Contact structures on the indicatrix of a complex Finsler space

Contact structures on the indicatrix of a complex Finsler space Elena Popovici Abstract. Continuing the study of the complex indicatrix Iz M , approached as an embedded CR hypersurface on the punctual holomorphic tangent bundle of a complex Finsler space, we study in this paper the almost contact structures that can be introduced on Iz M . The Levi form and characteristic direction of the complex indicatrix are given and the CR distributions integrability is studied. Using these we construct a natural contact structure subordonated to the CR-structure of the complex indicatrix, which is also normal. Moreover, with respect to the natural contact structure, the associated connections on Iz M , such as Tanaka and Tanaka Webster connections, are determined and the Bochner type tensor field of the complex indicatrix is introduced. M.S.C. 2010: 53B40, 53C60, 53C40, 53B25. Key words: Complex Finsler space; complex indicatrix; contact structure; associated connections; Bochner type tensor. 1 Introduction The study of unit tangent sphere, or indicatrix, in real Finsler spaces is one of interest ([8, 13, 16, 18], etc.), mainly because it is a compact and strictly convex set surrounding the origin. In the present paper, considering the indicatrix of a complex Finsler space as a CR hypersurface of the holomorphic tangent space in a fixed point, we introduce and analyze almost contact structures on the complex indicatrix and several of them properties are obtained. Firstly, in Section 1, we recall some basic notions regarding complex Finsler geometry. By taking z ∈ M an arbitrary point, the punctured holomorphic tangent bundle Tz′ M can be locally viewed as a Kähler manifold and the complex indicatrix is a real hypersurface, i.e. a CR hypersurface of Tz′ M . Thus, the characteristic direction, Levi distribution and integrability of the CR distributions are studied in Section 2. By using these, we introduce in Section 3 the natural contact structure associated to the ∗ Proceedings 23. The International Conference ”Differential Geometry - Dynamical Systems” BSG DGDS-2015, October 8-11, 2015, Bucharest-Romania, pp. 56-67. c Balkan Society of Geometers, Geometry Balkan Press 2016. ⃝ Contact structures on the indicatrix of a complex Finsler space 57 CR-structure of the complex indicatrix and we prove that any almost contact structure subordonated to the CR structure is normal. Hence, the complex indicatrix is a Sasakian manifold. However, the natural connection is not the only almost contact structure on the complex indicatrix and using basic transformations we can determine other almost contact structures subordonated to the indicatrix CR structure. In the last Section we describe the existence of some fundamental connections related to the natural contact structure subordonated to the complex indicatrix CR structure. In this sense we will present the action of Tanaka and Tanaka Webster connections on the tangent vectors of the complex indicatrix and with their help we are able to analyze geometric invariants of the complex indicatrix under certain transformations. Most of them are obtained from curvature tensor fields of linear connection and for a pseudo-convex CR-structure, the complex indicatrix in particular, it is defined the Bochner type curvature tensor field which is invariant under gauge transformation of almost contact structure associated to the CR-structure. Now, we make a short overview of the concepts and terminology used in complex Finsler geometry, as in [1, 15]. Let M be an n - dimensional complex manifold, with z := (z k ), k = 1, .., n, the complex coordinates on a local chart (U, φ). The complexified of the real tangent bundle TC M splits into the sum of holomorphic tangent bundle T ′ M and its conjugate T ′′ M , i.e. TC M = T ′ M ⊕ T ′′ M . The holomorphic tangent bundle T ′ M is in its turn a 2n-dimensional complex manifold and the local coordinates in a local chart in u ∈ T ′ M are u := (z k , η k ), k = 1, .., n. Definition 1.1. A complex Finsler space is a pair (M, F ), with F : T ′ M → R+ , F = F (z, η) a continuous function that satisfies the following conditions: ′ M := T ′ M \ {0}; (i) F is a smooth function on T] (ii) F (z, η) ≥ 0, the equality holds if and only if η = 0; (iii) F (z, λη) = |λ|F (z, η), ∀λ ∈ C; ( ) 2 (iv) the Hermitian matrix gij̄ (z, η) is positive definite, where gij̄ = ∂η∂i ∂Lη̄j is the fundamental metric tensor, with L := F 2 the complex Lagrangian associated to the complex Finsler function F . The positivity of the fundamental tensor assures the convexity of the Lagrangian L and the strongly pseudoconvex property of the complex indicatrix in a fixed point Iz M = {η | gij̄ (z, η)η i η̄ j = 1}, for any z ∈ M . Moreover, from iii. it takes that L is homogeneous with respect to the complex norm, L(z, λη) = λλ̄L(z, η), ∀λ ∈ C, and by applying Euler’s formula we get that: (1.1) ∂L k ∂L k η = η̄ = L; ∂η k ∂ η̄ k ∂gij̄ k ∂gij̄ k η = η̄ = 0 ∂η k ∂ η̄ k and L = gij̄ η i η̄ j . An immediate consequence concerns the following Cartan complex tensors: Cij̄k := ∂gij̄ ∂η k and Cij̄ k̄ := ∂gij̄ . ∂ η̄ k which have the following properties: (1.2) Cij̄k = Ckj̄i ; Cij̄ k̄ = Cik̄j̄ ; Cij̄k = Cj īk̄ ; Cij̄k η k = Cij̄ k̄ η̄ j = Cij̄k η i = Cij̄ k̄ η̄ k = 0. 58 Elena Popovici The geometry of complex Finsler spaces consists of the study of geometric objects on the complex manifold T ′ M endowed with a Hermitian metric structure defined by gij̄ . We start by analyzing the sections of the complexified tangent bundle TC (T ′ M ) = T ′ (T ′ M ) ⊕ T ′′ (T ′ M ), where Tu′′ (T ′ M ) = Tu′ (T ′ M ). Let V (T ′ M ) = span{ ∂η∂ k } ⊂ T ′ (T ′ M ) be the vertical bundle and we introduce the complex non-linear connection, denoted by (c.n.c.), as the supplementary complex subbundle to V (T ′ M ) in T ′ (T ′ M ), i.e. T ′ (T ′ M ) = H(T ′ M ) ⊕ V (T ′ M ). The horizontal distribution Hu (T ′ M ) is locally spanned by { δzδk = ∂z∂k − Nkj ∂η∂ j }, where Nkj (z, η) represent the coefficients of a (c.n.c.). Thus, we call the pair {δk := δk , ∂˙k := ∂ k } the adapted frame of the δz ∂η (c.n.c.), which has the dual adapted base {dz k , δη k := dη k + Njk dz j }. One fundamental (c.n.c.) of a complex Finsler space is the Chern-Finsler (c.n.c.) lm̄ l ([1],[15]), with Njk = g m̄k ∂g ∂z j η , which determines the Chern-Finsler linear coni nection, locally given by the next set of coefficients ([15]) Lijk = g l̄i δk (gj l̄ ), Cjk = g l̄i ∂˙ (g ), Lī = 0, C ī = 0, where D δ = Li δ , D ∂˙ = Li ∂˙ , D ˙ ∂˙ = C i ∂˙ , k j l̄ j̄k j̄k δk j jk i δk j jk i ∂k j jk i i i j i k D∂˙k δj = Cjk δi and Cjk η = Cjk η = 0 from (1.1). Further we will use the following j j̄ notation η̄ =: η to denote a conjugate object. The CR structure attempts to describe intrinsically the property of being a hypersurface in complex space; thus, a CR manifold can be considered as an embedded CR manifold (hypersurface and edges of wedges in complex space) or as an abstract CR manifold. Cauchy-Riemann (CR) submanifolds of almost Hermitian manifolds, introduced by A. Bejancu [3, 4, 5], were extended to the Finsler geometry by S. Dragomir in [10, 11]. Thus, a real submanifold M̃ of an almost Hermitian Finsler space (M, g), is a CR-submanifold endowed with a pair of complementary Finslerian distributions D and D⊥ of T M̃ , such that D is invariant, J(Du ) = Du , and D⊥ is anti-invariant, J(Du⊥ ) ⊂ (Tu M̃ )⊥ , for each u ∈ M̃ , where J is an almost complex structure on M̃ . It can be easily noticed that any real hypersurface M̃ of M is a CR-submanifold, having Du⊥ = J(Tu M̃ )⊥ and D as the complementary orthogonal distribution of D⊥ . 2 The CR geometry of the complex indicatrix Let us take (M, F ) a complex Finsler manifold, Tz′ M its corresponding holomorphic tangent space and Fz the Finsler metric in an arbitrary fixed point z ∈ M . Then, (Tz′ M, Fz ) can be regarded as a locally complex n - dimensional Minkowski space, with (η i ) its complex coordinate system, η = (η i ) = η i ∂z∂ i |z . Let g be the Hermitian ′ M ) associated to F , which can be extended to a complex bilinear structure on T ′ (T] z ′ form G, which defines a Hermitian metric on T] z M , locally given as: (2.1) G := ∂ 2 Fz2 dη j ⊗ dη̄ k = gj k̄ (z, η)dη j ⊗ dη̄ k . ∂η i ∂ η̄ j As in [15], we can can extend, by linearity, a linear connection on M to TC M, which is isomorphic to VC (T ′ M ) via vertical lift. We require ∇ to be a compatible complex connection with respect to J, i.e. ∇J = 0, such that ∇ conserves the holomorphic tangent space. Here J represents the natural complex structure √ (2.2) J(∂˙k ) = i∂˙k , J(∂˙k̄ ) = −i∂˙k̄ , with i := −1. Contact structures on the indicatrix of a complex Finsler space 59 We can choose ∇ to be the Levi-Civita connection, which is a metrical and symmetric connection and using (1.2) we get the following components: i (η); Γijk = g h̄i Cj h̄k =: Cjk Γīj̄k = 0; Γij̄k = 0; Γījk = 0. Since Γij̄k = Γīj̄k = 0, it takes that the Levi-Civita connection is Hermitian, and i i i j i k the non-zero coefficients satisfy Cjk = Ckj and Cjk η = Cjk η = 0. Taking into consideration that the Levi-Civita connection considered above is equivalent to the linear Chern connection on π ∗ T ′ M = span{ ∂z∂ i } [2], where π : T ′ M → M is the i i ′ = 0, we get that (T] − Ckj natural projection, and since Cjk z M , Fz ) is Kählerian and thus ∇ is Kählerian connection, i.e. ∇X (JY ) = J∇X Y . For an arbitrary fixed point z ∈ M , the unit sphere in (Tz′ M, Fz ), also called the complex indicatrix in z is: Iz M = {η ∈ Tz′ M | F (z, η) = 1} . Iz M is a strictly pseudo convex submanifold and since it has only one defining equation which involves the real valued Finsler function F , it is a real hypersurface of the holomorphic tangent bundle Tz′ M , and thus a CR-hypersurface, for any z ∈ M . Let (u1 , ..., u2n−1 ) be local coordinates on Iz M and η j = η j (u1 , .., u2n−1 ), ∀j ∈ i 1 j k̄ j ′ {1, .., n} the equations of inclusion Iz M ֒→ T] z M [11]. By taking l = F η , lj = gj k̄ l , from F (z, η(u)) = 1, by derivation after u, we get (2.3) lj ∂η j ∂η j̄ + l = 0, j̄ ∂uα ∂uα α ∈ {1, . . . , 2n − 1}, j ∈ {1, . . . , n}. ′ The tangent map i∗ : TR (Iz M ) → TC (T] z M ) acts on tangent vectors of Iz M as ) ( ∂η k ∂ ∂ η̄ k ∂ ∂ i∗ = X := + , α ∂uα ∂uα ∂η k ∂uα ∂ η̄ k where Xα is a tangent vector of the complex indicatrix expressed in terms of tangent vectors of TC (T ′ M ). Considering this and (2.3), we set (2.4) N = lj ∂˙j + lj̄ ∂˙ȷ̄ and thus we obtain GR (Xα , N ) = 0, where GR is the Riemannian metric applied to real vector fields as GR (X, Y ) = Re G(X ′ , Y ′ ). Here X ′ , Y ′ are the holomorphic, respectively, the anti-holomorphic part of tangent vectors X and Y of the complexified tangent bundle of T ′ M , obtained by X ′ = √ 1 ′ = 1 (Y + iJY ), i = (X − iJX), Y −1. Consequently, N ∈ TR (Iz M )⊥ and 2 2 GR (N, N ) = 1, so that N is the unit normal vector of the indicatrix bundle. Since each real orientable hypersurface of a Kähler manifold is a CR-submanifold with JDx⊥ = Tx⊥ , we get that the anti-invariant distribution D⊥ of the indicatrix of a complex Finsler space in a fixed point must satisfy JD⊥ = span{N }, where N is the unit normal vector field to Iz M given in (2.4) and J is the complex structure 60 Elena Popovici from (2.2). Then we take the characteristic direction of the complex indicatrix CR structure as ) ( √ ξ = JN = i lk ∂˙k − lk̄ ∂˙k̄ , i := −1, ¯ N = −Jξ, D⊥ = span{ξ} which is a real tangent unit vector on Iz M , with ξ = ξ, and GR (ξ, ξ) = 1. Let then D be the maximal J-invariant subspace of the tangent space of Iz M , also called the Levi distribution, which is orthogonal to D⊥ , such that (2.5) TR (Iz M ) = D ⊕ span{ξ}. Thus, dimR Iz M = 2n − 1 and dimR D = 2n − 2, since dimC M = n. Considering (2.5) and TR (Tz′ M ) = TR (Iz M ) ⊕ span{N } and TR (Iz M ) = D ⊕ span{ξ}, we can take D = TR (M̃ ), where M̃ is a complex hypersurface of Tz′ M , with dimC M̃ = n − 1 and complex unit normal vector N ′ = lj ∂˙j . Thus, we have D = Re{T ′ M̃ ⊕ T ′′ M̃ } and since T ′ (Tz′ M ) = span{∂˙j }, there exist the complex projection factors Pai such that T ′ M̃ = span{Ya′ := Paj ∂˙j }, a ∈ {1, . . . , n − 1}. Further, we denote by D′ := T ′ M̃ , D′′ := T ′′ M̃ , and so D ⊗ C = D′ ⊕ D′′ . Thus, having in mind that Ya := Ya′ + Ya′ and JYa = i(Ya′ − Ya′ ), we conclude that D = span{Ya := Paj ∂˙j + Pāȷ̄ ∂˙ȷ̄ , JYa = i(Paj ∂˙j − Pāȷ̄ ∂˙ȷ̄ )}. Moreover, since Ya′ and N ′ are complex tangent vectors, respectively the complex normal vector of the complex hypersurface M̃ , the following conditions are fulfilled with respect to the Hermitian metric G (2.1) Paj lj = 0, Pāȷ̄ lȷ̄ = 0 and lj lj = 1. In order to analyze if the real or complex integrability conditions of the complex indicatrix distributions D and D⊥ are fulfilled, we will need the following results regarding the Lie brackets of tangent vectors of Iz M [ξ, Ya ] = 2Re (ξ(Paj ) − 2i Paj )∂˙j ; [Ya , Yb ] = 2Re (Ya (Pbj ) − Yb (Paj ))∂˙j ; [ξ, JYa ] = 2Re(iξ(Paj ) + 21 Paj )∂˙j ; [Ya , JYb ] = 2Re (iYa (Pbj ) − JYb (Paj ))∂˙j ; [JYa , JYb ] = 2Re{i(JYa (Pbj ) − JYb (Paj ))∂˙j }. Moreover, the real metric GR action on these vectors is GR (Ya , Yb ) = GR (JYa , JYb ) = Re(gj k̄ Paj Pb̄k̄ ) =: Re(gab̄ ) GR (Ya , JYb ) = GR (JYa , Yb ) = −Re(igj k̄ Paj Pb̄k̄ ) =: −Re(igab̄ ), GR (ξ, Ya ) = GR (ξ, JYa ) = GR (N, Ya ) = GR (N, JYa ) = 0, m m and, by denoting Cab = Pak Pbj Cjk , we have the Levi-Civita connection action as (2.6) ∇Ya = F1 JYa , ∇JYa ξ = − F1 Ya , ∇ξ ξ = − F1 N ; ∇ξ Ya = 2Re{ξ(Pam )∂˙m }, m ∇Ya Yb = 2Re{(Cab + Ya (Pbm ))∂˙m }, m ∇JYa Yb = 2Re{(iCab + JYa (Pbm ))∂˙m }, ∇ξ Ya = 2Re{iξ(Pam )∂˙m }, m ∇Ya JYb = 2Re{i(Cab + Ya (Pbm ))∂˙m }, m ∇JYa JYb = 2Re{(−Cab + iJYa (Pbm ))∂˙m }. Contact structures on the indicatrix of a complex Finsler space 61 If we consider that Iz M a CR-hypersurface of the Kähler manifold Tz′ M and if we take into consideration the above results, we can study (as in [17]) (i) the complex involutivity condition [Γ(D′ ), Γ(D′ )] ⊂ Γ(D′ ) of D′ , which, according to [12], is equivalent to the integrability of the almost complex structure D [JX, Y ] + [X, JY ] ∈ Γ(D) and [JX, JY ] − [X, Y ] = J([JX, Y ] + [X, JY ]); (ii) the real integrability of D, h(X, JY ) = h(JX, Y ), ∀X, Y ∈ Γ(D) (cf. [5]), which is equivalent to GR (∇X JY, N ) = GR (∇JX Y, N ); (iii) the integrability of the anti-invariant distribution D⊥ (according to [5]): GR ((∇ξ J)ξ, X) = 0, for any X, Y ∈ Γ(D). Thus, we can state Theorem 2.1. Let (M, F ) be a complex Finsler manifold, z ∈ M an arbitrary fixed point and Iz M the complex indicatrix. Then the following affirmations take place with respect to the distributions of the CR-hypersurface Iz M of Tz′ M : (a) the anti-invariant distribution D⊥ is integrable; (b) even though the complex CR-structure D′ is integrable, the real invariant distribution D is neither involutive, nor integrable. 3 Almost contact structures on the complex indicatrix In order to introduce almost contact structures on the complex indicatrix Iz M , we will make a short overview of the fundamental notions from the general theory of almost contact structures. Thus, for an odd-dimensional manifold M̃ , dimR M̃ = 2n − 1, has an almost cotact structure if its structural group reduces to U (n) × 1. On the other hand, in terms of structure tensors, M̃ has an (ϕ, ξ, θ)-almost contact structure, if it admits a tensor field ϕ of type (1, 1), the Reeb vector field ξ and a 1-form η, satisfying ϕ2 = −I + θ ⊗ ξ, θ(ξ) = 1, ϕξ = 0, θ ◦ ϕ = 0, where I is the identity transformation [7]. For a contact manifold the 1-form θ satisfies in addition θ ∧ (dθ)n ̸= 0. An almost contact structure (ϕ, ξ, θ) is normal if N ≡ Nϕ + 2dθ ⊗ ξ = 0, where Nϕ is the Nijenhuis tensor field of ϕ given as Nϕ (X, Y ) = [ϕX, ϕY ] − ϕ[ϕX, Y ] − ϕ[X, ϕY ] + ϕ2 [X, Y ]. A Sasakian manifold is a normal contact metric manifold, and in some aspects it may be viewed as an odd-dimensional analogue of the Kähler manifold. 62 Elena Popovici If the almost contact manifold M̃ admits a Riemannian metric g such that g(ϕX, ϕY ) = g(X, Y ) − θ(X)θ(Y ), for any vector fields X, Y , we say M̃ has an almost contact metric structure and g is called a compatible metric. Setting Y = ξ, we have immediately that θ(X) = g(X, ξ). By denoting D = {X ∈ Γ(T M̃ ), θ(X) = 0}, dim D = 2n, we notice that the restriction of ϕ to D is an almost complex structure on D. In [6], A.Bejancu proved Proposition 3.1. Let M̃ be a manifold endowed with a normal almost contact structure. Then M̃ is a CR-manifold. Futher, we analyze the existence of almost contact structures on the complex indicatrix and the posibility to determine normal almost contact structures on it. Let us consider M a complex Finsler space, dimC M = n, z an arbitrary fixed point and Iz M the complex indicatrix in z. Then, dimR Iz M = 2n − 1 and, as in the pervious section, we can determine a CR-structure (D, J) on Iz M , such that D⊥ = span{ξ}, with ξ = JN . Since ξ ∈ / D, we can choose the Reeb vector of the almost contact structure to be the characteristic direction ξ and we define the 1-form θ(X) = GR (X, ξ) for any X ∈ Γ(TR (Iz M )), more precisely θ= i (l dη̄ k − lk dη k ). 2 k̄ It verifies θ(ξ) = 1 and θ(X) = 0, ∀X ∈ Γ(D), so ker θ = D, i.e. θ is a pseudoHermitian structure on M . Moreover, since Iz M is a pseudoconvex CR manifold, any 1-form θ having this properties is a contact form, such that θ ∧ (dθ)n−1 ̸= 0. By considering the decomposition X = P X + θ(X)ξ, ∀X ∈ Γ(TR (Iz M )), with P X ∈ D, we define the (1,1) tensor field ϕ as ϕX = J(P X) = JX + θ(X)N, ∀ X ∈ Γ(TR (Iz M )). We notice that ϕX = JX for X ∈ Γ(D) and it verifies ϕ2 X = −X + θ(X)ξ, ϕξ = 0 and θ(ϕX) = 0, such that we can state Proposition 3.2. On the complex indicatrix Iz M of a complex Finsler space it exists a contact structure associated to the CR structure (D, J), determined by (3.1) ϕ = J + θ ⊗ N, ξ = i(lk ∂˙k − lk̄ ∂˙k̄ ), θ= i (l dη̄ k − lk dη k ), 2 k̄ which is called the natural contact structure of the complex indicatrix Iz M . An almost contact structure (ϕ, ξ, θ) is subordonated to the CR-structure (Iz M, D) if it satisfies ϕ|D = J and ker θ = D. Moreover, for a subordonated almost contact structure we can always contruct another almost contact structure subordonated to the same CR-structure which satisfies in addition (3.2) [ξ, Γ(D)] ⊂ Γ(D), equivalent to ιξ dθ = 0 or Lξ θ = 0. Thus, we notice that the natural contact structure (3.1) is subordonated to the CR structure (Iz M, D), but is not unique. Another one can be constructed on Iz M by Contact structures on the indicatrix of a complex Finsler space 63 ˜ = 1, taking θ̃ a pesudohermitian structure, with θ̃(D) = 0, the Reeb vector ξ˜ as θ̃(ξ) ˜ and the (1, 1) tensor ϕ̃X = J(X − ιξ̃ dθ̃ = 0, such that TR (Iz M ) = D ⊕ spanR {ξ}, ˜ for any X ∈ χ(Iz M ). θ̃(X)ξ), Regarding the normality of the subordonated almost contact on Iz M , we mention the following Theorem from [14], which relates complex involutivity of the CRstructure to the normality condition Theorem 3.3. The complex involutivity condition for (Iz M, D) is equivalent to S ≡ 0, where S is a (1, 2) type tensor field on Iz M defined by S(X, Y ) = Nϕ̃ (X, Y ) + 2dθ̃(X, Y )ξ˜ + θ̃(X)ϕ̃(Lξ̃ ϕ̃)Y − η̃(Y )ϕ̃(Lξ̃ ϕ̃)X, where L represents the Lie derivative. Thus, for a normal almost contact structure, Lξ ϕ = 0 and the complex involutivity is obtained. Viceversa, besides the complex involutivity of D we also need Lξ ϕ = 0 to obtain a normal contact structure. However, we have Theorem 3.4 ([19]). For an almost contact metric manifold the contact structure (ϕ, ξ, θ, g) is normal if and only if D′ and D′′ ⊕ ⟨ξ⟩c (or D′′ and D′ ⊕ ⟨ξ⟩c ) are integrable and Lξ θ = 0, where D = ker θ and D ⊗ C = D′ ⊕ D′′ . By using this Theorem, relation (3.2) and the integrability of D′ , D′′ and D⊥ from Theorem 2.1, we can state ˜ η̃, g̃) subordonated to the Proposition 3.5. Any almost contact metric structure (ϕ̃, ξ, complex indicatrix CR structure (Iz M, D), in particular the natural one, is normal. Therefore, we have Theorem 3.6. Let M be a complex Finsler manifold and z an arbitrary fixed point. The complex indicatrix Iz M is a Sasakian manifold. Obviously, the 1-form θ which defines the invariant distribution D is not unique. So, starting from the natural contact structure (3.1), we can determine another almost contact structure associated to the same CR-distribution on (Iz M, D), denoted by (ϕ′ , ξ ′ , θ′ ), as Proposition 3.7. Two almost contact structures associated to the same CR-structure of Iz M are related by (3.3) θ′ = f θ, ξ ′ = A + 1/f ξ, ϕ′ + θ′ ⊗ JA = ϕ, where f is a differential function on Iz M and A ∈ Γ(D). If we require (ϕ′ , ξ ′ , θ′ ) to be subordonated to the CR-structure, i.e. [ξ ′ , X] ∈ Γ(D), ∀X ∈ Γ(D), vector A ∈ Γ(D) must also fulfil dθ(A, X) = 1 1 df (X) = 2 X(f ), f2 f ∀X ∈ Γ(D). A special case of transformation between two almost contact manifolds subordonated to the same almost complex distribution D is the gauge transformation of the 1-form θ, given as θ 7→ θ̃ = εef θ, with f ∈ C ∞ (Iz M ) and ε = ±1. It can be easily noticed that 1-forms θ and θ̃ define the same distribution D. 64 Elena Popovici ˜ θ̃) are subordonated Proposition 3.8. Two almost contact structures (ϕ, ξ, θ), (ϕ̃, ξ, to the same strict pseudoconvex CR-structure iff it exists a function f ∈ C ∞ (Iz M ) such that θ̃ = εef θ, ξ˜ = εe−f (ξ + ϕA), ϕ̃ = ϕ + θ ⊗ A, with A ∈ D defined by dθ(ϕA, X) = df (X) = X(f ), ∀X ∈ Γ(D). Remark 3.1. In general, the complex involutivity is invariant under gauge transformations. 4 Associated connections on the complex indicatrix In this section we will describe the existence and the action of some canonical connections closely related to the almost contact structures subordonated to the complex indicatrix CR-structure (Iz M, D). More precisely, we present the Tanaka and TanakaWebster connections. With respect to these connections, the Levi distribution D and the complex structure J are parallel. We start with the Tanaka connection associated to the natural contact structure (3.1), which is subordonated to the to the complex indicatrix CR-structure (Iz M, D). By adapting Tanaka’s result [20] for the complex indicatrix case we get Theorem 4.1. Let (ϕ, ξ, θ) be the natural almost contact structure (3.1) subordonated t to the CR-structure (Iz M, D). Then it exists an unique linear connection D such that (4.1) t t t t 1 Dϕ = 0, Dξ = 0, Dθ = 0, D|D GR = 0, TD = 0, τ = − ϕLξ ϕ, 2 where, Gθ (X, Y ) = dθ(X, JY ) = GR (X, Y ), ∀X, Y ∈ D is the Levi metric and τ (X) = T (ξ, X), ∀X ∈ χ(Iz M ). Here we used that dθ = i F ) ( 1 gj k̄ − lj lk̄ dη j ∧ dη̄ k 2 and thus dθ(X, Y ) = F1 GR (ϕX, Y ), which are equal for F = 1 on Iz M . In order to determine the Tanaka connection expression for the tangent vectors of Iz M , we take into consideration that, according to [15], a linear connection on M extends by linearity to Tc M , and implicitly on Iz M aswell, and is well defined by the next set of coefficients: t D∂˙k ∂˙ȷ̄ = Aiȷ̄k ∂˙i + Aīȷ̄k ∂˙ī ; t D∂˙k̄ ∂˙ȷ̄ = Aiȷ̄k̄ ∂˙i + Aīȷ̄k̄ ∂˙ī , D∂˙k ∂˙j = Aijk ∂˙i + Aījk ∂˙ī ; D∂˙k̄ ∂˙j = Aij k̄ ∂˙i + Aīj k̄ ∂˙ī ; t t where Aijk = Aīȷ̄k̄ , Aīȷ̄k = Aij k̄ , Aījk = Aiȷ̄k̄ , Aīj k̄ = Aiȷ̄k , which come from DX Y = DX̄ Ȳ . 65 Contact structures on the indicatrix of a complex Finsler space t Demanding D to satisfy the Tanaka connection conditions (4.1), we find the following non-zero coefficients Aijk i = Cjk − Aij k̄ = 1 F 1 i F lj δk gj k̄ li − − 1 i F lk δ j and 2 i F lj lk̄ l . Thus, Tanaka connection action on tangent vectors of Iz M is t D Ya ξ t D ξ Ya t Dξ JYa t (4.2) D Ya Y b t DYa JYb t DJYa Yb t DJYa JYb t t = DJYa ξ = Dξ ξ = 0, = 2Re{ξ(Pam ) − = 2Re{iξ(Pam ) + i F Pam }∂˙m ; 1 F Pam }∂˙m ; m = 2Re{Ya (Pbm ) + Cab + 1 F gbā lm }∂˙m ; 1 F = m + 2Re i{Ya (Pbm ) + Cab = m 2Re{JYa (Pbm ) + i[Cab − = m 2Re{iJYa (Pbm ) − Cab + gbā lm }∂˙m ; 1 F 1 F gbā lm ]}∂˙m ; gbā lm }∂˙m . m m with Cab = Paj Pbk Cjk , gab = Paj Pbk gjk and gbā = Pbj Pb̄k̄ gj k̄ . In the following we study the Tanaka Webster connection. Starting from the natural contact structure (3.1) subordonated to the complex indicatrix CR structure (Iz M, D), we adjust Tanaka-Webster’s result [12] for the Iz M case as Theorem 4.2. Let (ϕ, ξ, θ) be the natural almost contact structure (3.1) subordonated w to the CR-structure (Iz M, D). Then it exists an unique linear connection D such that w w (i) D is prallel w.r.t. D, i.e. DX Γ(D) ⊆ Γ(D), for any X ∈ χ(Iz M ); w w (ii) Dϕ = 0, DGR = 0; w (iii) the torsion T of D is pure, i.e. T (Z, W ) = 0, T (Z, W̄ ) = 2dθ(Z, W̄ ), ∀Z, W ∈ Γ(D′ ), and τ ◦ J + J ◦ τ = 0, with τ (X) = T (ξ, X). The Tanaka-Webster connection is pseudo-Hermitian, analogous to the Levi-Civita connection in Riemannian geometry and to the Chern connection in the Hermitian geometry. τ tensor in also called the pseudo-Hermitian torsion and measures the deviation from the normality of the almost contact structure subordonated to the indicatrix CR-structure. However, using Proposition 3.5, we have τ ≡ 0 on Iz M . By similar arguments as before, for the Tanaka connection case, we find that Webster connection is determined by the following non-zero coefficients i − Aijk = Cjk 1 1 1 1 i lj δk − lk δji and Aij k̄ = gj k̄ li − lj lk̄ li . F F F F 66 Elena Popovici Thus, we get that Webster connection action on tangent vectors of Iz M coincide with t w D action, the only exception being for Dξ ξ = F1 N . Moreover, using this observation, relations (2.6), (4.2) and θ(∇Ya Yb ) = θ(∇JYa JYb ) θ(∇JYa Yb ) = −θ(∇Ya JYb ) = θ(∇ξ Ya ) = θ(∇ξ JYa ) = 1 F 1 F Re(i gbā ), Re(gbā ), = 0, t we deduce the relation between the Levi-Civita connection ∇, Tanaka connection D w and Webster Tanaka connection D as: w t DX Y = DX Y + 1 F θ(X)θ(Y )N and w ∇X Y = DX Y + Ωθ (X, Y )ξ + θ(∇X ϕY )N + 1 F [θ(X)JY + θ(Y )JX], where Ωθ (X, Y ) = GR (X, ϕY ). At the end we analyze the Bochner type curvature tensor as being the complex analogous of the conform Weyl curvature tensor, which is a pseudoconform invariant of a CR manifold. More precisely, the Bochner tensor is the fourth curvature invariant given by S.S. Chern and J. Moser in [9]. Using the Tanaka connection, the (1,2) type Bochner curvature tensor on the CR structure of complex indicatrix (Iz M, D) is given by B(X, Y )Z = R(X, Y )Z + l(Y, Z)X − l(X, Z)Y + m(Y, Z)JX −m(X, Z)JY + GR (Y, Z)LX − GR (X, Z)LY +GR (JY, Z)M X − GR (JX, Z)M Y −2{m(X, Y )JZ + GR (JX, Y )M Z}, where l(X, Y ) m(X, Y ) 1 8n(n+1) ρGR (X, Y ), 1 + 8n(n+1) ρGR (JX, Y 1 s(X, Y ) + = − 2(n+1) 1 s(JX, Y ) = − 2(n+1) ), with s the usual Ricci tensor on Iz M , ρ = trace S, with GR (SX, Y ) = s(X, Y ), and GR (LX, Y ) = l(X, Y ), GR (M X, Y ) = m(X, Y ), for any X, Y ∈ D. Remark 4.1. The Bochner type tensor is invariant under gauge transformation. Moreover, we can easily verify that ∑ B(X, Y )Z = 0, B(JX, JY ) = B(X, Y ), B(X, Y )J = JB(X, Y ), (X,Y,Z) trace(X 7→ B(X, Y )Z) = 0. References [1] M. Abate, G. Patrizio, Finsler Metrics — A Global Approach, Springer-Verlag, Berlin 1994. Contact structures on the indicatrix of a complex Finsler space 67 [2] N. 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Hsu, On the integrability of almost contact structure, Tôhoku Math. J. 2, 14(1962), 167–176. [20] N. Tanaka, On non-degenerate real hypersurfaces, graded Lie algebras and Cartan connections, Japan. J. Math. (N.S.) 2, 1(1976), 131–190. Author’s address: Elena Popovici Transilvania University, Department of Mathematics and Informatics, Iuliu Maniu 50, Braşov 500091, Romania. E-mail: [email protected]