LAGRANGE GEOMETRY VIA COMPLEX LAGRANGE GEOMETRY

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1 Novi Sad J. Math. Vol. 32, No. 2, 2002, AGANGE GEOMETY VIA COMPEX AGANGE GEOMETY Gheorghe Munteanu 1 Abstract. Asking that the metric of a complex Finsler space should be strong convex, Abate and Patrizio ([1]) associate to the real tangent bundle a real Finsler metric for which they analyze the relation between Cartan (real) connection of the obtained space and the real image of Chern-Finsler complex connection. Following the same ideas, in the present paper we shall deal with the more general case of a complex agrange space (M, ). As distinct from these authors, we shall associate to the Hermitian metric g i j(z, η) of a complex agrangian its real representation g ab (x, y). The obtained real space (M, g ab ) is a generalized agrange space ([10]). Furthermore, the possibility of its reduction to one real agrange space, in particular the Finsler one, is studied. A comparative analysis of the elements of agrange geometry ([10]): nonlinear connection, N linear connection, metric canonical connection, and so on, and their corresponding real image from the complex agrange geometry ([11]) is made. AMS Mathematics Subject Classification (2000): 53B40, 53C60 Key words and phrases: complex agrange geometry 1. Introduction The study of complex agrange geometry was initiated by us starting with the paper [11]. A complex agrange space is the pair (M, ), where M is a complex manifold and (z, η) is a real agrangian differentiable function on the holomorphic bundle T M, which determines a nondegenerate metric g i j = 2 /η i η j. This geometry generalizes that of the known complex Finsler space ([1],[2],[5], [6],[7],[13],[14]), where, in addition, the homogeneity condition of complex agrangian in respect to η is required. In this paper, a complex agrange space determines a real structure of Hermitian manifold on the tangent real bundle. Everywhere, the indices i, j, k,... run in the interval 1, n, and a, b, c,... run in 1, 2n. We shall assume that the 1 Transilvania University, Faculty of Science, 2200 Brasov, omania, gh.munteanu@info.unitbv.ro

2 142 Gh. Munteanu reader is familiar with the geometry of T M, the holomorphic tangent bundle, and with the agrange geometry ([10]). et M be a complex manifold, dim C M = n, (U, z k ) the local coordinates in a local chart, z k = x k + ix n+k. M is also a real manifold, dim M = 2n, (U, x a ) is a real chart, and is endowed with the complex structure J, J 2 = I, acting on T,x M by J ( x ) =, a x J ( ) =. n+k x n+k x k et us consider the well-known Poincaire operators: (1.1) z k = 1 2 ( x k i ); xn+k from which clearly results that: (1.2) x k = z k + z k ; z k = 1 2 ( x k + i x n+k ) = i( xn+k z k z k ) The complex structure J is extended to the complexification T C M of the tangent bundle, obtaining the complex structure J(X + iy ) = J (X)+ i J (Y ), J 2 = I, behaving on Poincare operators as follows: J( ) = i ; J( ) = z k z k z k i. The eigenspaces of J determines two subbundles of T z k C M denoted by T M, the (1, 0) type vectors, and respectively T M = T M, the (0, 1) type vectors and T C M = T M T M. The bundle T M is holomorphic and as a manifold it is the geometric support of the complex agrange geometry. The bundle T M is isomorphic with the real tangent bundle T M by the map that acts on the corresponding tangent space as follows ([1]): (1.3) with the inverse: (1.4) : X X = X + X C : X X C = 1 (X ijx) 2 ocally, if X = η k, with η k = y k +iy n+k, then X = y k +y n+k, z k x k x n+k and conversely, if X = y a x, then X C = (y k + iy n+k ). a z k et us consider π : T M M the holomorphic bundle, u = (z k, η k ) T M and p : T M M the tangent real bundle, u = (x a, y a ) T M. Now, taking T M as a base manifold, arguing as before, we obtain: T C (T M) = T (T M) T (T M). The bundle π T : T (T M) T M is holomorphic and Ker π T = V (T M) is called the vertical bundle, a local base in V u (T M) being { }. Through conjugation, a local base { } in V η k η k u (T M) is obtained. et us denote by V C (T M) = V (T M) V (T M) the vertical complexified bundle. Analogously, we can consider the real vertical bundle V (T M) = Ker p T, where p T : T (T M) T M is the tangent map. A local base in V u (T M)

3 agrange geometry via complex agrange geometry 143 is indeed { y a } and because the vertical bundle is isomorphic with the tangent bundle T M, we have: (1.5) and conversely, (1.6) η k = 1 2 ( y k i ); yn+k y k = ( η k ) = η k + η k ; η k = 1 2 ( y k + i y n+k ) = (i yn+k η k ) = i( η k η k ) Therefore, if V = V k is a vertical complex field then V = U a η k y, with a U k = ev k and U n+k = ImV k, is a real vertical field, and conversely. Hence, (V C (T M)) = V (T M). We denote the same here by the isomorphism of passing to real on T M. 2. The induced real nonlinear connection As is known, in the study of tangent bundles it is very useful to use the notion of a nonlinear connection that determines the adapted base in which the study is linearized : many of the computations are made similarly as on the base manifold M. A nonlinear connection can be given by a splitting in an exact sequence that determines a supplementary subbundle to V (T M) in T (T M), i.e. T (T M) = H(T M) V (T M), called the horizontal subbundle. This determines the distribution N : u = (z k, η k ) H u (T M), called the complex nonlinear connection, shortly (c.n.c.). A local base on H u (T M) is { z = j z N k j j }, where N k η k j are the coefficients of (c.n.c.) and they are transforms at the local change of charts after the rule: (2.1) N k i z k z j = z i z k N k j 2 z i z j z k ηk and then the base { z }, called the adapted base of N j i i following rule of transformation: (2.2) z i = z j z i z j (c.n.c.), satisfies the, η k, z k η k } Through conjugation is obtained an adapted base {, z k on T C (T M), shortly denoted by { k, k, k, k}. The dual adapted base is denoted by {dz k, η k, d z k, η k }. From (2.2) it results that there exists an isomorphism ([1]) C θ: V C (T M) H C (T M), locally given by C θ ( η ) = i z and C θ ( i η ) = i z, where H i C (T M) = H(T M) H(T M) is the complexified horizontal bundle.

4 144 Gh. Munteanu As a complex manifold T M has the natural complex structure, still denoted by J, and locally given by: J(/z k ) = i/z k ; J(/η k ) = i/η k ; J(/z k ) = i/z k J(/η k ) = i/η k. Since, J( k ) = i k and J( k) = i k we deduce that J(H(T M)) = ih(t M) and J(H(T M)) = ih(t M). The same reasonings can be made on T (T M). A real nonlinear connection, shortly (r.n.c.), is given by the splitting T (T M) = H(T M) V (T M). The H(T M) bundle is for the moment not unique, being only supplementary to V (T M). We shall fix H(T M) acting analogously to [1]: We see that T M and T M bundles are isomorphic by C and. The same isomorphism is between T (T M) and T (T M). On the other hand, the complex horizontal lift, locally expressed by lc h ( ) = determines an isomorphism between T M and H(T M). Then the image of map l h = lc h C z k z k : T M T (T M) defines a real horizontal lift. et us consider now the local base x = l h a ( x ) in H(T a M) = l h (T M) which determines in turn an (r.n.c.). As it is known that the local expression of a real horizontal lift is ([10]): x = a x a Na b, where N b y b a are the coefficients of (r.n.c.). Taking into account the local expression of a complex vertical field, we can deduce that: (2.3) N h k = en h k ; N n+h k = ImN h k and therefore : Nk h = Nk h +i N n+h k and k = N h z k k h. Note that the map = θ C θ 1 : H C (T M) H(T M) is an isomorphism, where θ : V (T M) H(T M) is the corresponding real isomorphism to C θ. Because the horizontal bundle H C (T M) is J invariant, applying the operator it follows that H(T M) is J invariant. Hence, i z k (2.4) and in consequence x n+k corresponds to Nn+k a corresponds to in k h. So, we deduce that: N h n+k= ImN h k ; N n+h n+k = en h k Thus we obtain an (r.n.c.), Nb a, on T M, determined by the given (c.n.c.) Nk h on T M, whose coefficients are in fact the real representation of the complex matrix Nk h. Now, taking into account the action of the operator on the adapted base { k, k }, we get in addition to (1.6) that: (2.5) x k = ( z k ) ; = (i xn+k z k )

5 agrange geometry via complex agrange geometry 145 and the inverse through C = C θ θ 1. For the next computation it will be useful to have the following consequences of (1.6) and (2.5): ( η k ) = x k = y k ; ( z k ) = x k and z k + z k ; = i( xn+k z k z k ) Further, let us consider (M, ) a complex agrange space ([11]), where : T M is a agrangian function such that g j k = 2 /η j η k is a nondegenerate metric on T M. At each point u = (z k, η k ) T M, (u) = (z k, η k ) is a differentiable function. Because z k = x k + ix n+k and η k = y k + iy n+k, it follows that (u) = (x a, y a ) is a real differentiable function. et as note that, in general, g ab = /y a y b might be degenerate and hence the pair (M, ) is not always a real agrange space. In the special case when (g j k) determines a nondegenerate matrix ( g ab ), (A-P) ([1]) calls the metric g j k as being strongly convex. Moreover, let us note that if (z, λη) = λ 2 (z, η), then (x, λy) = λ 2 (x, y) and conversely. So, if (M, ) is a strong convex Finsler complex space then (M, ) is a Finsler real one. From now on we shall act in a manner different from [1]. We shall consider the real metric structure determined by the real representation of g j k. Proposition 2.1. et g j k(z, η) be the Hermitian metric of a complex agrange space (M, ). Then the pair (M, g ab (x, y)), where : (2.6) gjk = e g j k = 1 2 (g j k + g k j) ; gn+jk = Im g j k gjn+k = Im g j k = i 2 (g j k g k j) ; gn+jn+k = e g j k determines a (real) generalized agrange space([10]). ( ) For proof it suffices to remark that g ab = gab g ba and det 0 Thanks to (2.6) and g j kg kl = j l it results that g ab, the inverse of g ab, is the real representation of g kl jk, i.e. g = e g jk jn+k ; g = Im g jk n+jn+k ; g = e g jk ; and g n+jk = Im g jk. Now, considering a fixed (c.n.c.)nk h and {dzk, η k, d z k, η k } the dual adapted base determined by it, then ([11]): (2.7) G = g ij dz i dz j + g ij η i η j

6 146 Gh. Munteanu gives a Hermitian metric on T M with respect to the complex structure J and both to the almost Hermitian structure J N, locally given by J N ( k ) = k ; J N ( k ) = k ; J N ( k) = k; J N ( k) = k and globally defined. eplacing g i j = e g i j +iim g i j ; dz j = dx j +idx n+j and η j = y j +iy n+j in (2.7), it results that: Proposition 2.2. The structure (2.8) G= e G = g ab dx a dx b + g ab y a y b is a Hermitian metric on T M with respect to the complex structure J, and an almost Hermitian metric with respect to J N structure, ( J N ) 2 = I, locally given by J N ( x a ) = y a and J N ( y a ) = x a. The integrability of J N and J N structures depends only on the vanishing of torsion of (c.n.c.) and respectively (r.n.c.). et as note that G= Im G also defines a metric structure on T M. From the computation g j k = 1 4 ( y j y k y n+j y k ) we deduce just the real rep- ) + i y n+j y n+k 4 ( 2 resentation of the matrix g j k : 2 η j η k y j y n+k 2 = 1 4 ( + i )( y k y n+k y j i y n+j ) = gjk = 1 4 ( 2 y j y k + 2 y n+j y n+k ) ; gjn+k = 1 4 ( 2 (2.9) y j y n+k 2 y n+j y k ) et us remark that, in general, the tensor C abc = 1 2 { g bc y a + g ac y b g ab y } is c not totally symmetric and therefore the generalized agrange space (M, g ab ) is not always reducible to a argange space ([10]). Moreover, the space is neither weakly regular, hence the known procedures of agrange (real) geometry to obtain a (r.n.c.) cannot be applied, remaining in principle the method described above. The question is, however, when the generalized agrange space is a agrange one, particularly a Finsler space. This means that the tensor h ab = y a y b nondegenerated. In particular, if is a complex Finsler metric then becomes a real Finsler metric. A sufficient condition is when h ab = g ab, which in view of (2.9) is equivalent to: is (2.10) 2 y j y k = 2 y n+j y n+k ; 2 y j y n+k = 2 y n+j y k

7 agrange geometry via complex agrange geometry 147 and that happens if and only if : (2.11) g jk = 2 η j = 0, j, k = 1, n ηk Obviously, by conjugation from (2.11) it results also g j k = 0. Definition 2.1. In the condition (2.11) we call the complex agrange space (M, ) being with pure Hermitian metric. In particular, the Finsler complex space with pure Hermitian metric is obtained. Proposition 2.3. The generalized agrange space (M, g ab ) associated to a complex agrange space with pure Hermitian metric is reductible to a real agrange space (M, ). As shown in [10], the variational method in the real agrange space (M, ) gives an (r.n.c.): (2.12) 0 Nb a = Ga y b, where G a = 1 ac 2 g 4 y c x d yd x c = 1 4 ac g Φ c On the other hand, by the variational method a (c.n.c.), called canonical, in a complex agrange space (M, ) is obtained ([11]): (2.13) c N k j = Hk η j, where H k = 1 2 g mk 2 z h η m ηh Our next goal is to determine the circumstances when N c b a, the (r.n.c.) induced by Nj k 0 c, coincides with Nb a or, in an equivalent way, when the complex 0C image Nj k 0 0 = Nj k +i N n+k 0 j of Nb a c coincides with Nj k. For this reason we shall calculate the difference d tensor of two (c.n.c.) : (2.14) 2D k l = ( η l + η l )(Gk + ig n+k ) Hk η l First, we make the computation:

8 148 Gh. Munteanu 4(G k +ig n+k ) = e g mk Φ m Im g mk Φ n+m +i(im g mk Φ m +e g mk Φ n+m ) = g mk (Φ m + iφ n+m ). eplacing Φ m and Φ n+m from (2.12) and recalling that y = m η + m η m and y k = 1 2 (ηk + η k ) ; y = i( n+m η m η ) and y n+k = i m 2 (ηk η k ), a long but trivial computation gives: (2.15) 4Dl k = η l [g mk 2 ( η m z p ηp z m )] + η l [g mk 2 ( η m z p ηp + 2 η m z p ηp z m )] Hence, we have: Proposition 2.4. In the complex agrange space (M, ) the induced (r.n.c.) of c the Nj k 0 (c.n.c.) from (2.13) coincides with the Nb a (r.n.c.) given by (2.12) if and only if Dl k = 0. We recall here that a complex agrange space is called local Minkowski ([2],[3]) if there exist local charts in any u = (z, η) such that the agrange function depends only on the direction, i.e., = (η, η). The above Proposition and (2.15) yields: Proposition 2.5. If (M, ) is a complex agrange local Minkowski space there exist local charts in any u = (z, η) T M such that N c b a= N 0 b a. More interesting results are obtained in the particular case of complex Finsler space, when (z, λη) = λ 2 (z, η) and the consequences from it ([1],[12]). Then the formulas (2.12) lead to ([10]): (2.16) 0F Nb a = 1 γ00 a 2 y b, where γa 00 = γbcy a b y c and γ a bc = 1 2 g da { g dc x b gbd + x c gbc x d } 0F N a b (2.17) Γ k ij is the well-known Cartan (r.n.c.). And the formulas (2.13) give the Cartan (c.n.c.) ([11],[12]): cf Nj k = 1 Γ k 00 2 η j, where Γk 00 = Γ k ijη i η j with Γ k ij = 1 2 g mk { g j m z i + g i m z j } being the first complex Christoffel symbol.

9 agrange geometry via complex agrange geometry 149 Now, acting as before, after a long computation of passing from real to complex, we obtain that N 0F b a coincides with cf N b a, the real image of complex Cartan connection, if and only if the difference d tensor Dl k given by: (2.18) Dl k = η l (Γk i j ηi η j ) + η l (Γk ijη i η j + Γ k i j ηi η j ) is vanishing, where Γ k i j = 1 2 g mk gi m { z j of the evi-civita connection on T M. g i j z m } is the second Chritoffel symbol Clearly, if Γ k = 0, that is the evi-civita connection is of (1, 0) type, or i j equivalently the fact that g i j is a Kähler metric, then the difference d tensor is reduced to D k l = Γk ij η l η i η j. Therefore, we can state: Proposition 2.6. If g i j is a Kähler metric of the complex Finsler space (M, ) and the coefficients Γ k ij of the evi-civita linear connection on T M are holomorphic functions, then 0F N a b cf coincides with Nb a. Also, let us note that if g i j locally depends on T M only on z, i.e. g i j(z) (the point is called normal cf. [3]), then the metric comes from a Hermitian metric on M. In the Kählerian situation such metric is called Hermitian-Kähler ([1], [12]). So, from the local expression of Γ k ij, we have: Proposition 2.7. If g i j is a Hermitian-Kähler metric of the complex Finsler space (M, ), then N 0F b a cf coincides with Nb a. 3. The induced N real linear connection et us study now the real image of other geometric elements on T M induced from a (c.n.c.) on T M. et D : χ(t M) χ(t M) χ(t M) be a normal complex linear connection (shortly, N (c.n.c.)), that is a derivative law on T M which preserves the distributions V (T M), H(T M) and their conjugates. ocally, an N (c.l.c.) is characterized by its coefficients ( i jk ; i j k ; Ci jk ; Ci ), where: j k D k j = i jk i ; D k j = C i jk i (3.1) D k j = i j k i ; D k j = C i j k i As was proved in [11], D is an N (c.l.c.) iff DJ = DF = DF = 0, where J is the complex structure on T M, F is the natural tangent structure and F is the adjoint tangent structure of F with respect to the adapted base determined by a given (c.n.c.)n. ocally, F behaves on the adapted base as

10 150 Gh. Munteanu follow: F ( k ) = 0; F ( k ) = k ; F ( k) = 0; F ( k) = k, and globally defined. Correspondingly, on T M we have in addition to the complex structure J, the natural tangent structure F defined by F (X ) = (F X), and its adjoint tangent structuref in respect to an adapted base. ocally, their actions are given by: F ( x ) = a y ; F ( a y ) = 0; F ( a x ) = 0; F ( a y ) = a x a Then a derivative law D on T M is an N real linear connection, shortly N (r.l.c.), conformity to [10] iff D J= D F = D F = 0. Theorem 3.1. If D is an N (c.l.c.) on T M, then the following derivative law: (3.2) DA B = D A B, A, B χ(t M) or in other words: (3.3) D X Y = (D X Y ) + (D X Ȳ ), X, Y χ(t M) is an N (r.l.c.) on T M. Proof. et us remark that D is a linear connection on T M. Since J(A+iB) = J (A) + i J (B) and taking into account the definitions of F and F structures it is verified that D J= D F = D F = 0. Moreover, if D is of (1, 0) type, because D JX Y = D X JY, it results that D B = DAJ B. JA If the N (r.l.c.) D is given, obviously then D is obtained by linearity. Now, let us suppose that D is given in the local base by its coefficients ([10]): D x c x b = a bc x a ; D y c x b = C a bc x a (3.4) D x c y b = a bc y a ; D y c y b = C a bc y a Then making the computations in (3.3), thanks to (1.6) and (2.5) formulas, we find the relation between the coefficients of induced N (r.l.c.) D and N (c.l.c) D : (3.5) i jk= n+jk = e (i jk + i J k )

11 agrange geometry via complex agrange geometry 151 jk = i n+jk= Im ( i jk + i J k ) i jn+k= jn+k = n+jn+k = Im (i j k i jk) i n+jn+k= e ( i jk i J k ) Definition 3.1. An N (r.l.c.) D whose coefficients are connected by the relations: i jk = n+jk ; called of Hermitian type. jk = i n+jk ; i jn+k = n+jn+k ; jn+k = i n+jn+k will be The definition is justified by the fact that for a fixed index c the coefficient a bc is the real representation of a Hermitian matrix. Proposition 3.1. D is an N (r.l.c.) of Hermitian type if and only if D J= J D. If D is of (1, 0) type then formulas (3.5) are simplified, because i j k = Ci j k = 0. The calculus of bracket gives that [ X, Y ] = [X, Y ] and hence we have [ X, Y ] = [X, Y ]. So, the curvature and the torsion of the induced N (r.l.c.) D are expressed as a function of the curvature and respectively the torsion of N (c.l.c.)d as follows: (3.6) (X, Y )Z = (X, Y )Z ; T (X, Y ) = T(X, Y ) The components of this curvature and torsion are directly obtained from (3.6) as a function of the real and imaginary parts of the complex curvatures and torsions. As we have seen, the Hermitian metric G on T M is : G = e G+iIm G = G +i G. If D is a metrical N (c.l.c.), i.e. (D X G)(Y, Z) = XG(Y, Z) G(D X Y, Z) G(Y, D X Z) = 0, replacing X, Y, Z to their real parts, we obtain that D G= D G = 0. Therefore, D is metrical with respect to both real metric induced by G from T M. In a real case ([10]), and in the complex one ([11]), are known metrical N linear connections. In a real agrange geometry one has a special meaning, the so-called real canonical, or Miron s metric N (r.l.c.) : (3.7) c a bc= 1 da g { g dc 2 x b gbd + x c gbc x d }

12 152 Gh. Munteanu c Cbc= a 1 da g { g dc 2 y b gbd + y c gbc y d } In the complex case we know the next metric N (c.l.c.) : -The complex canonical connection([11]): cc i jk= 1 2 gli ( g jl z k + g cc kl z j ) ; Cjk= i 1 2 gli ( g jl η k + g kl η j ) = g li g j l η k (3.8) cc ī jk = 1 2 gīl ( g l j z k g k j z l ) ; cc Cī jk = 1 2 gil ( g l j η k g k j η l ) = 0 -The Chern-Finsler complex connection ([1],[2]...), for the special Finsler case: KF i jk= g li g KF j l z k ; i KF j k = C i KF j k = 0 ; Cjk i cc (3.9) = Cjk i (A comparative analysis of these is to be find in [12]) et N be the induced (r.n.c.) of the (c.n.c.)n. Because g j k = g jk +i g jn+k, g li = g li +i g ln+i and = 1 z k 2 { i }, developing the computation in x k x n+k (3.8) and then suitably grouping of terms, after a straightforward computation we obtain that: cc e( i jk) = 1 c 2 ( i c (3.10) jk i n+jn+k) cc Im( i jk) = 1 c 2 ( jk c n+jn+k ) cc e( i j k ) = 1 c 2 ( i c jk jn+k ) cc Im( i j k ) = 1 c 2 ( jk + c i jn+k) cc and analogous formulas for the Cjk i cc and C i, where in addition we have that j k c c Cjk i = C n+i c jn+k and C n+i c jk = Cjn+k i. Now, corroborate the (3.10) and (3.5) formulas, we have the next correspondence between the real image of canonical (c.l.c.) and canonical (r.l.c.) : cc i c jk= cc jk c i jk 1 2 ( i n+jn+k + = c jk 1 c 2 ( n+jn+k c jn+k ) c i jn+k)

13 agrange geometry via complex agrange geometry 153 cc i jn+k= 1 c 2 ( i c jn+k + n+jn+k ) (3.11) cc jn+k = 1 2 ( c c jn+k i n+jn+k) and analogous formulas for the coefficients cc C a bc. Proposition 3.2. The real image cc D of the canonical N (c.l.c.) cc D coincides with the real canonical (Miron s) connection c D,in respect to the adapted base of induced (r.n.c.) N, if and only if c D is of Hermitian type. Finally, let us consider again the complex structure J on T M and the metric structure given by (2.8) for a fixed (c.n.c.). It is easy to verify that (T M, J, G) is a Hermitian manifold in which c D is a metrical N (r.l.c.). Next we assume in addition that (T M, J, G, c D) is a real Kähler space. Then, by the fact that c D has zero torsion and because c D coincides in this circumstances with cc D, it results that cc D has too zero torsion and hence, (M, ) is a complex Hermitian- Kähler space ([1],[12]). By the above Proposition and because the complex torsion vanishes if and only if the torsion of the induced (r.l.c.) vanishes, the converse assertion is true. So, really we have proved that: Theorem 3.2. (M, ) is a complex Hermitian-Kähler space if and only if (T M, is a real Kähler manifold. eferences [1] Abate, M., Patrizio, G., Finsler Metrics-A Global Approach,ecture Notes in Math., 1591, Springer-Verlag, [2] Aikou, T., On Complex Finsler Manifolds, ep.of Kagoshima Univ. No. 24, (1991), 9-25, [3] Aikou, T., Compex manifolds modeled on a complex Minkowski Space, J. of Math of Kyoto Univ., Vol. 35, nr.1 (1995), [4] Gherghiev, G., Oproiu, V., Varietati Diferentiabile Finit si Infinit Dimensionale, Ed. Acad. omane, Vol. 2, [5] Faran, J. J.,Hermitian Finsler metric and the Kobayashi metric, J. Diff. Geom. 31 (1990), [6] Fukui, M., Complex Finsler manifolds, J. Math. Kyoto Univ (1987), [7] Kobayashi, S., Negative vector bundles and complex Finsler structures, Nagoya Math. J. 57 (1975),

14 154 Gh. Munteanu [8] Kobayashi, S., Wu, H., Complex Diff. Geometry, DMV Seminar Band 3, Berkhause Verlag, Basel-Boston, [9] Matsumoto, M., Foundation of Finsler Geomety and Special Finsler Space, Kaiseisha Press,Otsu, Japan, [10] Miron,., Anastasiei, M., The Geometry of agrange Spaces; Theory and Applications, Kluwer Acad. Publ., no 59, FTPH, [11] Munteanu, G., Complex agrange Spaces, Balkan J. of Gem. and its Appl. Vol 3, no 1 (1998), [12] Munteanu, G., On Cartan complex nonlinear connection, to appear in Publicatione Mathematicae, Debrecen. [13] oyden, H. I., Complex Finsler Spaces, Contemporany Math. Am. Math. Soc. 49(1986), [14] und, H., The curvature theory of direction -dependent connection on complex manifolds, Tensor N.S. 24 (1972), eceived by the editors September 15, 2002.