Proc. Conf. Opava (Czechoslovakia), August 24–28, 1992 Silesian University, Opava, 1993, 137–141

HOLOMORPHICALLY–PROJECTIVE MAPPINGS

OF T–QUASISEMISYMMETRIC AND GENERALLY

SYMMETRIC KÄHLERIAN SPACES

Michel Haddad

Abstract. The conditions on T-quasisemisymmetric Kählerian spaces to admit holomorphically-projective mappings are found. It is proved that m-symmetric spaces do not admit nontrivial holomorphically-projective mappings. Key words: holomorphically-projective mapping, Kählerian spaces, quasisemisymmetric spaces. MS classification: 53B35.

1. Introduction

Investigation of holomorphically-projective mappings of symmetric and semisymmetric spaces was initiated by T. Sakaguchi [6] and continued in works of V.V. Domašev and J. Mikeš [1], J.Mikeš [4], [6], [8], I.N. Kurbatova [3].

A Riemannian space K_{n} is said to be Kählerian if there exists, together with the metric tensor g_{ij}(x), a structure F_{i}^{h}(x) with the properties:

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

where e = ±1; F_{ij} ≡ g_{iα}F_{j}^{α}, δ_{i}^{h} is the Kronecker symbol, and comma denotes the covariant derivative with respect to the connection on K_{n}.

If e = −1, then K_{n} is called an (elliptical) Kählerian space and denoted by K_{n}^{−}, and if e = +1 then K_{n} is called a hyperbolic Kählerian space and denoted by K_{n}^{+}.

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

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 functions of the argument t.

A bijective correspondence between points of Kählerian spaces K_{n} and K̄_{n} is called a holomorphically-projective mapping, if any analytically planar curve in K_{n} is transformed into an analytically planar curve in K̄_{n}.

Typeset by AMS-TEX

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

A Kählerian space K_{n} admits a holomorphically-projective mapping onto K̄_{n} if and only if one of the following conditions is satisfied (via a diffeomorphism K_{n} → K̄_{n}):

ḡ_{ij,k} = 2ḡ_{ij}ψ_{k} + ψ_{(i}ḡ_{j)k} − eψ̄_{(i}F̄_{j)k};  (2)
a_{ij,k} = λ_{(i}g_{j)k} − eλ̄_{(i}F_{j)k},  (3)

where ḡ_{ij} is a metric tensor of K̄_{n}, ψ_{i}, λ_{i} are gradient vectors, a_{ij} is a tensor,

a_{ij} = a_{ji} = −ea_{αβ}F_{i}^{α}F_{j}^{β}; det∥a_{ij}∥ ≠ 0; ψ̄_{i} ≡ ψ_{α}F_{i}^{α}; λ̄_{i} ≡ λ_{α}F_{i}^{α}; F̄_{ij} ≡ ḡ_{iα}F_{j}^{α}.

The vector λ_{i} has the following properties: λ_{i,j} = λ_{j,i} = −eλ_{α,β}F_{i}^{α}F_{j}^{β}.

2. Holomorphically-projective mappings of T-quasisemisymmetric Kählerian spaces

In what follows T will be an m-covariant tensor, which cannot be expressed as a tensorial sum of products of invariants, the metric and the structure tensor. Later on let us define the tensor

Z_{ijk}^{h} ≡ R_{ijk}^{h} − B(δ_{k}^{h}g_{ij} − δ_{j}^{h}g_{ik} + eF_{k}^{h}F_{ij} − eF_{j}^{h}F_{ik} + 2eF_{i}^{h}F_{jk}),

where R_{ijk}^{h} is the Riemann tensor, B is a function.

It takes place the following

Theorem 1. Let in a Kählerian space K_{n} the condition

∑_{r=1}^{m} T_{i_{1}...i_{r−1}αi_{r+1}...i_{m}}Z_{i_{r}jk}^{α} = 0  (4)

be satisfied.

Then if K_{n} (n > 4(m − 1)) admits holomorphically-projective mappings, the basic equations of the theory of holomorphically-projective mappings take the form

(a) a_{ij,k} = λ_{(i}g_{j)k} − eλ̄_{(i}F_{j)k}; (b) λ_{i,j} = μg_{ij} + Ba_{ij},  (5)

where μ is a function.

Proof. Let T in K_{n} be as above, satisfying the conditions (4). Further, we shall suppose K_{n} to admit a holomorphically-projective mapping. According to [1],[5] the equations (3) hold and the corresponding integrability conditions are

a_{α(i}R_{j)kl}^{α} = λ_{l,(i}g_{j)k} − λ_{k,(i}g_{j)l} − eλ̄_{l,(i}F_{j)k} − λ̄_{k,(i}F_{j)l}.  (6)

In [4] the above theorem is proved for the case e = −1, m = 1, 2. The proof for the case e=+1, m=1,2, is analogous.

By induction we shall prove our theorem for an arbitrary m > 2 assuming that it is correct for 1, 2, ..., m − 1. Let us suppose the opposite, i.e. that for a holomorphicallyprojective mapping the conditions (5) do not hold. Evidently, any convolution of T with g^{ij} (∥g^{ij}∥ ≡ ∥g_{ij}∥^{−1}), or F^{ij} ≡ F_{α}^{i}g^{αj} satisfies the relations (4) and by our assumption these convolutions are of the form of a tensorial sum of products of invariants, metric and the structure tensor. For m odd the pointed convolutions are equal to zero. For m even and n ≥ 2m − 1 one easily can construct a tensor T^{∗}, being a sum of the tensor T and the tensorial sums of products of definite invariants, metric and the structure tensor, such that any convolution of T^{∗} with g^{ij} or F^{ij} is zero. The tensor T^{∗} satisfies the conditions (4); hence, without loss of generality, in what follows we shall consider T^{∗} instead of T.

We shall convolve the relation (4) with a_{kα}g^{αj}, and then symmetrize respectively to l and k. Taking into account the structure of the tensor Z_{ijk}^{h}, the conditions (6), and the properties of the Riemann tensor, after a corresponding grouping we get

∑_{r=1}^{m}{Λ_{h_{r}(i}T_{|h_{1}...h_{r−1}|j)h_{r+1}...h_{m}} − eΛ_{h_{r}α}F_{(i}^{α}F_{j)}^{β}T_{h_{1}...h_{r−1}βh_{r+1}...h_{m}}−

− g_{h_{r}(i}T_{|h_{1}...h_{r−1}|j)h_{r+1}...h_{m}}^{r} + eF_{h_{r}(i}F_{j)}^{β}T_{h_{1}...h_{r−1}βh_{r+1}...h_{m}}^{r}} = 0,  (7)

where

Λ_{ij} ≡ λ_{i,j} − μ g_{ij} − B a_{ij};  (8)
μ ≡ (λ_{α,β} − Ba_{αβ})g^{αβ}/n;
T_{h_{1}...h_{m}}^{r} ≡ Λ_{αh_{r}}g^{αβ}T_{h_{1}...h_{r−1}βh_{r+1}...h_{m}}.

Contracting (7) in turn with g^{ih_{p}} g^{jh_{q}}, g^{ih_{p}} F^{jh_{q}}, F^{ih_{p}} F^{jh_{q}}, where p, q = 1, 2, ..., m (p ≠ q), we shall obtain equations which for n > 4(m − 1) give rise to

\begin{matrix}g^{αβ}T_{h_{1}...h_{p−1}αh_{p+1}...h_{r−1}βh_{r+1}...h_{m}}^{r} = 0; \\ F^{αβ}T_{h_{1}...h_{p−1}αh_{p+1}...h_{r−1}βh_{r+1}...h_{m}}^{r} = 0. \end{matrix}  (9)

Then we contract (7) with g^{ih_{p}} and F^{ih_{p}}. From the obtained relations taking into account (9) we shall get, that T_{h_{1}...h_{m}}^{r} = 0. Then the formulas (7) are simplified. From them one can show that either Λ_{ij} = 0, or T_{h_{1}...h_{m}} = 0. Taking into account formulas (8), the theorem is completely proved.

It takes place

Theorem 2. Let T in a Kählerian space K_{n} (n ≥ 4(m−1)) be as above, and satisfying the conditions T_{,lm} − T_{ml} = 0. If a K_{n} admits a holomorphically-projective mapping then it holds

λ_{i,j} = μg_{ij},  (10)

where μ = const, i.e. there exists a convergent vector field in K_{n}.

The proof follows from Theorem 1 for B = 0. But it remains to prove that μ is constant. The latter easily follows from the integrability conditions of equations (10) and (6).

Note that Theorems 1 and 2 have been proved earlier by J. Mikeš [4], [6] for m=1,2; n ≥ 4, and for a special case where m=4, n ≥ 4 and T is the Riemann tensor. These theorems a priori are incorrect for n = m.

In the paper [4],[6] the spaces K_{n}, admitting holomorphically-projective mappings satisfying the condition (5) are called K_{n}[B]-spaces. Spaces where the conditions (4) hold are called T -quasisemisymmetric {T P s_{n}[B]}-spaces and the spaces T P s_{n}[0] are called T -semisymmetric {T P s_{n}}-spaces.The mentioned classes of spaces naturally develop the idea of a semisymmetric space.

Taking into account [7], one can show easily that the compact T P s_{n}-spaces, the tensor T of which at each point cannot be represented as a sum of products of invariants, metric and the structure tensor for n ≥ 4(m−1), do not admit a global holomorphicallyprojective mapping.

3. Holomorphically-projective mappings of generally symmetric spaces

Let us prove the following theorems

Theorem 3. Let T in K_{n} be as above, and suppose that it satisfies the following condition:

T_{h_{1}h_{2}...h_{m},l_{1}l_{2}...l_{r}} = 0,  (11)

for n ≥ 4(m + r − 3), r > 1 or n ≥ 4(m − 1), r = 1.

Then K_{n} admits a holomorphically-projective mapping satisfying the condition (10).

Proof.

a) First suppose that m + r is odd and r > 1. Then alternate (11) with respect to l_{r−1} and l_{r}. In virtue of Theorem 2 either the conditions (10) hold, or

T_{h_{1}h_{2}...h_{m},l_{1}l_{2}...l_{r−2}} = 0.

Applying this procedure consecutively we shall get either

T_{h_{1}h_{2}...h_{m}} = 0 or T_{h_{1}h_{2}...h_{m},l} = 0.

b) Consider the case m + r is even and r > 1. Then by an analogy of the above we can see that T_{h_{1}h_{2}...h_{m},l_{1}l_{2}...l_{r−2}} is a tensorial sum of products of invariants, metric and the structure tensor. The obtained expression shall be covariantly differentiated with respect to x^{r−1} and x^{r}. Then we shall convince that the second covariant derivative of the mentioned invariants is zero. Hence, either these invariants are constant, or the conditions (10) hold. But then

T_{h_{1}h_{2}...h_{m},l_{1}l_{2}...l_{r−1}} = 0,

i.e. the question under consideration is reduced to a).

c) It remains to investigate the case T_{h_{1}h_{2}...h_{m},l} = 0. Let us covariantly differentiate this relation respectively to x^{k} and then alternate with respect to k and l. For m odd the tensor T is vanishing by Theorem 2, and Theorem 3 is proved. In case of even m even (as in b)) we get that the tensor T is tensorial sum of products of constants, the metric and the structure tensor, what was required to prove.

Let us recall that the space where

R_{ijk,l_{1}l_{2}...l_{m}}^{h} = 0.

is called m-symmetric [2].

Theorem 4. An m−symmetric Kählerian space K_{n} (n ≥ 4(m + 3)) of nonconstant holomorphically sectional curvature does not admit a nontrivial holomorphically-projec-tive mapping.

In virtue of Theorem 3 it follows that such spaces could admit a holomorphicallyprojective mapping only if the condition (6) be satisfied. But this is impossible by results of J. Mikeš [8].

References

1. V.V.Domašev, J.Mikeš, Of the theory holomorphically-projective mapping of Kählerian space, Mat. Zametki 23 (1978), 297-304 (in Russian).

2. V.R.Kaygorodov, Riemannian spaces ^{∗}D_{n}^{s}. The structure of curvature of spaces type A, Izv. Vuzov, Math. (1974), 117-127 (in Russian).

3. I.N.Kurbatova, To question about the quasiholomorhically-projective mapping K-spaces, Dep. in VINITI N 2429-79 (in Russian) (1979).

4. J.Mikeš, On geodesic and holomorphically-projective mappings of special Riemannian spaces, Dissertation (Odessa, 1979), 107 (in Russian).

5. J.Mikeš, On holomorphically-projective mappings of Kählerian space, Ukraine Geom. Sb. 23 (1980), 45-55 (in Russian).

6. J.Mikeš, F -planar mappings and transformations, Diff. Geom. and Its Appl., Proc. Conf., Brno (Czechosl.) (1986), 245-254.

7. J.Mikeš, On global concircular vector fields in compact Riemannian spaces, Dep. in UkrNIINTI, N. 615-Uk88 (in Russian) (1988).

8. J.Mikeš, Geodesic and holomorhically-projective mapping of generalized m-recurrent spaces, Abst. Colloq. on Diff. Geom., Eger (Hungary). J. Bolyai Math. Soc. (1989), 35-36 (in Russian).

9. T.Sakaguchi, On the holomorphically projective correspondence between Kählerian spaces preserwing complex structure, Hokkaido Math. J. 3, N. 2 (1974), 203-212.

10. K.Yano, Differential Geometry on Complex and almost Complex Spaces, Oxford Pergamon Press (1965), 326.

Michel Haddad, Odessa University, Petra Velikogo 2, Odessa, Ukraine

Damascus University, Syria

Received 8 October 1992 Revised 30 January 1993