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The Focal Locus of a Riemannian 4-Symmetric Space

Published online by Cambridge University Press:  20 November 2018

J. Alfredo Jimenez
J. Alfredo Jimenez*
Affiliation:
The Pennsylvania State University, The Hazleton CampusHazleton, Pennsylvania18201
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Abstract

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compact Riemannian 4-symmetric space M can be regarded as a fibre bundle over a Riemannian 2-symmetric space with totally geodesic fibres isometric to a 2-symmetric space. Here the result of R. Crittenden for conjugate and cut points in a 2-symmetric space is extended to the focal points of the fibres of M. Also the restriction of the exponential map of M up to the first focal locus in the normal bundle of a fibre is proved to yield a covering map onto its image. It is shown that for the noncompact dual M*, the fibres have no focal points and hence the exponential map of M* restricted to the normal bundle of a fibre is a covering map. The classification of the compact simply connected 4-symmetric spaces G/L with G classical simple provides a large class of examples of these fibrations.

Keywords

53C3553C3055R1057R22
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 31 , Issue 2 , 01 June 1988 , pp. 175 - 181
Copyright
Copyright © Canadian Mathematical Society 1988

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