Proc. Conf. Opava (Czechoslovakia), August 24–28, 1992 Silesian University, Opava, 1993, 157–160

ON THE THEORY OF

HOLOMORPHICALLY–PROJECTIVE MAPPINGS

OF PARABOLICALLY–KÄHLERIAN SPACES

Mohsen Shiha

Abstract. General aspects of the theory of the holomorphically-projective mappings of parabolically-Kählerian spaces are considered. Spaces that do not admit these mappings are found. Key words: Holomorphically-projective mapping, parabolically-Kählerian spaces. MS classification: 53B35.

1. Introduction

In the paper holomorphically-projective mappings of parabolically-Kählerian spaces K_{n}^{o(m)} are considered. A basis for these investigations was laid by V.V. Vishnevsky [3]. Some facts from the theory of holomorphically-projective mapping K_{n}^{o(m)} were published in [1],[2].

A Riemannian space K_{n}^{o(m)} is said to be m-parabolically-Kählerian spaces if together with a metric tensor g_{ij}(x) it possesses a structure F_{i}^{h}(x) of a rank m ≥ 2 satisfying the following relations

a) F_{α}^{h}F_{i}^{α} = 0; b) F_{(ij)} = 0; c) F_{i,j}^{h} = 0,  (1)

where comma denotes covariant differentiation, (i, j) denotes symmetrization with respect to i and j, F_{ij} ≡ g_{iα}F_{j}^{α}.

By an analytically planar curve in K_{n}^{o(m)} we shall mean a curve defined by the equations x^{h} = x^{h}(t) such that the following condition holds

dλ^{h}/dt + Γ_{αβ}^{h}λ^{α}λ^{β} = ρ_{1}(t)λ^{h} + ρ_{2}(t)F_{α}^{h}λ^{α}

where λ^{h} ≡ dx^{h}/dt; ρ_{1}, ρ_{2} are function of the argument t, Γ_{ij}^{h} is the Christoffel symbol.

A diffeomorphism f: K_{n}^{o(m)} → K̄_{n}^{o(m)} is said to be holomorphically-projective mapping, if all analytically planar curves in K_{n}^{o(m)} are mapped into analytically planar curves in K̄_{n}^{o(m)}.

Typeset by AMS-TEX

This paper is in final form and no version of it will be submitted for publication elsewhere.

It is proved in [2] that K_{n}^{o(m)} admits a holomorphically-projective mapping onto some K̄_{n}^{o(m)} if and only if there exists a solution of the following equations

a_{ij,k} = λ̄_{i}g_{jk} + λ̄_{j}g_{ik} + λ_{i}F_{jk} + λ_{j}F_{ik}  (2)

on an unknown tensor a_{ij} and a vector λ_{i}, λ̄_{i} ≡ λ_{α}F_{i}^{α}, satisfying

a_{ij} = a_{ji}; a_{α(i}F_{j)}^{α} = 0; det∥a_{ij}∥ ≠ 0.  (3)

If λ_{i} ≢ 0, then the holomorphically-projective mapping will be called non-trivial. Contracting (2) with g^{ij} (∥g^{ij}∥ = ∥g_{ij}∥^{−1}) we see that the vector λ̄_{i} is a gradient: λ̄_{i} = \frac{1}{4}∂_{i}(a_{αβ}g^{αβ}) ≡ λ̄_{,i}. Thus, the conditions λ̄_{i,j} = λ̄_{j,i} hold. It follows that λ̄_{α,β}F_{i}^{α}F_{j}^{β} = 0.

2. Main equation

Theorem 1. A diffeomorphism f: K_{n}^{o(m)} → K̄_{n}^{o(m)} is a holomorphically-projective mapping if and only if there exists a solution of the following linear Cauchy-like system

\begin{aligned}a) & a_{ij,k} & = λ̄_{(i}g_{j)k} + λ_{(i}F_{j)k}; \\ b) & λ_{i,j} & = τF_{ij} + a_{αβ}M_{ij}^{αβ}; \\ c) & τ_{,i} & = λ_{α}M_{i}^{α} + a_{αβ}M_{i}^{αβ} \end{aligned}  (4)

on an unknown tensor a_{ij}, a vector λ_{i}, and a function τ, satisfying conditions (3). Here M_{ij}^{αβ}, M_{i}^{α}, M_{i}^{αβ} are tensors determined by formulas (8)-(12) below and dependent on g_{ij} and F_{i}^{h} of the space K_{n}^{o(m)}.

Proof. Consider the integrability conditions of the equation (2):

a_{α(i}R_{j)kl}^{α} = g_{k(i}λ̄_{j),l} − g_{l(i}λ̄_{j),k} − F_{k(i}λ_{j),l} + F_{l(i}λ_{j),k}.  (5)

Contracting (5) with F_{k′}^{k} and also with F_{l′}^{l} we obtain two expressions. After removing primes we sum them up. Since in K_{n}^{o(m)} it holds R_{iαk}^{h}F_{j}^{α}+R_{ijα}^{h}F_{k}^{α} = 0, we get F_{k(i}Λ_{j)l}− F_{l(i}Λ_{j)k} = 0 where Λ_{ij} ≡ λ̄_{i,j} + λ_{i,α}F_{j}^{α}.

It follows that Λ_{ij} = ΛF_{ij}, i.e. λ̄_{i,j}+λ_{i,α}F_{j}^{α} = ΛF_{ij} where Λ is a function. Contracting (5) with g^{jk} we then get

nλ̄_{i,j} = μg_{ij} + νF_{ij} + a_{αi}R_{j}^{α} − a_{αβ}R_{.ij.}^{α..β}  (6)

where R_{ijk}^{h} and R_{ij} are Riemann and Ricci tensors, respectively, the operations of lifting and lowering of indices are induced by the metric tensor, and μ, ν are certain functions.

After symmetrizing (6) we get

nλ̄_{i,j} = μg_{ij} + \frac{1}{2}a_{α(i}R_{j)}^{α} − a_{αβ}R_{.ij.}^{α..β}.  (7)

Substituting (7) into (5) we obtain

a_{αβ}M_{ijkl}^{αβ} = F_{li}λ_{j,k} + F_{lj}λ_{i,k} − F_{ki}λ_{j,l} + F_{kj}λ_{i,l}.  (8)

Here and in what follows M are tensors determined by g_{ij} and F_{i}^{h} on K_{n}^{o(m)}. More concretely

M_{ijkl}^{αβ} ≡ δ_{(i}^{α}R_{j)kl}^{β} + M_{1|k(i}^{αβ}g_{j)l} − M_{1|l(i}^{αβ}g_{j)k}; nM_{1|li}^{αβ} ≡ \frac{1}{2}δ_{(i}^{α}R_{j)}^{β} − R_{.ij.}^{α..β}

where δ_{i}^{h} is the Kronecker symbol.

Let ε^{j} and ν^{k} be vectors such that ε^{j}ν^{k}F_{jk} = 1. Denote M_{i} ≡ ε^{α}F_{αi}.

Contracting (8) with ε^{i}ε^{j}ν^{k} we get

ε^{α}λ_{α,l} = τM_{l} + a_{αβ}M_{2|l}^{αβ}  (9)

where τ ≡ λ_{α,β}ε^{α}ν^{β}. Contracting (8) with ε^{j}ν^{k} and using (9) we arrive at

λ_{i,j} = τF_{ij} + Λ_{i}M_{j} + a_{αβ}M_{3|ij}^{αβ}  (10)

where Λ_{i} is a vector. Substituting (10) into (8) we obtain

Λ_{i}(M_{k}F_{lj} − M_{l}F_{kj}) + Λ_{j}(M_{k}F_{li} − M_{l}F_{ki}) = a_{αβ}M_{4|ijkl}^{αβ}.  (11)

Contracting (11) with ν^{i}ν^{j}ν^{k}ε^{l} we get ν^{α}Λ_{α} = a_{αβ}M_{5|}^{αβ}, and contracting (11) with

ν^{j}ν^{k}ε^{l} we get Λ_{i} = a_{αβ}M_{6|i}^{αβ}. Then (10) assumes the form (4b). Integrability conditions of (4b) are

F_{ij}τ_{,k} − F_{ik}τ_{,j} = a_{αβ}M_{7|ijk}^{αβ} + λ_{α}M_{8|ijk}^{αβ}.  (12)

Contracting (11) with ε^{i}ε^{j}ν^{k} we get τ_{,a}ε^{α} = a_{αβ}M_{9|}^{αβ} + λ_{α}M_{10|}^{αβ}. Then contracting (11) with ε^{j}ν^{i} we get finally (4c). The theorem is proved.

The system (4) has at most one solution for the initial values in a point x_{o}: a_{ij}(x_{o}), λ_{i}(x_{o}) and τ(x_{o}). Hence, the general solution of this system depends on no more than (n + 2)(n + 1)/2 − m(n − m + 1) essential parameters.

The integrability conditions of the system (4) and their differential prolongations are linear algebraic equations on the components of the unknown tensors a_{ij}, λ_{i} and τ with coefficients from K_{n}^{o(m)}.

Thus, one can, in principle, solve the problem whether a given m-parabolically-Kählerian space K_{n}^{o(m)} admits or not a holomorphically-projective mapping and its uniqueness.

3. Holomorphically-projective mappings of Ricci-recurrent spaces

As in the theory of the holomorphically-projective mappings of the elliptical and hyperbolical Kählerian spaces, there exist generally-recurrent m-parabolical Kählerian spaces, which do not admit these mappings.

Recall that a space is Ricci-recurrent if its Ricci tensor satisfies the conditions

R_{ij,k} = φ_{k}R_{ij}

where φ_{k} is a vector.

Theorem 2. A Ricci-recurrent m-parabolically Kählerian space K_{n}^{o(m)} whose Ricci tensor satisfies the condition

R_{iα}F_{j}^{α} ≠ ρF_{ij},

does not admit non-trivial holomorphically-projective mappings.

Proof. Let a Ricci-recurrent spaces K_{n}^{o(m)} admits non-trivial holomorphically-projec-tive mappings. Hence in K_{n}^{o(m)} there exists a solution of equation (2) satisfying λ_{i} ≢ 0, and formulas (6) hold. Alternating (6) we obtain

2νF_{ij} + a_{αi}R_{j}^{α} − a_{αj}R_{i}^{α} = 0.  (13)

Differentiate covariantly (13) with respect to x^{k}. On the basis if (1), (2) and (13) we get

v_{k}F_{ij} + λ̄_{i}R_{jk} + λ̄_{α}R_{j}^{α}g_{ik} − λ̄_{j}R_{ik} − λ̄_{α}R_{i}^{α}g_{jk}+
+λ_{i}R_{j}^{α}F_{αk} + λ_{α}R_{j}^{α}F_{ik} − λ_{j}R_{i}^{α}F_{αk} − λ_{α}R_{i}^{α}F_{jk} = 0  (14)

where v_{k} is a vector. Contracting (14) with F_{l}^{i} we get

λ̄_{α}R_{j}^{α}F_{lk} + λ̄_{j}F_{l}^{α}R_{αk} + λ̄_{α}R_{l}^{α}F_{jk} + λ̄_{l}F_{j}^{α}R_{αk} = 0.  (15)

It follows that λ̄_{α}R_{i}^{α} = αλ̄_{i} or F_{i}^{α}R_{αj} = ρF_{ij}. But then it follows from (15) that λ̄_{i} = 0 and (14) reduces to

v_{k}F_{ij} + λ_{i}R_{j}^{α}F_{αk} + λ_{α}R_{j}^{α}F_{ik} − λ_{j}R_{i}^{α}F_{αk} − λ_{α}R_{i}^{α}F_{jk} = 0.  (16)

Symmetrizing (16) with respect to j and k we get

F_{ij}a_{k} + F_{ik}a_{j} − R_{i}^{α}F_{αj}λ_{k} − R_{i}^{α}F_{αk}λ_{j} = 0

where a_{k} ≡ 2v_{k} + λ_{α}R_{k}^{α}. Analogously, a_{i} = λ_{i} = 0.

Similarly, the following theorem is proved.

Theorem 3. If a Ricci-recurrent space K_{n}^{o(m)} (m > 2) satisfying Rg∥R_{ij} −R/n g_{ij}∥ ≥ 1 admits nontrivial holomorphically-projective mappings, then the following conditions hold

R_{iα}F_{j}^{α} = ρF_{ij} and λ̄_{i} = 0.

References

1. M. Shiha, J. Mikesh, The holomorphically-projective mappings of m-parabolically-Kählerian spaces, Pontryaginskie Chtenia, Kemerovo (1990), 90 (in Russian).

2. M. Shiha, J. Mikesh, The holomorphically-projective mappings of parabolically-Kählerian spaces, Dep. in UkrNIINTI, N 1128-Uk91 (1991 (in Russian)).

3. V.V. Vishnevski, A.P. Shirokov, V.V. Shurigin, Spaces over algebras. Kazan, 1985, pp. 262 (in Russian).

Mohsen Shiha Odessa university, Petra Velikogo 2, Odessa, Ukraine / Homs university, Syria

Received 8 October 1992 Revised 30 January 1993