O N G E O D E S I C M A P P I N G S and THEIR GENERALIZATIONS Josef MIKEˇ S Department of Algebra and Geometry Palacky University Olomouc Czech Republic mikes@inf.upol.cz VARNA (ODESSOS) 2 0 0 7
1. Introduction Diffeomorphisms and automorphisms of geometrically generalized spaces consti- tute one of the contemporary actual directions in differential geometry. A large number of works is devoted to geodesic, quasigeodesic, holomorphically projective, almost geodesic, F -planar and other mappings, transformations and deformations. This lecture is dedicated to some results concerning the fundamental equations of these mappings and deformations. Obviously the existence of a solution of these fundamental equations imply the existence of the mentioned mappings, transformations and deformations. These fundamental equations were found in several forms. Among these forms there is the particularly important form of a system of differential equations of Cauchy type. For their linear forms the question of solvability can be answered by algebraic methods.
2. On systems of differential equations of Cauchy type Here we introduce the basic notions of the theory of systems of differential equa- tions of Cauchy type. We restrict ourselves to the local theory. Assume a smooth domain D ⊂ R n with coordinates x = ( x 1 , x 2 , . . . , x n ) and smooth functions F A i ( x, y ) , i = 1 , . . . , n ; A = 1 , . . . , N , on D ⊂ D × R N . The system of differential equations of Cauchy type has the following form ∂y A ( x ) A, B = 1 , . . . , N, = F A i ( x, y ( x )) , (1) ∂x i i = 1 , . . . , n, y ( x ) = ( y 1 ( x ) , . . . , y N ( x )) where are unknown functions. For initial Cauchy conditions: y A ( x o ) = y A , A = 1 , . . . , N, (2) o A ∈ R N , then the system (1) has at most one solution. where x o ∈ D and y o For this reason the general solutions of the system (1) depends on r ≤ N real parameters. The system (1) may be written in terns of covariant derivatives. A fundamental investigation of (1) consists in a check of the integrability conditions, which are essentially algebraic equations for the unknown variables y A . In the case when they are identifically fulfilled, we have r = N .
A homogeneous system of linearly differential equations of Cauchy type has the following for ∂y A ( x ) A, B = 1 , . . . , N, = f A B i ( x ) y B ( x ) , (3) ∂x i i = 1 , . . . , n, where f A B i ( x ) are functions on D . The integrability conditions of the homogeneous linear system (3): ∂ 2 y A ( x ) ∂x i ∂x j = ∂ 2 y A ( x ) ∂x j ∂x i constitute a system of homogeneous linear algebraic equations with respect to the unknown functions y A ( x ) . Differential continuation of their integrability conditions forms also a system of homogeneous linear algebraic equations with respect to the unknown functions y A ( x ) . This means that with the aid of linear algebra we may convince ourselves, whether or not the system (3) has solutions and determine on how many pa- rameters r ≤ N it depends
Many problems of differential geometry have been successfully solved by homo- geneous systems of linearly differential equations of Cauchy type, for example: • isometric and homothetic transformations of Riemannian spaces, • affine and projective transformations of Riemannian spaces and spaces with affine connections, • holomorphically projective transformations of K¨ ahlerian spaces. • affine mappings of Riemannian spaces and spaces with affine connections, The above results were found in the years 1900 – 1960 and shown in many monographs – L.P. Eisenhart, S. Kobayashi, A.Z. Petrov, K. Yano, . . . .
Now I want to introduce new results which were obtaind in the last 40 years and are connected which the mentioned systems of Cauchy type. This means that for the mentioed types of geometrical problems regular methods of solution were found. • geodesic mappings of Riemannian spaces (N.S. Sinyukov, 1967), • geodesic mappings of spaces with affine connections onto Riemannian spaces (V.E. Berezovsky and J. Mikeˇ s, 1989), • geodesic deformation of Riemannian hypersurfaces in Riemannian spaces (M.L. Gavrilchenko, V.A. Kiosak and J. Mikeˇ s, 2004), • conformal mappings of Riemannian spaces onto Einstein spaces (M.L. Gavrilchenko, E. Gladysheva and J. Mikeˇ s, 1992),
• holomorphically projective mappings of K¨ ahlerian spaces (V.V. Domashev and J. Mikeˇ s, 1976), • holomorphically projective mappings of hyperbolically K¨ ahlerian spaces (I.N. Kurbatova, 1980), • holomorphically projective mappings of parabolically K¨ ahlerian spaces (M. Shiha, 1994), • F -planar mappings of spaces with affine connections onto Riemannian spaces (J. Mikeˇ s, 1994, 1999),
3. Spaces with affine connection, Riemannian and K¨ ahlerian spaces If the contrary is not specified, the present review is given locally in tensor form in the class of real sufficiently smooth functions. The dimension n of the spaces under consideration, as a rule, is greater than 2, and is not mentioned specially. All the spaces are assumed to be connected. Let us give the basic notions of the theory for space with affine-connected ( A n ) , Riemannian ( V n ), and K¨ ahlerian ( K n ) spaces. 3.1 Space with affine connection ( A n ) . In a space A n with an affine connec- tion without torsion covered by a local coordinate system x = ( x 1 , x 2 , . . . , x n ) together with an object of the affine connection Γ h ij ( x ) ( h, i, j, · · · = 1 , n ) the Riemannian tensor and Ricci tensor are defined in the following way: R h ijk ≡ ∂ j Γ h ki + Γ α ki Γ h jα − ∂ k Γ h ji − Γ α ji Γ h R ij ≡ R α ∂ i ≡ ∂/∂x i . kα , ijα , An equiaffine space is defined as A n , with R ij = R ji . The spaces where the conditions R h ijk = 0 ( R ij = 0 ) hold are called flat (Ricci-flat, respectively). The space A n belongs to the class C r ( A n ∈ C r ) if Γ h ij ( x ) ∈ C r .
3.2 Riemannian Spaces ( V n ) . In the Riemannian space V n , determined by the symmetric and nondegenerate metric tensor g ij , Christoffel symbols of types I and II are introduced by the formulas Γ ijk ≡ 1 2( ∂ i g jk + ∂ j g ik − ∂ k g ik ) and Γ h ij ≡ g hα Γ ijα , where g ij are elements of the inverse matrix to g ij . The signature of the metrics is assumed, in general, to be arbitrary. Christof- fel symbols of type II are the natural connection (the Levi-Civita connection) of Riemannian spaces, with respect to which the metric tensor is covariantly constant, i.e., g ij,k = 0 . Hereafter “,” denotes the covariant derivative with respect to the connection of the space V n (or A n ).
A Riemannian space is equiaffine. The space V n belongs to the class C r ( V n ∈ C r ) if g ij ∈ C r . Using g ij and g ij , we introduce in V n the operation of lowering and rising indices, for example: R hijk ≡ g hα R α ijk ; R h k .ij. ≡ g kα R h ijα ; R h i ≡ g hα R αi . Together with the tensors of Riemann, Ricci, and the projective Weyl tensor, the latter is simplified in V n : 1 W h ijk ≡ R h n − 1( δ h k R ij − δ h ijk − j R ik ) , where δ h i is the Kronecker symbol, in V n we introduce into consideration the scalar curvature R ≡ R αβ g αβ and the Brinkmann and Weyl tensors of conformal curvature: 1 R L ij ≡ n − 2 ( R ij − 2( n − 1) g ij ) and C hijk ≡ R hijk − g hk L ji + g ik L jh + g hj L ki − g ij L kh .
3.3 K¨ ahlerian Spaces ( K n ) . In the present lecture, by a K¨ ahlerian space we mean a wide class of spaces defined as follows: A Riemannian space is called a K¨ ahlerian space K n if, together with the metric tensor g ij ( x ) , an affine structure F h i ( x ) is defined that satisfies the relations F h α F α i = eδ h i ; F α ( i g j ) α = 0; F h i,j = 0 , where e = ± 1 , 0 . ahlerian space K − ∗ If e = − 1 , then K n is said to be an elliptical K¨ n , ahlerian space K + ∗ if e = +1 , then K n is said to be a hyperbolic K¨ n , and ∗ if e = 0 and Rg � F h i � = m ≥ 2 , then ahlerian space K o ( m ) K n is said to be an m -parabolic K¨ . n ∗ The space K o ( n/ 2) ahlerian space K o is called a parabolic K¨ n . n The spaces K + n , K − n and K o n must be of even dimension. The spaces K − n were first considered by P.A. Shirokov, the spaces K + n were considered by P.K. Rashevskii, and the spaces K o ( m ) were studied by V.V. Vish- n nevskii. In the investigations mentioned these spaces are referred to as A -spaces. Independently from P.A. Shirokov the spaces K − n were studied by E. K¨ ahler. In the international literature these spaces are preferably referred to as K¨ ahlerian spaces.
4. Conformal mappings onto Einstein spaces 4.1 Conformal mappings of Riemannian spaces Considering concrete mappings of spaces, for example, f : V n → ¯ V n , both spaces are referred to the general coordinate system x with respect to this mapping. This is a coordinate system where two corresponding points f ( M ) ∈ ¯ M ∈ V n and V n have equal coordinates x = ( x 1 , x 2 , . . . , x n ) ; the corresponding geometric objects in V n will be marked with a bar. For example, ¯ ij are the Christoffel symbols in ¯ Γ h V n . The mapping from V n onto ¯ V n is conformal if and only if, in the common coordinate system x with respect to the mapping, the conditions g ij ( x ) = e 2 ψ ( x ) g ij ( x ) , ¯ g ij are metric tensors of V n and ¯ where ψ ( x ) is a function on V n , g ij and ¯ V n , respectively.
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