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Diagonal Lifts of Metrics to Coframe Bundle Proceedings of the Institute of Mathematics and Mechanics, National Academy of Sciences of Azerbaijan Volume 44, Number 2, 2018, Pages 328{337 DIAGONAL LIFTS OF METRICS TO COFRAME BUNDLE HABIL FATTAYEV AND ARIF SALIMOV Abstract. In this paper the diagonal lift Dg of a Riemannian metric g ∗ of a manifold Mn to the coframe bundle F (Mn) is defined, Levi-Civita connection, Killing vector fields with respect to the metric Dg and also an almost paracomplex structures in the coframe bundle are studied. 1. Introduction The Riemannian metrics in the tangent bundle firstly has been investigated by the Sasaki [14]. Tondeur [16] and Sato [15] have constructed Riemannian metrics on the cotangent bundle, the construction being the analogue of the metric Sasaki for the tangent bundle. Mok [7] has defined so-called the diagonal lift of metric to the linear frame bundle, which is a Riemannian metric resembles the Sasaki metric of tangent bundle. Some properties and applications for the Riemannian metrics of the tangent, cotangent, linear frame and tensor bundles are given in [1-4,7-9,12,13]. This paper is devoted to the investigation of Riemannian metrics in the coframe bundle. In 2 we briefly describe the definitions and results that are needed later, after which the diagonal lift Dg of a Riemannian metric g is constructed in 3. The Levi-Civita connection of the metric Dg is determined in In 4. In 5 we consider Killing vector fields in coframe bundle with respect to Riemannian metric Dg. An almost paracomplex structures in the coframe bundle equipped with metric Dg are studied in 6. 2. Preliminaries We shall summarize briefly the basic definitions and results which be used later. 1 ∗ Let Mn be an n−dimensional differentiable manifold of class C and F (Mn) ∗ its coframe bundle (see, [10, 11]). The coframe bundle F (Mn) over Mn consists ∗ ∗ of all pairs (x; u ), where x is a point of Mn and u is a basis (coframe) for the ∗ ∗ cotangent space Tx M. We denote by π the natural projection of F (Mn) to Mn defined by π(x; u∗) = x. If (U; x1; x2; :::; xn) is a system of local coordinates in ∗ α 1 2 n ∗ Mn, then a coframe u = (X ) = (X ;X ; :::; X ) for Tx Mn can be expressed α α i uniquely in the form X = Xi (dx )x and hence −1 1 2 n 1 1 n π (U); x ; x ; :::; x ;X1 ;X2 ; :::; Xn 2010 Mathematics Subject Classification. 53C25, 55R10. Key words and phrases. Riemannian metric, coframe bundle, diagonal lift, Levi-Civita con- nection, Killing vector field, almost paracomplex structure. 328 DIAGONAL LIFTS OF METRICS TO COFRAME BUNDLE 329 ∗ is a system of local coordinates in F (Mn) (see, [10]). Indices i; j; k; :::; α; β; γ; ::: have range in f1; 2; :::; ng, while indices A; B; C; ::: have range in 1; :::; n; n + 1; :::; n + n2 : We put hα = α · n + h. Summation over repeated indices is always implied. r We denote by =s(Mn) the set of all differentiable tensor fields of type (r; s) i i on Mn. Let V = V @i and ! = !idx be the local expressions in U ⊂ Mn of a 1 0 vector and a covector (1-form) fields V 2 =0(Mn) and ! 2 =1(Mn), respectively. C H 1 ∗ Then the complete and horizontal lifts V; V 2 =0(F Mn) of V and the β−th Vβ 1 ∗ vertical lifts ! 2 =0(F Mn)(β = 1; 2; :::; n) of ! have respectively, i V i 0 C V H Vβ V = α j ; V = α j k ; ! = α (2.1) −Xj @iV Xj ΓikV δβ !i n @ @ o with respect to the natural frame f@i;@iα g = i ; α ; (see [10] for more @x @Xi details). V ∗ The vertical lift of a smooth function f on Mn is a function f on F (Mn) defined by V f = f ◦ π. i Let (U; x ) be a coordinate system in Mn. In U 2 Mn, we put @ X = ; θ(i) = dxi; i = 1; 2; :::; n: (i) @xi H Vα (i) Taking account of (2.1), we easily see that the components of X(i) and θ are respectively, given by h H H δi Di = X(i) = Ai = α j ; (2.2) Xj Γih Vα (i) H 0 Diα = θ = Aiα = α i (2.3) δβ δh H Vα (i) with respect to the natural frame f@i;@iα g. We call the set X(i); θ the frame adapted to the Levi-Civita connection rg. On putting H Vα (i) Di = X(i);Diα = θ ; we write the adapted frame as fDI g = fDi;Diα g. From equations (2.2), (2.3) and (2.1) we see that H V and Vα ! have respectively, components V i H V = V iD ; H V = H V I = ; (2.4) i 0 0 Vα X β Vα Vα I ! = !iδαDiα ; ! = ! = β (2.5) δα!i i i with respect to the adapted frame fDI g, where V and !i being local components 1 0 of V 2 =0(Mn) and ! 2 =1(Mn), respectively. Let us consider local 1−formsη ~I in π−1(U) defined by I ¯I J η~ = A J dx ; where ! ! A¯i A¯i δi 0 A−1 = (A¯I ) = j jβ = j : (2.6) J A¯iα A¯iα −Xα Γm δαδj j jβ m ij β i 330 HABIL FATTAYEV AND ARIF SALIMOV The matrix (2.6) is the inverse of the matrix A j A j ! j ! J k kγ δk 0 A = (AK ) = jβ jβ = β m β k (2.7) A A XmΓ δγ δ k kγ jk j J of the transformation DK = AK @J ( see (2.2) and (2.3)). It is easy to establish I that the set η~ is the coframe dual to the adapted frame fDK g, i.e. I ¯I J I η~ (DK ) = A J AK = δK : 3. Diagonal lift Dg of a Riemannian metric g to the coframe bundle On putting locally n D i j X ij iα jβ g = gijη~ ⊗ η~ + δαβ g η~ ⊗ η~ (3.1) i;j ∗ D in the coframe bundle F (Mn), we see that g defines a tensor field of type (0; 2) ∗ ∗ in F (Mn) which called the diagonal lift of the tensor field g to F (Mn) with k D respect to Γij. From (2.6), (2.7) and (3.1) we prove that g has components of the form D gij 0 g = ij (3.2) 0 δαβg ∗ with respect to the adapted frame fDI g in F (Mn) and components Pn ks α α m l js β l ! D gij + α=1 g XmXl ΓikΓjs −g Xl Γis g = is α l ij (3.3) −g Xl Γjs δαβg ij with respect to the natural frame f@i;@iα g, where g denote contravariant com- ponents of g. D From (3.2) it easily follows that if g is a Riemannian metric in Mn, then g ∗ D is a Riemannian metric in F (Mn). The metric g is similar to that of the Rie- mannian metric studied by Sasaki in tangent bundle T (Mn)[14] (for the cotangent ∗ bundle T (Mn)and linear frame bundle F (Mn) see [15],[7], respectively). From (2.1) and (3.3) we have Dg(H X; H Y ) = V (g(X; Y )) = g(X; Y ) ◦ π: (3.4) Therefore we have as follows. 1 Theorem 3.1. Let X; Y 2 =0(Mn). Then the inner product of the horizontal H H ∗ D lifts Xand Y to F (Mn)with the metric g is equal to the vertical lift of the inner product of Xand Y in Mn. From (2.1) and (3.3) we have also Dg(Vα !; Vβ θ) = δαβV (g−1(!; θ)) = δαβ(g−1(!; θ) ◦ π); (3.5) Dg(H X; Vβ θ) = 0 (3.6) 1 0 for all X 2 =0(Mn) and !; θ 2 =1(Mn). We recall that any element 0 ∗ t 2 =2(F (Mn)) is completely determined by its action on vector fields of type DIAGONAL LIFTS OF METRICS TO COFRAME BUNDLE 331 H X and Vα !. From this it follows that Dg is completely determined by its eqs (3.4), (3.5) and (3.6). 4. Levi-Civita connection of Dg In section 2 we see that the components of the adapted frame fDI g are given by (2.2) and (2.3). On the other hand the components of the dual coframe η~I are given by matrix (2.6). Since the adapted frame is non-holonomic, we put K [DI ;DJ ] = ΩIJ DK from which we have K L L ¯K ΩIJ = DI AJ − DJ AI A L: According to (2.2), (2.3), (2.6) and (2.7), the components of non-holonomic object K ΩIJ are given by ( k k Ω γ = −Ω γ = −δγΓj ; ijβ jβ i β ik (4.1) kγ γ m Ωij = XmRijk m all the others being zero, where Rijk local components of the curvature tensor field R of rg. Let Dr be the Levi-Civita connection determined by the metric Dg on the ∗ coframe bundle F (Mn). We put D D K rDI DJ = ΓIJ DK : From the equation D D 1 ∗ rX Y − rY X = [X; Y ] ; 8X; Y 2 =0(F (Mn)) we have D K D K K ΓIJ − ΓJI = ΩIJ : (4.2) The equation D D rX g (Y; Z) = 0 has form D D K D D K D DL gIJ − ΓLI gKJ − ΓLJ gIK = 0 (4.3) with respect to the adapted frame fDK g.
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