Holomorphically projective mappings and their generalizations

Abstract

Diffeomorphisms and automorphisms of geometrically generalized spaces constitute one of the current main directions in differential geometry. A large number of works are devoted to geodesic, holomorphicalty projective, almost geodesic, and other mappings.On the other hand, one line of thought is now the most important one, namely, the investigation of special, affine-eonnected Riemannian, and Ks spaces. Symmetric spaces a~d their generalizations play the most significant part among them.Symmetric and, in implicit form, semisymmetric spaces were introduced into consideration by Shirokov [991 and Cartan [115,116].Many ~vorks are devoted to the geometry of symmetric spaces. Cartan made a considerable contribution to their st~ dy.In this article, I present some new results obtained for holomorphically projective, almost geodesic, and conformal mappings of symmetric, semisymmetric, and more general spaces. I give new results for holomor-phicaUy projective mappings of Ks hyperbohcally and parabolically Ks spaces, conformal mappings onto Einstein spaces, almost geodesic, and F-planar mappings. This research is supported by the Czech Grant Agency~ grant No. 201/93/0469.The problems connected with these themes have been considered in many monographs and reviews, such as [3, 13-15, 21, 23, 24, 34, 52, 57, 59, 60, 66, 67, 69, 80, 98, 99, 112, 116, 118, 125, 126, 154].Unless otherwise specified, the investigation is carried out in tensor form, locally, in the class of real, sufficiently smooth functions. The dimension n of the spaces being considered is higher than three, as a rule. This fact is not specially stipulated. All spaces are assumed to be connected.This article is a direct continuation of [52]. The same notations are used here. In this article, the concept of a K~aler space is understood as a wider class of spaces in accordance with the following definition.A K~hIer space K,~ is a Riemannian space in which, along with the metric tensor gij(x), an affme struc~ ture Fib(x) is defined that satisfies the relations F~F'-.h ~ = esh; Fiigj),~,~ = 0; F .h-,,~ = 0, where e = -t-1,0.Here and in what follows "," denotes a covariant derivative. If e = -1, then Kn is an elliptically KShlerian space K~, if e =-+1, then Kn is a hyperbolically KShleriau space It "+, andif e =0 and Rg IIF, II --m < 5, then K,~ is an m-parabolicalIy KShlerian space K ~ The space K ('~/2) is known as a parabolically K5hlerian space K ~ Necessarily, the spaces K +, g~-, and K ~ are of an even dimension. The Spaces K~-were first considered by Shirokov [99], the spaces K + by Rashevsky [65], and the spaces K ~ by Vishnevsky [15]. In the indicated works these spaces are called A -spaces. Independently of Shirokov, the spaces K~ were studied by KKhler [124]. In the available literature these spaces are also called Kfihlerian. In the KKhlerian spaces Kn we shall introduce the operation of conjugation of indices as follows: A~ = Ag Fff ; A r = A~ i In the spaces K~ we can consider the holomorphically projective curvature tensor and the Bochner tensor ) Ph...

On holomorphically projective mappings of $e$-Kähler manifolds

“…Further we notice that for holomorphically projective mappings between e-Kähler manifolds K n andK n of class C 3 holds the following third set of equations [6,8,9,15,10,16]:…”

“…We present well known facts, which were proved by Domashev, Kurbatova, Mikeš, Prvanović, Otsuki, Tashiro etc., see [2,3,6,7,8,9,10,11,12,15,16,17,18,19]. In these results no details about the smoothness class of the metric were stressed.…”

Harmonic and Projective Diffeomorphisms

“…It is known that Riemannian spaces of constant curvature form a closed class with respect to geodesic mappings (Beltrami's theorem, see [36]). In 1978 (see [14,15,22,23]), J. Mikeš proved that under the conditions V n ,V n ∈ C 3 the following theorems hold (locally). Theorem 3.9.…”

“…Concircular fields play an important role in the theories of geodesic mappings and projective and conformal transformations. They were studied by a number of geometers: N. S. Sinyukov [35], A. V. Aminova [1], J. Mikeš [22,23], I. G. Shandra [31][32][33][34], etc.…”

Geodesic mapping onto Kählerian spaces of the first kind

“…Kählerian spaces and their mappings were investigated by many authors, for example T. Otsuki and Y. Tasiro [18], [25], K. Yano [26], J. Mikeš, V. V. Domashev [1], [6], [7], [8], [9], [10], [11], [22], M. Prvanović [19], N. Pušić [21], S. S. Pujar [20], M. S. Stanković at al. [16], [24], and many others.…”