International Electronic Journal of Geometry Volume 6 No.1 pp. 129-146 (2013) °c IEJG g-NATURAL METRICS ON THE COTANGENT BUNDLE FIL_ IZ_ AGCA¸ (Communicated by Arif SALIMOV) Abstract. The main aim of this paper is to investigate curvature properties and geodesics of the g-natural metric on the cotangent bundle of Riemannian manifold. 1. Introduction Let (M n; g) be an n-dimensional Riemannian manifold, T ¤M n its cotangent bundle and ¼ the natural projection T ¤M n ! M n. A system of local coordi- nates (U; xi); i = 1; :::; n on M n induces on T ¤M n a system of local coordinates ¡1 i ¹i ¹i (¼ (U); x ; x = pi), ¹i := n + i (i = 1; :::; n), where x = pi are the components ¤ n of covectors p in each cotangent space Tx M , x 2 U with respect to the natural coframe fdxig, i = 1; :::; n. r n r ¤ n n ¤ n 1 We denote by =s(M )(=s(T M )) the module over F (M )(F (T M )) of C tensor ¯elds of type (r; s), where F (M n)(F (T ¤M n)) is the ring of real-valued C1 functions on M n(T ¤M n). i @ i n Let X = X @xi and ! = !idx be the local expressions in U ½ M of a vector 1 n 0 n and a covector (1-form) ¯eld X 2 =0(M ) and ! 2 =1(M ), respectively. Then C H 1 ¤ n 1 n the complete and horizontal lifts X; X 2 =0(T M ) of X 2 =0(M ) and the V 1 ¤ n 0 n vertical lift ! 2 =0(T M ) of ! 2 =1(M ) are given, respectively, by X C i @ h @ (1.1) X = X ¡ ph@iX ; @xi @x¹i i X H i @ h j @ (1.2) X = X + ph¡ X ; @xi ij @x¹i i X V @ (1.3) ! = !i ; @x¹i i Date: December 11, 2012 and Accepted: March 3, 2013. 2000 Mathematics Subject Classi¯cation. 53C07, 53C22, 53C25. Key words and phrases. g-natural metric, cotangent bundle, vertical and horizontal lift, cur- vature tensor, geodesics. 129 130 FIL_ IZ_ AGCA¸ with respect to the natural frame f @ ; @ g, where ¡h are the components of the @xi @x¹i ij n Levi-Civita connection rg on M (see [19] for more details). Theorem 1.1. Let M n be a Riemannian manifold with metric g, r be the Levi- Civita connection and R be the Riemannian curvature tensor. Then the Lie bracket of the cotangent bundle T ¤M n of M n satis¯es the following i)[V !; V θ] = 0; H V V (1.4) ii)[ X; !] = (rX !); iii)[H X; H Y ] = H [X; Y ] + γR(X; Y ) = H [X; Y ] + V (pR(X; Y )) 1 n 0 n for all X; Y 2 =0(M ) and !; θ 2 =1(M ). (See [19, p.238, p.277] for more details) De¯nition 1.1. Let M n be a Riemannian manifold with metric g. A Riemannian metricg ¹ on cotangent bundle T ¤M n is said to be natural with respect to g on M n if i)g ¹(H X; H Y ) = g(X; Y ); ii)g ¹(H X; V !) = 0(1.5) 1 n 0 n for all X; Y 2 =0(M ) and !; θ 2 =1(M ). Theorem 1.2. Let M n be a Riemannian manifold with metric g and T ¤M n be the cotangent bundle of M n. If the Riemannian metric g¹ on T ¤M n is natural with respect to g on M n then the corresponding Levi-Civita connection r¹ satis¯es ¹ H H i)g ¹(rH X Y; Z) = g(rX Y; Z); H V 1 V V ii)g ¹(r¹ H Y; !) = g¹( !; (pR(X; Y ))); X 2 V H 1 V V iii)g ¹(r¹ H !; Z) = g¹( (pR(Z; X)); !; ); X 2 V V 1¡H V V V V iv)g ¹(r¹ H !; θ) = X(¹g( !; θ)) ¡ g¹( !; (r θ)) X 2 X V V ¢ +¹g( θ; (rX !)) ; H H 1 V V (1.6) v)g ¹(r¹ V Y; Z) = ¡ g¹( !; (pR(Y; Z))); ! 2 H V 1¡H V V V V vi)g ¹(r¹ V Y; θ) = Y (¹g( !; θ)) ¡ g¹( !; (r θ)) ! 2 Y V V ¢ ¡g¹( θ; (rY !)) ; V H 1¡ H V V V V vii)g ¹(r¹ V θ; Z) = ¡ Z(¹g( !; θ)) +g ¹( !; (r θ)) ! 2 Z V V +¹g( θ; (rZ !))); V V 1 V V V V V V V V V viii)g ¹(r¹ V θ; ») = ( !(¹g( θ; ») + θ(¹g( »; !) ¡ »(¹g( !; θ)) ! 2 1 n 0 n for all X; Y 2 =0(M ) and !; θ 2 =1(M ) [4]. Corollary 1.1. Let M n be a Riemannian manifold with metric g and g¹ be a natural metric on the cotangent bundle T ¤M n of M n. Then the Levi-Civita connection r¹ satis¯es H H 1 V (1.7) r¹ H Y = (r Y ) + (pR(X; Y )) X X 2 g-NATURAL METRICS ON THE COTANGENT BUNDLE 131 1 n for all X; Y 2 =0(M ) [4]. For each x 2 M n the scalar product g¡1 = (gij) is de¯ned on the cotangent ¡1 ¤ n space ¼ (x) = Tx M by ¡1 ij g (!; θ) = g !iθj 0 n for all !; θ 2 =1(M ). De¯nition 1.2. A g-natural metricg ~ is de¯ned on T ¤M n by the following three equations ¡ ¢ ¡ ¢ (1.8) g~ H X; H Y = V g(X; Y ) = g(X; Y ) ± ¼; ¡ ¢ (1.9) g~ V !; H Y = 0; ¡ ¢ g~ V !; V θ = '(z)g¡1(!; θ) + Ã(z)g¡1(!; p)g¡1(θ; p)(1.10) 1 n 0 n for any X; Y 2 =0(M ) and !; θ 2 =1(M ). Here ' and à are some functions of 1 1 ¡1 argument z = 2 jpj = 2 g (p; p) such that ' > 0 and ' + 2zà > 0. Since any tensor ¯eld of type (0,2) on T ¤M n is completely determined by its ac- tion on vector ¯elds of type H X and V !, it follows thatg ~ is completely determined by its equations (1.8), (1.9) and (1.10). The Sasaki metric is obtained for '(z) = 1 and Ã(z) = 0, while the Cheeger- 1 2 ¡1 Gromoll metric for '(z) = Ã(z) = 1+r2 , r = g (p; p). Sasaki, Cheeger-Gromoll and g-natural metrics are in the class of natural metric. C 1 n We now see, from (1.1) and (1.2), that the complete lift X of X 2 =0(M ) is expressed by ¡ ¢ (1.11) C X = H X ¡ V p(rX) ; i h where p(rX) = pi(rhX )dx . Using (1.8), (1.9), (1.10) and (1.11), we have ¡ ¢ ¡ ¢ ¡ ¢ g~ C X; C Y = V g(X; Y ) + '(z) g¡1(p(rX); p(rY )) (1.12) + Ã(z)g¡1(p(rX); p)g¡1(p(rY ); p); ¡1 ij l k ¡1 ij where g (p(rX); p(rY )) = g (plriX )(pkrjY ), g (p(rX); p) = g pi(p(rX))j. 0 ¤ n Since the tensor ¯eldg ~ 2 =2(T M ) is completely determined also by its action on vector ¯elds type C X and C Y (see[19, p.237]), we have an alternative charac- terization ofg ~ on T ¤M n:g ~ is completely determined by the condition (1.12). The main purpose of this paper is to introduce Levi-Civita connection of g- natural type metric on the cotangent bundle T ¤M n of Riemannian manifold M n and investigate curvature properties and geodesics on T ¤M n with respect to the Levi-Civita connection ofg ~. Since the construction of lifts to the cotangent bun- dle is not similar to the de¯nition of lifts to the tangent bundle, we have some di®erences for g-natural metrics on cotangent bundles. g-natural metric includes the Sasaki metric ([7], [12], [13]) and the Cheeger-Gromoll metric (see also [2], [4], [5], [6], [9], [11], [14], [15], [16], [18]) as a special cases. In [1]-[3] Abbasi and Sarih characterized the g-natural metric on the tangent bundle. In [17] Sukhova studied a class of Riemannian almost product metrics on the tangent bundle of a smooth manifold and investigated the scalar curvature of the tangent bundle. In [10] Munteanu computed the Levi-Civita connection, the curvature tensor, the 132 FIL_ IZ_ AGCA¸ sectional curvature and the scalar curvature of the g-natural metric on the tangent bundle. 2. Levi-Civita connection of g~ We put @ X = ; θ(i) = dxi; i = 1; :::; n: (i) @xi H V (i) Then from (1.2) and (1.3) we see that X(i) and θ have respectively local expressions of the form X H @ a @ (2.1) e~ = X = + pa¡ ; (i) (i) @xi hi @xh¹ h V (i) @ (2.2) e~ ¹ = θ = : (i) @x¹i H V (i) We call the set fe~(®)g = fe~(i); e~(¹i)g = f X(i); θ g the frame adapted to Levi- Civita connection rg. The indices ®; ¯; ::: = 1; :::; 2n indicate the indices with respect to the adapted frame. We now, from the equations (1.2), (1.3), (2.1) and (2.2) see that H X and V ! have respectively components µ ¶ Xi (2.3) H X = Xie~ ; H X = (H X®) = ; (i) 0 X µ ¶ V V V ® 0 (2.4) ! = !ie~(¹i); ! = ( ! ) = !i i i with respect to the adapted frame fe~(®)g, where X and !i being local components 1 n 0 n of X 2 =0(M ) and ! 2 =1(M ), respectively.