Proc. Conf. Opava (Czechoslovakia), August 24–28, 1992 Silesian University, Opava, 1993, 151–156

GEODESIC AND HOLOMORPHICALLY–PROJECTIVE

MAPPINGS OF CONFORMALLY–KÄHLERIAN SPACES

Želko Radulovič and Josef Mikeš

Abstract. It is proved that conformally Kählerian spaces do not admit nontrivial geodesic and holomorphically-projective mappings, preserving the structure, onto almost Hermitian spaces. Examples of conformally Kählerian spaces admitting nontrivial geodesic mapping onto a conformally Kählerian spaces are found. Key words: Geodesic mapping, holomorphically-projective mapping, conformally-Käh-lerian space, almost Hermitian space. MS classification: 53C35.

Geodesic and holomorphically-projective mappings were studied by many authors [1],[14]. W.Y. Westlake [13], K. Yano and T. Nagano [15] have shown that between Kählerian space one cannot establish nontrivial geodesic mappings preserving the structure. These works were further developed by A.V. Karmasina and I.N. Kurbatova [4], and it has been shown that K-spaces do not admit nontrivial and structure-preserving geodesic mappings onto almost Hermitian spaces.

In works of J. Mikeš [7]–[10] results are given on geodesic mappings of Kählerian spaces not preserving the structure.

Conformally Kählerian spaces are a natural generalization of Kählerian spaces. The interest in investigation of these spaces recently grows due to the possibility of using them as a model of Kaluza–Klein theory [3]. Investigations of geometric properties of conformally-Kählerian space are carried out, for example in [2],[5],[6].

1. Introduction

For the beginning we shall give basic definitions. A Riemannian space H_{n} is called almost Hermitian if both metric tensor g_{ij}(x) and almost Hermitian structure F_{i}^{h}(x) are determined in it, and F_{i}^{h}(x) satisfies the conditions

F_{α}^{h}F_{i}^{α} = −δ_{i}^{h}; F_{(i}^{α}g_{j)α} = 0,  (1)

δ_{i}^{h} being the Kronecker symbol, (i, j) denoting symmetrization without division.

An almost Hermitian space with covariantly constant structure is called Kählerian space. Riemannian space that can be conformally mapped onto Kählerian space is called conformally-Kählerian space K_{n}. Evidently, a conformally-Kählerian space K_{n} is almost

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Hermitian. Conformally-Kählerian spaces K_{n} are characterized by the existence of almost Hermitian structure (1), satisfying the following conditions [2]:

F_{i,j}^{h} = δ_{j}^{h}φ_{i} − φ^{h}g_{ij} + F_{j}^{h}φ̄_{i} + F_{ij}φ̄^{h}  (2)

where comma denotes the covariant derivative with respect to the connection on K_{n},

φ_{i} ≡ \frac{1}{n − 2}F_{i,α}^{α}; φ^{h} ≡ g^{hα}φ_{α}; φ̄_{i} ≡ φ_{α}F_{i}^{α}; φ̄^{h} ≡ φ^{α}F_{α}^{h}; F_{ij} ≡ g_{iα}F_{j}^{α}.

From the integrability conditions of relations (2) the following formula, for the Riemann tensor follows:

R_{hiαβ}F_{j}^{α}F_{k}^{β} − R_{hijk} = g_{h[k}φ̄_{j]i} − g_{i[k}φ̄_{j]h} + F_{h[k}φ_{j]i} − F_{i[k}φ_{j]h},  (3)

where

φ̄_{ij} ≡ φ_{αi}F_{j}^{α}; φ_{ij} ≡ φ_{i,j} − φ̄_{i}φ̄_{j} + αF_{ij},

α is an invariant, expressible in terms of g_{ij} and F_{i}^{h}, [j, k] denoting alternation without division.

A diffeomorphism of a Riemannian space V_{n} into V̄_{n} which maps all geodesics in V_{n} into geodesics in V̄_{n} is called a geodesic mapping [13]. A geodesic mapping which is not affine is said to be nontrivial.

V_{n} admits a geodesic mappings onto V̄_{n} if and only if one of the following conditions is satisfied [12]:

ḡ_{ij,k} = 2ψ_{k}ḡ_{ij} + ψ_{i}ḡ_{jk} + ψ_{j}ḡ_{ik},  (4)

where ψ_{i} is a gradient vector, ḡ_{ij} is a metric tensor of space V̄_{n}. In [12] it is proved, that V_{n} admits geodesic mappings if V_{n} there exists a solution of the following equations

a_{ij,k} = λ_{i}g_{jk} + λ_{j}g_{ik}  (5)

on an unknown symmetric regular tensor a_{ij} and a vector λ_{i}.

When ψ_{i} ≢ 0 ( ⟺ λ_{i} ≢ 0) the geodesic mapping is nontrivial. The conditions of integrability of these equations have the form

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

where λ_{ij} ≡ λ_{i,j}.

2. Geodesic mappings of conformally-Kählerian spaces onto almost Hermitian spaces

Let H_{n} and H̄_{n} are almost Hermitian spaces with structures F_{i}^{h} and F̄_{i}^{h}, respectively, and let a geodesic mapping be established between them. One can choose, in the neighborhoods U_{p} ⊂ H_{n}, Up̄ ⊂ H̄_{n} of each pair of corresponding points, a ”common” coordinate system. If, with respect to such a coordinate system, the condition

F_{i}^{h}(x) ≡ F̄_{i}^{h}(x),  (7)

is satisfied, then the mapping is said to be structure-preserving. As we have mentioned above, the geodesic mappings preserving the structure of almost Hermitian spaces has been investigated in [4],[7],[13],[15]. In the paper of A.V. Karmasina and I.N. Kurbatova [7] geodesic mappings of conformally-Kählerian space onto almost Hermitian spaces structure preserving were studied. There investigations were completed by the following theorem:

Theorem 1. A conformally-Kählerian space K_{n} (n > 2) does not admit nontrivial structure-preserving geodesic mappings onto an almost Hermitian space.

Proof. Let us suppose the opposite. Let K_{n} (g_{ij}, F_{i}^{h}) admits a nontrivial geodesic structure-preserving mapping onto almost an Hermitian space H̄_{n}(ḡ_{ij}, F̄_{i}^{h}).

The condition ḡ_{α(i}F̄_{j)}^{α} = 0, that takes place in H̄_{n} can be covariantly differentiated in K_{n}. After taking into account (2), (4) and (7), as a result we have

χ_{i}ḡ_{jk} + χ_{j}ḡ_{ki} + χ_{k}ḡ_{ij} + Θ_{i}g_{jk} + Θ_{j}g_{ki} + Θ_{k}g_{ij} = 0,  (8)

where χ_{i} ≡ φ + ψ_{α}F_{i}^{α}; Θ ≡ ḡ_{iα}φ^{α}.

From n > 2 one can see, that (8) implies either ḡ_{ij} = α g_{ij} or χ_{i} = Θ_{i} = 0. The first case means that K_{n} admits conformal mapping onto H_{n} in contradiction with nontriviality of the geodesic mapping. The second case is reduced to φ_{i} = 0, which means that K_{n} is Kählerian space, and for this case an analogous theorem is proved in [7]. Thus the theorem was proved.

3. Geodesic mappings of conformally-Kählerian spaces onto Riemannian spaces

A Riemannian space is called a V_{n}(B)–space, if it admits a geodesic mapping satisfying the equations (5) and

λ_{i,j} = μg_{ij} + Ba_{ij}  (9)

where μ and B are functions.

It takes place the following

Theorem 2. If a conformally–Kählerian space K_{n} (n > 10) admits a geodesic mapping onto a Riemannian space, then either space K_{n} is a V_{n}(B)-space, or the solution of equations (5) satisfies one of the conditions:

(a) Rg∥a_{ij} − ρg_{ij}∥ ≤ 3, or (b) a_{ij} = βφ_{(ij)} + γφ_{α(i}F_{j)}^{α} + δg_{ij}  (10)

where ρ, β, γ, δ are invariants.

Proof. Let a conformally–Kählerian space K_{n} (n > 10) admit a geodesic mapping. Let us contract the integrability condition (6) of the equations (5) with F_{j′}^{j}F_{k′}^{k}. After omitting primes, from the obtained relation we shall subtract (6). On the basis of (3) one gets

a_{[k|h}φ̄_{j]i} + a_{[k|i}φ̄_{j]h} + ā_{[k|h}φ_{j]i} + ā_{[k|i}φ_{j]h}+
+g_{[k|h}Φ̄_{j]i} + g_{[k|i}Φ̄_{j]h} + F_{[k|h}Φ_{j]i} + F_{[k|i}Φ_{j]h} = 0  (11)

where

Φ_{ji} ≡ λ_{iα}F_{j}^{α} − φ_{jα}a_{i}^{α}; Φ̄_{ji} ≡ Φ_{αi}F_{j}^{α}; ā_{ji} ≡ a_{αi}F_{j}^{α}.

A detailed analysis of the conditions (11) for n > 10 leads us to a conclusion that either conditions (10a) or one of conditions

(a) φ_{ij} = α_{1}g_{ij} + α_{2}F_{ij} + α_{3}a_{ij} + α_{4}ā_{ij},
or  (12)

(b) Φ_{ij} = β_{1}g_{ij} + β_{2}F_{ij} + β_{3}a_{ij} + β_{4}ā_{ij},

are satisfied, where α_{1}, ...,β_{4} are invariants.

If (α_{3})^{2} + (α_{4})^{2} ≠ 0 then from (12a) condition (10b) follows. In the opposite case (12a) turns into φ_{ij} = α_{1}g_{ij} + α_{2}F_{ij}, and from (12b) one obtains

λ_{i,j} = μg_{ij} + Ba_{ij} + α_{5}F_{ij} + α_{6}ā_{ij},  (13)

where μ,B,α_{5},α_{6} are invariants. Alternating (13) we obtained

2α_{5}F_{ij} + α_{6} ā_{[ij]} = 0.

Differentiating covariantly the latter, we see after some analysis that either conditions (10a) are satisfied, or α_{5} = α_{6}. Then formula (13) takes the form of (9), and, consequently, K_{n} is a V_{n}(B)-space which completes the proof.

4. Conformally-Kählerian spaces, admitting nontrivial geodesic mappings

It is easy to show that the space V_{n} with a metric form

ds^{2} = e(dx^{1})^{2} + f(x^{1})ds̃^{2},  (14)

where e = ±1, f (≠ 0) is a function of the argument x^{1} and ds̃^{2} is a metric of a Sasaki space, is a conformally-Kählerian space with concircular vector field.

In [9] it is shown, that (14) for f ≡ const (x^{1})^{2} determines the metric of a Kählerian space.

The direct analysis of the basic equations of the geodesic mappings makes it possible to prove

Theorem 3. The conformally-Kählerian space K_{n} with the metric (14) admits a geodesic mapping onto the conformally-Kählerian space K̄_{n} with the metric

ds^{2} = eα(1 + βf)^{−2}(dx^{1})^{2} + αf(1 + βf)^{−1}ds̃^{2},

where α, β are constants, satisfying the conditions α ≠ 0, 1 + βf ≠ 0. When βf′ ≢ 0, then the mapping is nontrivial one.

It is not difficult to show that if f = cos^{2} αx^{1}, (or sh^{2}αx^{1}), α ≠ 0, then conformally-Kählerian space with the metric (14) admits nontrivial projective transformations.

One should note that the spaces with metrics as in (14) are a particular case of V_{n}(B)–spaces.

5. Holomorphically-projective mappings of a conformally-Kählerian space onto an almost Hermitian space

We recall the concept of holomorphically-projective mapping of a conformally-Käh-lerian space K_{n}(g_{ij}, F_{i}^{h}) onto an almost Hermitian space H̄_{n}(ḡ_{ij}, F̄_{i}^{h}).

A curve L : x^{h} = x^{h}(t) in K_{n} will be called an analytically planar one, if its tangential vector λ^{h} ≡ dx^{h}/dt under a parallel motion is always complanar to the tangential vector λ^{h} and the conjugate vector λ^{α}F_{α}^{h}. Similarly we introduce this notion in H̄_{n}.

A diffeomorphism f:K_{n} → H̄_{n} will be called a holomorphically-projective mapping, if with respect to f, all the analytically planar curves in the Kählerian space are transformed into analytically planar curves in H̄_{n}.

Taking into account the modified results reported in [11], one can show easily that, in a general ”common” coordinate system,

Γ̄_{ij}^{h} = Γ_{ij}^{h} + δ_{(i}^{h}ψ_{j)} + F_{(i}^{h}Θ_{j)} and F̄_{i}^{h} = ±F_{i}^{h},

where ψ_{i}, Θ_{i} are vectors, Γ_{ij}^{h}, Γ̄_{ij}^{h} are the Christoffel symbols of K_{n} and H̄_{n}. These conditions are equivalent to the equations

ḡ_{ij,k} = 2ψ_{k}ḡ_{ij} + ψ_{(i}ḡ_{j)k} + Θ_{(i}F̃_{j)k}  (15)

where F̃_{ji} ≡ ḡ_{jα}F_{i}^{α}.

Differentiate covariantly the relation ḡ_{α(i}F̄_{j)}^{α} = 0, which holds in H̄_{n}. After the substitution from (2) and (15) we symmetrize the obtained expression. In the result we get

ξ_{i}g_{jk} + ξ_{j}g_{ki} + ξ_{k}g_{ij} + ξ̄_{i}ḡ_{jk} + ξ̄_{j}ḡ_{ki} + ξ̄_{k}ḡ_{ij} = 0,

where

ξ_{i} ≡ −ḡ_{iα}φ^{α}; ξ̄_{i} ≡ ψ_{α}F_{i}^{α} + Θ_{i} + φ_{i}.

Hence it follows, that either ξ_{k} = ξ̄_{k} = 0, or K_{n} is Kählerian space. The latter case leads to the homothetic mappings K_{n} → H̄_{n} that we consider as trivial. In result, we have proved

Theorem 4. Conformally-Kählerian spaces K_{n}, different from Kählerian ones, do does not admit nontrivial holomorphically-projective mappings onto almost Hermitian spaces.

References

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Želko Radulovič and Josef Mikeš, Odessa university, Petra Velikogo 2, Odessa, Ukraine

Received 8 October 1992 Revised 30 January 1993