Osaka Journal of Mathematics

Totally geodesic submanifolds of the exceptional Riemannian symmetric spaces of rank 2

Sebastian Klein

Full-text: Open access

Abstract

The present article is the final part of a series on the classification of the totally geodesic submanifolds of the irreducible Riemannian symmetric spaces of rank 2. After this problem has been solved for the 2-Grassmannians in my papers [7] and [8], and for the space $\mathrm{SU}(3)/\mathrm{SO}(3)$ in Section 6 of [9], we now solve the classification for the remaining irreducible Riemannian symmetric spaces of rank 2 and compact type: $\mathrm{SU}(6)/\mathrm{Sp}(3)$, $\mathrm{SO}(10)/\mathrm{U}(5)$, $E_{6}/(\mathrm{U}(1) \cdot \mathrm{Spin}(10))$, $E_{6}/F_{4}$, $G_{2}/\mathrm{SO}(4)$, $\mathrm{SU}(3)$, $\mathrm{Sp}(2)$ and $G_{2}$. Similarly as for the spaces already investigated in the earlier papers, it turns out that for many of the spaces investigated here, the earlier classification of the maximal totally geodesic submanifolds of Riemannian symmetric spaces by Chen and Nagano ([5], \S9) is incomplete. In particular, in the spaces $\mathrm{Sp}(2)$, $G_{2}/\mathrm{SO}(4)$ and $G_{2}$, there exist maximal totally geodesic submanifolds, isometric to 2- or 3-dimensional spheres, which have a ``skew'' position in the ambient space in the sense that their geodesic diameter is strictly larger than the geodesic diameter of the ambient space. They are all missing from [5].

Article information

Source
Osaka J. Math., Volume 47, Number 4 (2010), 1077-1157.

Dates
First available in Project Euclid: 20 December 2010

Permanent link to this document
https://projecteuclid.org/euclid.ojm/1292854318

Mathematical Reviews number (MathSciNet)
MR2791563

Zentralblatt MATH identifier
1215.53046

Subjects
Primary: 53C35: Symmetric spaces [See also 32M15, 57T15]
Secondary: 53C17: Sub-Riemannian geometry

Citation

Klein, Sebastian. Totally geodesic submanifolds of the exceptional Riemannian symmetric spaces of rank 2. Osaka J. Math. 47 (2010), no. 4, 1077--1157. https://projecteuclid.org/euclid.ojm/1292854318


Export citation

References

  • J.F. Adams: Lectures on Exceptional Lie Groups, Univ. Chicago Press, Chicago, IL, 1996.
  • K. Atsuyama: The connection between the symmetric space $E_{(6)}/\mathrm{SO}(10)\cdot\mathrm{SO}(2)$ and projective planes, Kodai Math. J. 8 (1985), 236–248.
  • K. Atsuyama: Projective spaces in a wider sense II, Kodai Math. J. 20 (1997), 41–52.
  • T. Bröcker and T. tom Dieck: Representations of Compact Lie Groups, Springer, New York, 1985.
  • B.-Y. Chen and T. Nagano: Totally geodesic submanifolds of symmetric spaces II, Duke Math. J. 45 (1978), 405–425.
  • S. Helgason: Differential Geometry, Lie Groups, and Symmetric Spaces, Academic Press, New York, 1978.
  • S. Klein: Totally geodesic submanifolds of the complex quadric, Differential Geom. Appl. 26 (2008), 79–96.
  • S. Klein: Totally geodesic submanifolds of the complex and the quaternionic 2-Grassmannians, Trans. Amer. Math. Soc. 361 (2009), 4927–4967.
  • S. Klein: Reconstructing the geometric structure of a Riemannian symmetric space from its Satake diagram, Geom. Dedicata 138 (2009), 25–50.
  • A.W. Knapp: Lie Groups Beyond an Introduction, second edition, Birkhäuser Boston, Boston, MA, 2002.
  • D.S.P. Leung: On the classification of reflective submanifolds of Riemannian symmetric spaces, Indiana Univ. Math. J. 24 (1974/75), 327–339.
  • D.S.P. Leung: Errata: “On the classification of reflective submanifolds of Riemannian symmetric spaces” (Indiana Univ. Math. J. 24 (1974/75), 327–339), Indiana Univ. Math. J. 24 (1975), 1199.
  • D.S.P. Leung: Reflective submanifolds III, Congruency of isometric reflective submanifolds and corrigenda to the classification of reflective submanifolds, J. Differential Geom. 14 (1979), 167–177.
  • O. Loos: Symmetric Spaces II: Compact spaces and classification, W.A. Benjamin, Inc., New York, 1969.
  • J.A. Wolf: Elliptic spaces in Grassmann manifolds, Illinois J. Math. 7 (1963), 447–462.
  • I. Yokota: Realizations of involutive automorphisms $\sigma$ and $G^{\sigma}$ of exceptional linear Lie groups $G$ I, $G=G_{2}$, $F_{4}$ and $E_{6}$, Tsukuba J. Math. 14 (1990), 185–223.