Monica PURCARU 1. Communicated to: Finsler Extensions of Relativity Theory, August 29 - September 4, 2011, Braşov, Romania

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1 Bulletin of the Transilvania University of Braşov Vol 4(53), No Series III: Mathematics, Informatics, Physics, ON RCOMPLEX FINSLER SPACES WITH KROPINA METRIC Monica PURCARU Communicated to: Finsler Extensions of Relativity Theory, August 29 - September 4, 20, Braşov, Romania Abstract In the present paper the notion of Rcomplex Finsler space with Kropina metric is defined. The fundamental tensor fields g ij and g i j are determined and the determinant and the inverse of these tensor fields are given. Also some properties of these spaces are studied. A special aproach is dedicated to the non-hermitian Rcomplex Finsler space with Kropina metric. Some examples of Hermitian and non-hermitian Rcomplex Finsler space with Kropina metric are given. 200 Mathematics Subject Classification: 53B40, 53C56. Key words: R - complex Finsler space, Rcomplex Finsler space with (α, β)-metric, Rcomplex Finsler space with Kropina metric. R - complex Finsler spaces In a previous paper [4], we extended the known definition of a complex Finsler space ([, 2, 3, 6]), reducing the scalars to λ R. The outcome was a new class of Finsler space called by us the R- complex Finsler spaces [4]. Our interest in this class of Finsler spaces issues from the fact that the Finsler geometry means, first of all, distance and this refers to curves depending on the real parameter. In the present paper, following the ideas from real Finsler spaces with Kropina metrics ([6, 6, 0,, 2]), we introduce the similar notions on R complex Finsler spaces. In this section we keep the general setting from [3, 4] and subsequently we recall only some needed notions. Let M be a complex manifold with dim C M = n, ( z k) be local complex coordinates in a chart (U, ϕ) and T M its holomorphic tangent bundle. It has a natural structure of complex manifold, dim C T M = 2n and the induced coordinates in a local chart on u T M are denoted by u = ( z k, η k). The changes of local coordinates in u are given by the rules z k = z k (z) ; η k = z k z j ηj. (.) Faculty of Mathematics and Informatics, Transilvania University of Braşov, Romania, mpurcaru@unitbv.ro

2 80 Monica Purcaru The natural frame { z k, η k } z k + 2 z j z j of T u (T M) transforms with the Jacobi matrix of (.) changes, = z j ; = z j z k η j z k A complex nonlinear connection, briefly (c.n.c.), is a supplementary distribution H (T M) to the vertical distribution V (T M) in T (T M). The vertical distribution is { η k } z k z h η h η k η j. spanned by and an adapted frame in H (T δ M) is = N j δz k z k k, where N j n j k are the coefficients of the (c.n.c.) and they have a certain rule of change at (.) so that δ transform like vectors on the base manifold M(d-tensor in [4] terminology). Next we δzk use the abbreviations: k = z k, δ k = δ δz k, k = η k and k, k, δ k for their conjugates. The dual adapted basis of {δ k, } } k are {dz k, δη k = dη k + Nj kdzj and { d z k, δ η k} theirs conjugates. We recall that the homogeneity of the metric function of a complex Finsler space ([, 2, 3, 6]) is with respect to all complex scalars and the metric tensor of the space, is a Hermititian one. In [4] we slightly changed the definition of a complex Finsler space. An R complex Finsler metric on M is a continuous function F : T M R + satisfying: i) L := F 2 is smooth on T M (except the 0 sections); ii) F (z, η) 0, the equality holds if and only if η = 0; iii) F (z, λη, z, λ η) = λ F (z, η, z, η), λ R; It follows that L is (2, 0) homogeneous with respect to the real scalars λ, and in [4] we proved that the following identities are fulfilled: L η i ηi + L η i ηi = 2L; g ik η j ηj + g ik η j ηj = 0; g ij := are the metric tensors of the space. g ij η i + g ji η i = L η j ; (.2) g i k η j ηj + g i k η j ηj = 0. 2L = g ij η i η j + gī j η i η j + 2g i jη i η j ; 2 L η i η j ; g i j := 2 L η i η j ; gī j := 2 L η i η j (.3) 2 Rcomplex Finsler spaces with Kropina metric As noticed in paper [4] an R complex Finsler space produce two tensor fields g ij and g i j. For a properly Hermitian geometry g i j be invertible is a compulsory requirement, but from some physicist point of view, for which Hermitian condition is an impediment, it seems more appropriate that g ij be an invertible metric tensor. These problems led us to in [4] to speak about Hermitian R complex Finsler spaces (i.e. det ( g i j) 0) and non - Hermitian R complex Finsler spaces ( i.e. det (g ij ) 0). The present section applies our

3 On RComplex Finsler spaces with Kropina metric 8 results to R complex Finsler spaces with Kropina metric, better illustrating the interest for this work. As in [2] we have: Definition 2.. An Rcomplex Finsler space (M, F ) is called with (α, β) metric if the fundamental function F (z, η, z, η) is R homogeneous by means of functions α(z, η, z, η) and β(z, η, z, η) depend by z i, η i, z i and η i, (i =,..., n) by means of α (z, η, z, η) and β (z, η, z, η), i.e. : F (z, η, z, η) = F (α (z, η, z, η), β (z, η, z, η)) (2.) where α 2 (z, η, z, η) = ( aij η i η j + aī j η i η j + 2a 2 i jη i η j) = Re { a ij η i η j + a i jη i η j}, β (z, η, z, η) = ( bi η i + bī η i) = Re { b i η i}, 2 (2.2) with: b i (z)dz i is a form on the complex manifold M. We denote: a ij = a ij (z), a i j = a i j (z), b i = b i (z), (2.3) L (α (z, η, z, η), β (z, η, z, η)) = F 2 (α (z, η, z, η), β (z, η, z, η)). (2.4) Remark 2. F 2 is a R (α, β) complex Finsler metric. Definition 2.2. An Rcomplex Finsler space with (α, β)-metric is called an Rcomplex Kropina space or a Rcomplex Finsler space with Kropina metric if: It follows that F (α, β) = α2 β, β 0. Taking into account the 2homogeneity condition of L : ( ) α 2 2 L (α, β) =, β 0. (2.4) β L (α (z, λη, z, λ η), β (z, λη, z, λ η)) = λ 2 L (α (z, η, z, η), β (z, η, z, η)), λ R +, (2.5) we have: Proposition 2.. ([5]) In an Rcomplex Finsler space with (α, β)-metric the following equalities hold: αl α + βl β = 2L, αl αα + βl αβ = L α, αl αβ + βl ββ = L β, α 2 L αα + 2αβL αβ + β 2 L ββ = 2L, (2.6) L α := L α, L β = L β, L αβ = 2 L αβ, L αα = 2 L α 2, L ββ = 2 L β 2. (2.6)

4 82 Monica Purcaru Particular case: For a Rcomplex Finsler space with Kropina metric, we have: L α = 4α3 β 2, L β = 2α4 β 3, αl α + βl β = 2L (2.6) L αα = 2α2 β 2, L αβ = 8α3 β 3, L ββ = 6α4 β 4. In the following, we propose to determine the metric tensors of an R complex Finsler space with Kropina metric, i.e. g ij := 2 L(z, η, z, λ η)/ η i η j ; g i j := 2 L(z, η, z, λ η)/ η i η j, each of these being of interest in the following. We consider: α ( aij η j + a i j η j) = β We find immediately: η i = 2α α η i = 2α 2α l i, ( aī j η j + a i jη j) = 2α l ī, η i = 2 b i, β η i = 2 b ī, (2.7) l i := a ij η j + a i j η j, l j := aī j η i + a i jη i. (2.7) l i η i + l j η j = 2α 2. (2.8) We denote: Consequently, we obtain: η i = L η i. (2.9) η i = ρ 0 l i + ρ b i, (2.0) ρ 0 = 2 α L α (0 homogeneity), ρ = 2 L β ( homogeneity), (2.0) Differentiating (2.0) by η j and η j respectively we obtain: ρ 0 η j = ρ 2l j + ρ b j, ρ 0 η j = ρ 2l j + ρ b j, ρ η i = ρ l j + µ 0 b i, ρ η i = ρ l ī + µ 0 bī, (2.) ρ 2 = αl αα L α 4α 3, ρ = L αβ 4α, µ 0 = L ββ 4. (2.) Proposition 2.2. The invarians of an Rcomplex Finsler space with Kropina metric: ρ 0, ρ, ρ 2, ρ are given by: ρ 0 := 2F β ; ρ := F 2 β, β o; ρ 2 := 2 ; ρ β 2 := 2F ; µ β 2 0 := 3F 2 ; β o. 2β 2 (2.)

5 On RComplex Finsler spaces with Kropina metric 83 Subscripts 2,, 0, give us the degree of homogeneity of these invariants. We have immediately: Proposition 2.3. The fundamental tensor fields of an Rcomplex Finsler space with Kropina metric are given by: g ij = 2F β a ij + 2 β 2 l il j + 3F 2 2β 2 b ib j + 2F β 2 (b j l i + b i l j ). (2.2) or, in the equivalent form: g i j = 2F β a i j + 2 β 2 l il j + 3F 2 2β 2 b ib j + 2F β 2 ( ) b jl i + b i l j. (2.3) g ij = 2F β a ij 2 β 2 l il j + F 2 2β 2 b ib j + F 2 η iη j, (2.2 ) g i j = 2F β a i j 2 β 2 l il j + F 2 2β 2 b ib j + F 2 η iη j, (2.3 ) Proof. Using the relations (2. ) in Theorem 2. [5] by direct calculus we have the results The next objective is to obtain the determinant and the inverse of the tensor field g ij. The solution is obtained by adapting Proposition.2., p. 287 from [6] and Proposition 2.2 from [4] for an arbitrary non-singular non-hermitian matrix (Q ij ). The result is: Proposition 2.4. Suppose: (Q ij ) is a non-singular n n complex matrix with inverse (Q ji ); C i and Cī := C i, i =,.., n, are complex numbers; C i := Q ji C j and its conjugates; C 2 := C i C i = C i Cī; H ij := Q ij ± C i C j Then i) det(h ij ) = ( ± C 2 ) det(q ij ) ii) Whenever ± C 2 0, the matrix (H ij ) is invertible and in this case its inverse is H ji = Q ji ±C 2 C i C j. Theorem 2.. For a non-hermitian Rcomplex Finsler space with Kropina metric, ) 2 (L (α, β) =, β 0, with ai j = 0), we have: ( α 2 β i) the contravariant tensor g ij of the fundamental tensor field g ij is: g jl = β 2F ajl + 2β(F 3 ω β 3 ) η j η l + F 2 β(4f γ 3β 3 ) b j b l + M 2M + 2α2 (β 3 + F 2 ε) M b j η l + 2F β(β3 F 2 δ) η j b l, M (2.4)

6 84 Monica Purcaru M = 4F 4 (εδ γω) + 4α 2 β(βγ α 2 ε α 2 δ) + 3α 6 ω + β 6, l i = a ij η j, η i = 2F β a ijη j F β 2 b i, (2.5) γ = a jk η j η k = l k η k, ε = b j η j, ω = b j b j, b k = a jk b j, b l = b k a kl, δ = a jk n j b k = l k b k, l j = a ji l i = n j. ii) det(g ij ) = (2q 2 ) n B 4 + q2 A 4 α 2 γ α 2 det(a ij ), A = ωα2 ωγ + ε 2 α 2 γ B = 2 2βε ω α 2 2α 2 β 2 + 2(α 2 γ) α 2 q 2 ε 2 2(α 2 γ) 2 (4 + q 2 A) q2 (2βε ω) 2 2α 2 β 2 (4 + q 2 A) εq2 (2βε ω) β(α 2 γ)(4 + q 2 A) Proof. To prove the claims we apply the above Proposition in a recursive algorithm in three steps. We write g ij from 2.2 in the form: g ij = 2q 2 (a ij α 2 l il j + q2 4 b ib j + 2q 4 α 2 η iη j ) I. In the first step, we set Q ij = a ij and c i = 2 l i. By applying the Proposition 2.4 we obtain Q ji = a ji, c 2 = γ 2, c 2 = α2 γ α α 2 H ij = a ij α 2 l il j is invertible with H ji = a ji + 0 and c i = ηi. So, the matrix α α 2 γ ηi η j and det(a ij α 2 l il j ) = α 2 γ α 2 det(a ij ). II. Now, we consider Q ij = a ij α 2 l il j and c i = q 2 b i. By applying the Proposition 2.4 we obtain this time: Q ji = a ji + + c 2 = + q2 (α 2 ω γω + ε 2 ) 4(α 2 γ) q 2 (bi + ε α 2 γ ηi ). = 4 + q2 A 4 α 2 γ ηi η j, c 2 = q2 4 (ω + α 2 γ ε2 ), 0, where A = α2 ω γω + ε 2 α 2 γ and c i = It results that the inverse of H ij = a ij α 2 l il j + q2 4 b ib j exists and it is H ji = a ji + α 2 γ ( q 2 ε 2 (4 + q 2 A)(α 2 γ) )ηi η j q2 4 + q 2 A bi b j q 2 ε (4 + q 2 A)(α 2 γ) (ηi b j + b i η j ) and α 2 γ α 2 det(a ij ). det(a ij α 2 l il j + q2 4 b ib j ) = 4 + q2 A 4 III. Finally, we put Q ij = a ij α 2 l il j + q2 From here, we obtain: 4 b ib j and c i = 2αq 2 η i.

7 On RComplex Finsler spaces with Kropina metric 85 Q ji = a ji + α 2 γ ( q 2 ε 2 (4 + q 2 A)(α 2 γ) )ηi η j q2 4 + q 2 A bi b j q 2 ε (4 + q 2 A)(α 2 γ) (ηi b j + b i η j ), c 2 = [ a ji + α 2 γ ( q 2 ε 2 (4 + q 2 A)(α 2 γ) )ηi η j q q 2 A bi b j q 2 ε ] (4 + q 2 A)(α 2 γ) (ηi b j + b i η j ) 2α 2 q 4 η iη j = 2βε ω α 2 2α 2 β 2 + 2(α 2 γ) α 2 q 2 ε 2 2(α 2 γ)(4 + q 2 A) q 2 (4 + q 2 A)2α 2 (2ε ω εq 2 β )2 (4 + q 2 A)(α 2 γ) (2ε ω β ), + c 2 = 2 2βε ω α 2 2α 2 β 2 + 2(α 2 γ) α 2 q 2 ε 2 2(α 2 γ)(4 + q 2 A) q2 (2βε ω) 2 2α 2 β 2 (4 + q 2 A) εq2 (2βε ω) β(α 2 γ)(4 + q 2 A) 0 and c i = 2αq 2 (Mηi + Nb i ), where M = 2 α + + α 2(α 2 γ) ( q 2 ε 2 (4 + q 2 A)(α 2 γ) ) q 4 ε(2βε ω) 2β(4 + q 2 A)(α 2 γ) and N = 2αβ q2 (2εβ ω) 2(4 + q 2 A)βα αεq 2 2(4 + q 2 A)(α 2 γ). is By applying the Proposition 2.4 we obtain that the inverse of H ij = a ij α 2 l il j + q2 4 b ib j + 2α 2 q 4 η iη j H ji = a ji + α 2 γ ( q 2 ε 2 (4 + q 2 A)(α 2 γ) )ηi η j q q 2 A bi b j q 2 ε (4 + q 2 A)(α 2 γ) (ηi b j + b i η j ) B 2α 2 q 4 (Mηi + Nb i )(Mη j + Nb j ) and det(a ij α 2 l il j + q2 4 b ib j + 2α 2 q 4 η iη j ) = B 4 + q2 A α 2 γ α 2 det(a ij ), where 4 B = + c 2. But, g ij = 2q 2 H ij, with H ij from III. Thus, g ij = (2q 2 ) n B 4 + q2 A 4 α 2 γ α 2 det(a ij ). From here, immediately results i) and ii). 2q 2 Hij and det(g ij ) =

8 86 Monica Purcaru Proposition 2.5. In a non-hermitian R complex Finsler space with Kropina metric we have the following properties: γ + γ = l i η i + l jη j = a ij η j η i + a j k η kη j = 2α 2 (2.6) ε + ε = b j η j + b jη j = 2β, δ = ε, (2.7) l i = a ij η j, η i = 2F β a ijη j F β 2 b i, γ = a jk η j η k = l k η k, ε = b j η j, ω = b j b j, b k = a jk b j, b l = b k a kl, δ = a jk η j b k = l k b k, l j = a ji l i = η j. Example. We set α as α 2 (z, η) := ( + ε z 2 ) n k= Re(ηk ) 2 εre < z, η > 2 ( + ε z 2 ) 2, (2.8) where z 2 := n k= zk z k, < z, η >:= n { } k= zk η k, defined over the disk n r = z C n, z < r, r := ε if ε < 0, on C n if ε = 0 and on the complex projective space P n (C) if ε > 0. By computation, we obtain a ij = δ ij ε zi z j and a i j = 0 and so, α 2 (z, η) = ( 2 aij η i η j + aī j η i η j). Now, taking β(z, η) := Re <z,η>, where b +ε z 2 i := zi, +ε z 2 we obtain some examples of non - Hermitian R - complex Kropina metrics: +ε z 2 ( +ε z 2 ) F ε := (+ε z 2 ) n k= Re(ηk ) 2 εre<z,η> 2 (+ε z 2 ) 2 Re <z,η> +ε z 2. (2.9) Theorem 2.2. For a Hermitian Rcomplex Finsler space with Kropina metric (F = α 2 β, β 0, a ij = 0) we have: g jk = β 2F a jk β(4β+f ω) + η j η k + β(3f β+α2 ) 2F 2 N 2F N b j b k + + β(4βε) 2F N b j η k + β(4βε) 2F N η j b k, (2.20) N = ε 2 α 2 ω + 3F βω + 8β 2 8βRe(ε) α 2 = a i jη i η j = l ī ηī, l j = aīj l ī = η j (2.2) b k = a jk b j, ε = l bī, ī ε = b ηī, ī ω = b bī. ī Proof. Assuming a ij = 0, from (2.7), (2.0) and Proposition 2.2 it follows: l i = a i jη j, η i = 2F β ai jη j F 2 β b i

9 On RComplex Finsler spaces with Kropina metric 87 Considering: g jk = β 2F a jk + C η j η k + D b j b k + E b j η k + F η j b k and on the condition: g i jg jk = δ k i, by direct calculus we obtain: C = β(4β + F ω) 2F 2 N where N is given in (2.2), D = β(3f β + α2 ), E β(4β ε) = 2F N 2F N, F β(4β ε) = 2F N Example 2. We consider α given by η 2 + ε ( z 2 η 2 < z, η > 2) α 2 (z, η) := ( + ε z 2 ) 2, (2.22) { } defined over the disk n r = z C n, z < r, r := ε if ε < 0, on C n if ε = 0 and on the complex projective space P n (C) if ε > 0, where < z, η > 2 :=< z, η > < z, η >. By computation, we obtain a ij = 0 and a i j = δ ij ε zi z j and so, α 2 (z, η) = +ε z 2 ( +ε z 2 ) a i j(z)η i η j. Thus it determines purely Hermitian metrics which have special properties. They are Kähler with constant holomorphic curvature K α = 4ε. Particularly, for ε = we obtain the Bergman metric on the unit disk n := n ; for ε = 0 the Euclidean metric on C n, and for ε = the Fubini-Study metric on P n (C). Setting β(z, η) as in Example, we obtain some examples of Hermitian R - complex Kropina metrics: F ε := η 2 +ε( z 2 η 2 <z,η> 2 ) (+ε z 2 ) 2 Re <z,η> +ε z 2. (2.23) References [] Abate, M., Patrizio, G., Finsler Metrics - A Global Approach, Lecture Notes in Math., 59, Springer-Verlag, 994. [2] Aikou, T., Projective Flatness of Complex Finsler Metrics, Publ. Math. Debrecen, 63 (2003), [3] Aldea, N., Complex Finsler spaces of constant holomorphic curvature, Diff. Geom. and its Appl., Proc. Conf. Prague 2004, Charles Univ. Prague (Czech Republic) 2005, [4] Aldea, N., Munteanu, G., On complex Finsler spaces with Randers metric. J. Korean Math. Soc. 46 (2009), No.5,

10 88 Monica Purcaru [5] Aldea, N., Purcaru, M., R Complex Finsler Spaces with (α, β)-metric. Novi Sad, J.Math., 38 (2008), No., -9. [6] Bao, D, Chern, S.S, Shen, Z., An Introduction to Riemannian Finsler Geom., Graduate Texts in Math., 200, Springer-Verlag, [7] Bao, D, Robles, C., On Randers spaces of constant flag curvature, Rep. Math. Phys., 5 (2003), [8] Fukui, Complex Finsler manifold, J. Math. Kyoto Univ., 29 (989), No.4, [9] Kobayashi, S., Horst, C., Wu H.H., Complex Differential Geometry, Birkhäuser Verlag, 983. [0] Matsumoto, M., Theory of Finsler spaces with (α, β)-metric, Rep. on Math. Phys., 3 (99), [] Miron, R., General Randers spaces, Kluwer Acad.Publ., FTPH No. 76 (996), [2] Miron, R., Hassan, B.T., Variational problem in Finsler Spaces with (α, β)-metric, Algebras Groups and Geometries, 20 (2003), [3] Munteanu, G., Complex Spaces in Finsler, Lagrange and Hamilton Geometries, Kluwer Acad. Publ., FTPH 4, [4] Munteanu, G., Purcaru, M., On R Complex Finsler Spaces, Balkan J. Geom. Appl. 4 (2009), No., [5] Wong, P.M., A survey of complex Finsler geometry. Finsler geometry, Sapporo in memory of Makoto Matsumoto, Adv. Stud. Pure Math., 48, Math. Soc. Japan, Tokyo, , [6] Sabãu, V.S., Shimada, H., Remarkable classes of (α, β)-metric spaces, Rep. on Math. Phys., 47 (200), No., 3-48.