R-COMPLEX FINSLER SPACES

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1 Novi Sad J. Math. Vol. 38, No. 1, 008, 1-9 R-COMPLEX FINSLER SPACES WITH, β)-metric Nicoleta Aldea 1, Monica Purcaru 1 Abstract. In this paper we introduce the class of R-complex Finsler spaces with, β)-metrics and study some important exemples: R-complex Randers spaces, R-complex Kropina spaces. The metric tensor field of a R-complex Finsler space with, β)-metric is determined ). A special approach is dedicated to the R-complex Randers spaces 3). AMS Mathematics Subject Classification 000): 53B40, 53C60 Key words and phrases: R-complex Finsler space,, β)-metric 1. R-complex Finsler spaces In a previous paper [14], we expanded the known definition of a complex Finsler space [1,, 13, 17]), reducing the scalars to λ R. The outcome was a new class of Finsler space called by us the R-complex Finsler spaces, [14]). Our interest in this class of Finsler spaces issues from the fact that the Finsler geometry means, first of all, a distance, and this refers to the curves depending on the real parameter. In the present papers, following the ideas from real Finsler spaces with, β)- metrics [6, 18, 10, 11, 1]), we introduce a similar notion on R-complex Finsler spaces. In this section we keep the general setting from [13, 14], and subsequently we recall only some necessary 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 = n 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 1.1) z k = z k z) ; η k = z k z j ηj. { The natural frame z k, z k η k } of T u T M) with the changes of Jacobi matrix z + z j j η h z k z h η ; j of 1.1) changes into = z j = z j z k 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 spanned by and an adapted frame in H T M) is } η k η k η j. 1 Faculty of Mathematics and Informatics, Transilvania Univ. Braşov, Iuliu Maniu 50, Braşov , Romania, nicoleta.aldea@lycos.com, mpurcaru@unitbv.ro

2 N. Aldea, M. Purcaru δ = N j δz k z k k n, N j j k are the coefficients of the c.n.c.) and they have a certain rule of change in 1.1). 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 [1,, 13, 17]) is with respect to all complex scalars and the metric tensor of the space is one Hermititian. In [14] we changed a bit the definition of a complex Finsler metric. A R-complex Finsler metric on M is a continuous function F : T M R + satisfying: i) L := F 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, 0) homogeneous with respect to the real scalars λ, and in [14] we proved that the following identities are fulfilled: L η i ηi + L η i ηi = L; g ij η i + g ji η i = L η j ; 1.) gik η j ηj + g ik η j ηj = 0; 1.3) g ij := L η i η j ; g i j := are the metric tensors of the space. g i k η j ηj + g i k η j ηj = 0; L = g ij η i η j + gī j η i η j + g i jη i η j, L η i η j ; g ī j := L η i η j. R-complex Finsler spaces with, β)-metric Following the ideas from the real case, [6, 18, 10, 11, 1], we shall introduce R-complex Finsler spaces with, β)-metrics. Let us consider z M, and η T zm, η = η i z, a section in a holomorphic tangent space. i Definition.1. A R-complex Finsler space M, F ) is called, β)-metric if the fundamental function F is R-homogeneous by means of the functions z, η, z, η) and βz, η, z, η),.1) F z, η, z, η) = F z, η, z, η), βz, η, z, η)) : = 1 aij η i η j + aī j η i η j + a i jη i η j) = Re{a ij η i η j + a i jη i η j };.) β : = 1 bi η i + bī η i) = Re{b i η i }, with a ij = a ij z), a i j = a i j z), both invertible or one of them being zero, and b := b i dz i, b i = b i z) a differential 1-form on M.

3 R-complex Finsler spaces with, β)-metric 3 If a ij = 0 and a i j) invertible, then the space is said to be of Hermitian type. If a i j = 0 and a ij ) invertible, then the space is called non-hermitian. Moreover, a ij = / η i η j and a i j = / η i η j. Indeed, and β are homogeneous with respect to η and η, i.e. z, λη, z, λ η) = λz, η, z, η), βz, λη, z, λ η) = λβz, η, z, η) for any λ R +, thus Lz, λη, z, λ η) = λ Lz, η, z, λ η) for any λ R, and so the homogeneity property implies.3) η i ηi + β η j ηj = ; η i ηi + β η j ηj = β; L + βl β = L ; L + βl β = L ; L β + βl ββ = L β ; L + βl β + β L ββ = L, L := L, L β := L β, L := L, etc. In the following, we propose to determine the metric tensors of a R-complex Finsler space with, β) metric, i.e. g ij := Lz, η, z, λ η)/ η i η j ; g i j := Lz, η, z, λ η)/ η i η j, each of these being of interest in the following. We consider.4).5) η i = 1 ρ 0 η j = ρ l j + ρ 1 b j ; aij η j + a i j η j) = 1 l i; β η i = 1 b i; ρ 1 η i = ρ 1l j + µ 0 b i ; l i : = a ij η j + a i j η j ; l i η i + l j η j = ; b k := a jk b j + a jk b j; b l : = b k a kl + b ka l k; ε := b j η j ; ω := b j b j ; ε + ε = β; η i : = L η i = ρ 0l i + ρ 1 b i ; ρ 0 := 1 1 L ; ρ 1 := 1 L β; ρ : = L L 4 3 ; ρ 1 := L β 4 ; µ 0 := L ββ 4 and their conjugates, with a jk) and a jk ) are the inverse of a ij ) and a i j), respectively. The functions ρ 0, ρ 1, ρ, ρ 1, µ 0 are called invariants of the R-complex Finsler space with, β)-metric as in [18]). Subscripts, 1, 0, 1 give us the degree of homogeneity of these invariants. Note that η i = ρ 0 l i + ρ 1 b i is uniquely represented in this form. Indeed, if fz, η)l i + gz, η)b i = 0, contracting it by η i, we obtain fz, η) + gz, η)β = 0. Deriving the last relation with respect to β, it results gz, η) = 0, and from here fz, η) = 0. So, 0 leads to fz, η) = 0. Theorem.1. are given by.6) The tensor fields of R-complex Finsler space with, β)-metric g ij = ρ 0 a ij + ρ l i l j + µ 0 b i b j + ρ 1 b j l i + b i l j, ) g i j = ρ 0 a i j + ρ l i l j + µ 0 b i b j + ρ 1 b jl i + b i l j

4 4 N. Aldea, M. Purcaru or in the equivalent form g ij = ρ 0 a ij + ρ ρ ) 0ρ 1 l i l j + µ 0 ρ 1ρ 1.7) g i j = ρ 0 a i j + ρ 1 ρ ρ 0ρ 1 ρ 1 ) l i l j + ρ 0 µ 0 ρ 1ρ 1 ρ 0 ) b i b j + ρ 1 ρ 0 ρ 1 η i η j, ) b i b j + ρ 1 ρ 0 ρ 1 η i η j. Proof. Taking into account.3), we have g ij = L η i η j = ) L η j η i = η j ρ 0l i + ρ 1 b i ) = ρ 0 η j l l i i + ρ 0 η j + ρ 1 η j b i and g i j = L η i η j = η j ) L η i = η j ρ 0l i + ρ 1 b i ) = ρ 0 η j l l i i + ρ 0 η j + ρ 1 η j b i. From here, immediately results.6). To prove.7), we compute η i η j, and obtain l i b j + l j b i = 1 ρ 0ρ 1 η i η j ρ 0l i l j ρ 1b i b j ). As in real case, we may consider the following examples of R-complex Finsler spaces with, β)-metric: 1. R-complex Randers spaces: L, β) = + β) with.8) L = L β = + β) =, L = L ββ = L β =. ). R-complex Kropina spaces: L, β) = β, β 0) with.9) L = 43 β ; L β = 4 β 3 ; L + βl β = L L = 1 β ; L β = 83 β 3 ; L ββ = 64 β R-complex Randers spaces As noticed in [14], an R-complex Finsler space produces two tensor fields g ij and g i j. For a properly Hermitian geometry g i j to be invertible is mandatory requirement, but from some physicist point of view, for which Hermitian condition is an impediment; it seems more attractive that g ij be an invertible metric tensor. These problems lead us in [14] to speak about R-complex Hermitian Finsler spaces i.e. det g i j) 0) and R-complex non-hermitian Finsler spaces i.e. det g ij ) 0). The present section applies our results to R-complex Randers spaces, better illustrating the interest for this work. A first question is when a R-complex Randers metric is positively defined on T M, i.e. > β, equivalently with 3.1) [ Re{aij η i η j + a i jη i η j } ] 1 > Re{b i η i }, for all η 0.

5 R-complex Finsler spaces with, β)-metric 5 The answer comes namely. If we suppose that 3.1) is true, then by substituting η i = b i into 3.1) we obtain ω+ ω =: b 0, 1). Conversely, b 0, 1) imply that b 1 b ) < 0. Setting b i = λη i, λ R, it results 3.1). So, we have proved the following proposition. Proposition 3.1. A R-complex Randers metric is positively defined on T M if and only if b 0, 1). 3.) The invariants of a R-complex Randers space are: ρ 0 : = 1 F ; ρ 1 := F ; ρ : = β 3 ; ρ 1 := 1 ; µ 0 := 1 and substituting them into.6) and.7), we obtain Proposition 3.. given by The metric tensor fields of a R-complex Randers space are 3.3) or equivalently 3.4) g ij = F a ij β 3 l il j + 1 b ib j + 1 b jl i + b i l j ); g i j = F a i j β 3 l il j + 1 b ib j + 1 b jl i + b i l j) g ij = F a ij F 3 l il j + 1 L η iη j, g i j = F a i j F 3 l il j + 1 L η iη j. The next objective is to obtain the determinant and the inverse of the tensor fields g ij and g i j. For this we apply Proposition 11..1, p. 87 from [6] and Proposition. from [4] for an arbitrary non-singular Hermitian matrix Q i j) : Proposition 3.3. [4] Suppose: Q i j) is a non-singular n n complex matrix with inverse Q ji ); C i and Cī := C i, i = 1,.., n, are complex numbers; C i := Q ji C j and its conjugates; C := C i C i = C i Cī; H i j := Q i j ± C i C j Then i) deth i j) = 1 ± C ) detq i j) ii) Whenever 1 ± C 0, the matrix H i j) is invertible and in this case its inverse is H ji = Q ji 1 1±C C ic j. Proposition 3.4. For the R-complex non-hermitian Randers space F := + β, with a i j = 0), we have i) g ij = F aij + β+ω) F M ηi η j + γ F M bi b j ε+) F M ηi b j + b i η j );

6 6 N. Aldea, M. Purcaru ii) det g ij ) = ) F n M 4 F det a ij), 3.5) γ : = a jk η j η k = l k η k ; l i = a ij η j ; γ + γ = ; b k = a jk b j ; b l = b k a kl ; M : = γω + β) + + ε). Proof. Applying Proposition 11..1, p. 87 from [6] in two steps, we are yields g ij and det g ij ). For the beginning we write g ij in the form g i j = ) F aij 1 l i l j + η 3 i η j. 1) In the first applications we set Q ij := a ij and C i := 1 l i. We obtain Q ji = a ji, C = γ, 1 C = γ and C i = 1 ηi. So, the matrix H ij = a ij 1 l i l j is invertible with H ji = a ji 1 γ ηi η j and det ) a ij 1 l i l j = γ det a ij ). ) Now, we consider Q ij := a ij 1 l i l j and C i := F η i. Hence Q ji = ] a ji + 1 γ ηi η j, C = a ji + 1 γ ηi η j η i η j = M γf 0 and C i = F 3 [ [a ji + 1 γ ηi η j ] η j = of H ij = a ij 1 l i l j + η 3 i η j exists. It ) is H ji = a ji + 1 γ ηi η j γ +ε M γ η i + b i γf, 1 + C = M γf ) +ε γ η i + b i. It results that the inverse +ε γ η j + b j )and det ) a ij 1 l i l j + η 3 i η j = M γf det ) a ij 1 l i l j = M 4 F det a ij). Taking into account that g ij = F H ij, with H ij from ), we obtain g ji = F Hji and det g ij ) = ) F n det Hij ). From here, immediately results i) and ii). Example 1. We set as 3.6) z, η) := 1 + ε z ) n k=1 Reηk ) εre < z, η > 1 + ε z ), z := n k=1 zk z k, < z, η >:= n k=1 zk η k, defined over the disk { } n r = z C n 1, 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. 1 By computation we obtain a ij = δ ij ε zi z j and a i j = 0 and so, z, η) = 1 aij η i η j + aī j η i η j). Now, taking βz, η) := Re <z,η>, b i := z i, we obtain some examples of R-complex non-hermitian Randers metrics: 1 + ε z ) n k=1 3.7) F ε := Reηk ) εre < z, η > 1 + ε z + Re < z, η > 1 + ε z.

7 R-complex Finsler spaces with, β)-metric 7 Proposition 3.5. For the R-complex Hermitian Randers space F := + β, with a ij = 0), we have i) g jī = aj j F + β+ω F H ηi η j 3 F H bi b [ j F H ε + ) η i bj + ε + ) b i η j] ; ii) det g i j) = F ) n H 4F det a i j), 3.8) = a j kη j η k = l k η k ; l i = a i j η j ; b k = a jk b j; b l = b k a k l; H : = 4F + β + ω) + ε ε. Proof. We apply Proposition 3.3 two times. Again, we write g i j = F ai j 1 l i l j + η 3 i η j). 1) Setting Q i j := a i j and C i := 1 l i we obtain Q ji = a ji, C = 1, 1 C = 1 and Ci = 1 ηi. So, the matrix H i j = a i j 1 l i l j is invertible with H ji = a ji + 1 η i η j and det ) a i j 1 l i l j = 1 det a i j). ) Considering Q i j := a i j 1 l i l j and C i := F η i, it gives Q ji = a ji + 1 η i η j, C = F [ 3 [a ji + 1 η i η j ] η j = a ji + 1 η i η ] j η i η j = H F F, 1 + C = H F 0 and Ci = + ε η i + b i). So, H i j = a i j 1 l i l j + η 3 i η j is invertible and its inverse is H ji = a ji + 1 η i η j + ε H η i + b i) +ε η j + b j) and det ) a i j 1 l i l j + η 3 i η j = H F det ) a i j 1 l i l j = H 4F det a i j). But g i j = F H i j, with H i j from ), so that we obtain g ji = H ji F and det g i j) = ) n ) det Hi j. It results in i) and ii). F Example. We consider given by 3.9) z, η) := η + ε z η < z, η > ) 1 + ε z ), defined over the disk { } n r = z C n 1, 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, < z, η > :=< z, η > ) < z, η >. By computation we obtain a ij = 0 and 1 a i j = δ ij ε zi z j and so, z, η) = a i jz)η i η j. Thus it determines a purely Hermitian metrics which have special properties. They are Kähler with constant holomorphic curvature K = 4ε. Particularly, for ε = 1 we obtain the Bergman metric on the unit disk n := n 1 ; for ε = 0 the Euclidean metric on C n, and for ε = 1 the Fubini-Study metric on P n C). Setting βz, η) as in

8 8 N. Aldea, M. Purcaru Example 1, we obtain some examples of R-complex Hermitian Randers metrics: η + ε z η < z, η > ) 3.10) F ε := 1 + ε z + Re < z, η > 1 + ε z. Similar considerations can be done for the R-complex Kropina spaces. We are aware of the fact that the subject offers much other working paths with various applications. References [1] Abate, M., Patrizio, G., Finsler Metrics - A Global Approach. Lecture Notes in Math. 1591, Springer-Verlag, [] Aikou, T., Projective Flatness of Complex Finsler Metrics. Publ. Math. Debrecen ), [3] Aldea, N., Complex Finsler spaces of constant holomorphic curvature. Diff. Geom. and its Appl., Proc. Conf. Prague 004, Charles Univ. Prague Czech Republic) 005, [4] Aldea, N., Munteanu, G., On complex Finsler spaces with Randers metric. submitted). [5] Aldea, N., Munteanu, G.,, β)-complex Finsler metrics. Proc. MENP-4, Bucharest 006, Geom. Balk. Press 007, 1-6. [6] Bao, D., Chern, S. S., Shen, Z., An Introduction to Riemannian Finsler Geom. Graduate Texts in Math. 00, Springer-Verlag, 000. [7] Bao, D., Robles, C., On Randers spaces of constant flag curvature. Rep. Math. Phys ), 9-4. [8] Fukui, Complex Finsler manifold. J. Math. Kyoto Univ. 9 No ), [9] Kobayashi, S., Horst, C., Wu H. H., Complex Differential Geometry, Birkhäuser Verlag, [10] Matsumoto, M., Theory of Finsler spaces with, β)-metric. Rep. on Math. Phys ), [11] Miron, R., General Randers spaces. Kluwer Acad. Publ., FTPH No ), [1] Miron, R., Hassan, B. T., Variational problem in Finsler Spaces with, β)- metric. Algebras Groups and Geometries 0 003), [13] Munteanu, G., Complex Spaces in Finsler, Lagrange and Hamilton Geometries. Kluwer Acad. Publ. FTPH 141, 004. [14] Munteanu, G., Purcaru, M., About R-complex Finsler Spaces. submitted). [15] Royden, H.L., Complex Finsler Metrics. Contemporary Math ), [16] Spiro, A., The Structure Equations of a Complex Finsler Manifold. Asian J. Math ), [17] Wong, P.-Mann, A survey of complex Finsler geometry. Advanced Studied in Pure Math. Vol.48 Math. Soc. of Japan 007),

9 R-complex Finsler spaces with, β)-metric 9 [18] Sabãu, V. S., Shimada, H., Remarkable classes of, β)-metric spaces. Rep. on Math. Phys. 47 No ), Received by the editors December 14, 006