On the volume of the indicatrix of a complex Finsler space

Turk J Math (2017) 41: 267 – 281 Turkish Journal of Mathematics http://journals.tubitak.gov.tr/math/ c TÜBİTAK ⃝ doi:10.3906/mat-1510-66 Research Article On the volume of the indicatrix of a complex Finsler space Elena POPOVICI∗ Department of Mathematics and Informatics, Faculty of Mathematics and Informatics, Transilvania University of Braşov, Romania Received: 17.10.2015 • Accepted/Published Online: 20.04.2016 • Final Version: 03.04.2017 Abstract: Following the study on volume of indicatrices in a real Finsler space, in this paper we are investigating some volume properties of the indicatrix considered in an arbitrary fixed point of a complex Finsler manifold. Since for each point of a complex Finsler space the indicatrix is an embedded CR-hypersurface of the punctured holomorphic tangent bundle, by means of its normal vector, the volume element of the indicatrix is determined. Thus, the volume function is pointed out and its variation is studied. Conditions under which the volume is constant are also determined and some classes of complex Finsler spaces with constant indicatrix volume are given. Moreover, the length of the complex indicatrix of Riemann surfaces is found to be constant. In addition, considering submersions from the complex indicatrix onto almost Hermitian surfaces, we obtain that the volume of the submersed manifold is constant. Key words: Complex Finsler space, complex indicatrix, volume element, volume variation, indicatrix length, submersion volume 1. Introduction The study of the unit tangent sphere, or indicatrix, in real Finsler spaces is one of interest ([15, 19, 21, 22], etc.), mainly because it is a compact and strictly convex set surrounding the origin. For example, the indicatrix plays a special role in the volume definition of a Finsler space. However, in the present paper, based on some ideas from the real case, the volume element and volume function of the indicatrix in a complex Finsler manifold (M, F ) are introduced and several of their properties are obtained. First we recall some basic notions about complex Finsler geometry in Section 1. By taking z ∈ M as an arbitrary point, the punctured holomorphic tangent bundle Tz′ M can be locally viewed as a Kähler manifold and the complex indicatrix, considered in the fixed point z , can be approached as a CR-hypersurface of Tz′ M . Thus, regarding the volume element of a complex Finsler space, in Section 2 we establish the volume element of complex indicatrices. In Section 3 we will then be able to express the volume function of the complex indicatrix and obtain its variation by the z variable. Thus, some conditions under which the volume of complex indicatrices is constant (i.e. it does not depend on the choice of z ∈ M ) are given. Even so, these conditions do not offer much information about the type of the complex Finsler spaces, but for certain classes of complex Finsler manifolds, as complex Berwald manifolds, the constant volume property is verified. Furthermore, by taking the particular case of Riemann surfaces, we study the length of the indicatrix and we obtain that it has constant value, equal to the unit circle value. This contrasts sharply with the situation from two-real-dimensional Finsler ∗Correspondence: [email protected] 2010 AMS Mathematics Subject Classification: 53B40, 53C60, 53C40, 53B25. 267 POPOVICI/Turk J Math geometry, where the indicatrix length is in general a function of x . In the last section, using an example of a submersion from the indicatrix Iz M onto the complex projective bundle Pz M and the result presented in [23], we state that the volume of any Hermitian manifold, on which exists a submersion from the complex indicatrix, has constant value. Now we provide a short overview of the concepts and terminology used in complex Finsler geometry (for more see [1, 20]). 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 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 holding if and only if η = 0 ; iii. F (z, λη) = |λ|F (z, η), ∀λ ∈ C ; ( ) iv. The Hermitian matrix gij̄ (z, η) is positive definite, where gij̄ = ∂2L ∂η i ∂ η̄ j is the fundamental metric tensor, with L := F 2 the complex Lagrangian associated to the complex Finsler function F . The positivity from the fourth condition is equivalent to the convexity of L and to the strongly pseudo- convex property of the complex indicatrix in a fixed point Iz M = {η | gij̄ (z, η)η i η̄ j = 1}, for any z ∈ M . It also ensures the existence of the inverse (g j̄i ) , with g j̄i gik̄ = δk̄j̄ . Moreover, condition iii represents the fact 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: ∂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 . (1) 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 . A first step represents the analysis of the ′ sections on 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 the complex nonlinear connection, briefly (c.n.c.), is 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) are the coefficients of a (c.n.c.). Then we will call the adapted frame of the (c.n.c.) the pair {δk := δ , δz k ∂˙k := ∂ }, ∂η k with the dual adapted base {dz k , δη k := dη k + Njk dz j }. One basic (c.n.c.) of a complex Finsler space is the Chern–Finsler (c.n.c.) ([1, 20]), with Njk = lm̄ l g m̄k ∂g ∂z j η . By means of the adapted base determined by the Chern–Finsler (c.n.c) it can be determined 268 POPOVICI/Turk J Math a Hermitian connection D of (1, 0)-type, named in [1] the Chern–Finsler linear connection. In notations i ı̄ , Cȷ̄k ), locally given by the next set of coefficients Lijk = g l̄i δk (gj l̄ ), from [20] it is DΓN = (Lijk , Lı̄ȷ̄k , Cjk ı̄ i i ˙ i Cjk = g l̄i ∂˙k (gj l̄ ), Lı̄ȷ̄k = 0, Cj̄k = 0, with Dδk δj = Lijk δi , Dδk ∂˙j = Lijk ∂˙i , D∂˙k ∂˙j = Cjk ∂i , D∂˙k δj = Cjk δi , etc., i j i k and Cjk η = Cjk η = 0 from (1). Further we will use the notation η̄ j =: η j̄ to denote a conjugate object. i = 0, Kähler iff In the terminology of [1], the complex Finsler space (M, F ) is strongly Kähler iff Tjk i j i j l i Tjk η = 0, and weakly Kähler iff gil̄ Tjk η η̄ = 0, where Tjk := Lijk − Likj . However, the notions of strongly Kähler and Kähler actually coincide (cf. [14]). A complex Finsler space that is Kähler and Lijk = Lijk (z) is named a complex Berwald space [3]. An extension of this is the generalized Berwald space, which has the coefficients of a complex spray Gi = 1 N i η j holomorphic in η , i.e. ∂˙ Gi = 0 [7]. 2 h̄ j Cauchy–Riemann (CR) submanifolds of almost Hermitian manifolds, introduced by Bejancu[10–12], were extended to the Finsler geometry by Dragomir in [16, 17]. A real submanifold M̃ of an almost Hermitian Finsler space (M, g), is a CR-submanifold if it carries a pair of complementary Finslerian distributions D and D⊥ of T M̃ , such that D is invariant, J(Du ) = Du , and D⊥ is antiinvariant, J(Du⊥ ) ⊂ (Tu M̃ )⊥ , for each u ∈ M̃ , where J is an almost complex structure on M̃ . Any real hypersurface M̃ of M is a CR-submanifold, with Du⊥ = J(Tu M̃ )⊥ and D the complementary orthogonal distribution of D⊥ . 2. The volume element of the complex indicatrix Consider Tz′ M the corresponding holomorphic tangent space of a complex Finsler manifold (M, F ) 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 ′ M ) associated to F , which is usually extended in Hermitian geometry the Hermitian structure on T ′ (T] z to a complex bilinear form G by G(Z, W̄ ) = g(Z, W ), G(Z, W ) = G(Z̄, W̄ ) = 0, G(Z̄, W ) = G(Z, W̄ ), ′ ∀Z, W ∈ T ′ (Tz′ M ). Thus, G defines a Hermitian metric on T] z M , which is smooth at η = 0 if and only if Fz is a Hermitian norm and has the explicit form: G := ∂ 2 Fz2 dη j ⊗ dη̄ k = gj k̄ (z, η)dη j ⊗ dη̄ k . ∂η i ∂ η̄ j (2) For a 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 pseudoconvex 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 ∈ {1, .., n} the equations i ′ of inclusion Iz M ֒→ T] z M [17]. Then F (z, η(u)) = 1 yields by derivation after u variables to lj ∂η j ∂η j̄ + lj̄ α = 0, α ∂u ∂u α ∈ {1, . . . , 2n − 1}, j ∈ {1, . . . , n}, (3) 269 POPOVICI/Turk J Math ∂F k̄ j where lj = 2 ∂η = j = gj k̄ l , with l tangent vectors of Iz M as i∗ 1 F ′ η j . Since the inclusion tangent map i∗ : TR (Iz M ) → TC (T] z M ) acts on ( ∂ ∂uα ) = Xα := ∂η k ∂ ∂ η̄ k ∂ + , ∂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 ) , from (3), we set N = lj ∂˙j + lj̄ ∂˙ȷ̄ (4) 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 ′ ). (5) Here X ′ , Y ′ are the holomorphic, respectively, the antiholomorphic part of tangent vectors X and Y of the √ complexified tangent bundle of T ′ M , obtained by X ′ = 12 (X −iJX), Y ′ = 12 (Y +iJY ), i = −1. Consequently, N ∈ TR (Iz M )⊥ and GR (N, N ) = 1, so that N is the unit normal vector of the indicatrix bundle. Remark. A similar result of the complex indicatrix normal vector was obtained in [24] by considering Tz′ M as √ a 2n -dimensional vector space VR , via the diffeomorphism I : VR → Tz′ M , (xk , y k ) 7→ (η k ) = (xk + −1y k ). Then g induces a Riemannian metric on VR \{0} , defined by ĝv (X, Y ) := RegI(v) (I∗ (X), I∗ (Y )), ∀X, Y ∈ Tv VR , v ̸= 0, (6) where Re(τ ) = 12 (τ + τ̄ ) , for any given form τ ∈ Ap,q (M ) . Set the indicatrix IR := I −1 (Iz M ), I −1 (η) = (v k ), with (v k ) taken as v 2k−1 := xk and v 2k := y k the coordinates of VR and let X(t), t ∈ (−ϵ, ϵ) be a smooth curve on IR with X(0) = v k ∂v∂ k . Then, by taking Ẋ(0) = ∂L ∂η k dv k ∂ dt ∂v k |t=0 , a tangent vector of the curve, and using = gkj̄ η j̄ , it is obtained that ĝX(0) (X(0), Ẋ(0)) = 0. Hence, n|(vk ) = X(0) = v k ∂v∂ k is the normal vector to the indicatrix, which expressed in terms of the Iz M tangent vector field is exactly N = η j ∂η∂ j + η j̄ ∂∂η̄j if ′ η ∈ Iz M , or, equivalently, N = lj ∂η∂ j + lj̄ ∂∂η̄j for an arbitrary η ∈ T] zM . By considering the volume form of (Tz′ M, Fz ) induced by the Hermitian metric G as dµ = 1 n! (√ −1 g dη i ∧ dη̄ j 2 ij̄ )n n2 = (−1) 2 g 1 dη ∧ dη 2 ∧ .. ∧ dη n ∧ dη̄ 1 ∧ dη̄ 2 ∧ .. ∧ dη̄ n , 2n with g = det(gij̄ ), we get the volume element of the induced metric on Iz M by a contraction of dµ with the outward-pointing normal N = η j ∂˙j + η ȷ̄ ∂˙ȷ̄ (restricted to Iz M ) in the first slot [9], i.e. the interior product with respect to N . Thus, we have dVz |η = = 270 ( ) dµ η j ∂˙j + η ȷ̄ ∂˙ȷ̄ , ... = ιN (dµ) = ( n2 (−1) 2 g ∑n j−1 j dj ∧ .. ∧ dη n ∧ dη̄ 1 ∧ .. ∧ dη̄ n + η dη 1 ∧ .. ∧ dη n j=1 (−1) 2 ) ∑n dj ∧ .. ∧ dη̄ n , + j=1 (−1)n+j−1 η ȷ̄ dη 1 ∧ .. ∧ dη n ∧ dη̄ 1 ∧ .. ∧ dη̄ POPOVICI/Turk J Math dj we denote absence of the dη j term in the exterior product where ι is the interior product operator and by dη above. Using that τ + τ̄ = 2Re(τ ) for any given form τ , we obtain: Proposition 1 For each η ∈ Iz M , the volume element of the complex indicatrix is dVz |η = g 2n−1 Re[(−1) n2 2 n ∑ i=1 ci ∧ ... ∧ dη n ∧ dη̄ 1 ∧ ... ∧ dη̄ n ]. (−1)i−1 η i dη 1 ∧ ... ∧ dη ′ Considering that Iz M is the unit sphere in (Tz′ M, Fz ), we take the map r : T] z M → Iz M , r(η) = η Fz (η) ∗ j ′ and define a form Θz on T] z M by Θz := r dVz . Then, without any constraint on the η s, the normal vector ′ is N = lj ∂˙j + lȷ̄ ∂˙ȷ̄ , and for any η ∈ T] z M the 2n − 1 form Θz has the expression Θz |η = n ∑ n2 g ci ∧ .. ∧ dη n ∧ dη̄ 1 ∧ .. ∧ dη̄ n ], 2 Re[(−1) (−1)i−1 η i dη 1 ∧ .. ∧ dη 2n−1 F 2n i=1 (7) ′ for any η ∈ T] z M . The Θz form has an important role in calculating the local degree of a holomorphic vector field at a nondegenerate zero. We also notice that the same form of volume element dVz and the Θz form were obtained in [24], Proposition 3.1 and Proposition 3.3, respectively, by taking dVz := I −1∗ [ι(n)dµĝ ] , where dµĝ is the Riemannian volume form induced by the Riemannian metric ĝ from (6). 3. Some properties of the indicatrices volume In this section we study the volume of unit tangent spheres Iz M in complex Finsler manifolds (M, F ), using the volume form obtained in the previous section in Proposition 1. Thus, we obtain conditions for a space to have indicatrix of constant volume, i.e. independent of the position variables z ∈ M , and we study this property for several cases of complex Finsler spaces. The volume function of the indicatrix Iz M is considered to be V ol(z) := V ol(Iz M ) = ∫ dVz . (8) Iz M This definition is natural since the indicatrices volume of Hermitian manifolds is always equal to the volume of unit Euclidean sphere S2n−1 in R2n ≡ Cn , as we can see from the following example. Example 1. 2 The Euclidean norm, F |z = ∥·∥ , i.e. ∥η∥ := |η|2 = ∑ η i η̄ i , on the complex Minkowski space (Tz′ Cn , F |z ) considered in an arbitrary point z ∈ Cn . Since the volume element dV is independent of the choices of (η i ) coordinates and the Hermitian structure satisfies g(∂˙i , ∂˙j ) = δij , so that g = 1, and by applying (8) and Proposition 1, we have ∫ √ 2 n n −1 ∑ ci ∧ .. ∧ dη n ∧ dη̄ 1 ∧ .. ∧ dη̄ n ). Re( n−1 (−1)i−1 η i dη 1 ∧ .. ∧ dη V ol(z) = 2 ∥η∥=1 i=1 271 POPOVICI/Turk J Math By taking each η k = v 2k−1 + ∑2n i=1 (−1) √ 2 −1v 2k , we obtain ∥η∥ = 2n ∑ (v k )2 and the integrand volume form ω = k=1 ci ∧ ... ∧ dv 2n , which is exactly the volume element of a (2n − 1) -sphere with radius v dv 1 ∧ ... ∧ dv i−1 i R = 1. Thus, we obtain that V ol(z) = V ol(S2n−1 ) . As we could expect, indicatrices of complex Finsler manifolds have different properties from those considered in real Finsler manifolds. For instance, if for real Finsler manifolds the volume of the indicatrix can never exceed V ol(Sn−1 ) if the real Finsler metric is positively homogeneous of degree 1, with equality for the case of the Euclidean norm (cf. [8], Proposition 14.9.1), in the complex Finsler case this result cannot be generalized, as we can see from the next example. Example 2. Take the complex version of the Antonelli–Shimada metric, 1 2 FAS = LAS (z, w; p, q) := e2σ (|p|4 + |q|4 ) 2 , with p, q ̸= 0, ′ M , dim M = 2, such that its metric tensor is nondegenerated. Here, z, w, p, q represent on a domain D from T] C a relabel of the local coordinates z 1 , z 2 , η 1 , η 2 , respectively; the exponent σ(z, w) is a real valued function; and |η k |2 := η k η̄ k , η k ∈ {p, q}. A direct computation leads to (cf. [7]) g = det(gj k̄ ) = 2e4σ(z,w) |p|2 |q|2 |p|4 + |q|4 and by taking n = 2 in Proposition 1, the volume element of the complex indicatrix becomes dVz = 1 g(pdq ∧ dp̄ ∧ dq̄ − qdp ∧ dp̄ ∧ dq̄ + p̄dq̄ ∧ dp ∧ dq − q̄dp̄ ∧ dp ∧ dq). 4 If we consider the algebraic form of the complex coordinates p = x + iy, q = u + iv , i = volume function V ol(z) = ∫ Σ √ −1, we obtain the g(xdy ∧ du ∧ dv − ydx ∧ du ∧ dv + udx ∧ dy ∧ dv − vdx ∧ dy ∧ dv), (9) where g = 2e8σ(z,w) (x2 + y 2 )(u2 + v 2 ) and the indicatrix condition F (z, η) = 1 becomes the equation of the surface Σ : (x2 + y 2 )2 + (u2 + v 2 )2 = e−4σ(z,w) . If we denote r = e−σ(z,w) , the coordinates of Σ can be changed √ √ √ √ using a type of Hopf coordinates: x = r cos α sin θ, y = r sin α sin θ, u = r cos β cos θ, v = r sin β cos θ, where θ ∈ [0, π2 ], α, β ∈ [0, 2π] . Thus, we easily obtain that V ol(z) ≡ V ol(S3 ) . However, in the general case V ol(z) does vary with z , as can be verified by the following example. Example 3. Consider the case n = 2, M = C2 , of local coordinates z = (z 1 , z 2 ) and η = (p, q), and we define √ √ F (z, η) := λ(z)(|p|2 + |q|2 ) + |p|4 + |q|4 , where λ is an arbitrary smooth nonnegative function of the z variables only and |η k |2 := η k η̄ k , η k ∈ {p, q} . When λ ≡ constant the Minkowskian structure is obtained. It can be checked that L = F 2 is actually a smooth 272 POPOVICI/Turk J Math ′ M and F is a complex Finsler function. Calculations give function on T]  (gij̄ ) =  λ+ |p|2 (|p|4 +2|q|4 ) (|p|4 +|q|4 )3/2 2 2 |q| − (|p|pq̄|p| 4 +|q|4 )3/2 2 2 |q| − (|p|p̄q|p| 4 +|q|4 )3/2 λ+ |q|2 (2|p|4 +|q|4 ) (|p|4 +|q|4 )3/2  . Hence, 2|p|2 |q|2 (|p|2 + |q|2 )3 − |p|2 |q|2 (|p|2 + |q|2 ) + . |p|4 + |q|4 (|p|4 + |q|4 )3/2 √ If we consider the algebraic form p = x + iy, q = u + iv , i = −1, we obtain volume function (9), with √ Σ : λ(x2 + y 2 + u2 + v 2 ) + (x2 + y 2 )2 + (u2 + v 2 )2 = 1. By changing the coordinates of Σ as g = det(gij̄ ) = λ2 + λ x= y= √ √ cos θ 1+λ(cos θ+sin θ) sin α, u= cos θ 1+λ(cos θ+sin θ) cos α, v= √ √ sin θ 1+λ(cos θ+sin θ) cos β, sin θ 1+λ(cos θ+sin θ) sin β, with θ ∈ [0, π2 ], α, β ∈ [0, 2π] , the volume function of the complex indicatrix will have the form ∫π/2 dθ sin 2θ . V ol(z) = 2π ) + sin 2θ] [λ2 + λ(cos θ + sin θ)(1 + 2 [1 + λ(cos θ + sin θ)]2 2 0 It can be seen that the result depends on λ(z) and thus V ol(z) is nonconstant. In the following, using the ideas from [9], we study the condition under which a complex Finsler space has indicatrix with constant volume. Thus, we take η 7→ σ(η) a Minkowski norm on Cn . Since the restriction of F to Tz′ M gives a Minkowski norm Fz , any Finsler manifold may be viewed as a smoothly varying family of Minkowski spaces {(Tz′ M, Fz )} . Thus, if we assign the same σ to each tangent space of Cn , identifiable with Cn itself, every Minkowski space (Cn , σ) is a Finsler manifold and the resulting Finsler manifold is an example of locally Minkowski space. In each z ∈ M the indicatrix of F is strongly convex and σ : Cn ≃ Tz′ M → R, σ(η) = F (z, η), is a strictly convex function, cf. [1], p. 167. The embedding φz : I → Iz M, φz (η 1 , ..., η n ) := η i ∂z∂ i z , with I = {η ∈ Cn : σ(η) = 1} the indicatrix of the Minkowski space (Cn , σ) and image just Iz M , allow us to switch the integrating domain in the volume function (8 ) over to I . Thus, V ol(z) = ∫ Iz M dVz = ∫ φ∗z dVz . I Using Proposition 1 and the tangential map φ∗z , we can express V ol(z) as V ol(z) = 1 2n−1 ∫ I gRe[(−1) n2 2 n ∑ i=1 ci ∧ .. ∧ dη n ∧ dη̄ 1 ∧ .. ∧ dη̄ n ]. (−1)i−1 η i dη 1 ∧ .. ∧ dη (10) Therefore, the volume of the indicatrix is independent of z or z̄ variables if the variational formula for V ol(z) vanishes: ∂V ol(z) ∂V ol(z) br (z, z̄) = 0 and br (z, z̄) =0 (11) ∂z r ∂ z̄ r 273 POPOVICI/Turk J Math for any vector fields br ∂z∂ r and br ∂∂z̄r on M , where the br = br (z, z̄)s are functions of z and z̄ only on M . Since we work in local coordinates (z i ) and their induced global coordinates (η i ) on each tangent space Tz′ M and the domain of integration is thus I , which does not vary with z or z̄ , using (10) the conditions from (11) become ∂g ∂g br (z, z̄) r = 0 and br (z, z̄) r = 0. ∂z ∂ z̄ ( ) By applying the Jacobi formula for the determinant of the positive definite Hermitian matrix gij̄ (z, η) , the above equalities are rewritten as br (z, z̄)gΓiir = 0 and br (z, z̄)gΓiir̄ = 0, i m̄i ∂gim̄ r i r i im̄ with Γiir = g m̄i ∂g ∂z r and Γir̄ = g ∂ z̄ r . Since g ̸= 0, V ol(z) = const. if and only if b (z, z̄)Γir = b (z, z̄)Γir̄ = 0. Moreover, we can state: Theorem 1 Let (M, F ) be a complex Finsler manifold, z ∈ M an arbitrary point, and V ol(z) the volume function of the complex indicatrix Iz M . Then, for any vector fields br ∂z∂ r and br ∂∂z̄ r on M , with br = br (z, z̄) , we have = br (z, z̄) ∂V∂zol(z) r 1 2n−1 ∫ gbr Γiir Re[(−1) n2 2 I ∑n i=1 ci (−1)i−1 η i dη 1 ∧ ... ∧ dη ∧... ∧ dη n ∧ dη̄ 1 ∧ ... ∧ dη̄ n ] and, respectively, br (z, z̄) ∂V∂ol(z) = z̄ r 1 2n−1 ∫ gbr Γiir̄ Re[(−1) I n2 2 ∑n i=1 ci (−1)i−1 η i dη 1 ∧ ... ∧ dη ∧... ∧ dη n ∧ dη̄ 1 ∧ ... ∧ dη̄ n ] i m̄i ∂gim̄ im̄ where Γiir = g m̄i ∂g ∂z r and Γir̄ = g ∂ z̄ r . Corollary 1 Let (M, F ) be a complex Finsler manifold. If Γiir = Γiir̄ = 0 , the volume function V ol(z) of the complex indicatrix is constant. Remark. Any local Minkowski metric verifies the corollary’s conditions; thus, V ol(z) is constant (see Example 1 and Example 2 with σ a constant function). However, in Example 2, V ol(z) is constant for any function σ(z, w) and thus there are other types of complex Finsler metrics with constant V ol(z). For instance, further we will provide an example of a pure Hermitian metric (i.e. g = g(z)) for which we obtain V ol(z) ≡ V ol(S 2n−1 ) . Example 4. A pure Hermitian metric. Take M = C2 endowed with a purely Hermitian metric F 2 (z, η) = a(z, w)|p|2 + b(z, w)|q|2 , with a, b real valued functions and local coordinates z = (z, w), η = (p, q). It can be easily verified that g11̄ = a(z, w) , g12̄ = g21̄ = 0, g22̄ = b(z, w) , such that F represents a pure Hermitian metric, and thus g = ab . Considering the algebraic form p = x + iy, q = u + iv , we obtain the volume (9) with the surface Σ : a(x2 + y 2 ) + b(u2 + v 2 ) = 1. Taking x = a−1/2 cos α sin θ , y = a−1/2 sin α sin θ , u = b−1/2 cos β cos θ , v = 274 POPOVICI/Turk J Math b−1/2 sin β cos θ, where θ ∈ [0, π2 ], α, β ∈ [0, 2π], we easily obtain V ol(z) ≡ V ol(S3 ) . The same result is obtained if we choose to use the spherical coordinates x = a−1/2 cos α , y = a−1/2 sin α cos β , u = b−1/2 sin α sin β cos θ , v = b−1/2 sin α sin β sin θ, with θ ∈ [0, π], α, β ∈ [0, 2π] . By generalizing to an n-dimensional complex manifold M , take the purely Hermitian metric F 2 (z, η) = n ∏n ∑ ai (z)|η i |2 , with ai (z)s real valued functions and g = i=1 ai (z) . Using η j = xj + iy j , condition F = 1 i=1 is equivalent to −1/2 a1 ∑n i=1 −1/2 cos α1 , y 1 = a1 .., xn = −1/2 an ai (xi + y i ) = 1. If we apply an adapted form of the spherical coordinates as x1 = −1/2 sin α1 cos α2 , x2 = a2 −1/2 sin α1 sin α2 cos α3 , y 2 = a2 sin α1 .. sin α2n−2 cos α2n−1 and y n = α1 , .., α2n−2 ∈ [0, 2π] , we obtain that V ol(z) = V ol(S −1/2 an 2n−1 ). sin α1 sin α2 sin α3 cos α4 , sin α1 .. sin α2n−2 sin α2n−1 , with α2n−1 ∈ [0, π] , However, the variational formula of V ol(z) does not offer much information about classes of complex Finsler manifolds that may verify the constant volume conditions. The following result slightly improves the one from [24]. Proposition 2 Let (M, F ) be a weakly Kähler manifold, with Lijk = Lijk (z) . Then V ol(z) is constant. Proof Since (M, F ) is a weakly Kähler manifold, according to [1], p. 101, Corollary 2.4.2, for any p ∈ M ′ and Xp ∈ T] p M , there exists a unique geodesic γ through (p, Xp ), such that γ(0) = p and γ̇(0) = Xp ∈ Ip M . Thus, given p and q in M , we can take γ(t), t ∈ [0, 1] as a unique geodesic such that γ(0) = p and γ(1) = q , which for the weakly Kähler metric and its Chern–Finsler connection D satisfies ([1], p. 101, [20], p. 79) Dγ̇ h +γ̇ h γ̇ h = 0, with γ̇ the tangent map of γ and γ̇ h = lh (γ̇(t)) the horizontal lift of the vector γ̇(t). In local coordinates it becomes [γ̈ i + γ̇ j Lijk (γ, γ̇)γ̇ k ]δi = 0. (12) ′ M, γ̇(0) 7→ γ̇(t), be the parallel translation induced by (12). By taking the Chern–Finsler Let Pt : Tp′ M → Tγ(t) horizontal coefficients Lijk to be dependent only on the position z , the parallel translation Pt is a complex linear isomorphism and an isometry, so we have ||γ̇(t)|| = F (γ(t), γ̇(t)) = F (p, γ̇(0)) = ||γ̇(0)|| . ′ M , respectively. In a local chart consider γ̇(0) = Set (z i , η i ), (ξ i , ζ i ) local coordinates on Tp′ M , Tγ(t) dγ i ∂ dt (0) ∂z i |γ(0) and denote X0i := dγ i dt (0) and by {ei }i=1,n the natural basis { ∂z∂ i }i=1,n in the fixed point p . If ′ we take Ei (t) = Pt (ei ) to be the translated basis for Tγ(t) M , there exists a nonsingular matrix (Aji (t)) such that Ei (t) = Aji (t) ∂ξ∂ j . If we consider γ̇(t) = X i (t) ∂z∂ i |γ(t) = X0i Ei (t) along γ , where X i (t) = condition from (12) becomes dX dt i dγ i dt (t), the + X j Lijk (ξ, X0i Aji (t) ∂ξ∂ j )X k = 0. We observe that, in general, the coefficients Lijk are dependent on the translated frame, but in the considered case, Lijk depending only on position, that does not happen. Thus, by applying it to Em (t) , we obtain dAim = −Ajm Akm Lijk (ξ). dt 275 POPOVICI/Turk J Math Now we follow the same steps as in [24]. Since ζ i = Aij η j , the tangent and cotangent application of Pt satisfies Pt∗ ∂ ∂ = Aji j , ∂η i ∂ζ Pt∗ dζ i = Aij dη j . Relative to the Hermitian metric G from (2), using the last three relations, it can be easily checked that [ ( )] ] d ∂ ∂ d [ GPt (η) Pt∗ l , Pt∗ m̄ = gij̄ (γ(t), γ̇(t))Ail Aj̄m̄ = 0. dt ∂η ∂η dt This implies GPt (η) (Pt∗ (∂˙i ), Pt∗ (∂˙ȷ̄ )) = Gη (∂˙i , ∂˙ȷ̄ ) , det[Gγ̇(t) (Pt∗ (∂˙i ), Pt∗ (∂˙ȷ̄ ))] = det[Gη (∂˙i , ∂˙ȷ̄ )]. ′ Take Bγ(t) (r) := {ζ ∈ Tγ(t) M : F (γ(t), ζ) < r} , Bγ(t) (r) = Bγ(0) (r) , and using Stokes’ formula, the above results, and the fact that d(dVz ) = 2ndµ|z for any z ∈ M , we get V ol(γ(t)) = V ol(p). If t = 1, we obtain that V ol(p) = V ol(q) , for any p, q ∈ M . ✷ Recall that, cf. [3], a complex Finsler space is Berwald iff it is Kähler and Lijk depends only on position. Since the Kähler condition implies weakly Kähler, we can state: Corollary 2 Let (M, F ) be a complex Berwald manifold. Then V ol(z) is constant. Remark. There are two senses regarding the definition of a complex Berwald space. The first definition of the complex Berwald manifold was introduced by Aikou in [2], as a complex Finsler manifold whose Chern–Finsler coefficients Lijk are functions of position (z i ) alone. However, in a later paper, [3], Aikou approached this problem again and considered the complex Finsler spaces having Lijk (z) as being modeled on a Minkowski space, and by a complex Berwald manifold, he defined a complex Finsler manifold modeled on a Minkowski space, together with the Kähler condition of its associated Hermitian metric (which is equivalent to the Kähler condition of the complex Finsler function). The latter one is the sense that we are using in this paper, while the first one leads to the generalized Berwald spaces according to [4, 5]. We remark that conditions from Proposition 2 are only sufficient. Next, we want to exemplify this. First, if we take into consideration that in Example 2 according to [7] the Lijk nonzero coefficients of the Chern–Finsler connection of the Antonelli–Shimada complex metric are L111 = L221 = 2 ∂σ , ∂z L112 = L222 = 2 ∂σ , ∂w we observe that the local coefficients Lijk depend only on z and w , but the metric LAS is not Berwald, because in general, it is not Kähler. However, the metric LAS is one generalized Berwald, since the spray coefficients ∂σ ∂σ ∂σ w Gz = ( ∂σ ∂z η + ∂w θ)η and G = ( ∂z η + ∂w θ)θ are holomorphic in η . Thus, we found an example of a generalized Berwald manifold for which V ol(z) is constant, another one being given in the following example. Example 5. A complex Finsler space with Randers metric. We adjust the example from [6] and we consider a two-dimensional manifold M = C2 a purely Hermitian metric α2 = ez+z̄ |p|2 + ez+z̄+w+w̄ |q|2 , 276 (13) POPOVICI/Turk J Math and we choose the (1, 0) -differential form β as β = ez p. (14) Then a11̄ = ez+z̄ , a22̄ = ez+z̄+w+w̄ , a12̄ = a21̄ = 0, det(aij̄ ) = e2(z+z̄)+w+w̄ , a1̄1 = e−(z+z̄) , a2̄2 = e−(z+z̄+w+w̄) , a1̄2 = a2̄1 = 0, |β|2 = ez+z̄ |p|2 and thus b1 = ez , b1 = e−z , b2 = b2 = 0, and ||b|| = 1, where we used bi = aj̄i bj̄ and ||b||2 = aj̄i bi bj̄ . In addition, we calculate Γlr̄m̄ = Γlr̄m̄ = 0, except for the coefficients Γ21̄2̄ = −Γ22̄1̄ = e z+z̄+w+w̄ ∂alm̄ ∂ z̄ r − ∂alr̄ ∂ z̄ m and we obtain ̸= 0. Thus, metric (13) is not Kähler. Now we are able to construct the complex Randers metric F = √ √ ez+z̄ |p|2 + ez+z̄+w+w̄ |q|2 + ez+z̄ |p|2 , (15) which is nonpurely Hermitian, since α2 ||b||2 ̸= |β|2 . Some computations lead to the conclusion that metric (15) satisfies the condition ) ( ( ) ∂br̄ ∂br̄ r ∂b1̄ 1 ∂b1̄ j β̄lr̄ j + β j η̄ η = β̄l1̄ j + β j η̄ η j = 0, ∂z ∂z ∂z ∂z with lr̄ = air̄ η i , and so, from [4], Theorem 4.2, this metric is generalized Berwald. However, there exists Γ22̄1̄ b1̄ = −ez+w+w̄ ̸= 0, and thus the condition Γlr̄m̄ bm̄ = 0 cannot be fulfilled, and according to [6] it is not a complex Douglas metric, even if Ωm̄ = 0, m = 1, 2. By direct calculus, using a11̄ |p|2 = |β|2 and a22̄ |q|2 = α2 − |β|2 , we obtain a11̄ a a F4 √11̄ + 2F 2 121̄ , 2 α2 α2 |β| α g11̄ = F2√ g12̄ = g21̄ = g22̄ = 1 a22̄ |q|2 |β| a √ , F √22̄ − 2 α2 α2 α2 α2 |β| − F a11̄ a22̄ p̄q 2 √ (α − |β|), 2 α2 α2 |β| and g = det(gij̄ ) = F 4 det(aij̄ ) √ > 0, 2 α2 α2 |β| i, j = 1, 2. We obtain the volume function V ol(z) as in (9) for the algebraic form of the complex coordinates √ √ √ p = x + iy, q = u + iv , i = −1 and surface Σ : a11̄ (x2 + y 2 ) + a22̄ (u2 + v 2 ) + a11̄ (x2 + y 2 ) = 1. If we denote a11̄ (x2 + y 2 ) + a22̄ (u2 + v 2 ) = cos4 θ and a11̄ (x2 + y 2 ) = sin4 θ , θ ∈ [0, π4 ] , the coordinates of Σ can be √ √ −1/2 −1/2 −1/2 −1/2 changed as x = a11̄ cos α sin2 θ, y = a11̄ sin α sin2 θ, u = a22̄ cos β cos 2θ, v = a22̄ sin β cos 2θ, where θ ∈ [0, π4 ], α, β ∈ [0, 2π] . Thus, we easily obtain g = a11̄ a22̄ 1 2 cos6 θ sin2 θ and by direct calculus we get that the volume function has constant value V ol(z) = 2π 2 . Remark. In [6] it can be found that any complex Randers–Douglas space of dimension two is a complex Berwald space, and thus, according to Corollary 2, we can state: 277 POPOVICI/Turk J Math Corollary 3 Let (M, F ) be a complex Randers–Douglas space with dimC M = 2 . Then the volume function V ol(z) of its complex indicatrix is constant. Application: The length of the complex indicatrix A Riemann surface is a connected one-complex dimensional analytic manifold, i.e. a two-real-dimensional connected manifold M , with a complex structure on it. Take the local chart (U, z) , the holomorphic tangent ∂ space determined by (z, η), where the η coordinate comes from z through η ∂z , and a Finsler complex function ∂2L ∂η∂ η̄ F with g = . By the (0, 0)-homogeneity condition (1) of the metric tensor g , it follows that for the Riemannian surface (M, F ) the complex Finsler metric is Hermitian. Then we consider the indicatrix as a one-real-dimensional curve of the one-complex-dimensional Finsler space, I = {η ∈ T ′ M : gη η̄ = 1} , and we introduce the notion of the indicatrix length as the volume function V ol(z) from (8) with n = 1 in Proposition 1. The Hermitian metric of the Minkowski plane Tz′ M is G = gdη ⊗ dη̄ , which induces G = ReG on I , N = 1 F ∂ η ∂η + 1 F η̄ ∂∂η̄ , and the indicatrix volume form dV = i 2F 2 g(ηdη̄ of arc length on the standard unit circle S = {η ∈ C : |η|2 = 1}, ds = that L= ∫ i g(ηdη̄ − η̄dη) = 2F 2 ∫ − η̄dη). Using it, we define the element g F 2 Re(iηdη̄) and it can be easily verified g (xdy − ydx) = 2π, F2 x2 +y 2 =1 S √ where we used the algebraic form of the complex coordinate η = x + iy , i = −1. ∫ Then the length of the indicatrix I is L(z) = 2Fi 2 g(ηdη̄ − η̄dη) . Since from (1) we have F 2 = gη η̄ , the I length formula becomes L(z) = ∫ i (ηdη̄ − η̄dη). 2|η|2 (16) F =1 As dV |η = i 2|η|2 (ηdη̄ − η̄dη) is constant along any complex ray that emanates from the origin, i.e. it is invariant under rescaling in η , dV |λη = dV |η , ∀λ ∈ C , we expect its integrals over I and S to give the same answer. Our intuition is borne out by the fact that the new integrand is a closed 1-form on C∗ and, together with an application of Stokes’ theorem, by integrating dV over I and S the same answer of the length function is obtained. By direct calculus, if we take η = x + iy , the formula of the indicatrix length becomes L(z) = ∫ 1 (xdy − ydx), (x2 + y 2 ) g(z,x,y)(x2 +y 2 )=1 and if x = √ cos θ g(z,x,y) , y = √ sin θ g(z,x,y) , θ ∈ [0, 2π) , we get L(z) = 2π , as deduced above. Thus, we can state: Proposition 3 The length of the complex indicatrix of any Riemannian Finsler surface has constant value equal to 2π . 278 POPOVICI/Turk J Math 4. On the volume of submersed manifolds of Iz M In [18] we find that a submersion from a CR-submanifold M̃ of a Kähler manifold M onto an almost Hermitian manifold M ′ is a Riemannian submersion ϕ : M̃ → M ′ with: (i) D⊥ is the kernel of ϕ∗ ; (ii) ϕ∗ : Du → Tϕ(u) M ′ is complex isometry for every u ∈ M̃ . This definition is given for the case (T M̃ )⊥ = J(D⊥ ). Since Iz M is a CR-hypersurface of the Kähler manifold Tz′ M , from [12], we deduce D⊥ = span{JN } √ on Iz M , with N the unit normal vector field from (4), J(∂˙k ) = i∂˙k , J(∂˙k̄ ) = −i∂˙k̄ , i := −1 the complex structure defined on Tz′ M , and D the maximal J -invariant subspace, such that TR (Iz M ) = D ⊕ D⊥ . Then we take the tangent unit vector ( ) √ ξ = JN = i lk ∂˙k − lk̄ ∂˙k̄ , i := −1. Consider then the submersion ϕ′ : Iz M → M ′ onto an almost Hermitian manifold M ′ , named submersed manifold, such that ϕ′∗ : TR (Iz M ) → TR M ′ fulfills ϕ∗ (ξ) = 0 and ϕ∗ : Dη → Tϕ(η) M ′ is a complex isometry for every η ∈ Iz M . ( ) Considering that F = 1 on the complex indicatrix, ξ = i η k ∂˙k − η k̄ ∂˙k̄ is the outward-pointing normal to any submersed manifold M ′ of Iz M , with dimR M ′ = 2n − 2, and thus the volume element dω of the induced metric on M ′ is given by the contraction of the indicatrix volume form dVz with ξ in the first slot dVz (N, ...) , as ( ( ) ) dω = dVz i η k ∂˙k − η k̄ ∂˙k̄ , ... = ιξ (dVz ) = g 2n−1 Re[(−1) + (−1) n ∑ n2 +1 2 dk ∧ .. ∧ dη dj ∧ .. ∧ dη n ∧ dη̄ 1 ∧ .. ∧ dη̄ n (−1)j+k η j η k dη 1 ∧ .. ∧ dη j,k=1 k̸=j (n+1)2 2 n ∑ dk ∧ .. ∧ dη̄ n ]. dj ∧ .. ∧ dη n ∧ dη̄ 1 ∧ .. ∧ dη̄ (−1)j+k η j η̄ k dη 1 ∧ .. ∧ dη j,k=1 Since the volume element is quite complicated, it is difficult to calculate its volume. However, we can construct ϕ : Iz M → Pz M , ϕ(η/F ) := [η] a submersion onto the punctured complex projective bundle Pz M , with the tangential map ϕ∗ : TR (Iz M ) → TR (Pz M ), where TR (Iz M ) = span{vα }, vα = 1 F vα 7→ eα := Re{(vαj √ i 2 j ˙ η )∂j }, − 2 (vαj ∂˙j + vαȷ̄ ∂˙ȷ̄ ), with vαj , vαȷ̄ 1-homogeneous functions in η variables, with α ∈ {1, .., 2n − 1} , which verify GR (vα , vβ ) = 1 j k̄ L Re{vα vβ gj k̄ } . j We take ξ = v2n−1 , which implies v2n−1 = iη j , vαj ηj = 0, v ȷ̄ ηȷ̄ = 0, and ϕ∗ (ξ) = 0. Since the Sasaki type lift of the metric tensor gij̄ on Tz′ M G = gij̄ dz i ⊗ dz̄ k + gij̄ δη i ⊗ δ η̄ j descends on the complex projective space P M = T ′ M/C∗ to the metric [13] G̃ = gij̄ dz i ⊗ dz̄ j + (log L)ij̄ δη i ⊗ δ η̄ j , 279 POPOVICI/Turk J Math we can state that the Hermitian metric on Tz′ M given by G descends to the metric h = (log L)j k̄ dη j ⊗ dη̄ k on Pz M , which is equivalent to 1 1 h = ( gj k̄ − 2 ηj ηk̄ )dη j ⊗ dη̄ k . L L Therefore, we obtain h(ea , eb ) = i.e. ϕ∗ is an isometry. 1 j k̄ L Re{vα vβ gj k̄ } , so that h(ϕ∗ (va ), ϕ∗ (vb )) = GR (va , vb ) , ∀a, b ∈ {1, .., 2n − 2} , Thus, between the tangent bundle of any submersed manifold M ′ of Iz M and the tangent bundle of Pz M exists the isometry ϕ′∗ ◦ ϕ−1 ∗ . Considering that according to [23] the volume function of the projectivized tangent bundle Pz M is constant, we state: Proposition 4 Let (M, F ) be a complex Finsler manifold with Iz M the complex indicatrix. The volume function of any almost Hermitian submersed manifold from Iz M has constant value. References [1] Abate M, Patrizio G. Finsler Metrics — A Global Approach. Berlin, Germany: Springer-Verlag, 1994. [2] Aikou T. On complex Finsler manifolds. Rep Fac Sci Kagoshima Univ Math Phys Chem 1991; 24: 9-25. [3] Aikou T. Some remarks on locally conformal complex Berwald spaces. In: Finsler Geometry (Seattle, WA, 1995). 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