Revista de la
Unión Matemática Argentina
A property of homogeneous isoparametric submanifolds
Cristián U. Sánchez

Volume 69, no. 1 (2026), pp. 269–279    

Published online (final version): February 24, 2026

https://doi.org/10.33044/revuma.4941

Download PDF

Abstract

The present paper contains a new result concerning the second fundamental form of a compact, connected, homogeneous, isoparametric submanifold of codimension $h\geq 2$ in a Euclidean space.

References

  1. A. Agrachev, D. Barilari, and U. Boscain, A comprehensive introduction to sub-Riemannian geometry, Cambridge Stud. Adv. Math. 181, Cambridge University Press, Cambridge, 2020.  DOI  MR  Zbl
  2. A. A. Agrachev and A. V. Sarychev, Sub-Riemannian metrics: Minimality of abnormal geodesics versus subanalyticity, ESAIM Control Optim. Calc. Var. 4 (1999), 377–403.  DOI  MR  Zbl
  3. J. Berndt, S. Console, and C. Olmos, Submanifolds and holonomy, second ed., Monogr. Res. Notes Math., CRC Press, Boca Raton, FL, 2016.  DOI  MR  Zbl
  4. M. Gromov, Carnot–Carathéodory spaces seen from within, in Sub-Riemannian geometry, Progr. Math. 144, Birkhäuser, Basel, 1996, pp. 79–323.  MR  Zbl
  5. E. Hakavuori, Sub-Riemannian geodesics, Ph.D. thesis, University of Jyväskylä, 2019. Available at https://urn.fi/URN:ISBN:978-951-39-7810-5.
  6. S. Helgason, Differential geometry, Lie groups, and symmetric spaces, Pure Appl. Math. 80, Academic Press, New York-London, 1978.  MR  Zbl
  7. R. Montgomery, Survey of singular geodesics, in Sub-Riemannian geometry, Progr. Math. 144, Birkhäuser, Basel, 1996, pp. 325–339.  MR  Zbl
  8. R. Montgomery, A tour of subriemannian geometries, their geodesics and applications, Math. Surveys Monogr. 91, American Mathematical Society, Providence, RI, 2002.  DOI  MR  Zbl
  9. R. Monti, The regularity problem for sub-Riemannian geodesics, in Geometric control theory and sub-Riemannian geometry, Springer INdAM Ser. 5, Springer, Cham, 2014, pp. 313–332.  DOI  MR  Zbl
  10. C. Olmos and C. Sánchez, A geometric characterization of the orbits of $s$-representations, J. Reine Angew. Math. 420 (1991), 195–202.  DOI  MR  Zbl
  11. R. S. Palais and C.-L. Terng, A general theory of canonical forms, Trans. Amer. Math. Soc. 300 no. 2 (1987), 771–789.  DOI  MR  Zbl
  12. C. U. Sánchez, A canonical distribution on isoparametric submanifolds I, Rev. Un. Mat. Argentina 61 no. 1 (2020), 113–130.  DOI  MR  Zbl
  13. C. U. Sánchez, A canonical distribution on isoparametric submanifolds II, Rev. Un. Mat. Argentina 62 no. 2 (2021), 491–513.  DOI  MR  Zbl
  14. G. Thorbergsson, Isoparametric foliations and their buildings, Ann. of Math. (2) 133 no. 2 (1991), 429–446.  DOI  MR  Zbl
  15. G. Thorbergsson, A survey on isoparametric hypersurfaces and their generalizations, in Handbook of differential geometry. Vol. I, North-Holland, Amsterdam, 2000, pp. 963–995.  MR  Zbl
  16. G. Thorbergsson, From isoparametric submanifolds to polar foliations, São Paulo J. Math. Sci. 16 no. 1 (2022), 459–472.  DOI  MR  Zbl
  17. F. W. Warner, Foundations of differentiable manifolds and Lie groups, Scott, Foresman, and Co., Glenview, Ill.-London, 1971.  MR  Zbl