Conformal holomorphically projective mappings satisfying a certain initial condition

Abstract

In this paper we study conformal holomorphically projective mappings between conformal e-Kähler manifolds K n =.M; g; F / and x K n =. x M ; x g; x F /, i. e. diffeomorphisms f : M 3 x M satisfying f h f 1 f 2 f 3 , where f 1 ; f 3 are conformal mappings and f 2 is a holomorphically projective mapping between e-Kähler manifolds (i. e. Kähler, pseudo-Kähler and hyperbolic Kähler manifolds). Suppose that the initial condition f £ x g h k ¡ g is satisfied at a point x 0 P M and that at this point the Weyl conformal tensor satisfies a certain inequality. We prove that the mapping f is then necessarily conformal.

Holomorphically projective mappings of (pseudo-) Kähler manifolds preserve the class of differentiability

“…The above mentioned formulae can be found in the papers [6,28,35]. The integrability conditions of equations (7) have the following form…”

On holomorphically projective mappings of parabolic K\"ahler manifolds

“…First we study the general dependence of holomorphically-projective mappings of parabolic Kähler manifolds in dependence on the smoothness class of the metric. We present well known facts, which were proved by M. Shiha, J. Mikeš et al, see [2,3,14,[17][18][19][20][21]. I. Hinterleitner [5] has solved the analogically problems for classical, pseudo-and hyperbolic Kähler manifolds.…”

On Fε2-planar mappings with function ε of (pseudo-)Riemannian manifolds

“…Many papers are devoted to geodesic, almost geodesic, quasigeodesic, holomorphically projective, F-planar mappings and many others. Study of special manifold with affine connection, (pseudo-) Riemannian, e-Kählerian and e-Hermitian spaces, give one of the most important area, see [1]- [33]. For example, T. Levi-Civita [15] used geodesic mappings for modeling mechanical processes, A.Z.…”

“…The PQ ε -projective equivalence between n-dimensional Riemannian manifolds were introduced by Topalov [32], P and Q are tensors of type (1,1) for which PQ = ε Id, ε ∈ R, ε 1, 1 + n. Moreover, these mappings are special cases of F 2 -planar mappings, [8], studied in [19], see [24, p. 225 -231].…”