Research Article
BibTex RIS Cite

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

Year 2017, Volume: 10 Issue: 2, 11 - 19, 29.10.2017

Bang-yen Chen

https://doi.org/10.36890/iejg.545041
Cited By: 1

Abstract


Keywords

Borsuk-Ulam theorem maximal antipodal set continuous function 2-number compact symmetric space

References

  • [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.

Year 2017, Volume: 10 Issue: 2, 11 - 19, 29.10.2017

Bang-yen Chen

https://doi.org/10.36890/iejg.545041
Cited By: 1

Abstract

References

  • [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.
There are 33 citations in total.

Details

Primary Language English
Journal Section Research Article
Authors

Bang-yen Chen

Publication Date October 29, 2017
Published in Issue Year 2017 Volume: 10 Issue: 2

Cite

APA Chen, B.-y. (2017). Borsuk-Ulam Theorem and Maximal Antipodal Sets of Compact Symmetric Spaces. International Electronic Journal of Geometry, 10(2), 11-19. https://doi.org/10.36890/iejg.545041
AMA Chen By. Borsuk-Ulam Theorem and Maximal Antipodal Sets of Compact Symmetric Spaces. Int. Electron. J. Geom. October 2017;10(2):11-19. doi:10.36890/iejg.545041
Chicago Chen, Bang-yen. “Borsuk-Ulam Theorem and Maximal Antipodal Sets of Compact Symmetric Spaces”. International Electronic Journal of Geometry 10, no. 2 (October 2017): 11-19. https://doi.org/10.36890/iejg.545041.
EndNote Chen B-y (October 1, 2017) Borsuk-Ulam Theorem and Maximal Antipodal Sets of Compact Symmetric Spaces. International Electronic Journal of Geometry 10 2 11–19.
IEEE B.-y. Chen, “Borsuk-Ulam Theorem and Maximal Antipodal Sets of Compact Symmetric Spaces”, Int. Electron. J. Geom., vol. 10, no. 2, pp. 11–19, 2017, doi: 10.36890/iejg.545041.
ISNAD Chen, Bang-yen. “Borsuk-Ulam Theorem and Maximal Antipodal Sets of Compact Symmetric Spaces”. International Electronic Journal of Geometry 10/2 (October 2017), 11-19. https://doi.org/10.36890/iejg.545041.
JAMA Chen B-y. Borsuk-Ulam Theorem and Maximal Antipodal Sets of Compact Symmetric Spaces. Int. Electron. J. Geom. 2017;10:11–19.
MLA Chen, Bang-yen. “Borsuk-Ulam Theorem and Maximal Antipodal Sets of Compact Symmetric Spaces”. International Electronic Journal of Geometry, vol. 10, no. 2, 2017, pp. 11-19, doi:10.36890/iejg.545041.
Vancouver Chen B-y. Borsuk-Ulam Theorem and Maximal Antipodal Sets of Compact Symmetric Spaces. Int. Electron. J. Geom. 2017;10(2):11-9.

Cited By

https://doi.org/
 Download Cover Image