1 Introduction. FILOMAT (Niš) 16 (2002), GEODESIC MAPPINGS BETWEEN KÄHLERIAN SPACES. Josef Mikeš, Olga Pokorná 1 and Galina Starko

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1 FILOMAT (Niš) 16 (2002), GEODESIC MAPPINGS BETWEEN KÄHLERIAN SPACES Josef Mikeš, Olga Pokorná 1 and Galina Starko Abstract Geodesic mappings from a Kälerian space K n onto a Kälerian space K n will be investigated in tis paper. We present a construction of Kälerian space K n wic admits non-trivial geodesic mapping onto Kälerian space K n. 1 Introduction Geodesic mappings of special Riemannian spaces were studied by many autors (see e.g. [9], [13], [15]). Geodesic mappings of Kälerian spaces were investigated namely by N. Coburn [1], K. Yano [17], V.J. Westlake [16], K. Yano, T. Nagano [18] and oters. In tese works te autors prove tat in te case of preserving te structure of Kälerian spaces by geodesic mappings tese mappings are trivial (i.e. affine). Koga Mitsuru [5] as found more general conditions for te structure of Kälerian spaces forcing any geodesic mapping to be trivial. Similar questions for geodesic mappings of almost Hermitian spaces were investigated by A. Karmazina and I.N. Kurbatova [3]. Te geodesic mappings from a Kälerian space K n onto a Riemannian space V n were studied by J. Mikeš (see [6], [7], [8], [9]). Papers of J. Mikeš, G. Starko, M. Sia were devoted to geodesic mappings of yperbolical and parabolical Kälerian spaces wic are generalizations of classical Kälerian spaces (see [9], [11],[12]). In te sequel, by Kälerian space we mean bot classical (i.e. elliptical) as well as yperbolical and parabolical Kälerian space. In tis paper, we investigate geodesic mappings from a Kälerian space K n onto a Kälerian space K n. We present a construction of non-trivial Kälerian spaces K n wic are geodesically mapped onto Kälerian spaces K n. 1 Supported by grant No. 201/02/0616 of Te Grant Agency of Czec Republic. 43

2 44 Josef Mikeš, Olga Pokorná and Galina Starko 2 Geodesic mappings of Kälerian spaces A diffeomorpism f from a Riemannian space V n onto a Riemannian space V n is called a geodesic mapping if f maps any geodesic line of V n into a geodesic line of V n (see [2], [9], [13], [15]). A mapping from V n onto V n is geodesic if and only if, in te common coordinate system x wit respect to te mapping te conditions Γ ij(x) = Γ ij(x) + δ i ψ j + δ j ψ i (1) old, were ψ i (x) is a covector, Γ ij and Γ ij are te Cristoffel s symbols of V n and V n, respectively, δi is te Kronecker symbol. Conditions (1) are equivalent to ḡ ij,k = 2ψ k ḡ ij + ψ i ḡ jk + ψ j ḡ ik, (2) were ḡ ij is te metric tensor of V n and, denotes te covariant derivative wit respect to te connection of te space V n. Conditions (1) and (2) are called te Levi-Civita equations. Te covector ψ i is gradient-like, i.e. ψ i = ψ,i. If ψ i 0 ten a geodesic mapping is called non-trivial; oterwise it is said to be trivial or affine. If a mapping f: V n V n is geodesic ten te following conditions old: were a) R ijk = Rijk + δ k ψ ij δj ψ ik, b) Rij = R ij + (n + 1)ψ ij, (3) c) W ijk = Wijk, ψ ij ψ i,j ψ i ψ j, (4) Rijk ( R ijk ) are te Riemannian tensors ofv n ( V n ), R ij ( R ij ) are te Ricci tensors of V n ( V n ), Wijk ( W ijk ) are te Weyl tensors of te projective curvature of V n ( V n ). Te Weyl tensor of te projective curvature is an invariant object of te geodesic mapping. In te present paper, by a Kälerian space we mean a wide class of spaces defined as follows [9]: a Riemannian space is called a Kälerian space K n if, togeter wit te metric tensor g ij (x), an affine structure Fi (x) is defined on K n wic satisfies te relations F α F α i = e δ i ; F α i g αj + F α j g αi = 0; F i,j = 0, (5)

3 Geodesic Mappings Between Kälerian Spaces 45 were e = ±1, 0. If e = 1 ten K n is said to be an elliptic Kälerian space Kn, if e = +1 ten K n is said to be a yperbolic Kälerian space K n +, and if e = 0 and Rg Fi = m 2 ten K n is said to be an m-parabolic Kälerian space Kn o(m). Te space Kn o(n/2) is called te parabolic Kälerian space Kn. o As in K n te structure F is covariantly constant, from [6] follows tat on a Kälerian space K n admitting a nontrivial geodesic mapping, tere exists a nonzero convergent vector field (see [6], [7], [8], [9], [12], [11]). In every space Kn wit a covariantly nonconstant convergent vector field tere exists a coordinate system x wit te following metrics and structure (see [7], [8], [9]): g ab = g a+m b+m = ab f + a+m b+m f; g a b+m = a b+m f a+m b f; were a, b = 1, m, m = n/2, F a+m b = F a b+m = δ a b ; F a b = F a+m b+m = 0, f = exp(2x 1 ) G(x 2, x 3,..., x m, x 2+m, x 3+m,..., x n ). If G C 3, ten tese formulas generate (provided g ij = 0) te metric of a Kälerian space Kn, were a non-constant convergent vector field exists. A similar property olds (see [12]) for te yperbolic Kälerian spaces wit metrics and structure of te type K + n were a, b = 1, m, m = n/2, g ab = g a+m b+m = 0; g a b+m = a b+m f; F a+m b = F a b+m = 0; F a b = F a+m b+m = δa b, f = exp(x 1 + x 1+m ) G(x 2 + x 2+m, x 3 + x 3+m,..., x m + x n ). Metrics of parabolically Kälerian spaces K o n, admitting covariantly nonconstant convergent vector fields were found by J. Mikeš and M. Sia [11]. 3 Geodesic mappings onto Kälerian spaces In tis section, we determine conditions wic are necessary and sufficient for a Riemannian space V n to admit a nontrivial geodesic mapping onto a Kälerian space K n satisfying te formulas (5). Te following teorem olds:

4 46 Josef Mikeš, Olga Pokorná and Galina Starko Teorem 1 Te Riemannian space V n admits a nontrivial geodesic mapping onto a Kälerian space K n if and only if, in te common coordinate system x wit respect to te mapping, te conditions a) ḡ ij,k = 2ψ k ḡ ij + ψ i ḡ jk + ψ j ḡ ik, b) F i,k = F k ψ i δk F i αψ α old, were ψ i 0 and tensors ḡ ij and F i (6) satisfy te following conditions: ḡ ij = ḡ ji, det ḡ ij = 0, F α F α i = ēδ i, F α i ḡ αj + F α j ḡ αi = 0. (7) Ten ḡ ij and F i are te metric tensor and te structure of K n, respectively. Proof. Te Levi-Civita equation (6a) (2) guarantees te existence of geodesic mappings from a Riemannian space V n onto a Riemannian space V n wit metric tensor ḡ ij. Te formula (6b) implies tat te structure F i in V n is covariantly constant. Furter, te algebraic conditions (7) guarantee tat ḡ ij and F i are te metric tensor and te structure of te same Kälerian space K n, respectively. Te system (6) is a system of partial differential equations wit respect to te unknown functions ḡ ij (x), F i (x) and ψ i (x) wic moreover must satisfy algebraic conditions (7). 4 Geodesic mappings between Kälerian spaces As was said in te introduction, a geodesic mapping between Kälerian spaces K n and K n wic preserves te structure (i.e. in te common coordinate system x wit respect to te mapping te conditions F i (x) = F i (x) old, were Fi (i.e. affine). and F i Since te structures F i are structures of K n and K n, respectively) is trivial and F i are covariantly constant in K n and K n, respectively, we can deduce from te results of [6], [9] tat for te tensor ψ ij under a geodesic mapping K n onto K n te relation ψ ij = 0 olds, i.e. ψ i,j = ψ i ψ j. (8) It follows from te relations (3) and (8) tat te Riemannian tensor for a geodesic mapping of K n onto K n is invariant. We sall construct a Kälerian space K n admitting a nontrivial geodesic mapping; of course, te structure of K n is not preserved.

5 Geodesic Mappings Between Kälerian Spaces 47 Obviously, te existence of a nontrivial geodesic map between (pseudo-) euclidean spaces E n and Ēn follows from te Beltrami teorem. On te oter and, under some specific conditions on te dimension and te signature of metrics, te spaces E n and Ēn are Kälerian spaces K n and K n in our sense. For example, E 2m is K2m and K+ 2m, too, were g = (I); F = ( 0 ) I I 0 and g = ( ) ( ) 0 I I 0 ; F = I 0 0 I old, respectively. We now construct a nontrivial example of a geodesic mapping between Kälerian spaces. Let K n be a product of Riemannian spaces wit te metric ds 2 = d s 2 + d s 2, (9) were d s 2 is te metric of te euclidean Kälerian space Kñ wit te metric tensor g ab and te structure F a b, (a, b, c = 1, 2,..., ñ); d s 2 is te metric of a Kälerian space Kñ wit te metric tensor g AB and te structure F A B, (A, B, C = ñ+1,..., ñ+ñ), and suc tat a noncovariantly constant concircular vector field ξ exists on K n. Tis space is a Kälerian space, and g = ( ) g 0 0 g are its metrics and structure, respectively. ( ) F 0 and F = 0 F Te spaces Kñ and Kñ must be of te same type, i.e. bot of tem must be eiter elliptic or yperbolic or parabolic. We prove te following result. Teorem 2 Te Kälerian space K n, constructed above, admits a nontrivial geodesic mapping onto a Kälerian space K n. Proof. In te space Kñ we sall investigate te equations (analogical to (6)): q ab c = 2 ψ c q ab + ψ a q bc + ψ b q ac, B b c a = B c a ψ b δc a B b d ψ d, (10) ψ a b = ψ a ψb, ψa = ψ a 0,

6 48 Josef Mikeš, Olga Pokorná and Galina Starko were is te covariant derivative in Kñ; q ab, Ba b, ψa are some tensors satisfying te algebraic conditions B a c B c b = eδ a b, Bc a q cb + B c b q ca = 0, q ab = q ba, q ab = 0. (11) Te solution of te equations (10) satisfying (11) exists, because te equations (10) are completely integrable in te euclidean space Kñ. On te oter and, since tere exists a noncovariantly constant concircular vector field in Kñ, we can find a function ξ satisfying te conditions 2 ξ = ξa ξa, ξa B = δ A B, (12) were 2 ξa ξb g AB, ξa ξ A, g AB = g AB 1 and denotes te covariant derivative of Kñ. We put ḡ ab = 2 k exp(2 ψ) ξ ψa ψb + q ab, ḡ ab = k exp(2 ψ) ξb ψa, ḡ AB = k exp(2 ψ) g AB, F a b = B a b F, B = 0, F A B = F A B, F A b = F A B ξ ψb ξa Bc b ξc, were k is a constant suc tat g ij 0. Putting ψ = ψ, we can verify te formulas (6) and (7). Hence te tensors ḡ ij and F i constructed by Teorem 1 are te metric and structure tensors of te Kälerian space K n, respectively, and K n is a geodesic image of K n. References [1] N. Coburn, Unitary spaces wit corresponding geodesic, Bull. Amer. Mat. Soc (1941). [2] L.P. Eisenart, Riemannian Geometry, Princeton Univ. Press, [3] A. Karmasina, I.N. Kurbatova, On some problems of geodesic mappings of almost Hermitian spaces, Odessk. Ped. Inst. (1990), Deposited at UkrNIINTI, Kiev, , No. 458-Uk90 (1990).

7 Geodesic Mappings Between Kälerian Spaces 49 [4] S. Kobayasi, Transformation Groups in Differential Geometry, Springer-Verlag, Berlin, [5] Koga Mitsuru, On projective diffeomorpismus not necessarily presering complex structure, Mat. J. Okayama Univ. 19, No. 2, (1977). [6] J. Mikeš, On geodesic mappings of Ricci-2-symmetric Riemannian spaces, Mat. Zametki 28, No. 2, (1980). [7] J. Mikeš, On equidistant Kälerian spaces, Mat. Zametki 38, No. 4, (1985). [8] J. Mikeš, On Sasakian and equidistant Kälerian spaces, Dokl. Acad. Nauk SSSR 291, No. 1, (1986). [9] J. Mikeš, Geodesic mappings of affine-connected and Riemannian spaces, J. Mat. Sci., New York 78, 3, (1996). [10] J. Mikeš, Holomorpically projective mappings and teir generalizations, J. Mat. Sci., New York 89, 3, (1998). [11] J. Mikeš, M. Sia, On equidistant parabolically Kälerian spaces, Tr. Geom. Semin. 22, (1994). [12] J. Mikeš, G.A. Starko, On yperbolically Sasakian and equidistant yperbolic Kälerian spaces, Ukr. Geom. Sb. 32, Karkov, (1989). [13] A.Z. Petrov, New Metods in Teory of General Relativity, Nauka, Moscow, [14] Z. Radulović, J. Mikeš, Geodesic mappings of conformally Kälerian spaces, Russ. Mat. 38, No. 3, (1994). [15] N.S. Sinyukov, Geodesic mappings of Riemannian spaces, Nauka, Moscow, [16] W.J. Westlake, Hermitian spaces in geodesic correspondence, Proc. Amer. Mat. Soc. 5, No. 2, (1954). [17] K. Yano, Differential Geometry on Complex and Almost Complex Spaces, Pergamon Press, Oxford, 1965.

8 50 Josef Mikeš, Olga Pokorná and Galina Starko [18] K. Yano, T. Nagano, Some teorems on projective and conformal transformations, Koninkl. Nederl. Akad. Wet. A60, No. 4, (1957). Josef Mikeš: Dept. of Algebra and Geometry, Fac. Sci., Palacky Univ., Tomkova 40, Olomouc, Czec Republic Olga Pokorná: Dept. of Matematics, Czec University of Agriculture, Kamýcká 129, Praa 6, Czec Republic Galina Starko: Dept. of Matematics, Building Academy of Odessa, Ukraine.