Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-09T22:34:17.020Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

References

1. Bolton, J., On Riemannian immersions without focal points, J. London Math. Soc. (2) 26 (1982), pp. 331–334.Google Scholar
2. Cheeger, J. and Ebin, D. G., Comparison Theorems in Riemannian Geometry, North-Holland, Amsterdam 1975.Google Scholar
3. Graham, P. J. and Ledger, A. J., Sur une classe de s-variétés riemanniennes ou affines, C. R. Acad. Se. Paris 267 (1968), pp. 105–107.Google Scholar
4. Helgason, S., Differential Geometry, Lie Groups and Symmetric Spaces, Academic Press, London 1978.Google Scholar
5. Hermann, R., Focal points of closed submanifolds of Riemannian spaces, Nederl. Akad. Wetensch. Proc. Serv. A 66 (1963), pp. 613–628.Google Scholar
6. Jimenez, J. A., Riemannian 4-symmetric spaces, Ph.D. Thesis, University of Durham, England 1983.Google Scholar
7. Kawalski, O., Existence of generalized symmetric Riemannian spaces of arbitrary order. J. Differential Geom. 12 (1977), pp. 203–208.Google Scholar
8. Ledger, A. J., Espaces de Riemann symétriques generalises, C. R. Acad. Sc. Paris 264 (1967), pp. 947–948.Google Scholar
9. Ledger, A. J. and Obata, M., Affine and Riemannian s-manifolds, J. Differential Geom. 2 (1968), pp. 451–459.Google Scholar
10. Loos, O., Spiegelungsraume und homogène symmetrische Raume, Math. Z. 99 (1967), pp. 141–170.Google Scholar
11. O'Neill, B., Submersions and Geodesies, Duke Math. J. 34 (1967), pp. 363–373.Google Scholar