Geodesic
The Focal Locus of a Riemannian 4-Symmetric Space
Canadian mathematical bulletin, Tome 31 (1988) no. 2, pp. 175-181

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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.
DOI : 10.4153/CMB-1988-026-6
Mots-clés : 53C35, 53C30, 55R10, 57R22 
Jimenez, J. Alfredo. The Focal Locus of a Riemannian 4-Symmetric Space. Canadian mathematical bulletin, Tome 31 (1988) no. 2, pp. 175-181. doi: 10.4153/CMB-1988-026-6
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     title = {The {Focal} {Locus} of a {Riemannian} {4-Symmetric} {Space}},
     journal = {Canadian mathematical bulletin},
     pages = {175--181},
     year = {1988},
     volume = {31},
     number = {2},
     doi = {10.4153/CMB-1988-026-6},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1988-026-6/}
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