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Minimum and Conjugate Points in Symmetric Spaces

Published online by Cambridge University Press:  20 November 2018

Richard Crittenden
Richard Crittenden*
Affiliation:
Massachusetts Institute of Technology, Northwestern University
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The purpose of this paper is to discuss conjugate points in symmetric spaces. Although the results are neither surprising nor altogether unknown, the author does not know of their explicit occurrence in the literature.

Briefly, conjugate points in the tangent bundle to the tangent space at a point of a symmetric space are characterized in terms of the algebraic structure of the symmetric space. It is then shown that in the simply connected case the first conjugate locus coincides with the minimum (cut) locus. The interest in this last fact lies in its identification of a more or less locally and analytically defined set with one which includes all the topological interest of the space.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 14 , 1962 , pp. 320 - 328
Copyright
Copyright © Canadian Mathematical Society 1962

References

1. Borel, A., Lectures on symmetric spaces, M.I.T. notes (1958).Google Scholar
2. Bott, R., An application of the Morse theory to the topology of Lie groups, Bull. Soc. Math. France, 84 (1956), 251–281.Google Scholar
3. Bott, R., The stable homotopy of the classical groups, Ann. Math., 70 (1959), 313–337.Google Scholar
4. Bott, R. and Samelson, H., Application of the theory of Morse to symmetric spaces, Amer. J. Math., 80 (1958), 964–1029.Google Scholar
5. Bott, R. and Samelson, H., Seminar on Symmetric Spaces, Chicago University notes (1958).Google Scholar
6. Bott, R. and Samelson, H., Séminaire “Sophus Lie,” Théorie des algebres de Lie, Théorie des groupes de Lie, E. N. S. (Paris, 1955).Google Scholar