On the quaternionic sectional curvature of an indefinite quaternionic Kähler manifold
type: Item
instance of: scholarly article
title: On the quaternionic sectional curvature of an indefinite quaternionic Kähler manifold
zbMATH Open document ID: 0853.53028
author: Eduardo García-Río
author: M. E. Vázquez-Abal
published in: Tsukuba Journal of Mathematics
publication date: 1997-01-09T00:00:00Z
review text: In this paper, the authors characterize indefinite quaternionic space forms by the condition: \(R(U, \varphi U,U, \varphi U)=0\) for all null vectors \(U\) and \(\varphi= I,J, K\) any local basis of the bundle of almost complex structures. By a completely different approach, the authors streighten out a proof given in [\textit{J. D. Pérez} and \textit{F. Santos}, Ann. Mat. Pura Appl., IV. Ser. 132, 383-398 (1982; Zbl 0513.53043)]. Notice that \(R(U,JU, U, JU) =0\) for all null vectors \(U\) is not sufficient for the constancy of the holomorphic sectional curvature of an indefinite Kähler manifold [\textit{A. Bonome}, \textit{R. Castro}, \textit{E. García-Río} and \textit{L. M. Hervella} [C. R. Acad. Sci., Paris, Sér. I 315, No. 11, 1183-1187 (1992; Zbl 0765.53051)]. This makes a substantial difference with respect to the complex case. Lastly, using the expression of the Jacobi operator along null geodesics, the authors show the nonexistence of conjugate points along null geodesics in indefinite quaternionic space forms.
Mathematics Subject Classification ID: 53C15
Mathematics Subject Classification ID: 53C55
Mathematics Subject Classification ID: 53C22
zbMATH DE Number: 895937
zbMATH Keywords: indefinite quaternionic space forms
zbMATH Keywords: bundle of almost complex structures
zbMATH Keywords: Jacobi operator
zbMATH Keywords: null geodesics
zbMATH Keywords: nonexistence of conjugate points
reviewed by: Luis A. Cordero
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.21099/tkbjm/1496162869
OpenAlex ID: W178561008
DOI: 10.21099/TKBJM/1496162869
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description: scientific article; zbMATH DE number 895937