Borsuk-Ulam Theorem and Maximal Antipodal Sets of Compact Symmetric Spaces

___

  • [1] Borel, A. and Serre, J.-P., Sur certains sousgroupes des groupes de Lie compacts. Comm. Math. Helv., 27 (1953), 128-139.
  • [2] Borsuk, K., Drei Satze uber die n-dimensionale euklidische Sphäre. Fund. Math., 20 (1933), 177-190.
  • [3] Burns, J. M., Homotopy of compact symmetric spaces. Glasgow Math. J., 34 (1992), no. 2, 221-228.
  • [4] Cartan, É. Sur une classe remarquable d’espaces de Riemann. Bull. Soc. Math. France, 54 (1926), 214-264.
  • [5] Chen, B.-Y., Geometry of submanifolds. Marcel Dekker, New York, NY, 1973.
  • [6] Chen, B.-Y., A new approach to compact symmetric spaces and applications. A report on joint work with Professor T. Nagano. Katholieke Universiteit Leuven, Louvain, 1987.
  • [7] Chen, B.-Y., The 2-ranks of connected compact Lie groups. Taiwanese J. Math., 17 (2013), no. 3, 815-831.
  • [8] Chen, B.-Y., Two-numbers and their applications - a survey. preprint, 2017.
  • [9] Chen, B.-Y. and Nagano, T., Totally geodesic submanifolds of symmetric spaces. I. Duke Math. J., 44 (1977), 745-755.
  • [10] Chen, B.-Y. and Nagano, T., Totally geodesic submanifolds of symmetric spaces II. Duke Math. J., 45 (1978), no. 2, 405-425.
  • [11] Chen, B.-Y. and Nagano, T., Un invariant géométrique riemannien. C. R. Acad. Sci. Paris Sér. I Math., 295 (1982), no. 5, 389-391.
  • [12] 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, 273-297.
  • [13] Console, S., Geodesics and moments maps of symmetric R-spaces. Dipartimento di Matematica - Universit‘a di Torino Quaderno N. 25.
  • [14] Helgason, S.. Differential geometry, Lie groups and symmetric spaces. Academic Press, New York, 1978.
  • [15] Ikawa, O., Tanaka, M. S. and Tasaki, H., The fixed point set of a holomorphic isometry, the intersection of two real forms in a Hermitian symmetric space of compact type and symmetric triads. Internat. J. Math., 26 (2015), no. 6, 1541005, 32 pp.
  • [16] Lyusternik, L. A. and Fet, A. I., Variational problems on closed manifolds. Doklady Akad. Nauk SSSR (N.S.), 81 (1951), 17-18.
  • [17] Lyusternik, L. A. and Shnirel’man, S., Topological Methods in Variational Problems. Trudy Inst. Math. Mech., Moscow State Univ, Moscow, 1930.
  • [18] Matou˘sek, J., Using the Borsuk-Ulam theorem. Springer-Verlag, Berlin, 2003.
  • [19] Nagano, T., The involutions of compact symmetric spaces. Tokyo J. Math., 11 (1988), 57-79.
  • [20] Nagano, T., The involutions of compact symmetric spaces, II. Tokyo J. Math. 15 (1992), 39-82.
  • [21] Rotman, J. J., An introduction to algebraic topology. Springer-Verlag, 1988.
  • [22] Sanchez, C. U., The invariant of Chen-Nagano on flag manifolds. Proc. Amer. Math. Soc., 118 (1993), no. 4, 1237-1242.
  • [23] Sanchez, C. U., The index number of an R-space: an extension of a result of M. Takeuchi’s. Proc. Amer. Math. Soc., 125 (1997), no. 3, 893-900.
  • [24] Sanchez, C. U. and Giunta, A., The projective rank of a Hermitian symmetric space: a geometric approach and consequences. Math. Ann., 323 (2002), no. 1, 55-79.
  • [25] Sanchez, C. U., Cali, A. L. and Moreschi, J. L., Spheres in Hermitian symmetric spaces and flag manifolds. Geom. Dedicata , 64 (1997), no. 3, 261-276.
  • [26] Steinlein, H., Borsuk’s antipodal theorem and its generalizations and applications: a survey. Méthodes topologiques en analyse non linéaire. Sém. Math. Supér. Montréal, Sém. Sci. OTAN (NATO Adv. Study Inst.), 95 (1985), 166-235.
  • [27] Takeuchi, M., Two-number of symmetric R-spaces. Nagoya Math. J., 115 (1989), 43-46.
  • [28] Tanaka, M. S., Antipodal sets of compact symmetric spaces and the intersection of totally geodesic submanifolds. Differential geometry of submanifolds and its related topics, 205-219, World Sci. Publ., 2014.
  • [29] Tanaka, M. S. and Tasaki, H., The intersection of two real forms in Hermitian symmetric spaces of compact type. J. Math. Soc. Japan, 64 (2012), no. 4, 1297-1332.
  • [30] Tanaka, M. S. and Tasaki, H., Antipodal sets of symmetric R-spaces. Osaka J. Math., 50 (2013), no. 1, 161-169.
  • [31] Tasaki, H., The intersection of two real forms in the complex hyperquadric. Tohoku Math. J., 62 (2010), no. 3, 375-382.
  • [32] Tasaki, H., Antipodal sets in oriented real Grassmann manifolds. Internat. J. Math., 24 (2013), no. 8, 1350061, 28 pp.
  • [33] Tasaki, H., Estimates of antipodal sets in oriented real Grassmann manifolds. Internat. J. Math., 26 (2015), no. 6, 1541008, 12 pp.