Complex Lagrange Spaces

Gheorghe Munteanu

Abstract The geometry of Lagrange spaces is a field of large interest that knows

a permanent development [6].

In the present paper our desire is to extend the results from the geomety of complex Finsler manifolds, recently obtained by T. Aikou [1], to the case of manifolds endowed with a complex Lagrangian. We shall make a study of holomorphic bundle of (1,0) vector fields using a nonlinear connection determined by the complex Lagrange metric. Finally, the discovered results will be applied to the case of complex Lagrangian of electromagnetic field.

Mathematics Subject Classification: 53C60, 53C55 Key Words: generalized Lagrange spaces, complex geometry.

1 The geometry of T′M holomorphic bundle

We consider M be a complex manifold, dim_{C}M = n, (U, (z^{i})) complex coordinates in a local map. The complexification T_{C}M of tangent bundle T M is decomposed at each point z ∈ U after the (1, 0) vector fields and their conjugate of (0,1) type ([3], [5]), T_{C}M = T′M ⊕ T′′M.

The bundle T′M is a holomorphic bundle and the projection π_{T} : T′M → M is a holomorphic map, dim_{C}T′M = 2n, and T′′M is the conjugate of T′M.

We denote by (π_{T}^{−1}(U), z^{i}, η^{i}) the induced complex coordinates on T′M. The transition functions of the holomorphic tangent bundle T′(T′M), of the complex manifold T′M, are given by

(1.1)  ( \begin{matrix}∂z^{′j}/∂z^{i} & 0 \\ (∂^{2}z^{′j}/∂z^{i}∂z^{k})η^{k} & ∂z^{′j}/∂z^{i} \end{matrix} ) .

According to [1], a vector bundle Ẽ over T′M is said to be a Finsler complex bundle if Ẽ = π_{T}^{−1}(E), i. e. Ẽ is the pull-back of a complex bundle E over M. If E is holomorphic, then Ẽ is called a holomorphic Finsler vector bundle.

The vertical subbundle

V (T′M) = {ξ ∈ T′(T′M) / π_{T ∗}(ξ) = 0}

Balkan Journal of Geometry and Its Applications, Vol.3, No.2, 1998, pp. 61-71 \begin{matrix}c \\ ◯ \end{matrix}Balkan Society of Geometers, Geometry Balkan Press

is a vector bundle of rank n over T′M; a base of local fields is {∂/∂η^{i}} and from (1.1) it follows that V (T′M) is a holomorphic bundle of Finsler type.

Let H(T′M) be a supplementary subundle of V (T′M) in T′(T′M) i. e. T′(T′M) = H(T′M) ⊕ V (T′M). This spliting determines a distribution N : u = (z^{i}, η^{i}) → H_{u}(T′M) which will be called a complex nonlinear connection. Let us consider

(1.2)  \frac{δ}{δz^{i}} = \frac{∂}{∂z^{i}} − N_{i}^{j} \frac{∂}{∂η^{j}} ,

a local base on H_{u}(T′M). The complex functions N_{i}^{j}(z, η) are called the coefficients of the complex nonlinear connection. After (1.1) the functions N_{i}^{j} satisfy the following transformation law

(1.2′)  N_{k}^{′i} \frac{∂z^{′k}}{∂z^{j}} = \frac{∂z^{′i}}{∂z^{k}} N_{j}^{k} − \frac{∂^{2}z^{′i}}{∂z^{j}∂z^{k}} η^{k}.

From the spliting of T′(T′M) it results that H(T′M) is a Finsler complex vector bundle of rank n therefore the adapted base {δ/δz^{i}} changes by the rule

(1.3)  \frac{δ}{δz^{i}} = \frac{∂z^{′j}}{∂z^{i}} \frac{δ}{δz^{′j}} .

The dual base of (δ/δz^{i}, ∂/∂η^{i}) will be denoted by (dz^{i}, δη^{i}).

Consider the conjugates \overline{H(T′M)} and \overline{V (T′M)}. These are subbundles of T′′(T′M). It results the following decomposition of the complexified tangent bundle (1.4) T_{C}(T′M) = T′(T′M) ⊕ T′′(T′M) = H(T′M) ⊕ V (T′M) ⊕ \overline{H(T′M)} ⊕ \overline{V (T′M)}

The corresponding projectors on the distributions are denoted by h, v, \overline{h} and \overline{v}.

Taking into account that \frac{∂}{∂η^{i}} = \frac{∂z^{′j}}{∂z^{i}} \frac{∂}{∂η^{′j}}, by conjugation it results Proposition 1.1. A local base on the conjugate T′′(T′M) is

(\frac{δ}{δ\overline{z}^{i}} = \frac{∂}{∂\overline{z}^{i}} − \overline{N}_{i}^{j} \frac{∂}{∂\overline{η}^{j}}; \frac{∂}{∂\overline{η}^{i}} )

and it is changed by the rules

(1.5)  \frac{δ}{δ\overline{z}^{i}} = \frac{∂\overline{z}^{′j}}{∂\overline{z}^{i}} \frac{δ}{δ\overline{z}^{′j}}; \frac{∂}{∂\overline{η}^{i}} = \frac{∂\overline{z}′}{∂\overline{z}^{i}} \frac{∂}{∂\overline{η}^{′j}} ,

where \overline{N}_{i}^{j}(z, η) are the conjugates of coefficients of the nonlinear connection N_{i}^{j} and are transformed by the rule obtained from (1.2′) by conjugation.

If E = T′M then T̃′M is a holomorphic bundle of Finsler type that can be identified with H(T′M) ⊕ V (T′M).

Since T′M is a holomorphic manifold, on the complexification T_{C}(T′M) there exists a natural complex strucure J defined by

J(∂/∂z^{k}) = i∂/∂z^{k} ; J(∂/∂η^{k}) = i∂/∂η^{k} ;
J(∂/\overline{z}^{k}) = −i∂/∂\overline{z}^{k} ; J(∂/∂\overline{η}^{k}) = −i∂/∂\overline{η}^{k} , (i^{2} = −1),

which is globaly defined.

For simplicity, in the following we shall use for the adapted bases the abreviations

δ_{k} = δ/δz^{k} ; ∂_{k} = ∂/∂η^{k} and δ\overline{k} = −iδ/δ\overline{z}^{k} ; ∂\overline{k} = ∂/∂\overline{η}^{k}.

Hence, in adapted bases the complex structure are written

(1.6)  J(δ_{k}) = iδ_{k} ; J(∂_{k}) = i∂_{k} ; J(δ\overline{k}) = −iδ\overline{k} ; J(∂\overline{k}) = −i∂\overline{k}.

Moreover, let us consider the adjoint almost tangent structures F and F^{∗}

(1.7)  \begin{matrix}F (δ_{k}) = ∂_{k}; F (∂_{k}) = 0; F (δ\overline{k}) = ∂\overline{k}; F (∂\overline{k}) = 0 \\ F^{∗}(δ_{k}) = 0; F^{∗}(∂_{k}) = δ_{k}; F^{∗}(δ\overline{k}) = 0; F^{∗}(∂\overline{k}) = δ\overline{k} \end{matrix}

globaly defined as it results from (1.3) and (1.5).

The following relations hold

F · F^{∗} = v + \overline{v} ; F^{∗} · F = h + \overline{h} ; J · F = F · J ; J · F^{∗} = F^{∗} · J

and

F^{2} = 0 F^{∗} = 0 J^{2} = −I,

i.e., the structures (J, F, J · F) and (J, F^{∗}, J · F^{∗}) are commutative almost semiqaternionic structures ([8]) , adjoint one to the other.

Let us consider D a derivative law on the complexification T_{C}(T′M).

Through analogy to the real case ([6]), we say that D is a N-linear complex connection (shortly N-l.c.c.) if it preserves the four distributions determined by the complex nonlinear connection N.

In adapted base (δ_{k}, ∂_{k,}δ\overline{k}, ∂\overline{k}) the N-l.c.c. D has the local expression (1.8)

D_{δ_{k}}δ_{j} =L^{1}_{jk}^{i} δ_{i}; D_{∂_{k}}δ_{j} =C^{1}_{jk}^{i} δ_{i}; D_{δ\overline{k}}δ_{j} =L^{3}_{j\overline{k}}^{i} δ_{i}; D_{∂\overline{k}}δ_{j} =C^{3}_{j\overline{k}}^{i} δ_{i}

D_{δ_{k}}∂_{j} =L^{2}_{jk}^{i} ∂_{i}; D_{∂_{k}}∂_{j} =C^{2}_{jk}^{i} ∂_{i}; D_{δ\overline{k}}∂_{j} =L^{4}_{j\overline{k}}^{i} ∂_{i}; D_{∂\overline{k}}∂_{j} =C^{4}_{j\overline{k}}^{i} ∂_{i}

D_{δ_{k}}δ\overline{j} =L^{3}\overline{i} ; \overline{j}k δ\overline{i}; D_{∂_{k}}δ\overline{j} =C^{3}\overline{i} ; \overline{j}k δ\overline{i}; D_{δ\overline{k}}δ\overline{j} =L^{1}\overline{i} ; \overline{j}\overline{k} δ\overline{i}; D_{∂\overline{k}}δ\overline{j} =C^{1}\overline{i} ; \overline{j}\overline{k} δ\overline{i}

D_{δ_{k}}∂\overline{j} =L^{4}\overline{i} ; \overline{j}k ∂\overline{i}; D_{∂_{k}}∂\overline{j} =C^{4}\overline{i} ; \overline{j}k ∂\overline{i}; D_{δ\overline{k}}∂\overline{j} =L^{2}\overline{i} ; \overline{j}\overline{k} ∂\overline{i}; D_{∂\overline{k}}∂\overline{j} =C^{2}\overline{i} ; \overline{j}\overline{k} ∂\overline{i}.

Because \overline{D_{X}Y} = D\overline{X} \overline{Y} it results that some of the coefficients in (1.8) are

the conjugate of the others, for instance L^{1}\overline{i} ; \overline{j}\overline{k}= \overline{L^{1}_{jk}^{i}}, etc.

Let us note that a N-l.c.c. is decomposed in D = D′ + D′′, with D′′ = \overline{D′} and in order D′ = D^{′h} + D^{′v} ; D′′ = D^{′′\overline{h}} + D^{′′\overline{v}}, where D^{′h} = D_{δ_{k}}, D^{′v} = D_{∂_{k}}, D^{′′\overline{h}} = D_{δ\overline{k}} , D^{′′\overline{v}} = D_{∂\overline{k}}. Proposition 1.2. D is a N-l.c.c. if and only if DJ = 0 and Dv = 0. Proof. DJ = 0 implies that D′ is a derivative law of (1,0)-type, and D′′ of (0,1)-type. Dv = 0 is the condition to preserve the H(T′M) and V (T′M) on T′(T′M). By conjugation it results D\overline{v} = 0.

The N-l.c.c. D is said to be normal if in addition the connection coefficients satisfy the following conditions

L^{1}_{jk}^{i}=L^{2}_{jk}^{i} ; L^{3}_{\overline{j}k}^{i}=L^{4}\overline{i} ; \overline{j}k ; C^{1}_{jk}^{i}=C^{2}_{jk}^{i} ; C^{3}\overline{i} ; \overline{j}k=C^{4}\overline{i} ; \overline{j}k

(and their conjugates). In the case of a normal N-l.c.c. we shall use a simple written form for the coefficients

L^{1}_{jk}^{i}= L_{jk}^{i} ; L^{3}_{\overline{j}k}^{i}= L\overline{i} ; \overline{j}k ; C^{1}_{jk}^{i}= C_{jk}^{i} ; and C^{3}\overline{i} ; \overline{j}k= C\overline{i} ; \overline{j}k.

Theorem 1.3. D is a normal N-l.c.c. if and only if it is a commutative semiquaternionic connection with respect to both structures, i.e.,

DJ = 0, DF = 0, DF^{∗} = 0.

The proof results from the fact F · F^{∗} = v and D_{X}(F Y) = F D_{X}Y for any X and Y vector fields of the adapted base. The general family of such connections is given in [8] for the anticommutative case.

For a normal N-l.c.c. we can write without any difficuly the components of torsions and curvatures. The nonvanishing torsions are

(1.9)  \begin{matrix}T_{ij}^{k}δ_{k} = hT (δ_{j}, δ_{i}) & Θ_{i\overline{j}}^{k}∂_{k} = vT (δ\overline{j}, δ_{i}) & τ_{i\overline{j}}^{k}δ_{k} = hT (δ\overline{j}, δ_{i}) \\ R_{ij}^{k}∂_{k} = vT (δ_{j}, δ_{i}) & Θ\overline{k} ; i\overline{j}∂\overline{k} = \overline{v}T (δ\overline{j}, δ_{i}) & τ\overline{k} ; i\overline{j}δ\overline{k} = \overline{h}T (δ\overline{j}, δ_{i}) \\ Q_{ij}^{k}δ_{k} = hT (∂_{j}, δ_{i}) & ρ_{i\overline{j}}^{k}∂_{k} = vT (∂\overline{j}, δ_{i}) & Υ_{i\overline{j}}^{k}δ_{k} = hT (∂\overline{j}, δ_{i}) \\ P_{ij}^{k}∂_{k} = vT (∂_{j}, δ_{i}) & Σ\overline{k} ; i\overline{j}∂\overline{k} = \overline{v}T (δ\overline{j}, ∂_{i}) & χ_{i\overline{j}}^{k}∂_{k} = vT (∂\overline{j}, ∂_{i}) \\ S_{ij}^{k}∂_{k} = vT (∂_{j}, δ_{i}) & Σ_{i\overline{j}}^{k}∂_{k} = vT (δ\overline{j}, ∂_{i}) & χ\overline{k} ; i\overline{j}∂\overline{k} = \overline{v}T (∂\overline{j}, ∂_{i}) \end{matrix}

and their conjugates.

The local expresions of torsions are obtained by direct calculation

(1.10)  \begin{matrix}T_{ij}^{k} = L_{ij}^{k} − L_{ji}^{k}, & Θ_{i\overline{j}}^{k} = \frac{δN_{i}^{k}}{δ\overline{z}^{j}} , & τ_{i\overline{j}}^{k} = L_{ij}^{k} \\ R_{ij}^{k} = \frac{δN_{j}^{k}}{δz^{i}} − \frac{δN_{i}^{k}}{δz^{j}} , & Θ\overline{k} ; i\overline{j} = −^{\frac{δ\overline{N}_{j}^{k}}{δz^{i}}} , & τ\overline{k} ; i\overline{j} = −L\overline{k} ; i\overline{j} \\ Q_{ij}^{k} = C_{ij}^{k}, & ρ_{i\overline{j}}^{k} = −\frac{∂N_{i}^{k}}{∂\overline{η}^{j}} & Υ_{i\overline{j}}^{k} = C_{i\overline{j}}^{k} \\ P_{ij}^{k} = \frac{∂N_{i}^{k}}{∂η^{j}} − L_{ji}^{k}, & Σ\overline{k} ; i\overline{j} = −^{\frac{∂\overline{N}_{j}^{k}}{∂η^{i}}} , & χ_{i\overline{j}}^{k} = C_{i\overline{j}}^{k} \\ S_{ij}^{k} = C_{ij}^{k} − C_{ji}^{k}, & Σ_{i\overline{j}}^{k} = −L_{i\overline{j}}^{k}, & χ\overline{k} ; i\overline{j} = −C\overline{k} ; \overline{j}i \end{matrix}

and their conjugates.

The nonvanishing curvatures are

(1.11) \begin{matrix}R_{jkh}^{i}δ_{i} = R(δ_{h}, δ_{k})δ_{j} & R\overline{i} ; \overline{j}khδ\overline{i} = R(δ_{h}, δ_{k})δ\overline{j} & R\overline{i} ; \overline{j}\overline{k}hδ\overline{i} = R(δ_{h}, δ\overline{k})δ\overline{j} \\ S_{jkh}^{i}∂_{i} = R(∂_{h}, ∂_{k})∂_{j} & S\overline{i} ; \overline{j}kh∂\overline{i} = R(∂_{h}, ∂_{k})∂\overline{j} & S\overline{i} ; \overline{j}\overline{k}h∂\overline{i} = R(∂_{h}, ∂\overline{k})∂\overline{j} \\ P_{jkh}^{i}∂_{i} = R(∂_{h}, δ_{k})∂_{j} & P \overline{i} ; \overline{j}kh∂\overline{i} = R(∂_{h}, δ_{k})∂\overline{j} & P \overline{i} ; \overline{j}\overline{k}h∂\overline{i} = R(∂_{h}, δ\overline{k})∂\overline{j} \\ Ω_{j\overline{k}h}^{i}∂_{i} = R(δ_{h}, δ\overline{k})∂_{j} & Ω\overline{i} ; \overline{j}\overline{k}h∂\overline{i} = R(δ_{h}, δ\overline{k})∂\overline{j} & Ω\overline{i} ; \overline{j}kh∂\overline{i} = R(δ_{h}, δ_{k})∂\overline{j} \\ Π_{j\overline{k}h}^{i}∂_{i} = R(δ_{h}, ∂\overline{k})∂_{j} & χ\overline{i} ; \overline{j}k\overline{h}∂\overline{i} = R(δ\overline{h}, ∂_{k})∂\overline{j} \end{matrix}

and conjugates.

The local expressions of curvatures are easily obtained from (1.11).

A system of functions W_{h_{1}...h_{q} \overline{h}_{1}...\overline{h}_{s}}^{k_{1}...k_{p}\overline{k}_{1}...\overline{k}_{k}}(z, η) is a complex d-tensor field if at (1,1) transformations are changed by the rule

(1.12)  W_{j_{1}.....j_{q} \overline{j}_{1}.....\overline{j}_{s}}^{′i_{1}.....i_{p}\overline{i}_{1}.....\overline{i}_{r}}(z′, η′) = \frac{∂z^{′i_{1}}}{∂z^{k_{1}}} ⋯\frac{∂z^{′i_{p}}}{∂z^{k_{p}}} · \frac{∂z^{h_{1}}}{∂z^{′j_{1}}} ⋯\frac{∂z^{h_{r}}}{∂z^{′j_{r}}} ·

·\frac{∂\overline{z}^{′i_{1}}}{∂\overline{z}^{k_{1}}} ⋯\frac{∂\overline{z}^{′i_{r}}}{∂\overline{z}^{k_{r}}} · \frac{∂\overline{z}^{h_{1}}}{∂\overline{z}^{′j_{1}}} ⋯\frac{∂\overline{z}^{h_{s}}}{∂\overline{z}^{′j_{s}}} W_{h_{1}...h_{q} \overline{h}_{1}...\overline{h}_{s}}^{k_{1}...k_{p}\overline{k}_{1}...\overline{kr}}(z, η).

Examples of complex d-tensor fields are: δ_{k} , δ\overline{k} , ∂_{k} , ∂\overline{k} , C_{jk}^{i} , C\overline{i} ; \overline{j}\overline{k} , C\overline{i} ; \overline{j}k the components of torsions and curvatures, etc.

A Hermitean metric structure on T_{C}(T′M) is a nondegenerate 2-field G(X, Y) = \overline{G(Y, X)} and with respect to the commutative almost semiquaternion structure (J , F) it satisfies the conditions

(1.13)  \begin{matrix}G(JX, JY) = G(X, Y) \\ G(F X, F Y) = G((h + \overline{h})X(h + \overline{h})Y). \end{matrix}

Expressing G in adapted base and taking into account the required conditions (1.13), we obtain Proposition 1.3. G defines a Hermitian metric structure on T_{C}(T′M) if and only if

(1.14)  G = g_{i\overline{j}}dz^{i} ⊗ d\overline{z}^{j} + g_{i\overline{j}}δη^{i} ⊗ δ\overline{η}^{j},

where g_{i\overline{j}}(z, η) is a complex d-tensor field of ( \begin{matrix}0 & \overline{0} \\ 1 & \overline{1} \end{matrix} ) type and nondegener-

ate.

From F · F^{∗} = v + \overline{v} it results that G(F^{∗}X, F^{∗}Y) = G((v + \overline{v})X, (v + \overline{v})Y). A complex N-linear conection D is said to be metric with respect to the

Hermitian metric structure G if DG = 0.

Let’s denote by ” † ” and by ” | ”the h- and respectively v-covariant derivatives of one complex d- tensor field with respect to D′, and by ” † †” and ” || ”, the \overline{h}- and respectively \overline{v}-derivatives with respect to D′′. It follows that D is a complex metric N-linear conection if and only if:

(1.15)  g_{i\overline{j} †k} = 0 , g_{i\overline{j} | k} = 0 , g_{i\overline{j} ††\overline{k}} = 0 , g_{i\overline{j} || \overline{k}} = 0

Analogous reasons as in [7] show that if D is a complex metric d-linear connection then its distributions are ortogonal, and if D is compatible with respect to one metric semiquaternionic structure (G, J, F), then D is also, with respect to the other (G, J, F^{∗}) structure. Therefore, D is also a normal d-l.c.c.

In [7] is given the general set of metric almost semiquaternionic connections for the anticommutative case. In the present commutative case similar calculus gives: Theorem 1.4. If ∇^{∘} is the Levi-Civita connection of the metric G, then a metric connection with respect to the structure (G, J, F) is:

(1.16) \begin{aligned}2 D^{c}_{X} Y & = & (v + \overline{v}) ∇^{∘}_{X} (v + \overline{v})Y − (v + \overline{v})J ∇^{∘}_{X} (v + \overline{v})JY + \\ & + & F^{∗} ∇^{∘}_{X} F Y − JF^{∗} ∇^{∘}_{X} JF^{∗}Y. \end{aligned}

It is easly to find that D^{c} is a normal d-l.c.c. and is metric because ∇^{∘} is metric. It’s local expression will be given in the next section.

2 Complex Lagrange spaces

The notion of complex Finsler space is already well-known from the papers of S. Kobayashi [4], H. Rund [10], H. Royden [9], M. Fukui [2], and T. Aikou [1]. Giving up to the homogeneity condition with respect to η, we shall define the notion of complex Lagrange metric. Definition 2.1. A complex Lagrangian on T′M is a smooth real function L : T′M → R, with the property that

(2.1)  g_{i\overline{j}} = \frac{1}{2} \frac{∂^{2}L}{∂η^{i}∂\overline{η}^{j}}

is a nondegenerate complex d-tensor field.

The pair (T′M, L) is called a complex Lagrange space. In the particular case when L : (z, η) → L(z, η) is absolutely homogeneous of degree 1 in η we obtain the notion of complex Finsler space.

As it is known, the existence of a Lagrange function involves the study of the variational problem on curves. Let L(z, η) be a complex Lagrangian on T′M and c : [0, 1] → M be a holomorphic curve. The Euler-Lagrange equation for a geodesic is ([1], [9]):

(2.2)  \frac{∂L}{∂z^{i}} − \frac{d}{dt} (\frac{∂L}{∂η^{i}} ) = 0 with η^{i} = \frac{dz^{i}}{dt} .

Developing the calculus in (2.2), we obtain

(2.3)  (2g_{i\overline{j}} \frac{d^{2}\overline{z}^{j}}{dt^{2}} + \frac{∂^{2}L}{∂\overline{z}^{j}∂η^{i}} \overline{η}^{j}) + (2a_{ij} \frac{d^{2}z^{j}}{dt^{2}} + \frac{∂^{2}L}{∂z^{j}∂η^{i}} η^{j} − \frac{∂l}{∂z^{i}}) = 0,

where a_{ij} = \frac{1}{2} \frac{∂^{2}L}{∂η^{i}∂η^{j}} is a symmetric d-tensor field.

In [1], [9] is defined the notion of complex geodesic in a complex Finsler space as being a curve γ(t) = φ(e^{iθ} · t) independent of θ, and φ is a holmorphic map on the small disk Δ(r) to M satisfying φ(0) = z and dφ(0) = η, where (z, η) is a fixed point.

Taking into account the independence on θ of γ, we shall define a geodesic complex Lagrange space as being a curve that satisfies the system of equations

(2.4)  2g_{i\overline{j}} \frac{d^{2}\overline{z}^{j}}{dt} + \frac{∂^{2}L}{∂\overline{z}^{j}∂η^{i}} \overline{η}^{j} = 0

(2.5)  2a_{ij} \frac{d^{2}z^{j}}{dt^{2}} + \frac{∂^{2}L}{∂z^{j}∂η^{i}} η^{j} − \frac{∂L}{∂z^{i}} = 0.

Theorem 2.1. Let’s consider the functions H^{j} given by

(2.6)  H^{j} = \frac{1}{4}g^{\overline{i}j} \frac{∂^{2}L}{∂z^{k}∂\overline{η}^{i}} η^{k}.

Then N_{i}^{j} = \frac{∂H^{j}}{∂η^{i}} are the coefficients of one complex nonlinear connection. The proof consists in verifying by direct calculus that N_{i}^{j} satisfies the law

(1.2), and hence these are the coefficients of a complex nonlinear connection on T′(T′M). The conjugate \overline{N}_{i}^{j} = \frac{∂\overline{H}^{j}}{∂\overline{η}^{i}}, where \overline{H}^{j} = \frac{1}{4}g^{\overline{i}j} \frac{∂^{2}L}{∂\overline{z}^{k}∂η^{i}} \overline{η}^{k} are the coefficients of a complex nonlinear connection on T′′(T′M).

Let’s note that if (a_{ij}) is a nondegenerate complex matrix, then N_{i}^{j} = \frac{∂G^{j}}{∂η^{i}}, with G^{j} = \frac{1}{4}a^{ij}(\frac{∂^{2}L}{∂z^{k}∂η^{i}} η^{k} − \frac{∂L}{∂z^{i}}) determine another complex nonlinear connection on T′(T′M).

In the real case it is known that a geodesic determines a spray and conversely ([6]). We shall extend this topic to the complex case.

Let be Γ =η^{i} \frac{∂}{∂η^{i}} + \overline{η}^{i} \frac{∂}{∂\overline{η}^{i}} the complex Liouville d-field and T the natural almost tangent structure on T′M , T (\frac{∂}{∂z^{i}}) = \frac{∂}{∂η^{i}} , T (\frac{∂}{∂η^{i}}) = 0 , T (\frac{∂}{∂\overline{z}^{i}}) = \frac{∂}{∂\overline{η}^{i}} , T (\frac{∂}{∂\overline{η}^{i}}) = 0 (globally defined because T′M is a holomorphic manifold). A complex spray is avector field S ∈ χ(T′M) with the property T (S) = Γ. It follows that a complex spray is locally given in the form

(2.7)  S = η^{i} \frac{∂}{∂z^{i}} − 2G^{i} \frac{∂}{∂η^{i}} + \overline{η}^{i} \frac{∂}{∂\overline{z}^{i}} − 2\overline{H}^{i} \frac{∂}{∂\overline{η}^{i}} ,

where G^{i} and \overline{H}^{i} are the coefficients of the spray and have the following rules of transformation:

(2.8) 2G^{′i} = 2\frac{∂z^{′i}}{∂z^{j}} G^{j} − \frac{∂^{2}z^{′i}}{∂z^{k}∂z^{j}} η^{k}η^{j} and 2\overline{H}^{′i} = 2\frac{∂\overline{z}^{′i}}{∂\overline{z}^{j}} \overline{H}^{j} − \frac{∂^{2}\overline{z}^{′i}}{∂\overline{z}^{k}∂\overline{z}^{j}} \overline{η}^{k}\overline{η}^{j}.

Proposition 2.2. The functions N_{j}^{i}^{(1)}= \frac{∂G^{i}}{∂η^{j}} and N_{j}^{i}^{(2)}= \frac{∂\overline{H}^{i}}{∂\overline{η}^{j}} define complex nonlinear connections. Remark. From (2.7) and (2.8) it results in particular that \overline{H}^{i} can be taken the conjugate of G^{i}. Proposition 2.3. A complex spray S determines a complex differential system

(2.9)  \frac{d^{2}z^{i}}{dt^{2}} + 2G^{i}(z, η, \overline{z}, \overline{η}) + \frac{d^{2}\overline{z}^{i}}{dt^{2}} + 2\overline{H}^{i}(z, η, \overline{z}, \overline{η}) = 0

and coversely. Proof. Let’s consider the complex spray S and the real curves c : t → (z + \overline{z}) with the tangent vectors \frac{d(z^{i}+\overline{z}^{i})}{dt} = η^{i} + \overline{η}^{i}. By derivation is obtained: \frac{d^{2}z^{i}}{dt^{2}} + \frac{d^{2}\overline{z}^{i}}{dt^{2}} = \frac{d}{dt}(η^{i} + \overline{η}^{i}) = −2x^{i}, where x^{i} is a real number that can be of the form x^{i} = G^{i} + \overline{H}^{i} (with ImG^{i} = −Im\overline{H}^{i}). Conversely, G^{i} and \overline{H}^{i} from (2.9) determines the complex spray given by (2.7). Proposition 2.4. A complex spray S determines a complex geodesic of the complex Lagrange space (M, L) if and only if there exist such local maps on T′M, in wich the coordinates η^{i} are real. Proof. The equation (2.3) is a complex spray iff g_{i\overline{j}} = a_{ij}, and therefore \frac{∂}{∂η^{i}} (\frac{∂L}{∂η^{j}} − \frac{∂L}{∂\overline{η}^{j}}) = 0 ⇒ \frac{∂}{∂η^{j}} = \frac{∂}{∂\overline{η}^{j}} ⇒ η^{j} ∈ R.

In the next we shall make some references to the particular case of complex Finsler spaces.

From the homogeneity condition L(z, λη) =| λ |^{2} L(z, η), it results the complex Euler Theorem, \frac{∂L}{∂η^{i}} η^{i} = 2L and the conjugate \frac{∂L}{∂\overline{η}^{i}} \overline{η}^{i} = 2L. Deriving, it results that

(2.10)  \frac{∂g_{k\overline{j}}}{∂η^{i}} η^{i} = \frac{∂g_{i\overline{j}}}{∂η^{k}} η^{i} = g_{k\overline{j}} and \frac{∂g_{j\overline{k}}}{∂\overline{η}^{i}} \overline{η}^{i} = \frac{∂g_{j\overline{i}}}{∂\overline{η}^{k}} \overline{η}^{i} = g_{j\overline{k}}.

Let us consider Γ_{jk}^{i} = \frac{1}{2}g^{\overline{l}j}(\frac{∂g_{j\overline{l}}}{∂z^{k}} + \frac{∂g_{k\overline{l}}}{∂z^{j}}) the first Christoffel symbols of the Hermitean metric g_{i\overline{j}} ([3], [5]) and let’s denote by E = g_{i\overline{j}}η^{i}\overline{η}^{j} the complex energy of complex Finsler space (M, L). Theorem 2.5. The functions

(2.11)  N_{i}^{j}^{c}= \frac{1}{2} \frac{∂Γ_{00}^{j}}{∂η^{i}} , where Γ_{00}^{j} = Γ_{kl}^{j}η^{k}η^{l},

are the coefficients of a complex nonlinear connection on T′M, called the Cartan nonlinear connection of the Finsler space (M, L). Proof. Using (2.10), by direct calculus it is found out that H^{j} from (2.6) is: H^{j} = \frac{1}{2}Γ_{00}^{j} = \frac{1}{2}g^{\overline{i}j} \frac{∂g_{l\overline{i}}}{∂z^{k}} η^{l}η^{k}.

From (2.4) and (2.5) it results that: Proposition 2.6. The autoparallel curves of the Cartan nonlinear connection of the complex Finsler space (M, L) are just the extremal curves of the energy action E.

Now, let’s return to the general case of the complex Lagrangian space (M, L), endowed with a fixed nonlinear connection, for example given by (2.6).

From Prop. 1.3 it results that G defined by (1.14) with the Lagrange metric g_{i\overline{j}} given by (2.1), is a Hermitean metric structure on T_{C}(T′M). Then the set (M, G, J, F) defined by (1.6), (1.7) and (1.13) is called the metric commutative almost semiquater nionic model of the complex Lagrange space (M, L).

We are interested to determine a metric d-l.c.c. of the complex Lagrange space (M, L). Theorem 2.7. The following normal N-linear complex connection D^{c} is metric

(2.12)  \begin{matrix}L_{jk}^{i} = \frac{1}{2}g^{\overline{l}i}(\frac{δg_{j\overline{l}}}{δz^{k}} + \frac{δg_{k\overline{l}}}{δz^{j}}) & C_{jk}^{i} = \frac{1}{2}g^{\overline{l}i}(\frac{∂g_{j\overline{l}}}{∂η^{k}} + \frac{∂g_{k\overline{l}}}{∂η^{j}}) \\ L\overline{i} ; \overline{j}k = \frac{1}{2}g^{\overline{i}l}(\frac{δg_{l\overline{j}}}{δz^{k}} − \frac{δg_{k\overline{j}}}{δz^{l}}) & C\overline{i} ; \overline{j}k = \frac{1}{2}g^{\overline{i}l}(\frac{∂g_{l\overline{j}}}{∂η^{k}} − \frac{∂g_{k\overline{j}}}{∂η^{l}}) \end{matrix}

Moreover, D^{c} has the following vanishing torsions: T_{jk}^{i} = 0 and S_{jk}^{i} = 0.

Proof. Let ∇^{∘} be the Levi-Civita connection of the metric G given by (1.14) with the complex Lagrange metric g_{i\overline{j}}. Since G is Hermitean with respect to J, the coefficients Γ_{jk}^{i} , Γ_{j\overline{k}}^{i} and Γ_{\overline{j}k}^{i} and their conjugates are known from complex manifolds ([3], [5]). Replacing this in (1.16) it results the coefficients (2.12) of D^{c} connection in the adapted base. The connection (2.12) is metric and normal. From the symmetry of Γ_{jk}^{i} it results that T_{jk}^{i} = S_{jk}^{i} = 0.

The connection (2.12) will be called the Miron canonic N-linear complex connection of the complex Lagrange space (M, L).

3 The complex Lagrangian of electrodynamics

In [6] it is given a Lagrangian model for real electrodinamycs. In the present section we shall consider a similar model for the complex case. The complex Lagrangians frequently appear in quantum mechanics and gauge theories ([11]). Inspired from these models, let’s consider the complex Lagrangian

(3.1)  L_{0}(z, η) = mcγ_{i\overline{j}}(z)η^{i}\overline{η}^{j} + \frac{e}{m}A_{i}(z)(η^{i} + \overline{η}^{i}),

where γ_{i\overline{j}} is a Hermitian metric on the complex manifold M ; m, c, e are real constants and A_{i} is a real covector.

The metric γ_{i\overline{j}} can be taken as the extension through linearity of Lorentz metric to the complexification T_{C}M.

From (2.1), it results that L_{0} determine the following Hermitian metric on T′M:

(3.1′)  g_{i\overline{j}} = mcγ_{i\overline{j}}(z).

The metric g_{i\overline{j}} determines a nonlinear connection N_{i}^{j} = \frac{∂H^{j}}{∂η^{i}}, where H^{j} is given by (2.6). So, we have:

(3.2)  N_{i}^{j} = \frac{1}{2} \frac{∂Γ_{00}^{j}}{∂η^{i}} + \frac{e}{4m^{2}c}γ^{\overline{k}j} \frac{∂A_{k}}{∂z^{i}} ,

where Γ_{jk}^{i} are the first Christoffel coefficients of γ_{i\overline{j}} metric and Γ_{00}^{i} = Γ_{jk}^{i}η^{j}η^{k}. Accordingly, the Miron canonic connection (2.12) has the components:

(3.3)  \begin{matrix}L_{jk}^{i} = Γ_{jk}^{i} = \frac{1}{2}γ^{\overline{l}i}(\frac{∂γ_{j\overline{l}}}{∂z^{k}} + \frac{∂γ_{k\overline{l}}}{∂z^{j}}) & C_{jk}^{i} = 0 \\ L\overline{i} ; \overline{j}k = Γ\overline{i} ; \overline{j}k = \frac{1}{2}γ^{\overline{i}l}(\frac{∂γ_{l\overline{j}}}{∂z^{k}} + \frac{∂γ_{k\overline{j}}}{∂z^{l}}) & C\overline{i} ; \overline{j}k = 0. \end{matrix}

From (1.10) we obtain the following vanishing torsions: T_{ij}^{k} = S_{ij}^{k} = Q_{ij}^{k} = Υ_{ij}^{k} = χ_{i\overline{j}}^{k} = χ\overline{k} ; i\overline{j} = 0 and their conjugates. Because C_{jk}^{i} = C\overline{I} ; \overline{j}k = 0, it results that the Miron canonic connection has the next nonvanishing components of curvatures:

R_{jkh}^{i} = r_{jkh}^{i}; R\overline{i} ; \overline{j}kh = r\overline{i} ; \overline{j}kh; R\overline{i} ; \overline{j}\overline{k}h = r\overline{i} ; \overline{j}\overline{k}h

(and the conjugates) where r_{jkh}^{i} , ..., are the curvatures of the linear connection Γ_{jk}^{i}.

For the real Lagrangian space from [6] we know the Einstein equations with respect to the canonic connection. They result from the Einstein equations

R_{αβ} − \frac{1}{2}RG_{αβ} = χT_{αβ} ( R_{αβ}=Ricci tensor, R=Ricci curvature, G_{αβ}=the metric structure, χ=universal constant, T_{αβ} = the energy-momentum tensor), written in the adapted base of the canonic nonlinear connection. Acting in a similar manner for the complex Lagrangian, it results the following set of Einstein equations

(3.4)  \begin{matrix}R_{i\overline{j}} − \frac{1}{2}ρg_{i\overline{j}} = χ T_{i\overline{j}}^{h\overline{h}}, & R_{ij} = χ T_{ij}^{hh}, & P_{ij} = χ T_{ij}^{hv} \\ S_{i\overline{j}} − \frac{1}{2}ρg_{i\overline{j}} = χ T_{i\overline{j}}^{v\overline{v}}, & S_{ij} = χ T_{ij}^{vv}, & π_{i\overline{j}} = χ T_{i\overline{j}}^{h\overline{v}}, \end{matrix}

where R_{i\overline{j}} = \overline{R}_{\overline{i}j} = −R_{i\overline{j}k}^{k}, e.t.c., ρ = r + s ∈ R , r = g^{\overline{i}j}R_{\overline{i}j}, s = g^{\overline{i}j}S_{\overline{i}j},

T^{h\overline{h}}_{i\overline{j}}, e.t.c.= the components of energy-momentum tensor, χ ∈ R. The conservation law with respect to the one X_{α} field of adapted base is

D^{c}_{X_{α}} (R_{β}^{α} − \frac{1}{2}Rδ_{β}^{α}) = 0, where R_{β}^{α} = G^{αγ}R_{γβ}.

In the particular case of the complex Lagrangian L_{0} for electrodynamics given by (3.1), taking into account the nonvanishing curvatures, the first (3.4) Einstein equation is just that for the Levi-Civita connection of γ_{i\overline{j}} metric, in the

secound equation S_{i\overline{j}} = 0, and the last three equations reduce to T^{vv}=T^{hv}=T^{h\overline{v}}= 0. The conservation law is identically satisfied.

Acknowledgements. A version of this paper was presented at the First Conference of Balkan Society of Geometers, Politehnica University of Bucharest, September 23-27, 1996.

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Gheorghe Munteanu Department of Geometry, Faculty of Sciences Transilvania University, Braşov Romania