LAGRANGE GEOMETRY VIA COMPLEX LAGRANGE GEOMETRY

Novi Sad J. Math.

Vol. 32, No. 2, 2002, 141-154

LAGRANGE GEOMETRY VIA COMPLEX LAGRANGE GEOMETRY

Gheorghe Munteanu¹

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 Lagrange space (M, L).

As distinct from these authors, we shall associate to the Hermitian metric g_i¯j(z, η) of a complex Lagrangian L its real representation ^Rg_ab (x, y).The obtained real space (M,^Rg_ab) is a generalized Lagrange space ([10]). Furthermore, the possibility of its reduction to one real Lagrange space, in particular the Finsler one, is studied.

A comparative analysis of the elements of Lagrange geometry ([10]):

nonlinear connection,N−linear connection, metric canonical connection, and so on, and their corresponding real image from the complex Lagrange geometry ([11]) is made.

AMS Mathematics Subject Classification (2000): 53B40, 53C60 Key words and phrases: complex Lagrange geometry

1. Introduction

The study of complex Lagrange geometry was initiated by us starting with the paper [11].

A complex Lagrange space is the pair (M, L), whereM is a complex mani- fold andL(z, η) is a real Lagrangian differentiable function on the holomorphic bundleT⁰M, which determines a nondegenerate metricg_i¯j=∂²L/∂ηⁱ∂¯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 La- grangian in respect toη is required.

In this paper, a complex Lagrange space determines a real structure of Her- mitian manifold on the tangent real bundle. Everywhere, the indicesi, j, k, ...

run in the interval 1, n, and a, b, c, ... run in 1,2n. We shall assume that the

1Transilvania University, Faculty of Science, 2200 Brasov, Romania, e-mail:[email protected]

reader is familiar with the geometry ofT⁰M, the holomorphic tangent bundle, and with the Lagrange geometry ([10]).

LetM be a complex manifold,dimCM =n, (U, z^k) the local coordinates in a local chart,z^k=x^k+ix^n+k. M is also a real manifold,dimRM = 2n,(U, x^a) is a real chart, and is endowed with the complex structureJ^R,

R

J²=−I, acting onT_R,xM byJ^R(_∂x^∂a) =_∂x^∂n+k,J^R(_∂x^∂n+k) =−_∂x^∂k.

Let us consider the well-known Poincaire operators:

∂

∂z^k = 1 2( ∂

∂x^k −i ∂

∂x^n+k); ∂

∂¯z^k = 1 2( ∂

∂x^k +i ∂

∂x^n+k) (1.1)

from which clearly results that:

∂

∂x^k = ∂

∂z^k + ∂

∂¯z^k; ∂

∂x^n+k =i( ∂

∂z^k − ∂

∂z¯^k) (1.2)

The complex structure J^R is extended to the complexification TCM of the tangent bundle, obtaining the complex structureJ(X+iY) =J^R(X)+iJ^R(Y), J²=−I, behaving on Poincare operators as follows: J(_∂z^∂k) =i_∂z^∂k;J(_∂¯^∂_zk) =

−i_∂^∂_z¯k. The eigenspaces of J determines two subbundles of TCM denoted by T⁰M, the (1,0)−type vectors, and respectively T⁰⁰M =T⁰M, the (0,1)−type vectors andTCM =T⁰M ⊕T⁰⁰M.

The bundleT⁰Mis holomorphic and as a manifold it is the geometric support of the complex Lagrange geometry.

The bundle T⁰M is isomorphic with the real tangent bundle TRM by the map that acts on the corresponding tangent space as follows ([1]):

R◦:X →X^R=X+X (1.3)

with the inverse:

C◦:X →X^C= 1

2(X−iJX) (1.4)

Locally, ifX =η^{k ∂}_∂zk, withη^k =y^k+iy^n+k, thenX^R=y^{k ∂}_∂xk+y^n+k_∂x^∂n+k, and conversely, ifX=y^{a ∂}_∂xa, thenX^C= (y^k+iy^n+k)_∂z^∂k.

Let us consider π : T⁰M → M the holomorphic bundle, u = (z^k, η^k) ∈ T⁰M and p: TRM →M the tangent real bundle, u= (x^a, y^a) ∈TRM. Now, taking T⁰M as a base manifold, arguing as before, we obtain: TC(T⁰M) = T⁰(T⁰M)⊕T⁰⁰(T⁰M). The bundle πT : T⁰(T⁰M) → T⁰M is holomorphic and Ker πT =V(T⁰M) is called the vertical bundle, a local base in Vu(T⁰M) being {_∂η^∂k }. Through conjugation, a local base {_∂^∂_η¯k} inVu(T⁰M) is obtained. Let us denote byVC(T⁰M) =V(T⁰M)⊕V(T⁰M) the vertical complexified bundle.

Analogously, we can consider the real vertical bundleV(TRM) =Ker pT, where pT : TR(TRM)→ TRM is the tangent map. A local base in Vu(TRM)

is indeed{_∂y^∂a}and because the vertical bundle is isomorphic with the tangent bundleTRM, we have:

∂

∂η^k = 1 2( ∂

∂y^k −i ∂

∂y^n+k); ∂

∂η¯^k = 1 2( ∂

∂y^k +i ∂

∂y^n+k) (1.5)

and conversely,

∂

∂y^k = ( ∂

∂η^k)^R= ∂

∂η^k + ∂

∂η¯^k; ∂

∂y^n+k = (i ∂

∂η^k)^R=i( ∂

∂η^k − ∂

∂η¯^k) (1.6)

Therefore, ifV =V^{k ∂}_∂ηk is a vertical complex field thenV^R=U^{a ∂}_∂ya, with U^k=ReV^k and U^n+k =ImV^k, is a real vertical field, and conversely. Hence, (VC(T⁰M))^R = V(TRM). We denote the same here by ^R◦ the isomorphism of passing to real onT⁰M.

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 manifoldM.

A nonlinear connection can be given by a splitting in an exact sequence that determines a supplementary subbundle toV(T⁰M) inT⁰(T⁰M),i.e. T⁰(T⁰M) = H(T⁰M)⊕V(T⁰M), called the horizontal subbundle. This determines the distri- butionN :u= (z^k, η^k)→Hu(T⁰M),called the complex nonlinear connection, shortly (c.n.c.). A local base onHu(T⁰M) is {_δz^δj = _∂z^∂j −N_j^k_∂η^∂k}, whereN_j^k are the coefficients of (c.n.c.) and they are transforms at the local change of charts after the rule:

N_k⁰ⁱ∂z^0k

∂z^j =∂z⁰ⁱ

∂z^kN_j^k− ∂²z⁰ⁱ

∂z^j∂z^kη^k (2.1)

and then the base {_δz^δi}, called the adapted base of N_i^j (c.n.c.), satisfies the following rule of transformation:

δ

δzⁱ =∂z^0j

∂zⁱ δ δz^0j (2.2)

Through conjugation is obtained an adapted base {_δz^δk, _∂η^∂k, _δ¯^δ_zk, _∂^∂_η¯k} onTC(T⁰M) , shortly denoted by{δk , ∂k , δ¯k , ∂¯k}. The dual adapted base is denoted by{dz^k, δη^k , d¯z^k, δη¯^k}.

From (2.2) it results that there exists an isomorphism ([1]) ^Cθ:VC(T⁰M)→ HC(T⁰M), locally given by^Cθ (_∂η^∂i) =_δz^δi and^Cθ(_∂^∂_η¯i) =_δ^δ_z¯i,whereHC(T⁰M) = H(T⁰M)⊕H(T⁰M) is the complexified horizontal bundle.

As a complex manifoldT⁰M has the natural complex structure, still denoted byJ, 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(δk) =iδkandJ(δ¯k) =−iδ¯kwe deduce thatJ(H(T⁰M)) =iH(T⁰M) andJ(H(T⁰M)) =−iH(T⁰M).

The same reasonings can be made onTR(TRM).A real nonlinear connection, shortly (r.n.c.), is given by the splittingTR(TRM) =H(TRM)⊕V(TRM).

TheH(TRM) bundle is for the moment not unique, being only supplemen- tary toV(TRM). We shall fix H(TRM) acting analogously to [1]:

We see thatTRM andT⁰M bundles are isomorphic by^C◦ and^R◦. The same isomorphism is betweenTR(TRM) andT⁰(T⁰M).On the other hand, the com- plex horizontal lift, locally expressed byl^h_C(_∂z^∂k) = _δz^δk determines an isomor- phism between T⁰M and H(T⁰M). Then the image of map l^h_R = ^R◦ ·l_C^h· ^C◦ : TRM → TR(TRM) defines a real horizontal lift. Let us consider now the local base _δx^δa =l^h_R(_∂x^∂a) in H(TRM) =l^h_R(TRM) which determines in turn an (r.n.c.).

As it is known that the local expression of a real horizontal lift is ([10]):

δ

δx^a = _∂x^∂a−

R

N_a^b _∂y^∂b, where

R

N_a^b are the coefficients of (r.n.c.). Taking into account the local expression of a complex vertical field, we can deduce that:

R

N_k^h=ReN_k^h ;

R

N_k^n+h=ImN_k^h (2.3)

and therefore : N_k^h=N^R_k^h+i

R

N_k^n+h andδ_k =_∂z^∂k−N_k^h∂_h.

Note that the map ^R∗ =^Rθ · ^R◦ · θ^C−1 : HC(T⁰M) → H(TRM) is an isomor- phism, where^Rθ :V(TRM)→H(TRM) is the corresponding real isomorphism to^Cθ .Because the horizontal bundleHC(T⁰M) isJ invariant, applying the op- erator^R∗ it follows that H(TRM) is J^Rinvariant. Hence, _δxn+k^δ corresponds to i_δz^δk and in consequenceN_n+k^Ra corresponds to iN_k^h. So, we deduce that:

R

N_n+k^h =−ImN_k^h ;

R

N_n+k^n+h=ReN_k^h (2.4)

Thus we obtain an (r.n.c.) ,N^R_b^a, onTRM, determined by the given (c.n.c.) N_k^honT⁰M, whose coefficients are in fact the real representation of the complex matrixN_k^h.

Now, taking into account the action of the^R∗ operator on the adapted base {δk, ∂k}, we get in addition to (1.6) that:

δ

δx^k = ( δ

δz^k)^R ; δ

δx^n+k = (i δ δz^k)^R (2.5)

and the inverse through^C∗ =^Cθ ·^R◦ ·

R

θ⁻¹.

For the next computation it will be useful to have the following consequences of (1.6) and (2.5):

( ∂

∂η¯^k)^R = ∂

∂y^k ; ( δ

δ¯z^k)^R= δ δx^k and δ

δx^k = δ δz^k + δ

δ¯z^k ; δ

δx^n+k =i( δ δz^k − δ

δz¯^k)

Further, let us consider (M, L) a complex Lagrange space ([11]), where L:T⁰M →R is a Lagrangian function such thatg_j¯k =∂²L/∂η^j∂η¯^k is a non- degenerate metric onT⁰M.At each pointu= (z^k, η^k)∈T⁰M, L(u) =L(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 thatL(u) =L^R(x^a, y^a) is a real differentiable function.

Let as note that, in general,^Rg_ab= ¹₂∂²L^R/∂y^a∂y^b might be degenerate and hence the pair (M, L^R) is not always a real Lagrange space. In the special case when (g_j¯k) determines a nondegenerate matrix (^Rg_ab), (A-P) ([1]) calls the metric g_jk¯ as beingstrongly convex.

Moreover, let us note that if L(z, λη) = |λ|²L(z, η), then L^R(x, λy) =

|λ|²L^R(x, y) and conversely. So, if (M, L) is a strong convex Finsler complex space then (M, L^R) 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 ofg_jk¯.

Proposition 2.1. Letg_j¯k(z, η)be the Hermitian metric of a complex Lagrange space(M, L). Then the pair(M,g^Rab(x, y)), where :

Rg_jk=Re g_j¯k= 1

2(g_j¯k+g_k¯j) ; ^Rg_n+jk=−Im g_j¯k

(2.6)

Rg_jn+k=Im g_jk¯= −i

2 (g_j¯k−g_k¯j) ; ^Rg_n+jn+k=Re g_jk¯

determines a (real) generalized Lagrange space([10]).

For proof it suffices to remark that ^Rg_ab=^Rg_baanddet µR

g_ab

¶ 6= 0

Thanks to (2.6) and g_j¯kg^¯kl = δ^l_j it results that ^Rg^ab, the inverse of ^Rg_ab, is the real representation of g^kl¯, i.e. ^Rg^jk=Re g^¯jk; ^Rg^jn+k=Im g^¯jk; ^Rg^n+jn+k= Re g^¯jk; and ^Rg^n+jk=−Im g^¯jk.

Now, considering a fixed (c.n.c.)N_k^h and {dz^k , δη^k , d¯z^k , δη¯^k} the dual adapted base determined by it, then ([11]):

G=g_ijdzⁱ⊗dz^j+g_ijδηⁱ⊗δη^j (2.7)

gives a Hermitian metric onT⁰M with respect to the complex structureJ and both to the almost Hermitian structure JN, locally given by JN(δk) = ∂k; JN(∂k) =−δk; JN(δ¯k) =∂¯k; JN(∂¯k) =−δk¯ and globally defined.

Replacingg_i¯j =Re g_i¯j+iIm g_i¯j;dz^j=dx^j+idx^n+j andδη^j=δy^j+iδy^n+j in (2.7), it results that:

Proposition 2.2. The structure

GR=Re G=^Rg_abdx^a⊗dx^b+^Rg_abδy^a⊗δy^b (2.8)

is a Hermitian metric on TRM with respect to the complex structure J^R, and an almost Hermitian metric with respect toJR

N structure, (JR

N)²=−I,locally given byJR

N(_δx^δa) = _∂y^∂a andJR

N(_∂y^∂a) =−_δx^δa. The integrability ofJN andJR

N structures depends only on the vanishing of torsion of (c.n.c.) and respectively (r.n.c.).

Let as note that

R˜

G=Im Galso defines a metric structure onTRM.

From the computation g_j¯k = ^∂2

R

∂η^j∂Lη¯^k = ¹₄(_∂y^∂k +i_∂y^∂n+k)(^∂

R

∂yL^j −i ^∂

R

∂y^n+jL ) =

1

4(_∂y^∂j²∂y^RLk +_∂yn+j^∂2∂y^LRn+k) +₄ⁱ(_∂yj^∂∂y^2RLn+k −_∂yn+j^∂2RL∂y^k) we deduce just the real rep- resentation of the matrixg_j¯k:

Rg_jk=1 4( ∂²L^R

∂y^j∂y^k + ∂^2RL

∂y^n+j∂y^n+k) ; ^Rg_jn+k= 1

4( ∂²L^R

∂y^j∂y^n+k − ∂²L^R

∂y^n+j∂y^k) (2.9)

Let us remark that, in general, the tensorCabc= ¹₂{^∂

Rg_bc

∂y^a +^∂

Rg_ac

∂y^b −^∂

Rg_ab

∂y^c} is not totally symmetric and therefore the generalized Lagrange space (M,^Rg_ab) is not always reducible to a Largange space ([10]). Moreover, the space is neither weakly regular, hence the known procedures of Lagrange (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 Lagrange space is a Lagrange one, particularly a Finsler space. This means that the tensorhab= ¹₂ ^∂2

R

∂y^a∂yL^b is nondegenerated. In particular, ifLis a complex Finsler metric then^RLbecomes a real Finsler metric. A sufficient condition is whenhab=^Rg_ab, which in view of (2.9) is equivalent to:

∂^2RL

∂y^j∂y^k = ∂^2RL

∂y^n+j∂y^n+k ; ∂²L^R

∂y^j∂y^n+k =− ∂^2RL

∂y^n+j∂y^k (2.10)

and that happens if and only if : gjk= ∂²L

∂η^j∂η^k = 0 , ∀j, k= 1, n (2.11)

Obviously, by conjugation from (2.11) it results alsog¯j¯k= 0.

Definition 2.1. In the condition (2.11) we call the complex Lagrange space (M, L)being with pure Hermitian metric.

In particular, the Finsler complex space with pure Hermitian metric is ob- tained.

Proposition 2.3. The generalized Lagrange space(M,^Rg_ab)associated to a com- plex Lagrange space with pure Hermitian metric is reductible to a real Lagrange space(M,L).^R

As shown in [10], the variational method in the real Lagrange space (M,L)^R gives an (r.n.c.):

N0_b^a = ∂G^a

∂y^b, where (2.12)

G^a = 1 4

Rg^ac





∂²L^R

∂y^c∂x^dy^d−∂^RL

∂x^c



= 1 4

Rg^acΦc

On the other hand, by the variational method a (c.n.c.), called canonical, in a complex Lagrange space (M, L) is obtained ([11]):

c

N_j^k = ∂H^k

∂η^j , where (2.13)

H^k = 1

2g^mk¯ ∂²L

∂z^h∂η¯^mη^h

Our next goal is to determine the circumstances when N^cR_b^a, the (r.n.c.) in- duced by

c

N_j^k, coincides with N⁰_b^a or, in an equivalent way, when the complex image

0C

N_j^k=

0

N_j^k +i

0

N_j^n+k ofN⁰_b^a coincides with

c

N_j^k.

For this reason we shall calculate the difference d−tensor of two (c.n.c.) : 2D^k_l = ( ∂

∂η^l + ∂

∂¯η^l)(G^k+iG^n+k)−∂H^k

∂η^l (2.14)

First, we make the computation:

4(G^k+iG^n+k) =Re g^mk¯ Φm−Im g^mk¯ Φn+m+i(Im g^mk¯ Φm+Re g^mk¯ Φn+m)

=g^mk¯ (Φm+iΦn+m).

Replacing Φm and Φn+mfrom (2.12) and recalling that _∂y^∂m = _∂η^∂m +_∂η^∂_¯m

andy^k = ¹₂(η^k+ ¯η^k) ; _∂yn+m^∂ =i(_∂η^∂m −_∂η^∂_¯m) and y^n+k = ₂ⁱ(η^k−η¯^k), a long but trivial computation gives:

4D^k_l = ∂

∂η^l[g^mk¯ ( ∂²L

∂η¯^m∂z¯^pη¯^p− ∂L

∂z¯^m)] + (2.15)

∂

∂η¯^l[g^mk¯ ( ∂²L

∂η¯^m∂z^pη^p+ ∂²L

∂η¯^m∂z¯^pη¯^p− ∂L

∂z¯^m)]

Hence, we have:

Proposition 2.4. In the complex Lagrange space(M, L)the induced(r.n.c.)of the

c

N_j^k (c.n.c.)from (2.13) coincides with theN⁰_b^a (r.n.c.)given by (2.12) if and only ifD^k_l = 0.

We recall here that a complex Lagrange space is called local Minkowski ([2],[3]) if there exist local charts in any u = (z, η) such that the Lagrange functionLdepends only on the direction, i.e.,L=L(η,η).¯

The above Proposition and (2.15) yields:

Proposition 2.5. If(M, L)is a complex Lagrange local Minkowski space there exist local charts in anyu= (z, η)∈T⁰M such thatN^cR_b^a=N⁰_b^a .

More interesting results are obtained in the particular case of complex Finsler space, whenL(z, λη) =|λ|²L(z, η) and the consequences from it ([1],[12]).

Then the formulas (2.12) lead to ([10]):

N0F_b^a = 1 2

∂γ₀₀^a

∂y^b , where γ₀₀^a =γ_bc^ay^by^c and (2.16)

γ_bc^a = 1 2

Rg^da{∂^Rg_dc

∂x^b +∂^Rg_bd

∂x^c −∂^Rg_bc

∂x^d } N0F_b^a is the well-known Cartan (r.n.c.).

And the formulas (2.13) give the Cartan (c.n.c.) ([11],[12]):

cF

N_j^k = 1 2

∂Γ^k₀₀

∂η^j , whereΓ^k₀₀= Γ^k_ijηⁱη^j with (2.17)

Γ^k_ij = 1

2g^mk¯ {∂gjm¯

∂zⁱ +∂gim¯

∂z^j } Γ^k_ij being the first complex Christoffel symbol.

Now, acting as before, after a long computation of passing from real to complex, we obtain that N^0F_b^a coincides with ^{cF R}N_b^a, the real image of complex Cartan connection, if and only if the differenced−tensorD^k_l given by:

D^k_l = ∂

∂η^l(Γ^k_i¯jηⁱη¯^j) + ∂

∂η¯^l(Γ^k_ijηⁱη^j+ Γ^k_i¯jηⁱη¯^j) (2.18)

is vanishing, where Γ^k_i¯j = ¹₂g^mk¯ {^∂g_∂¯z^imj^¯ −^∂g_∂z¯^i¯m^j } is the second Chritoffel symbol of the Levi-Civita connection onT⁰M.

Clearly, if Γ^k_i¯j = 0, that is the Levi-Civita connection is of (1,0)−type, or equivalently the fact thatg_i¯j is a K¨ahler metric, then the differenced−tensor is reduced toD^k_l = ^∂Γ_∂η¯^kijlηⁱη^j.Therefore, we can state:

Proposition 2.6. Ifg_i¯j is a K¨ahler metric of the complex Finsler space(M, L) and the coefficients Γ^k_ij of the Levi-Civita linear connection on T⁰M are holo- morphic functions, thenN^0F_b^a coincides with ^{cF R}N_b^a .

Also, let us note that ifg_i¯jlocally depends onT⁰M only onz, 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 K¨ahlerian situation such metric is called Hermitian-K¨ahler ([1], [12]). So, from the local expression of Γ^k_ij , we have:

Proposition 2.7. If g_i¯j is a Hermitian-K¨ahler metric of the complex Finsler space(M, L), then N^0F_b^a coincides with^{cF R}N_b^a .

3. The induced N −real linear connection

Let us study now the real image of other geometric elements onTRM induced from a (c.n.c.) onT⁰M.

LetD:χ(T⁰M)×χ(T⁰M)→χ(T⁰M) be a normal complex linear connection (shortly, N −(c.n.c.)), that is a derivative law on T⁰M which preserves the distributionsV(T⁰M), H(T⁰M) and their conjugates. Locally, anN−(c.l.c.) is characterized by its coefficients (Lⁱ_jk;Lⁱ_jk¯;C_jkⁱ ;C_jⁱ_¯k), where:

Dδkδj=Lⁱ_jkδi; D∂k∂j =C_jkⁱ ∂i

Dδ¯kδj=Lⁱ_j¯kδi; D∂¯k∂j =C_jⁱ¯k∂i

(3.1)

As was proved in [11], D is an N −(c.l.c.) iff DJ = DF = DF^∗ = 0, where J is the complex structure on T⁰M, F is the natural tangent structure andF^∗ is the adjoint tangent structure ofF with respect to the adapted base determined by a given (c.n.c.)N. Locally, F^∗ behaves on the adapted base as

follow: F^∗(δk) = 0; F^∗(∂k) = δk ; F^∗(δ¯k) = 0; F^∗(∂k¯) = δk¯ , and globally defined.

Correspondingly, on TRM we have in addition to the complex structureJ^R, the natural tangent structureF^Rdefined by F^R(X^R) = (F X)^R, and its adjoint tangent structureF^R∗ in respect to an adapted base. Locally, their actions are given by:

FR(_δx^δa) = _∂y^∂a; F^R(_∂y^∂a) = 0; F^R∗(_δx^δa) = 0; F^R∗ (_∂y^∂a) =_δx^δa

Then a derivative lawD^R onTRM is an N^R −real linear connection, shortly NR−(r.l.c.), conformity to [10] iffD^RJ^R=D^RF^R=D^RF^R∗= 0.

Theorem 3.1. If D is an N−(c.l.c.) on T⁰M, then the following derivative

law: R

DAB=DAB , ∀A, B∈χ(TRM) (3.2)

or in other words:

DRX^RY^R= (DXY)^R+ (DXY¯)^R, ∀X, Y ∈χ(T⁰M) (3.3)

is anN^R −(r.l.c.)on TRM.

Proof. Let us remark thatD^R is a linear connection onT⁰M.SinceJ(A+iB) =J^R (A) +iJ^R(B) and taking into account the definitions ofF^RandF^R∗structures it is verified thatD^RJ^R=D^RF^R=D^RF^R∗= 0.

Moreover, if D is of (1,0)−type, because DJXY =DXJY, it results that DRR

JAB=D^RA

J B.R

If theN^R−(r.l.c.)D^R is given, obviously thenD is obtained by linearity.

Now, let us suppose thatD^R is given in the local base by its coefficients ([10]):

DR ^δ

δxc

δ

δx^b =L^Ra_bc δ

δx^a ; D^R∂

∂yc

δ

δx^b =C^R_bc^a δ δx^a

DR ^δ

δxc

∂

∂y^b =L^Ra_bc ∂

∂y^a ; D^{R ∂}

∂yc

∂

∂y^b =C^R_bc^a ∂

∂y^a (3.4)

Then making the computations in (3.3), thanks to (1.6) and (2.5) formulas, we find the relation between the coefficients of inducedN^R−(r.l.c.)D^R and N− (c.l.c)D:

R

Lⁱ_jk=

R

Lⁿ⁺ⁱ_n+jk=Re(Lⁱ_jk+Lⁱ_J¯k) (3.5)

R

Lⁿ⁺ⁱ_jk =

R

−Lⁱ_n+jk=Im(Lⁱ_jk+Lⁱ_J¯k)

R

Lⁱ_jn+k=

R

Lⁿ⁺ⁱ_n+jn+k=Im(Lⁱ_j¯k−Lⁱ_jk)

R

Lⁿ⁺ⁱ_jn+k=

R

Lⁱ_n+jn+k=Re(Lⁱ_jk−Lⁱ_Jk¯)

Definition 3.1. AnN^R −(r.l.c.)D^R whose coefficients are connected by the rela- tions: L^Ri_jk=

R

Lⁿ⁺ⁱ_n+jk;

R

Lⁿ⁺ⁱ_jk =−L^Ri_n+jk; Lⁱ_jn+k^R =

R

Lⁿ⁺ⁱ_n+jn+k ;

R

Lⁿ⁺ⁱ_jn+k=Lⁱ_n+jn+k^R will be called of Hermitian type.

The definition is justified by the fact that for a fixed indexc the coefficient LR^a_bcis the real representation of a Hermitian matrix.

Proposition 3.1.D^R is anN^R −(r.l.c.)of Hermitian type if and only ifD^RJ^R=J^RD .^R IfDis of (1,0)−type then formulas (3.5) are simplified, becauseLⁱ_j¯k=C_jⁱ_k¯= 0.

The calculus of bracket gives that £ X, Y¤

= [X, Y] and hence we have

£X^R, Y^R¤

= [X, Y]^R. So, the curvature and the torsion of the induced N^R

−(r.l.c.) D^R are expressed as a function of the curvature and respectively the torsion ofN−(c.l.c.)D as follows:

RR (X^R, Y^R)Z^R=R(X^R,Y^R)Z^R; T^R(X^R,Y^R) =T(X^R,Y^R) (3.6)

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 metricGonT⁰Mis : G=Re G+iIm G=G^R +i

R˜ G .

IfDis a metricalN−(c.l.c.) , i.e. (DXG)(Y, Z) =XG(Y, Z)−G(DXY, Z)−

G(Y, DXZ) = 0, replacing X, Y, Z to their real parts, we obtain thatD^RG=^R D^R

R˜ G

= 0. Therefore, D^R is metrical with respect to both real metric induced by G fromT⁰M.

In a real case ([10]), and in the complex one ([11]), are known metrical N−linear connections. In a real Lagrange geometry one has a special meaning, the so-called real canonical, or Miron’s metricN^R −(r.l.c.) :

LcR^a_bc=1 2

Rg^da{δ^Rg_dc

δx^b +δ^Rg_bd

δx^c −δ^Rg_bc δx^d } (3.7)