Holomorphically projective mappings of almost Hermitian spaces and their Gray-Hervella classification
by
Josef Mikeš
Palacky University Olomouc, Czech Republic
Coauthors: V. Malícková, O. Pokorná

Oral Communication

By the term almost Hermitian spaces H_n, we denote all (pseudo-) Riemannian spaces, where the affinor structure F^h_i exists, for which the conditions

Fh\alphaF\alphai=-\deltahi,     g\alpha(iF\alphaj) = 0

hold. Here g_ij is the metric tensor H_n, \delta^h_i is the Kronecker symbol, (i, j) denotes a symmetrization without division.

A natural classification containg 16 types of Hermitian spaces has been done by A. Gray and L.M. Hervella [1].

In many papers holomorphically projective mappings and transformations of Hermitian spaces H_n --> [`H]<sub>n</sub> are studied (see [3], [4], ... ). These are special cases of F<sub>1</sub>-planar mappings. In [2], [3], F<sub>1</sub>-planar mappings from the space with affine connection A<sub>n</sub> onto Riemannian space [`V]_n are defined and studied. These are characterized w.r.t. a common coordinate system x by the following equations

- \Gamma   h ij  (x)=\Gammahij(x)+\deltah(i\psij)+Fh(i\phij),     - g   \alpha(i  F\alphaj) = 0,

where \Gamma^h_ij and [`(\Gamma)]<sup>h</sup><sub>ij</sub> are the object of affine connection on A<sub>n</sub> and [`V]_n respectively, [`g]<sub>ij</sub> is the metric tensor of [`V]_n, \psi_i(x),  \phi_i(x) are covectors, and F^h_i(x) (Rank||F^h_i-\rho\delta^h_i|| > 1) are affinor structure on A_n and [`V]_n.

In [2], it is proved that a general solution of the system (1) for a given space A_n and a given structure F^h_i depends on finitely many parameters. This is surely valid also for holomorphically projective mappings between Hermitian spaces. When studying fundamental equations of holomorphically projective mappings of Hermitian spaces, we made them more accurate. The conditions implied from the classification given by Gray - Hervella are linear ones onto a solution of these fundamental equations and these have an influence on the number of essential parameters of a general solution.

This work was supported by Grant GA CR 201/99/0265.

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Gray A., Hervella L.M.: The sixteen classes of almost Hermitian manifolds and their linear invariants, Ann. Mat. Pura Appl. 123, 4 (1980), 35-58.

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Mikes J.: Special F-planar mappings of affinely connected spaces onto Riemannian spaces, Mosc. Univ. Math. Bull. 49, 3 (1994), 15-21.

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Mikes J.: Holomorphically projective mappings and their generalizations, J. Math. Sci. New York, 89, 3 (1998), 1334-1353.

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Yano K.: Differential Geometry on Complex and Almost Complex Spaces, Pergamon Press, Oxford, 1965.

Date received: June 30, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cafh-38.