Geometric realizations of curvature
type: Item
instance of: scholarly article
title: Geometric realizations of curvature
zbMATH Open document ID: 1184.53021
author: Peter B. Gilkey
author: M. Brozos-Vázquez
author: Stana Nikčević
published in: Nihonkai Mathematical Journal
publication date: 2009-11-30T00:00:00Z
full work available at URL: https://arxiv.org/abs/0904.1192
review text: Interesting problems involving the geometric realization of curvature occur in several areas of differential geometry. A detailed exposition of recent results and open question on the geometric realization of the curvature in different settings is the subject of this paper. Firstly, the authors treat the geometric realizability of an affine algebraic curvature operator acting on a linear space and of an algebraic curvature tensor on a linear space endowed with a non-degenerate symmetric bilinear form. Then, they consider realization problems related to affine structures in the presence of a pseudo-Riemannian metric. The same problems are treated in the context of almost Hermitian, almost para Hermitian, almost hyper Hermitian geometries and, in particular, Hermitian and para Hermitian structures are considered. In each of the mentioned cases, particular attention to the decomposition of the appropriate space of tensors into irreducible modules under a suitable structure group is given. Furthermore, the authors point their attention to Ivanov-Petrova and Osserman geometries and discuss several questions on curvature homogeneity. The paper, which is carefully written, is equipped with a wide bibliography.
reviewed by: Maria Falcitelli
Mathematics Subject Classification ID: 53B20
zbMATH DE Number: 5640680
zbMATH Keywords: geometric realization
zbMATH Keywords: affine and Riemannian geometry
zbMATH Keywords: almost Hermitian geometry
MaRDI profile type: Publication
arXiv ID: 0904.1192
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description: scientific article