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Centrioles in symmetric spaces

Published online by Cambridge University Press:  11 January 2016

Peter Quast*
Affiliation:
Institut für Mathematik, Universität Augsburg, Germany, peter.quast@math.uni-augsburg.de
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Abstract

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We describe all centrioles in irreducible simply connected pointed symmetric spaces of compact type in terms of the root system of the ambient space, and we study some geometric properties of centrioles.

Keywords

Type
Research Article
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2013

References

[BCO] Berndt, J., Console, S., and Olmos, C., Submanifolds and Holonomy, Chapman & Hall/RC Res. Notes Math. 434, Chapman & Hall, Boca Raton, Fla., 2003. MR 1990032. DOI 10.1201/9780203499153.Google Scholar
[BS] Borel, A. and Siebenthal, J. de, Les sous-groupes fermés de rang maximum des groupes de Lie clos, Comment. Math. Helv. 23 (1949), 200221. MR 0032659.CrossRefGoogle Scholar
[Bt] Bott, R., The stable homotopy of the classical groups, Ann. of Math. (2) 70 (1959), 313337. MR 0110104.Google Scholar
[Bo] Bourbaki, N., Éléments de mathématique, fasc. 34: Groupes et algébres de Lie, chapitres 46, Masson, Paris, 1981. MR 0647314.Google Scholar
[Bu1] Burns, J. M., Homotopy of compact symmetric spaces, Glasg. Math. J. 34 (1992), 221228. MR 1167338. DOI 10.1017/S0017089500008764.Google Scholar
[Bu2] Burns, J. M., Conjugate loci of totally geodesic submanifolds of symmetric spaces, Trans. Amer. Math. Soc. 337 (1993), no. 1, 411425. MR 1091705. DOI 10.2307/2154329.CrossRefGoogle Scholar
[Bu3] Burns, J. M., Conjugate loci in compact symmetric spaces, Ph.D. dissertation, University of Notre Dame, Notre Dame, Indiana, 1985. MR 2634291.Google Scholar
[C] Cartan, E., Sur certaines formes Riemanniennes remarquables des géométries à groupe fondamental simple, Ann. Sci. Éc. Norm. Supér. (3) 44 (1927), 345467. MR 1509283.CrossRefGoogle Scholar
[Ch1] Chen, B.-Y., A New Approach to Compact Symmetric Spaces and Applications, Katholieke Universiteit Leuven, Louvain, 1987. MR 0952556.Google Scholar
[Ch2] Chen, B.-Y., “Symmetries of compact symmetric spaces” in Geometry and Topology of Submanifolds (Marseille, 1987), World Scientific, Teaneck, N.J., 1989, 3848. MR 1116130.Google Scholar
[CN1] Chen, B.-Y. and Nagano, T., Totally geodesic submanifolds of symmetric spaces, II, Duke Math. J. 45 (1978), 405425. MR 0487902.Google Scholar
[CN2] Chen, B.-Y. and Nagano, T., A Riemannian geometric invariant and its applications to a problem of Borel and Serre, Trans. Amer. Math. Soc. 308 (1988), no. 1, 273297. MR 0946443. DOI 10.2307/2000963.Google Scholar
[EH] Eschenburg, J.-H. and Heintze, E., Extrinsic symmetric spaces and orbits of s-representations, Manuscripta Math. 88 (1995), 517524. MR 1362935. DOI 10. 1007/BF02567838.Google Scholar
[F1] Ferus, D., Immersionen mit paralleler zweiter Fundamentalform: Beispiele und Nicht-Beispiele, Manuscripta Math. 12 (1974), 153162. MR 0339015.Google Scholar
[F2] Ferus, D., Symmetric submanifolds of Euclidean space, Math. Ann. 247 (1980), 8193. MR 0565140. DOI 10.1007/BF01359868.Google Scholar
[Ha] Hatcher, A., Algebraic Topology, Cambridge University Press, Cambridge, 2002. MR 1867354.Google Scholar
[He] Helgason, S., Differential Geometry, Lie Groups and Symmetric Spaces, Pure Appl. Math. 80, Academic Press, New York, 1978. MR 0514561.Google Scholar
[HH] Hsiang, W.-T. and Hsiang, W.-Y., On the existence of codimension-one minimal spheres in compact symmetric spaces of rank 2, II, J. Differential Geom. 17 (1982), 583594. MR 0683166.Google Scholar
[HHT] Hsiang, W.-T., Hsiang, W.-Y., and Tomter, P., On the existence of minimal hyperspheres in compact symmetric spaces, Ann. Sci. Ec. Norm. Supér. (4) 21 (1988), 287305. MR 0956769.Google Scholar
[KN] Kobayashi, S. and Nagano, T., On filtered Lie algebras and geometric structures, I, J. Math. Mech. 13 (1964), 875907. MR 0168704.Google Scholar
[Le] Leung, D. S. P., The reflection principle for minimal submanifolds of Riemannian symmetric spaces, J. Differential Geom. 8 (1973), 153160. MR 0367872.CrossRefGoogle Scholar
[Lo] Loos, O., Symmetric Spaces, II: Compact Spaces and Classification, Benjamin W. A., New York, 1969. MR 0239006.Google Scholar
[LT] Lusztig, G. and Tits, J., The inverse of a Cartan matrix, An. Univ. Timi¸soara. Ser. Ştinţ. Mat. 30 (1992), 1723. MR 1329156.Google Scholar
[MQ1] Mare, A.-L. and Quast, P., On some spaces of minimal geodesics in Riemannian symmetric spaces, Q. J. Math. 63 (2012), 681694. MR 2967170. DOI 10.1093/qmath/har003.Google Scholar
[MQ2] Mare, A.-L. and Quast, P., Bott periodicity for inclusions of symmetric spaces, Doc. Math. 17 (2012), 911952.Google Scholar
[Ml] Milnor, J., Morse Theory, Ann. of Math. Stud. 51, Princeton University Press, Princeton, 1963. MR 0163331.Google Scholar
[Mi1] Mitchell, S. A., “The Bott filtration of a loop group” in Algebraic Topology (Barcelona, 1986), Lecture Notes in Math. 1298, Springer, Berlin, 1987, 215226. MR 0928835. DOI 10.1007/BFb0083012.Google Scholar
[Mi2] Mitchell, S. A., Quillen’s theorem on buildings and the loops on a symmetric space, Enseign. Math. (2) 34 (1988), 123166. MR 0960196.Google Scholar
[N1] Nagano, T., The involutions of compact symmetric spaces, Tokyo J. Math. 11 (1988), 5779. MR 0947946. DOI 10.3836/tjm/1270134261.Google Scholar
[N2] Nagano, T., The involutions of compact symmetric spaces, II, Tokyo J. Math. 15 (1992), 3982. MR 1164185. DOI 10.3836/tjm/1270130250.CrossRefGoogle Scholar
[NS] Nagano, T. and Sumi, M., The spheres in symmetric spaces, Hokkaido Math. J. 20 (1991), 331352. MR 1114410.Google Scholar
[NT1] Nagano, T. and Tanaka, M. S., The involutions of compact symmetric spaces, III, Tokyo J. Math. 18 (1995), 193212. MR 1334718. DOI 10.3836/tjm/1270043621.CrossRefGoogle Scholar
[NT2] Nagano, T. and Tanaka, M. S., The involutions of compact symmetric spaces, V, Tokyo J. Math. 23 (2000), 403416. MR 1806473. DOI 10.3836/tjm/1255958679.Google Scholar
[Q1] Quast, P., Homotopy of EVII, C. R. Math. Acad. Sci. Paris 350 (2012), 425426. MR 2922098. DOI 10.1016/j.crma.2012.04.008.Google Scholar
[Q2] Quast, P. Complex structures and chains of symmetric spaces, Habilitationsschrift, Universität Augsburg, Augsburg, 2010.Google Scholar
[S1] Sakai, T., On the structure of cut loci in compact Riemannian symmetric spaces, Math. Ann. 235 (1978), 129148. MR 0500710.Google Scholar
[S2] Sakai, T., “Cut loci of compact symmetric spaces” in Minimal Submanifolds and Geodesics (Tokyo, 1977), North-Holland, Amsterdam, 1979, 193207. MR 0574265.Google Scholar
[S3] Sakai, T., Riemannian Geometry, Transl. Math. Monogr. 149, Amer. Math. Soc., Providence, 1996. MR 1390760.Google Scholar
[Se] Serre, J.-P., Complex Semisimple Lie Algebras, Springer, New York, 1987. MR 0914496.Google Scholar
[Tk] Takeuchi, M., On the fundamental group and the group of isometries of a symmetric space, J. Fac. Sci. Univ. Tokyo Sect. I 10 (1964), 88123. MR 0170983.Google Scholar
[Tn] Tanaka, M. S., Stability of minimal submanifolds in symmetric spaces, Tsukuba J. Math. 19 (1995), 2756. MR 1346752.Google Scholar
[W] Wolf, J. A., Spaces of Constant Curvature, 5th ed., Publish or Perish, Houston, 1984. MR 0928600.Google Scholar