On the geometry of the tangent bundle with gradient Sasaki metric

Lakehal Belarbi (Department of Mathematics, Laboratory of Pure and Applied Mathematics, University of Mostaganem (U.M.A.B.), Mostaganem, Algeria)
Hichem Elhendi (University of Bechar, Bechar, Algeria)

Arab Journal of Mathematical Sciences

ISSN: 1319-5166

Article publication date: 3 May 2021

589

Abstract

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 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/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,gf) 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,gf) and several important results are obtained on curvature scalar and sectional curvatures.

Keywords

Citation

Belarbi, L. and Elhendi, H. (2021), "On the geometry of the tangent bundle with gradient Sasaki metric", Arab Journal of Mathematical Sciences, Vol. ahead-of-print No. ahead-of-print. https://doi.org/10.1108/AJMS-11-2020-0125

Publisher

:

Emerald Publishing Limited

Copyright © 2021, Lakehal Belarbi and Hichem Elhendi

License

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


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,xi)i=1n on M induces a local chart (π1(U),xi,yi)i=1n on TM. Denote by Γijk the Christoffel symbols of g and by 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(dπ(x,u))={aiyi|(x,u);ai}
H(x,u)={aixi|(x,u)aiyjΓijkyk|(x,u);ai}

where (x,u)TM, such that T(x,u)TM=H(x,u)V(x,u).

Let X=Xixi be a local vector field on M. The vertical and the horizontal lifts of X are defined by

XV=Xiyi
(1.1)XH=Xiδδxi=Xi{xiyjΓijkyk}

For consequences, we have (xi)H=δδxi and (xi)V=yi, then (δδxi,yi)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 gs, 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

(1.2)gs(XH,YH)=g(X,Y)πgs(XH,YV)=0gs(XV,YV)=g(X,Y)π
for all vector fields X and Y on M. More intuitively, the metric gs is constructed in such a way that the vertical and horizontal subbundles are orthogonal and the bundle map π:(TM,gs)(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 on (M,g). Later, the Lie bracket of the tangent bundle TM, the Levi-Civita connection s on TM and its Riemannian curvature tensor Rs have been obtained in Refs. [2, 3]. Furthermore, the explicit formulas of another natural metric gCG, which is called Cheeger-Gromoll metric, on the tangent bundle TM of a Riemannian manifold (M,g). It is uniquely determined by

(1.3)gCG(XH,YH)=g(X,Y)πgCG(XH,YV)=0gCG(XV,YV)=1α{g(X,Y)+g(X,u)g(Y,u)}π
where X,YΓ(TM), (x,u)TM, α=1+gx(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 CG and the Riemannian curvature tensor RCG of (TM,gCG) 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

(1.4)gfs(XH,YH)=f(p)gp(X,Y)gfs(XH,YV)=0gfs(XV,YV)=gp(X,Y)
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 gfs the rescaled Sasaki metric and studied the geometry of TM endowed with gfs.

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

(1.5)G(p,u)f(XH,YH)=gp(X,Y)G(p,u)f(XH,YV)=0G(p,u)f(XV,YV)=f(p)gp(X,Y)
for some strictly positive smooth function f in (M,g) and any vector fields X and Y on M. We call Gf vertical rescaled metric.

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

(1.6)G(p,u)f,h(XH,YH)=f(p)gp(X,Y)G(p,u)f,h(XV,YH)=0G(p,u)f,h(XV,YV)=h(p)gp(X,Y)
where f,h be strictly positive smooth functions on M and any vector fields X and Y on M. For h=1 the metric Gf,h is exactly the rescaled Sasaki metric. If f=1, the metric Gf,h is exactly the vertical rescaled metric. We call Gf,h the twisted Sasaki metric.

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

(1.7)gf(XH,YH)(x,u)=gx(X,Y)gf(XV,YH)(x,u)=0gf(XV,YV)(x,u)=gx(X,Y)+Xx(f)Yx(f)
where f be strictly positive smooth functions on M and any vector fields X and Y on M. If f is constant the metric gf 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[0,+]. then the gradient Sasaki metric gf on the tangent bundle TM of M is given by

gf(XH,YH)(x,u)=gx(X,Y)
gf(XV,YH)(x,u)=0
gf(XV,YV)(x,u)=gx(X,Y)+Xx(f)Yx(f)
for all vector fields X,YΓ(TM), (x,u)TM.

Remark 2.1.

  1. If f is constant, then gf is the Sasaki metric.

  2. gf(XH,(gradf)H)=g(X,gradf)=X(f)

  3. gf(XV,(gradf)V)=(1+gradf2)X(f)=αX(f), where α=1+gradf2.

  4. gf(XV,YV)gf(XH,YH)=X(f)Y(f), where X,YΓ(TM).

2.1 Levi-Civita connection of gf

Lemma 2.1.

Let (M,g) be a Riemannian manifold and (resp f) denote the Levi-Civita connection of (M,g) (resp(TM,gf)), then we have:

  1. gf(XHfYH,ZH)=gf((XY)H,ZH)

  2. gf(XHfYH,ZV)=12gf((R(X,Y)u)V,ZV)

  3. gf(XHfYV,ZH)=12gf((R(u,Y)X)H+Y(f)(R(u,gradf)X)H,ZH)

  4. gf(XHfYV,ZV)=gf((XY)V,ZV)+12Y(f)gf((Xgradf)V,ZV)+12α[g(Y,Xgradf)12X(α)Y(f)]gf((gradf)V,ZV)

  5. gf(XVfYH,ZH)=12gf((R(u,X)Y)H+X(f)(R(u,gradf)Y)H,ZH)

  6. gf(XVfYH,ZV)=12X(f)gf((Ygradf)V,ZV)+12α[g(X,Ygradf)12Y(α)X(f)]gf((gradf)V,ZV)

  7. gf(XVfYV,ZH)=12gf(X(f)(Ygradf)H+Y(f)(Xgradf)H,ZH)

  8. gf(XVfYV,ZV)=0

Using Lemma 2.1, we have the theorem

Theorem 2.1.

Let (M,g) be a Riemannian manifold and f be the Levi-Civita connection of the tangent bundle (TM,gf). Then, we have

(XHfYH)p=(XY)pH12(R(X,Y)u)pV
(XHfYV)p=12(R(u,Y)X)ph+12Y(f)(R(u,gradf)X)H+12Y(f)(Xgradf)V+(XY)V+12α[g(Y,Xgradf)12X(α)Y(f)](gradf)V
(XVfYH)p=12(R(u,X)Y)ph+12X(f)(R(u,gradf)Y)H+12X(f)(Ygradf)V+12α[g(X,Ygradf)12Y(α)X(f)](gradf)V
(XVfYV)p=12X(f)(Ygradf)H12Y(f)(Xgradf)H
for all vector fields X,YΓ(TM), p=(x,u)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,gf) its tangent bundle equipped with the gradient Sasaki metric. If R (resp Rf) denote the Riemann curvature tensor of M (resp TM), then we have the following formulas

  • (1)

    Rpf(XH,YH)ZH=(R(X,Y)ZpH+12(R(u,R(X,Y)u)Z)pH+14(R(u,R(X,Z)u)YpH14(R(u,R(Y,Z)u)X)pH14gx(R(Y,Z)u,gradf)(R(u,gradf)X)pH+14gx(R(X,Z)u,gradf)(R(u,gradf)Y)pH+12gx(R(X,Y)u,gradf)(R(u,gradf)Z)pH+12((ZR)(X,Y)u)pV+12gx(R(X,Y)u,gradf)(Zgradf)pV+14gx(R(X,Z)u,gradf)(Ygradf)pV14gx(R(Y,Z)u,gradf)(Xgradf)pV+14α[gx(R(X,Z)u,Ygradf)gx(R(Y,Z)u,Xgradf)+2gx(R(X,Y)u,Zgradf)](gradf)pV+18α[Xx(α)gx(R(Y,Z)u,gradf)Yx(α)gx(R(X,Z)u,gradf)2Zx(α)gx(R(X,Y)u,gradf)](gradf)pV

  • (2)

    Rpf(XH,YV)ZV=12Yx(f)(XZgradf)pH14Yx(f)(R(u,gradf)R(u,Z)X)pH12Zx(f)(XYgradf)pH14(R(u,Y)R(u,Z)X)pH12Zx(f)(R(Y,gradf)X)pH14Zx(f)(R(u,Y)R(u,gradf)X)pH14Yx(f)Zx(f)(R(u,gradf)(R(u,gradf)X)pH12(R(Y,Z)X)pH+14Yx(f)Zx(f)((Xgradf)gradf)pH+12Zx(f)((XY)gradf)pH12gx(Y,Xgradf)(Zgradf)pH+12Yx(f)((XZ)gradf)pH+Yx(f)[18αgx(Z,Xgradf)116αXx(α)Zx(f)](gradα)pH+[18αXx(α)Zx(f)1+α4αgx(Z,Xgradf)](Ygradf)pH+14Yx(f)(R(X,Zgradf)u)pV+14Zx(f)(R(X,Ygradf)u)pV14Yx(f)((R(u,Z)X)gradf)pV14Yx(f)Zx(f)((R(u,gradf)X)gradf)pV+[18αYx(f)gx(R(u,Z)X,gradα)14αgx(R(u,Z)X,Ygradf)+18αYx(f)Zx(f)gx(R(u,gradf)X,gradα)14αZx(f)gx(R(u,gradf)X,Ygradf)](gradf)pV

  • (3)

    Rpf(XV,YV)ZH=14Yx(f)(R(u,X)R(u,gradf)Z)pH+12Yx(f)(R(X,gradf)Z)pH14Xx(f)(R(u,Y)R(u,gradf)Z)pH12Xx(f)(R(Y,gradf)Z)pH+14(R(u,X)R(u,Y)Z)pH14(R(u,Y)R(u,X)Z)pH+(R(X,Y)Z)pH14Xx(f)(R(u,gradf)(R(u,Y)Z)pH14Yx(f)(R(u,gradf)(R(u,X)Z)pH[18αYx(f)Zx(α)+α14αgx(Y,Zgradf)](Xgradf)pH+[18αXx(f)Zx(α)+α14αgx(X,Zgradf)](Ygradf)pH+18α[Yx(f)gx(X,Zgradf)Xx(f)gx(Y,Zgradf)](gradα)pH+14Xx(f)((R(u,Y)Z)gradf)pV14Yx(f)((R(u,X)Z)gradf)pV+[18αYx(f)gx(R(u,X)Z,gradα)18αXx(f)gx(R(u,Y)Z,gradα)+14αgx(R(u,Y)Z,Xgradf)14αgx(R(u,X)Z,Ygradf)+14αYx(f)gx(R(u,gradf)Z,Xgradf)14αXx(f)gx(R(u,gradf)Z,Ygradf)](gradf)pV

  • (4)

    Rpf(XH,YV)ZH=12((XR)(u,Y)Z)pH+12Yx(f)((XR)(u,gradf)ZpH+12Yx(f)(R(u,Xgradf)Z)pH+12gx(Y,Xgradf)(R(u,gradf)Z)pH+14Yx(f)(R(u,Zgradf)X)pH+14gx(Y,Zgradf)(R(u,gradf)X)pH14Yx(f)((R(X,Z)u)gradf)pH14gx(R(X,Z)u,gradf)(Ygradf)pH+12(R(X,Z)Y)pV+12gx(Y,Xgradf)(Zgradf)pV+12Yx(f)(XZgradf)pV12Yx(f)((XZ)gradf)pV14(R(X,R(u,Y)Z)u)pV14Yx(f)(R(X,R(u,gradf)Z)u)pV+[α+14αgx(Y,Zgradf)18αYx(f)Zx(α)](Xgradf)pV+[14αYx(f)gx(Zgradf,Xgradf)12αgx(XZ,Ygradf)+12αgx(XZgradf,Y)14αYx(f)gx(Z,Xgradα)18α2Xx(α)Yx(f)Zx(α)14αZx(α)gx(Y,Xgradf)α+28α2Xx(α)gx(Y,Zgradf)](gradf)pV

  • (5)

    Rpf(XH,YH)ZV=12((XR)(u,Z)Y)pH+14gx(Z,Xgradf)(R(u,gradf)Y)pH12((YR)(u,Z)X)pH14gx(Z,Ygradf)(R(u,gradf)X)pH+12Zx(f)((XR)(u,gradf)Y)pH12Zx(f)((YR)(u,gradf)X)pH+14Zx(f)(R(u,Xgradf)Y)pH14Zx(f)(R(u,Ygradf)X)pH12gx(R(X,Y)u,gradf)(Zgradf)pH12Zx(f)((R(X,Y)u)gradf)pH+(R(X,Y)Z)pV+12Zx(f)(R(X,Y)gradf)pV14(R(X,R(u,Z)Y)u)pV14Zx(f)(R(X,R(u,gradf)Y)u)pV+14(R(Y,R(u,Z)X)u)pV+14Zx(f)(R(Y,R(u,gradf)X)u)pV[18αZx(f)Yx(α)+α14αgx(Z,Ygradf)](Xgradf)pV+[18αZx(f)Xx(α)+α14αgx(Z,Xgradf)](Ygradf)pV+[α28α2Xx(α)gx(Ygradf,Z)+12αgx(R(X,Y)gradf,Z)α28α2Yx(α)gx(Xgradf,Z)](gradf)pV

  • (6)

    Rpf(XV,YV)ZV=14Xx(f)(R(u,Y)(Zgradf))pH14Yx(f)(R(u,X)(Zgradf))pH+14Zx(f)(R(u,Y)(Xgradf))pH14Zx(f)(R(u,X)(Ygradf))pH+14Yx(f)Zx(f)(R(u,gradf)(Xgradf))pH14Xx(f)Zx(f)(R(u,gradf)(Ygradf))pH18αYx(f)Zx(f)gx(Xgradf,gradα)(gradf)pV+18αXx(f)Zx(f)gx(Ygradf,gradα)(gradf)pV14αYx(f)gx(Xgradf,Zgradf)(gradf)pV+14αXx(f)gx(Ygradf,Zgradf)(gradf)pV+14Yx(f)Zx(f)((Xgradf)gradf)pV14Xx(f)Zx(f)((Ygradf)gradf)pV

for all p=(x,u)TM and X,Y,ZΓ(TM).

2.3 Sectional curvature of the gradient Sasaki metric

Let V and W be two orthonormal tangent vectors V,WT(x,u)TM. The sectional curvatures of the tangent bundle (TM,gf) is given by

(2.1)Kf(V,W)=Gf(V,W)Qf(V,W)
where
Qf(V,W)=gf(V,V)gf(W,W)|gf(V,W)|2andGf(V,W)=gf(Rf(V,W)W,V)
Lemma 2.2.

Let (M,g) be a Riemannian manifold and (TM,gf) its tangent bundle equipped with the gradient Sasaki metric, then for any orthonormal vectors fields X,YΓ(TM), we have

  • (1)Qf(XH,YH)=1

  • (2)Qf(XH,YV)=1+|Y(f)|2

  • (3)Qf(XV,YV)=1+|X(f)|2+|Y(f)|2

  • (4)Gf(XH,YH)=g(R(X,Y)Y,X)34R(X,Y)u234|g(R(X,Y)u),gradf|2

  • (5)

    Gf(XH,YV)=Y(f)[g(Xgradf,XY)g(XYgradf,X)+14αX(α)g(Xgradf,Y)+12g(R(u,Y)X,R(u,gradf)X)]+|Y(f)|2[14R(u,gradf)X2+14Xgradf2116α|X(α)|214R(u,Y)X23α+14α|g(Xgradf,Y)|2

  • (6)

    Gf(XV,YV)=14|Y(f)|2Xgradf2+14|X(f)|2Ygradf212X(f)Y(f)g(Xgradf,Ygradf)

Proposition 2.1.

Let (M,g) be a Riemannian manifold and (TM,gf) its tangent bundle equipped with the gradient Sasaki metric. If K, (resp Kf) denotes the sectional curvature of (M,g) (resp.,(TM,gf)), then for any orthonormal vectors fields X,YΓ(TM), we have

  • (1)Kf(XH,YH)=K(X,Y)34R(X,Y)u234|g(R(X,Y)u,gradf|2

  • (2)

    Kf(XH,YV)=Y(f)1+|Y(f)|2[g(Xgradf,XY)g(XYgradf,X)+14αX(α)g(Xgradf,Y)+12g(R(u,Y)X,R(u,gradf)X)]+|Y(f)|21+|Y(f)|2[14R(u,gradf)X2+14||Xgradf||2116α|X(α)|2]+11+|Y(f)|2[14R(u,Y)X23α+14α|g(Xgradf,Y)|2]

  • (3)

    Kf(XV,YV)=11+|X(f)|2+|Y(f)|2[12X(f)Y(f)g(Xgradf,Ygradf)+14|Y(f)|2Xgradf2+14|X(f)|2Ygradf2]

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,gf) its tangent bundle equipped with the gradient Sasaki metric. If (E1,,Em) be a local orthonormal frame on M such that E1=gradfgradf. Then (F1,,F2m) is a local orthonormal on (TM,gf).

Where Fi=EiH, Fm+1=1αE1V and Fm+j=EjV, i=1,m,j=2,m.

Lemma 2.4.

Let (M,g) be a Riemannian manifold and (TM,gf) its tangent bundle equipped with the gradient Sasaki metric. If (E1,,Em) (resp (F1,,F2m)) are local orthonormal on M (resp., TM), then for all i,j=1,m et k,l=2,m, we have

  • (1)Kf(Fi,Fj)=K(Ei,Ej)34R(Ei,Ej)u234g(R(Ei,Ej)u,gradf)2

  • (2)

    Kf(Fi,Fm+1)=α+34αEigradf2α2α+416α2(α1)|Ei(α)|2+α4(α1)||R(u,gradf)Ei||212αg(Eigradα,Ei)

  • (3)Kf(Fi,Fm+l)=14R(u,El)Ei23α+14α|g(Eigradf,El)|2

  • (4)Kf(Fm+k,Fm+1)=α14αEkgradf2

  • (5)Kf(Fm+k,Fm+l)=0

Proof. Using proposition 2.1, we have

  • (1)direct application

  • (2)

    Kf(Fi,Fm+1)=Gf(EiH,1α(α1)(gradf)V)=1α(α1)Gf(EiH,(gradf)V)=1α(α1)[(α1)[g(Eigradf,Eigradf)g(Eigradfgradf,Ei)=+14αEi(α)g(Eigradf,gradf)+12g(R(u,gradf)Ei,R(u,gradf)Ei)]+(α1)2[14R(u,gradf)Ei2+14Eigradf2116α|Ei(α)|2+14R(u,gradf)Ei23α+14αg(Eigradf,gradf)]]=1αEigradf212αg(Eigradα,Ei)+18α2|Ei(α)|2+12αR(u,gradf)Ei2+α14αR(u,gradf)Ei2+α14αEigradf2
    α116α2|Ei(α)|2+14α(α1)R(u,gradf)Ei23α+116α2(α1)|Ei(α)|2=3+α4αEigradf2α2α+416α2(α1)|Ei(α)|2+α4(α1)R(u,gradf)Ei212αg(Eigradα,Ei)

  • (3)

    Kf(Fi,Fm+l)=Kf(EiH,ElV)=14R(u,El)Ei23α+14α|g(Eigradf,El)|2

  • (4)

    Kf(Fm+k,Fm+1)=Gf(EkV,1α1(gradf)V)=1α1Gf(EkH,(gradf)V)=1α(α1)(α1)24Ekgradf2=α14αEkgradf2

  • (5)direct application□

Lemma 2.5.

[18] Let (E1,,Em be local orthonormal frame on M, then for all i,j=1,m, we have

i,j=1mR(u,Ei)Ej2=i,j=1mR(Ei,Ej)u2

Proposition 2.2.

Let (M,g) be a Riemannian manifold and (TM,gf) its tangent bundle equipped with the metric of the gradient Sasaki metric. If σ (resp., σf denote the scalar curvature of (M,g) (resp, (TM,gf)), then for any local orthonormal frame (E1,,Em) on M, we have

σf=σ14i,j=1mR(Ei,Ej)u234i,j=1m|g(R(Ei,Ej)u,gradf)|23α+12αi,j=1m|g(Eigradf,Ej|2α2α+48α2(α1)gradα2+α+1αgradf212traceg((R(u,gradf))21αgradα)

Proof. Using Lemma 2.3, we have

σf=s,t=12mKf(Fs,Ft)=i,j=1,ijmKf(Fi,Fj)+2i,j=1mKf(Fi,Fm+j)+i,j=1,ijmKf(Fm+i,Fm+j)=i,j=1,ijmKf(Fi,Fj)+2i=1mKf(Fi,Fm+1)+2i=1,j=2mKf(Fi,Fm+j)+2i=1mKf(Fm+i,Fm+1)+i,j=2,ijmKf(Fm+i,Fm+j)σf=i,j=1,ijm[K(Ei,Ej)34R(Ei,Ej)u234|g(R(Ei,Ej)u,gradf|2]+i=1m[α+34αEigradf2α2α+416α2(α1)|Ei(α)|2+α4(α1)R(u,gradf)Ei212αg(Eigradα,Ei)]+2i=1,j=2m[14R(u,Ej)Ei23α+14α|g(Eigradf,Ej)|2]+i=2mα14αEigradf2=σ34i,j=1,ijmR(Ei,Ej)u234i,j=1,ijm|g(R(Ei,Ej)u,gradf)|2+α+32αi=1mEigradf2α2α+48α2(α1)i=1m|Ei(α)|2+α2(α1)i=1mR(u,gradf)Ei21αi=1mg(Eigradα,Ei)+12i=1,j=2mR(u,Ej)Ei23α+12αi,j=2,ijm|g(Eigradf,Ej)|2+α12αi=1mEigradf2=σ14i,j=1,ijmE(Ei,Ej)u234i,j=1,ijm|g(R(Ei,Ej)u,gradf)|2+α+1αi=1mEigradf2α2α+48α2(α1)i=1m|Ei(α)|2+α2(α1)i=1mR(u,gradf)Ei212i=1mR(u,E1)Ei21αi=1mg(Eigradα,Ei)3α+12αi,j=1,ijm|g(Eigradf,Ej)|2σf=σ14i,j=1,ijmR(Ei,Ej)u234i,j=1,ijm|g(R(Ei,Ej)u,gradf)|23α+12αi,j=1,ijm|g(Eigradf,Ej|2α2α+48α2(α1)gradα2+α+1αgradf212traceg((R(u,gradf)*)21α*gradα)

Corollary 2.1.

Let (M,g) be a Riemannian manifold of constant sectional curvature λ and (TM,gf) its tangent bundle equipped with the gradient Sasaki metric. If σf denote the scalar curvature of TM, then for any local orthonormal frame (E1,,Em) on M, we have

σf=m(m1)λ+λ2[2αm12u2|g(u,gradf)|2]3α+12αi,j=1m|g(Eigradf,Ej)|2α2α+48α2(α1)gradα2+α+1αgradf21αtraceg(gradα)

Proof. Taking account that σ=m(m1)λ and for any vector fields X,Y,ZΓ(TM)

R(X,Y)Z=λ(g(Z,Y)Xg(X,Z)Y)
then we obtain
i,j=1mR(Ei,Ej)u2=λ2i,j=1mg(u,Ej)Eig(Ei,u)Ej2=λ2i,j=1m[|g(u,Ej)|22g(u,Ej)g(Ei,u)δij+|g(Ei,u)|2=λ2[mu22u2+mu2]=2(m1)λ2u2

From Proposition 2.2, we deduce

i,j=1m|g(R(Ei,Ej)u,gradf)|2=λ2i,j=1m|g(g(u,Ej)Eig(Ei,u)Ej,gradf)|2=λ2i,j=1m|g(Ei,gradf)g(u,Ej)g(Ej,gradf)g(Ei,u)|2=0
and
tracegg(R(u,gradf))2=i=1mg(R(u,gradf)R(u,gradf)Ei,Ei)=i=1mR(u,gradf)Ei2)=λ2i=1mg(Ei,gradf)ug(u,Ei)gradf2=λ2[(α1)u22|g(u,gradf)|2+(α1)u2]=2λ2[(α1)u2|g(u,gradf)|2]

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Acknowledgements

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

Corresponding author

Lakehal Belarbi can be contacted at: lakehalbelarbi@gmail.com

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