On Complex Cartan Spaces

Gh. Munteanu

Dedicated to the Memory of Grigorios TSAGAS (1935-2003), President of Balkan Society of Geometers (1997-2003)

Abstract

In some recent articles ([13, 14]) we have studied the geometry of complex Hamilton spaces.

In brief, the geometry of a complex Hamilton space is the geometry of the dual holomorphic bundle (T′M)^{∗} endowed with a Hermitian metric derived from a Hamiltonian function. In this study the notion of complex nonlinear connection plays a special role. A significant result provides the complex nonlinear connection derived only from the Hamiltonian function.

If in addition a positive Hamiltonian satisfies the condition of homogeneity, then the notion of complex Cartan space is obtained. This is the correspondent of complex Finsler space on the manifold (T′M)^{∗}, and coincides with the notion of complex Finsler Hamiltonian introduced by S. Kobayashi ([7, 5]).

In the present paper we make a geometric study of the complex Cartan space and of some its immediate generalizations.

Mathematics Subject Classification: 53B40, 53C55

Key words: holomorphic bundle, Cartan space, Hamilton space.

1 The bundle (T′M)^{∗}

Let M be a complex manifold, dim_{C}M = n, and denote by (z^{k}) the complex coordinates in a local chart. T′M is the holomorphic bundle of (1, 0)−type vectors and (T′M)^{∗} is its dual bundle. In a local chart on the manifold (T′M)^{∗}, a point u^{∗} is characterized by the coordinates u^{∗} = (z^{k}, ζ_{k}), k = \overline{1, n}, and the change of local charts determines the following change of coordinates ([14]):

z^{′k} = z^{′k}(z) ; ζ′_{k} = \frac{∂z^{j}}{∂z^{′k}} ζ_{j} ; rank(\frac{∂z^{j}}{∂z^{′k}}) = n (1.1)

Now, let us consider the holomorphic bundle π_{T}^{∗} : T′(T′M)^{∗} → (T′M)^{∗}. A local frame in u^{∗} is \{\frac{∂}{∂z^{k}} , \frac{∂}{∂ζ_{k}} \} and its changes are imposed by the Jacobi matrix of (1.1).

The vertical subbundle V (T′M)^{∗} = ker π_{T}^{∗} is holomorphic too and a local base in the vertical distribution V^{∗} is \{\frac{∂}{∂ζ_{k}} \}. A complex nonlinear connection (in brief (c.n.c.)) on (T′M)^{∗} is a supplementary subbundle of V (T′M)^{∗} in T′(T′M)^{∗} , i.e. T′(T′M)^{∗} = H(T′M)^{∗} ⊕ V (T′M)^{∗}. If a (c.n.c.) is given, by conjugation a decomposition of the whole complexification T_{C}(T′M)^{∗} is obtained.

∗ Balkan Journal of Geometry and Its Applications, Vol.8, No.1, 2003, pp. 71-78.

\begin{matrix}c \\ ◯ \end{matrix} Balkan Society of Geometers, Geometry Balkan Press 2003.

In the horizontal distribution H^{∗} = H_{⊓^{∗}}(T′M)^{∗} a local basis has the form

\frac{δ}{δz^{k}} = \frac{∂}{∂z^{k}} + N_{jk}^{∗} \frac{∂}{∂ζ_{j}} (1.2)

and this basis is said to be adapted if it transforms under the rule:

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

The basis \{δ_{k} = \frac{δ}{δz^{k}} , ˙∂^{k} = \frac{∂}{∂ζ_{k}} \} is an adapted basis on T′_{u^{∗}}(T′M)^{∗}. The corre-

sponding dual basis {dz^{k}, δζ_{k} = dζ_{k}− N_{kj}^{∗} dz^{j}} is an adapted basis on T_{u^{∗}}^{′∗}(T′M)^{∗}.

Of course, the condition (1.3) involves that the coefficients N_{jk}^{∗} of (c.n.c.) obey

a certain rule of transformation. Let us note that if N_{jk}^{∗} is a (c.n.c.) then N_{kj}^{∗} and

\frac{1}{2}(N_{jk}+^{∗}N_{kj}^{∗}) are (c.n.c) too.

Proposition 1.1 If N_{jk}^{∗} is a (c.n.c.) then ^{\frac{∂N_{jk}^{∗}}{∂ζ_{m}}} ζ_{m} determines a (c.n.c.), called the

spray connection of N_{jk}^{∗} .

In our approach a special meaning have those geometrical objects, called d−complex tensors, which are transformed only by means of the matrices (∂z^{i}/∂z^{′j}) or (∂z̄^{i}/∂z̄^{′j}) for the bar indices, and with their inverses, in a similar way as on the base manifold M.

A linear connection D : χ_{C}(T′M)^{∗} × χ_{C}(T′M)^{∗} → χ_{C}(T′M)^{∗} is said to be a

N^{∗} −complex linear connection (shortly N^{∗} −(c.l.c.)) if for a given (c.n.c.) it preserves the four distributions of T_{C}(T′M)^{∗} and its coefficients coincide two by two ([14]). Note that for a d − (c.l.c.) D we have \overline{D_{X}Y} = D\overline{X} \overline{Y}, and so it is well defined in respect to the adapted base if the following local expression is given:

D_{δ_{k}}δ_{j} = H_{jk}^{i}δ_{i} ; D_{δ_{k}} ˙∂^{i} = −H_{jk}^{i} ˙∂^{j} ; D_{δ_{k}}δj̄ = Hī ; j̄kδī ; D_{δ_{k}} ˙∂ī = −Hī ; j̄k ˙∂j̄

D ˙∂^{k}δ_{j} = C_{j}^{ik}δ_{i} ; D ˙∂^{k} ˙∂^{i} = −C_{j}^{ik} ˙∂^{j} ; D ˙∂^{k}δj̄ = Cīk ; j̄δī ; D ˙∂^{k} ˙∂ī = −Cīk ; j̄ ˙∂j̄ (1.4)

Therefore, a N^{∗} −(c.l.c.) is characterized only by the set of coefficients (H_{jk}^{i} ; Hī ; j̄k; C_{j}^{ik}; Cīk ; j̄), and their conjugates. The covariant derivatives of a d−complex tensor in

respect to a N^{∗} −(c.l.c.) D will be denoted by “_{|k}”, “ |_{k} ” or “_{|k̄}”, “ |k̄ ”. The local

expressions of curvatures and torsions of a N^{∗} −(c.l.c.) are calculated in [14].

2 Complex Hamilton space

Let N^{∗} be a fixed (c.n.c.) and g_{ij̄}(z, ζ) a Hermitian metric on (T′M)^{∗}, i.e. g_{ij̄} is a d−complex tensor, \overline{g_{ij̄}} = g_{jī} and det(g_{ij̄}) ≠ 0. By (g^{īj}) we denote the inverse matrix of (g_{ij̄}). The following metric structure on T_{C}(T′M)^{∗}, G = g_{ij̄}dz^{i} ⊗ dz̄^{j} + g^{j̄i}δζ_{i} ⊗ δζ̄_{j} (2.1)

is called the N^{∗} −lift of the metric structure g_{ij̄} .

A N^{∗} −(c.l.c.) D is metrical, that is DG = 0, iff g_{ij̄|k} = g_{ij̄} |_{k}= g_{ij̄|k̄} = g_{ij̄} |k̄= 0.

A remarkable example of metrical N^{∗} −(c.l.c.) on (T′M)^{∗} is given by

Theorem 2.1 ([14]).The following N^{∗} −(c.l.c.), denoted by D^{c}, is metrical :

(2.2)  H_{jk}^{i}^{c} = \frac{1}{2}g^{h̄i}(\frac{δg_{jh̄}}{δz^{k}} + \frac{δg_{kh̄}}{δz^{j}}) ; C_{j}^{ik}^{c}= −\frac{1}{2}g_{jh̄}(\frac{∂g^{h̄i}}{∂ζ_{k}} + \frac{∂g^{h̄k}}{∂ζ_{i}})
Hī ; j̄k^{c} = \frac{1}{2}g^{īh}(\frac{δg_{hj̄}}{δz^{k}} − \frac{δg_{kj̄}}{δz^{h}})  ; Cīk ; j̄^{c}= −\frac{1}{2}g_{hj̄}(\frac{∂g^{īh}}{∂ζ_{k}} − \frac{∂g^{īk}}{∂ζ_{h}})

and has the following zero torsions hT (hX, hY) = vT (vX, vY) = 0.

The notion of Hermitian metric has a special signification if it is derived from a complex Hamiltonian. A complex Hamiltonian is given by a C^{∞}−differentiable function H : (T′M)^{∗} → R with the property that the following d−complex tensor is nondegenerate

g^{j̄i}(z, ζ) = \frac{∂^{2}H}{∂ζ_{i}∂ζ̄_{j}} , rank(g^{j̄i}) = n. (2.3)

The pair (M, H) is said to be a complex Hamilton space.

In [15] we made an extension of the well-known Legendre transformation to the complexified of (T′M)^{∗}. As a product, a special result gives a very simple form of a (c.n.c.)

Theorem 2.2 The following functions

N_{ji}^{∗c}= −g_{jh̄} \frac{∂^{2}H}{∂z^{i}∂ζ̄_{h}} (2.4)

are the coefficients of a (c.n.c.) on (T′M)^{∗}, depending only on the complex Hamiltonian function H.

A straight computation of the bracket [δ_{j}, δ_{k}] = Ω_{ijk} ˙∂^{i} yields to Ω_{ijk} = δ_{j}(N_{ik}^{∗c})

−δ_{k}(N_{ij}^{∗c}) = 0 and consequently, the N_{ji}^{∗c} (c.n.c.) plays a special role.

In respect to the adapted basis of the (c.n.c.) given by (2.4), we consider the

connection D^{c} from (2.2). So, the set ΓH^{c}= (N_{jk}^{∗c}, H_{jk}^{i}^{c}, Hī ; j̄k^{c}, C_{j}^{ik}^{c}, Cīk ; j̄^{c}) will be called the canonical (c.l.c.) of the complex Hamilton space (M, H).

In the next lines we shall describe another method to obtain a N^{∗} −(c.l.c.) which generalizes to the dual case the idea of vertical connections ([1]) from the theory of complex Finsler spaces.

Let ∇ : χ(T′M)^{∗} × V (T′M)^{∗} → V (T′M)^{∗} be a linear connection on the vertical bundle, locally given by its coefficients Γ_{ik}^{j} and C_{i}^{jk}, where

∇_{\frac{∂}{∂z^{k}}} ˙∂^{j} = −Γ_{ik}^{j} ˙∂^{i} ; ∇ ˙∂^{k} ˙∂^{j} = C_{i}^{jk} ˙∂^{i} .

By d_{i}^{k} is denoted the d−complex tensor d_{i}^{k} = δ_{i}^{k} − C_{i}^{jk}ζ_{j}. As in [4] we prove that Γ_{ik}^{0} = Γ_{ik}^{j}ζ_{j} are transformed by the rule:

Γ_{jk}^{′0} = \frac{∂z^{p}}{∂z^{′j}} \frac{∂z^{q}}{∂z^{′k}} Γ_{pq}^{0} + d_{j}^{′p}ζ_{q} \frac{∂^{2}z^{q}}{∂z^{′p}∂z^{′k}} (2.5)

Therefore, if there exists the inverse (d_{i}^{k})^{−1} = b_{i}^{k} then N_{ik}^{∗}= b_{i}^{k}Γ_{jk}^{0} satisfies the rule of change of a (c.n.c.) on (T′M)^{∗}. If there exist b_{i}^{k}, by analogy with [1], we say that ∇ is a good vertical connection on (T′M)^{∗}.

Based on (2.5), it follows

Proposition 2.1 Any good vertical connection determines a (c.n.c.) on (T′M)^{∗}.

Moreover, a good vertical connection determines a N^{∗} −(c.l.c.) of (1,0)-type as follows. The coefficients C_{i}^{jk} of a good vertical connection satisfy the same rule of

transformation as C_{i}^{jk} of one N^{∗} −(c.l.c.)D and H_{ik}^{j} is directly obtained from the

calculation of D_{δ_{k}} ˙∂^{j} = ∇_{( \frac{∂}{∂z^{k}} +N_{hk}^{∗} ˙∂^{h})} ˙∂^{j}. So we have that H_{ik}^{j} = Γ_{ik}^{j}+ N_{hk}^{∗} C_{i}^{jh} are

the horizontal coefficients of a N^{∗} −(c.l.c.) on (T′M)^{∗}. The coefficients Cj̄k ; ī, Hj̄ ; īk can be

zero (since they are d−tensors) and then the obtained N^{∗} −(c.l.c.)D is of (1, 0)−type.

Let us consider the whole vertical complexified bundle V (T′M)^{∗} ⊕ \overline{V (T′M)^{∗}} and let G = \}^{⪯ |⟩}⌈ζ_{⟩}⊗⌈{⪯ \atop ζ_{|}} be a Hermitian vertical metric. We assume that ∇ is a metric linear connection of (1.0)−type, i.e. (∇_{X}G)(U, V) = X G(U, V)−G(∇_{X} U, V)−G(U, ∇_{X} V) = ′ and Cj̄h ; ī = Γj̄ ; īh = 0. Then by choosing U = ˙∂^{j}, V = ˙∂k̄ and X = \frac{∂}{∂z^{h}} or \frac{∂}{∂z̄^{h}} it results that:

(2.6)  Γ_{ih}^{j} = −g_{ik̄} \frac{∂g^{k̄j}}{∂z^{h}}  ; C_{i}^{jh} = −g_{ik̄} \frac{∂g^{k̄j}}{∂ζ_{h}}
N_{ik}^{∗} = −b_{i}^{j}g_{jh̄} \frac{∂g^{h̄l}}{∂z^{k}} ζ_{l} ; H_{ik}^{j} = −g_{im̄} \frac{δg^{m̄j}}{δz^{k}} .

Thus, we have:

Theorem 2.3 A good vertical connection on a complex Hamilton space (M, H) de-

termines a N^{∗} −(c.l.c.) of (1, 0)−type, ΓH^{CH}= (N_{ik}^{∗}, H_{ik}^{j}, 0, C_{i}^{jh}, 0) given by (2.6), and called the Chern-Hamilton connection.

3 Complex Cartan spaces

In the geometry of complex Finsler spaces there already exists a large reference ([1, 2, 3, 6, 11, 17]), the geometric support of such geometry being the holomorphic bundle T′M.

Concerning the Lagrangian-Hamiltonian duality from the classical mechanics we have considered necessary to make a study of complex Hamilton spaces based on the manifold (T′M)^{∗}. The correspondent of complex Finsler spaces in (T′M)^{∗} are the complex Cartan spaces, defined as follows:

Definition 3.1 A complex Cartan space is a complex Hamilton space (M, H) for which the function H : (T′M)^{∗} − {0} → R_{+} satisfies the homogeneity condition:

H(z, λζ) =| λ |^{2} H(z, ζ) , ∀λ ∈ C. (3.1)

We see that this notion coincides with that of the complex Finsler Hamiltonian initially introduced by S.Kobayashi ([7]), but here we prefer to use the notion of complex Cartan space by analogy with the real known terminology ([8, 9, 10]).

Accordingly, the Hamilton metric g^{j̄i}(z, ζ) = ∂^{2}H/∂ζ_{i}∂ζ̄_{j} is 0−homogeneous and, applying the complex version of the Euler Theorem, a Cartan space is characterized by

Proposition 3.1 In a complex Cartan space the following terms are true:

\frac{∂H}{∂ζ_{i}} ζ_{i} = H ; \frac{∂H}{∂ζ̄_{i}} ζ̄_{i} = H (3.2)

g^{j̄i}ζ_{i} = \frac{∂H}{∂ζ̄_{j}} ; g^{j̄i}ζ̄_{j} = \frac{∂H}{∂ζ_{i}} ; g^{j̄i}ζ_{i}ζ̄_{j} = H (3.3)

\frac{∂g^{j̄i}}{∂ζ_{k}} ζ_{i} = \frac{∂g^{j̄k}}{∂ζ_{i}} ζ_{i} = 0 ; \frac{∂g^{j̄i}}{∂ζ̄_{k}} ζ̄_{j} = \frac{∂g^{j̄k}}{∂ζ̄_{i}} ζ̄_{i} = 0 (3.4)

\frac{∂g^{j̄i}}{∂ζ_{k}} ζ̄_{j} = g^{ik} ; \frac{∂^{2}H}{∂z^{k}∂ζ_{i}} ζ_{i} = \frac{∂H}{∂z^{k}} ; \frac{∂^{2}H}{∂z^{k}∂ζ̄_{i}} ζ̄_{i} = \frac{∂H}{∂z^{k}} (3.5)

g^{ij}ζ_{j} = 0 ; g^{ij}ζ_{i}ζ_{j} = 0 ; \frac{∂g^{ij}}{∂ζ_{k}} ζ_{j} = −g^{ik}. (3.6)

In view of (3.4) we note that the coefficients C_{i}^{jh} from (2.6) obey the condition C_{i}^{jk}ζ_{j} = 0 and then b_{i}^{k} = d_{i}^{k} = δ_{i}^{k}; therefore the vertical connection is good. Consequently, in a complex Cartan space, from (2.6) it results the following (c.n.c.)

N_{ji}^{∗K}= −g_{jh̄} \frac{∂g^{h̄l}}{∂z^{i}} ζ_{l} (3.7)

and taking into account (3.3), we remark that it coincides with N_{ji}^{∗c} .

Now we can consider the following (c.l.c.): the canonical metrical connection

ΓH^{c}= (N_{jk}^{∗c}, H_{jk}^{i}^{c}, Hī ; j̄k^{c}, C_{j}^{ik}^{c}, Cīk ; j̄^{c}) from (2.2), and in the same time the Chern-Cartan

metrical connection ΓH^{K}= (N_{ji}^{∗K}, H_{jk}^{i}^{K}, 0, C_{j}^{ik}^{K}, 0) with the coefficients given by (2.6). Like in the complex Finsler case ([13]), we can consider the transformations group of metrical connections and then express the d− tensors which ties this pair of connections (possible with others that may be considered: Rund, Berwald type complex connections).

We emphasize only the fact that, although the Chern-Cartan connection being of (1, 0)−type is simpler, the canonical connection is h− and v− symmetrical and therefore easy to use in calculations. For the complex Finsler space this aspect was clearly proved by us in a paper that will appear.

Now let us summarize some direct properties of the canonical metrical connection.

Proposition 3.2 The following assertions are true:

1. ΓH^{c} depends only on the Hamilton function H(z, ζ)

2. We have: H_{jk}^{i}^{K}= ˙∂^{i}(N_{jk}^{∗c})

3. C_{i}^{jk}^{c}=C_{i}^{jk}^{K} ; Cj̄k ; ī^{c}=Cj̄k ; ī^{K}= 0

4. C_{i}^{0k}^{c}=C_{i}^{jk}^{c} ζ_{j} = 0 ; C^{ījk}^{c}= −\frac{∂g^{īj}}{∂ζ_{k}} ; C^{¯0jk}^{c}=C^{ī0k}^{c}=C^{īj0}^{c}= 0

5. ΓH^{c} has only the following nonzero torsions

vT ( ˙∂^{k}, δ_{j}) = [H_{jk}^{i}^{c} − ˙∂^{k}(N_{ij}^{∗c})] ˙∂^{i} ; hT ( ˙∂^{k}, δ_{j}) =C_{j}^{ik}^{c} δ_{i}
vT ( ˙∂k̄, δ_{j}) = − ˙∂k̄(N_{ij}^{∗c}) ˙∂^{i}  ; hT (δk̄, δ_{j}) =H_{jk̄}^{i}^{c} δ_{i}
vT (δk̄, δ_{j}) = −δk̄(N_{ij}^{∗c}) ˙∂^{i}  ; h̄T (δk̄, δ_{j}) =−Hī ; k̄j^{c} δ_{i}
v̄T (δk̄, δ_{j}) = −δ_{j}(N̄_{ik}^{∗c}) ˙∂ī ; h̄T (δk̄, ˙∂^{j}) = − ˙∂^{j}(N̄_{ik}^{∗c}) ˙∂ī

6. θ = dz^{k} ∧ δζ_{k} + dz̄^{k} ∧ δζ̄_{k} is a symplectic form on (T′M)^{∗}.

It seems that the class of complex Cartan spaces is poor enough (as well as that of complex Finsler spaces). For the moment we have two classical examples: one provided from a Hermitian metric on the base manifold M and, the Kobayashi Finsler Hamiltonian metric ([7, 5]). The homogeneity condition (3.1) with λ ∈ C is more restrictive. If we consider (3.1) only for all λ ∈ R (which is not an uninteresting case for geometry, taking in account that the parameter on a curve is real, unlike for the complex function theory) the class of examples is wider. If α^{2} = a^{j̄i}(z)ζ_{i}ζ̄_{j} and β = 2Re{A^{i}(z)ζ_{i}}, where a^{j̄i}(z) is a Hermitian metric on M and A^{i}(z) is a vector field, then in analogy to the real case we can discuss on R−complex Randers-Cartan spaces, Kropina-Cartan spaces or, more general, on R−complex (α, β)−Cartan spaces.

A complex Hamilton space (M, H) is said to be an almost Cartan-Hamilton (a.C − H) space if the metric tensor g^{j̄i}(z, ζ) = ∂^{2}H/∂ζ_{i}∂ζj̄ is 0−

Let us note that in an (a.C − H) space we have C_{i}^{0k}^{c}= 0. Hence b_{i}^{k} = d_{i}^{k} = δ_{i}^{k}, and

then in an (a.C − H) a (c.n.c) is N_{ji}^{∗c} too.

Theorem 3.1 A complex Hamilton space (M, H) is an (a.C − H) space if and only if the Hamilton function has the form:

H(z, ζ) = g^{j̄i}(z, ζ)ζ_{i}ζ̄_{j} + 2Re{A^{i}(z)ζ_{i}} + B(z)

where A^{i}(z) is a vector and B(z) is a real valued function.

The proof is based on the fact that ˙∂^{i} ˙∂j̄(H − E) = 0 and by H(z, ζ) = \overline{H(z, ζ)}, where E = g^{j̄i}(z, ζ)ζ_{i}ζ̄_{j} is the complex energy.

A complex Hamilton space is said to be of local Minkowski type if at any point u^{∗} there exists a local chart where g^{j̄i} depend only on the variable ζ.

Particularly, the complex Cartan space of local Minkowski type is obtained. In a complex local Minkowski space there exists a local chart in which the coeffi-

cients of one (c.n.c.) obtained from a good vertical connection are zero, and therefore δ_{i} = ∂/∂z^{i}. For such a choice of local atlas one obtains simplified forms of torsions and curvatures of (c.l.c.).

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Transilvania University

Faculty of Mathematics and Informatics 2200, Brasov, Romania email: gh.munteanu@info.unitbv.ro