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Helgason spheres of compact symmetric spaces and immersions of finite type

Published online by Cambridge University Press:  17 April 2009

Bang-Yen Chen
Bang-Yen Chen
Affiliation:
Department of Mathematics, Michigan State University, East Lansing, MI 48824–1027, United States of America e-mail: bychen@math.msu.edu
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Abstract

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A unit speed curve γ = γ(s) in a Riemannian manifold N is called a circle if there exists a unit vector field Y(s) along γ and a positive constant k such that ∇sγ′(s) = kY(s), ∇sY(s) = −kγ′(s). A maximal totally geodesic sphere with maximal sectional curvature in a compact irreducible symmetric space M is called a Helgason sphere. A circle which lies in a Helgason sphere of a compact symmetric space is called a Helgason circle. In this article we establish some fundamental relationships between Helgason circles, Helgason spheres of irreducible symmetric spaces of compact type and the theory of immersions of finite type.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 63 , Issue 2 , April 2001 , pp. 243 - 255
Copyright
Copyright © Australian Mathematical Society 2001

References

[1]Baikoussis, C., Defever, F., Koufogiorgos, T. and Verstraelen, L., ‘Finite type immersions of flat tori into Euclidean spaces’, Proc. Edinburgh Math. Soc. (2) 38 (1995), 413–420.CrossRefGoogle Scholar
[2]Chen, B.-Y., Total mean curvature and submanifolds of finite type, Series in Pure Mathematics 1 (World Scientific, Singapore, 1984).Google Scholar
[3]Chen, B.-Y., ‘A report of submanifolds of finite type’, Soochow J. Math. 22 (1996). 117–337.Google Scholar
[4]Chen, B.-Y., Deprez, J. and Verheyen, P., ‘Immersions, dans un espace euclidien, d'un espace symétrique compact de rang un à géodésiques simples’, C. R. Acad. Sci. Paris Sér. I Math. 304 (1987), 567–570.Google Scholar
[5]Chen, B.-Y. and Nagano, T., ‘Totally geodesic submanifolds of symmetric spaces’, Duke Math. J. 44 (1977), 745–755; II C. R. Acad. Sci. Paris Sér. I Math. 45 (1978), 405–425.Google Scholar
[6]Deprez, J., Immersions of finite type of compact homogeneous Riemannian manifolds, Doctoral Thesis (Katholieke Universiteit Leuven, 1988).Google Scholar
[7]Helgason, S., ‘Totally geodesic spheres in compact symmetric spaces’, Math. Ann. 165 (1966), 309–317.Google Scholar
[8]Helgason, S., Lie groups, differential geometry and symmetric spaces, Pure and Applied Mathematics 80 (Academic Press, New York, London, 1978).Google Scholar
[9]Mashimo, K. and Tojo, K., ‘Circles in Riemannian symmetric spaces’, Kodai Math. J. 22 (1999), 1–14.Google Scholar
[10]Nomizu, K. and Yano, K., ‘On circles and spheres in Riemannian geometry’, Math. Ann. 210 (1974), 163–170.Google Scholar
[11]Wallach, N. R., ‘Minimal immersions of symmetric spaces into spheres’, in Symmetric space, Pure and Applied Mathematics (Marcel Dekker, New York, 1972), pp. 1–40.Google Scholar