

\begin{array}{l} σ^{f} = \overset{2 m}{\sum_{s, t = 1}} K^{f} (F_{s}, F_{t}) \\ = \overset{m}{\sum_{i, j = 1, i \neq j}} K^{f} (F_{i}, F_{j}) + 2 \overset{m}{\sum_{i, j = 1}} K^{f} (F_{i}, F_{m + j}) + \overset{m}{\sum_{i, j = 1, i \neq j}} K^{f} (F_{m + i}, F_{m + j}) \\ = \overset{m}{\sum_{i, j = 1, i \neq j}} K^{f} (F_{i}, F_{j}) + 2 \overset{m}{\sum_{i = 1}} K^{f} (F_{i}, F_{m + 1}) + 2 \overset{m}{\sum_{i = 1, j = 2}} K^{f} (F_{i}, F_{m + j}) \\ + 2 \overset{m}{\sum_{i = 1}} K^{f} (F_{m + i}, F_{m + 1}) + \overset{m}{\sum_{i, j = 2, i \neq j}} K^{f} (F_{m + i}, F_{m + j}) \\ σ^{f} = \overset{m}{\sum_{i, j = 1, i \neq j}} [K (E_{i}, E_{j}) - \frac{3}{4} {‖ R (E_{i}, E_{j}) u ‖}^{2} - \frac{3}{4} {| g (R (E_{i}, E_{j}) u, g r a d f |}^{2}] \\ + \overset{m}{\sum_{i = 1}} [\frac{α + 3}{4 α} {‖ \nabla_{E_{i}} grad f ‖}^{2} - \frac{α^{2} - α + 4}{16 α^{2} (α - 1)} {| E_{i} (α) |}^{2} \\ + \frac{α}{4 (α - 1)} ‖ R (u, grad f) E_{i} ‖^{2} - \frac{1}{2 α} g (\nabla_{E_{i}} grad α, E_{i})] \\ + 2 \overset{m}{\sum_{i = 1, j = 2}} [\frac{1}{4} ‖ R (u, E_{j}) E_{i} ‖^{2} - \frac{3 α + 1}{4 α} {| g (\nabla_{E_{i}} grad f, E_{j}) |}^{2}] \\ + \overset{m}{\sum_{i = 2}} \frac{α - 1}{4 α} {‖ \nabla_{E_{i}} grad f ‖}^{2} \\ = σ - \frac{3}{4} \overset{m}{\sum_{i, j = 1, i \neq j}} ‖ R (E_{i}, E_{j}) {u ‖}^{2} - \frac{3}{4} \overset{m}{\sum_{i, j = 1, i \neq j}} {| g (R (E_{i}, E_{j}) u, grad f) |}^{2} \\ + \frac{α + 3}{2 α} \overset{m}{\sum_{i = 1}} ‖ \nabla_{E_{i}} grad f ‖^{2} - \frac{α^{2} - α + 4}{8 α^{2} (α - 1)} \overset{m}{\sum_{i = 1}} {| E_{i} (α) |}^{2} \\ + \frac{α}{2 (α - 1)} \overset{m}{\sum_{i = 1}} {‖ R (u, grad f) E_{i} ‖}^{2} - \frac{1}{α} \overset{m}{\sum_{i = 1}} g (\nabla_{E_{i}} grad α, E_{i}) \\ + \frac{1}{2} \overset{m}{\sum_{i = 1, j = 2}} ‖ R (u, E_{j}) E_{i} ‖^{2} - \frac{3 α + 1}{2 α} \overset{m}{\sum_{i, j = 2, i \neq j}} {| g (\nabla_{E_{i}} grad f, E_{j}) |}^{2} + \frac{α - 1}{2 α} \overset{m}{\sum_{i = 1}} \nabla_{E_{i}} grad f^{2} \\ = σ - \frac{1}{4} \overset{m}{\sum_{i, j = 1, i \neq j}} ‖ E (E_{i}, E_{j}) {u ‖}^{2} - \frac{3}{4} \overset{m}{\sum_{i, j = 1, i \neq j}} {| g (R (E_{i}, E_{j}) u, grad f) |}^{2} \\ + \frac{α + 1}{α} \overset{m}{\sum_{i = 1}} ‖ \nabla_{E_{i}} grad f ‖^{2} - \frac{α^{2} - α + 4}{8 α^{2} (α - 1)} \overset{m}{\sum_{i = 1}} {| E_{i} (α) |}^{2} \\ + \frac{α}{2 (α - 1)} \overset{m}{\sum_{i = 1}} {‖ R (u, grad f) E_{i} ‖}^{2} - \frac{1}{2} \overset{m}{\sum_{i = 1}} {‖ R (u, E_{1}) E_{i} ‖}^{2} \\ - \frac{1}{α} \overset{m}{\sum_{i = 1}} g (\nabla_{E_{i}} grad α, E_{i}) - \frac{3 α + 1}{2 α} \overset{m}{\sum_{i, j = 1, i \neq j}} {| g (\nabla_{E_{i}} grad f, E_{j}) |}^{2} \\ σ^{f} = σ - \frac{1}{4} \overset{m}{\sum_{i, j = 1, i \neq j}} {‖ R (E_{i}, E_{j}) u ‖}^{2} - \frac{3}{4} \overset{m}{\sum_{i, j = 1, i \neq j}} {| g (R (E_{i}, E_{j}) u, grad f) |}^{2} \\ - \frac{3 α + 1}{2 α} \overset{m}{\sum_{i, j = 1, i \neq j}} {| g (\nabla_{E_{i}} grad f, E_{j} |}^{2} - \frac{α^{2} - α + 4}{8 α^{2} (α - 1)} {‖ grad α ‖}^{2} \\ + \frac{α + 1}{α} {‖ \nabla grad f ‖}^{2} - \frac{1}{2} {trace}_{g} ({(R (u, grad f) *)}^{2} - \frac{1}{α} \nabla_{*} grad α) \end{array}

□

Corollary 2.1.

Let $(M, g)$ be a Riemannian manifold of constant sectional curvature λ and $(TM, g^{f})$ its tangent bundle equipped with the gradient Sasaki metric. If $σ^{f}$ denote the scalar curvature of $TM$ ⁠, then for any local orthonormal frame $(E_{1}, \dots, E_{m})$ on M, we have

\begin{array}{l} σ^{f} = m (m - 1) λ + λ^{2} [\frac{2 α - m - 1}{2} ‖ u ‖^{2} - {| g (u, grad f) |}^{2}] \\ - \frac{3 α + 1}{2 α} \overset{m}{\sum_{i, j = 1}} {| g (\nabla_{E_{i}} grad f, E_{j}) |}^{2} - \frac{α^{2} - α + 4}{8 α^{2} (α - 1)} {‖ grad α ‖}^{2} \\ + \frac{α + 1}{α} \nabla grad f^{2} - \frac{1}{α} {trace}_{g} (\nabla_{*} grad α) \end{array}

Proof. Taking account that $σ = m (m - 1) λ$ and for any vector fields $X, Y, Z \in Γ (TM)$

R (X, Y) Z = λ (g (Z, Y) X - g (X, Z) Y)

then we obtain

\begin{array}{l} \overset{m}{\sum_{i, j = 1}} {‖ R (E_{i}, E_{j}) u ‖}^{2} = λ^{2} \overset{m}{\sum_{i, j = 1}} ‖ g (u, E_{j}) E_{i} - g (E_{i}, u) E_{j} ‖^{2} \\ = λ^{2} \overset{m}{\sum_{i, j = 1}} [{| g (u, E_{j}) |}^{2} - 2 g (u, E_{j}) g (E_{i}, u) δ_{i j} + {| g (E_{i}, u) |}^{2} \\ = λ^{2} [m {‖ u ‖}^{2} - 2 {‖ u ‖}^{2} + m {‖ u ‖}^{2}] \\ = 2 (m - 1) λ^{2} {‖ u ‖}^{2} \end{array}

From Proposition 2.2, we deduce

\begin{matrix} \overset{m}{\sum_{i, j = 1}} {| g (R (E_{i}, E_{j}) u, grad f) |}^{2} = λ^{2} \overset{m}{\sum_{i, j = 1}} {| g (g (u, E_{j}) E_{i} - g (E_{i}, u) E_{j}, grad f) |}^{2} \\ = λ^{2} \overset{m}{\sum_{i, j = 1}} {| g (E_{i}, grad f) g (u, E_{j}) - g (E_{j}, grad f) g (E_{i}, u) |}^{2} = 0 \end{matrix}

and

\begin{array}{l} {trace}_{g} g {(R (u, grad f))}^{2} = \overset{m}{\sum_{i = 1}} g (R (u, grad f) R (u, grad f) E_{i}, E_{i}) \\ = - \overset{m}{\sum_{i = 1}} {‖ R (u, grad f) E_{i} ‖}^{2}) \\ = - λ^{2} \overset{m}{\sum_{i = 1}} ‖ g (E_{i}, grad f) u - g (u, E_{i}) grad f ‖^{2} \\ = - λ^{2} [(α - 1) {‖ u ‖}^{2} - 2 {| g (u, grad f) |}^{2} + (α - 1) {‖ u ‖}^{2}] \\ = - 2 λ^{2} [(α - 1) {‖ u ‖}^{2} - {| g (u, grad f) |}^{2}] \end{array}

□

The authors are thankful the referee for helpful suggestions to improve the paper. The authors was supported by The National Agency Scientific Research (DGRSDT).

References

[1]

Sasaki

S

.

On the differential geometry of tangent bundles of Riemannian manifolds

.

Tohoku Math J

.

1958

;

10

(

3

):

338

-

54

.

Google Scholar

Crossref

[2]

Dombrowski

P

.

On the geometry of the tangent bundle

.

J Reine Angew Math

.

1962

;

210

:

73

-

88

.

Google Scholar

[3]

Kowalski

O

.

Curvature of the induced Riemannian metric of the tangent bundle of a Riemannian manifold

.

J Reine Angew Math

.

1971

;

250

:

124

-

29

.

Google Scholar

[4]

Musso

E

,

Tricerri

F

.

Riemannian metrics on tangent bundles

.

Ann Mat Pura Appl

.

1988

;

150

(

4

):

1

-

19

.

Google Scholar

[5]

Cheeger

J

,

Gromoll

D

.

On the structure of complete manifolds of non negative curvature

.

Ann Math

.

1972

;

96

(

2

):

413

-

43

.

Google Scholar

[6]

Gudmundsson

S

,

Kappos

E

.

On the geometry of the tangent bundle with the cheeger-Gromoll metric

.

Tokyo J Math

.

2002

;

25

(

1

):

75

-

83

.

Google Scholar

Crossref

[7]

Sekizawa

M

.

Curvatures of tangent bundles with Cheeger-Gromoll metric

.

Tokyo J Math

.

1991

;

14

(

2

):

407

-

17

.

Google Scholar

Crossref

[8]

Albuquerque

R

.

Notes on the Sasaki metric

.

Expo Math.

2019

;

37

:

207

-

24

.

Google Scholar

Crossref

[9]

Belarbi

L

,

Belarbi

M

,

Elhendi

H

.

Legendre curves on Lorentzian heisenberg space

.

Bull Transilv Univ Brasov SER. III

.

2020

;

13

(

62

). (

1

):

41

-

50

.

Google Scholar

[10]

Belarbi

L

,

Elhendi

H

,

Latti

F

.

On the geometry of the tangent bundle with vertical rescaled generalized Cheeger-Gromoll metric

.

Bull Transilv Univ Brasov SER. III

.

2019

;

12

(

61

). (

2

):

247

-

64

.

Google Scholar

[11]

Belarbi

L

,

Elhendi

H

.

Harmonic and Biharmonic maps between tangent bundles

.

Acta Math Univ Comenianae

.

2019

;

88

(

2

):

187

-

99

.

Google Scholar

[12]

Elhendi

H

,

Belarbi

L

.

Deformed diagonal metrics on tangent bundle of order two and harmonicity

.

Panamer Math J.

2017

;

27

(

2

):

90

-

106

.

Google Scholar

[13]

Elhendi

H

,

Belarbi

L

.

On paraquaternionic submersions of tangent bundle of order two

.

Nonlinear Stud.

2018

;

25

(

3

):

653

-

64

.

Google Scholar

[14]

Gudmundsson

S

,

Kappos

E

.

On the geometry of the tangent bundles, expo

.

Math.

2002

;

20

(

1

):

1

-

41

.

Google Scholar

[15]

Mazouzi

H

,

Elhendi

H

and

Belarbi

L

.

On the generalized Bi-f-harmonic map equations on singly warped product manifolds

.

Comm Appl Nonlinear Anal.

2018

;

25

(

3

):

52

-

76

.

Google Scholar

[16]

Vaisman

I

.

From generalized Kähler to generalized Sasakian structures

.

J Geom Symmetry Phys.

2010

;

18

:

63

-

86

.

Google Scholar

[17]

Abbassi

MTK

,

Sarih

M

.

Killing vector fields on tangent bundles with Cheeger-Gromoll Metric

.

Tsukuba J Math

.

2003

;

27

(

2

):

295

-

306

.

Google Scholar

Crossref

[18]

Anastasiei

M

.

Locally conformal Kaehler structures on tangent manifold of a space form

.

Libertas Math

.

1999

;

19

:

71

-

6

.

Google Scholar

[19]

Oproiu

V

.

Some new geometric structures on the tangent bundle

.

Publ Math Debrecen

.

1999

;

55

(

3-4

):

261

-

81

.

Google Scholar

Crossref

[20]

Oproiu

V

,

Papaghiuc

N

.

General natural Einstein Kahler structures on tangent bundles

.

Differential Geom Appl

.

2009

;

27

:

384

-

92

.

Google Scholar

Crossref

[21]

Abbassi

MTK

,

Sarih

M

.

On natural metrics on tangent bundles of riemannian manifolds

.

Arch Math.

2005

;

41

:

71

-

92

.

Google Scholar

[22]

Munteanu

MI

.

CheegerGromoll type metrics on the tangent bundle

.

Sci Ann Univ Agric Sci Vet Med

.

2006

;

49

(

2

):

257

-

68

.

Google Scholar

[23]

Salimov

AA

,

Akbulut

K

.

A note on a paraholomorphic CheegerGromoll metric

.

Proc Indian Acad Sci

.

2009

;

119

(

2

):

187

-

95

.

Google Scholar

[24]

Zayatuev

BV

.

On geometry of tangent Hermitian surface, Webs and Quasigroups

.

T.S.U

.

1995

:

139

-

43

.

Google Scholar

[25]

Wang

J

,

Wang

Y

.

On the geometry of tangent bundles with the rescaled metric

.

arXiv:1104.5584v1[math.DG] 29 Apr 2011

.

[26]

Dida

HM

,

Hathout

F

,

Azzouz

A

.

On the geometry of the tangent bundle with vertical rescaled metric

.

Commun Fac Sci Univ Ank Ser A1 Math Stat

.

2019

;

68

(

1

):

222

-

35

.

Google Scholar

Crossref

[27]

Belarbi

L

,

Elhendi

H

.

Geometry of twisted Sasaki metric

.

J Geom Symmetry Phys

.

2019

;

53

:

1

-

19

.

Google Scholar

Crossref

2021

Lakehal Belarbi and Hichem Elhendi

Published in Arab Journal of Mathematical Sciences. Published by Emerald Publishing Limited. This article is published under the Creative Commons Attribution (CC BY 4.0) licence. Anyone may reproduce, distribute, translate and create derivative works of this article (for both commercial and non-commercial purposes), subject to full attribution to the original publication and authors. The full terms of this licence may be seen at http://creativecommons.org/licences/by/4.0/legalcode

On the geometry of the tangent bundle with gradient Sasaki metric

1. Introduction

2. Gradient Sasaki metric

2.1 Levi-Civita connection of $g^{f}$

2.2 Curvature tensor of gradient Sasaki metric

2.3 Sectional curvature of the gradient Sasaki metric

References

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On the geometry of the tangent bundle with gradient Sasaki metric

1. Introduction

2. Gradient Sasaki metric

2.1 Levi-Civita connection of gf

2.2 Curvature tensor of gradient Sasaki metric

2.3 Sectional curvature of the gradient Sasaki metric

References

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2.1 Levi-Civita connection of $g^{f}$