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. 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 T∗M. We denote by π the natural projection of F∗(Mn) to Mn defined by π(x, u∗x) = x. If (U; x1, x2, ..., xn) is a system of local coordinates in Mn, then a coframe u∗ = (Xα) = (X1, X2, ..., Xn) for Tx∗Mn can be expressed uniquely in the form Xα = Xiα(dxi)x and hence <ul style="display: flex;"><li style="flex:1">ꢀ</li><li style="flex:1">ꢁ</li></ul><ul style="display: flex;"><li style="flex:1">π</li><li style="flex:1">−1(U); x1, x2, ..., xn, X11, X21, ..., Xnn </li></ul>2010 Mathematics Subject Classification. 53C25, 55R10. Key words and phrases. Riemannian metric, coframe bundle, diagonal lift, Levi-Civita connection, Killing vector field, almost paracomplex structure. 328 <ul style="display: flex;"><li style="flex:1">DIAGONAL LIFTS OF METRICS TO COFRAME BUNDLE </li><li style="flex:1">329 </li></ul>is a system of local coordinates in F∗(Mn) (see, [10]). Indices i, j, k, ..., α, β, γ, ... have range in {1, 2, ..., n}, while indices A, B, C, ... have range in ꢂ<ul style="display: flex;"><li style="flex:1">1, ..., n, n + 1, ..., n + n2 </li><li style="flex:1">.</li></ul>We put hα = α · n + h. Summation over repeated indices is always implied. rWe denote by =s(Mn) the set of all differentiable tensor fields of type (r, s) on Mn. Let V = V i∂i and ω = ωidxi be the local expressions in U ⊂ Mn of a <ul style="display: flex;"><li style="flex:1">1</li><li style="flex:1">0</li></ul>vector and a covector (1-form) fields V ∈ =0(Mn) and ω ∈ =1(Mn), respectively. Then the complete and horizontal lifts CV,H V ∈ =0(F∗Mn) of V and the β−th 1Vβ 1vertical lifts ω ∈ =0(F∗Mn) (β = 1, 2, ..., n) of ω have respectively, <ul style="display: flex;"><li style="flex:1">ꢃ</li><li style="flex:1">ꢄ</li><li style="flex:1">ꢃ</li><li style="flex:1">ꢄ</li><li style="flex:1">ꢃ</li><li style="flex:1">ꢄ</li></ul>V i V i 0Vβ <ul style="display: flex;"><li style="flex:1">CV = </li><li style="flex:1">,</li><li style="flex:1">HV = </li><li style="flex:1">,</li><li style="flex:1">ω = </li></ul>(2.1) XjαΓjikV k <ul style="display: flex;"><li style="flex:1">δβαωi </li><li style="flex:1">−Xjα∂iV j </li></ul><ul style="display: flex;"><li style="flex:1">n</li><li style="flex:1">o</li></ul><ul style="display: flex;"><li style="flex:1">∂</li><li style="flex:1">∂</li></ul>with respect to the natural frame {∂i, ∂i } = details). ,, (see [10] for more ∂Xiα ∂xi αThe vertical lift of a smooth function f on Mn is a function V f on F∗(Mn) defined by V f = f ◦ π. Let (U, xi) be a coordinate system in Mn. In U ∈ Mn, we put ∂ ∂xi X(i) =<ul style="display: flex;"><li style="flex:1">,</li><li style="flex:1">θ(i) = dxi, i = 1, 2, ..., n. </li></ul>HVα (i) Taking account of (2.1), we easily see that the components of X(i) and θare respectively, given by <ul style="display: flex;"><li style="flex:1">ꢃ</li><li style="flex:1">ꢄ</li></ul>ꢄδih <ul style="display: flex;"><li style="flex:1">ꢀ</li><li style="flex:1">ꢁ</li></ul>Di = HX(i) = AHi =,(2.2) (2.3) XjαΓjih ꢃ<ul style="display: flex;"><li style="flex:1">ꢀ</li><li style="flex:1">ꢁ</li></ul>0δβαδhi Vα (i) Di =θ= AHi =ααꢂwith respect to the natural frame {∂i, ∂i }. We call the set HX(i), Vα θ(i) the αframe adapted to the Levi-Civita connection ∇g. On putting Vα (i) Di = HX(i), Di =θ , αwe write the adapted frame as {DI} = {Di, Di }. From equations (2.2), (2.3) αVα and (2.1) we see that HV and ω have respectively, components <ul style="display: flex;"><li style="flex:1">ꢃ</li><li style="flex:1">ꢄ</li></ul>0δαβωi V i 0ꢃ<ul style="display: flex;"><li style="flex:1">ꢀ</li><li style="flex:1">ꢁ</li></ul>HV = V iDi, HV = HV I =,(2.4) (2.5) ꢄ X<ul style="display: flex;"><li style="flex:1">ꢀ</li><li style="flex:1">ꢁ</li></ul><ul style="display: flex;"><li style="flex:1">ω = </li><li style="flex:1">ωiδαβDi , ω = </li></ul>Vα ωI =<ul style="display: flex;"><li style="flex:1">Vα </li><li style="flex:1">Vα </li></ul>αiwith respect to the adapted frame {DI}, where V i and ωi being local components <ul style="display: flex;"><li style="flex:1">1</li><li style="flex:1">0</li></ul>of V ∈ =0(Mn) and ω ∈ =1(Mn), respectively. Let us consider local 1−forms η˜I in π−1(U) defined by η˜ = A J dxJ , <ul style="display: flex;"><li style="flex:1">I</li><li style="flex:1">I</li></ul>¯where <ul style="display: flex;"><li style="flex:1"> </li><li style="flex:1">!</li><li style="flex:1"> </li><li style="flex:1">!</li></ul><ul style="display: flex;"><li style="flex:1">i</li><li style="flex:1">i</li></ul>δji 0<ul style="display: flex;"><li style="flex:1">¯</li><li style="flex:1">¯</li></ul>A j A j Iβ<ul style="display: flex;"><li style="flex:1">A−1 = (A J ) = </li><li style="flex:1">=</li></ul>.(2.6) ¯−Xmα Γmij δβαδij iα jiα ¯A¯Ajβ <ul style="display: flex;"><li style="flex:1">330 </li><li style="flex:1">HABIL FATTAYEV AND ARIF SALIMOV </li></ul>The matrix (2.6) is the inverse of the matrix <ul style="display: flex;"><li style="flex:1"> </li><li style="flex:1">!</li><li style="flex:1"> </li><li style="flex:1">!</li></ul><ul style="display: flex;"><li style="flex:1">j</li><li style="flex:1">j</li></ul>γjβ δkj Xmβ Γmjk δγβδjk 0<ul style="display: flex;"><li style="flex:1">Ak </li><li style="flex:1">Ak </li></ul><ul style="display: flex;"><li style="flex:1">A = (AK J ) = </li><li style="flex:1">=</li><li style="flex:1">(2.7) </li></ul>jβ <ul style="display: flex;"><li style="flex:1">Ak </li><li style="flex:1">Ak </li></ul>γof the transformation DK = AK J ∂J ( see (2.2) and (2.3)). It is easy to establish ꢂthat the set η˜I is the coframe dual to the adapted frame {DK}, i.e. η˜ (DK) = A J AK = δKI . <ul style="display: flex;"><li style="flex:1">I</li><li style="flex:1">I</li><li style="flex:1">J</li></ul>¯D3. Diagonal lift g of a Riemannian metric g to the coframe bundle On putting locally nX<ul style="display: flex;"><li style="flex:1">Dg = gijη˜i ⊗ η˜j + δαβ </li><li style="flex:1">gijη˜iα ⊗ η˜jβ </li></ul>(3.1) i,j in the coframe bundle F∗(Mn), we see that Dg 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 respect to Γkij. From (2.6), (2.7) and (3.1) we prove that Dg has components of the form <ul style="display: flex;"><li style="flex:1">ꢃ</li><li style="flex:1">ꢄ</li></ul>gij 0 0Dg = (3.2) δαβgij with respect to the adapted frame {DI} in F∗(Mn) and components <ul style="display: flex;"><li style="flex:1"> </li><li style="flex:1">!</li></ul>Pgij + α=1 gksXmα XlαΓimkΓjl s −gjsXβΓils nDg = (3.3) l<ul style="display: flex;"><li style="flex:1">−gisXlαΓljs </li><li style="flex:1">δαβgij </li></ul>with respect to the natural frame {∂i, ∂i }, where gij denote contravariant com- αponents of g. From (3.2) it easily follows that if g is a Riemannian metric in Mn, then Dg is a Riemannian metric in F∗(Mn). The metric Dg is similar to that of the Riemannian 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(HX, HY ) = V (g(X, Y )) = g(X, Y ) ◦ π. (3.4) Therefore we have as follows. 1Theorem 3.1. Let X, Y ∈ =0(Mn). Then the inner product of the horizontal lifts HXand HY to F∗(Mn)with the metric Dg is equal to the vertical lift of the inner product of Xand Y in Mn. From (2.1) and (3.3) we have also Vβ Dg(V ω, θ) = δαβV (g−1(ω, θ)) = δαβ(g−1(ω, θ) ◦ π), (3.5) (3.6) αDg(HX, V θ) = 0 β<ul style="display: flex;"><li style="flex:1">1</li><li style="flex:1">0</li></ul>for all X ∈ =0(Mn) and ω, θ ∈ =1(Mn). We recall that any element t ∈ =2(F∗(Mn)) is completely determined by its action on vector fields of type 0<ul style="display: flex;"><li style="flex:1">DIAGONAL LIFTS OF METRICS TO COFRAME BUNDLE </li><li style="flex:1">331 </li></ul>HX and ω. From this it follows that Dg is completely determined by its eqs Vα (3.4), (3.5) and (3.6). D4. Levi-Civita connection of g In section 2 we see that the components of the adapted frame {DI} are given by ꢂ(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 [DI, DJ ] = ΩIJ KDK from which we have <ul style="display: flex;"><li style="flex:1">ꢀ</li><li style="flex:1">ꢁ</li></ul><ul style="display: flex;"><li style="flex:1">K</li><li style="flex:1">K</li></ul>ΩIJ = DIAJL − DJ AIL A L. ¯ According to (2.2), (2.3), (2.6) and (2.7), the components of non-holonomic object KΩIJ are given by (kγ kγ kγ <ul style="display: flex;"><li style="flex:1">= −Ωj i = −δβγΓjik, </li><li style="flex:1">Ωij </li></ul>Ωij

Diagonal Lifts of Metrics to Coframe Bundle

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