Translator Disclaimer
December 2010 Totally geodesic submanifolds of the exceptional Riemannian symmetric spaces of rank 2
Sebastian Klein
Osaka J. Math. 47(4): 1077-1157 (December 2010).

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].

Citation

Download Citation

Sebastian Klein. "Totally geodesic submanifolds of the exceptional Riemannian symmetric spaces of rank 2." Osaka J. Math. 47 (4) 1077 - 1157, December 2010.

Information

Published: December 2010
First available in Project Euclid: 20 December 2010

zbMATH: 1215.53046
MathSciNet: MR2791563

Subjects:
Primary: 53C35
Secondary: 53C17

Rights: Copyright © 2010 Osaka University and Osaka City University, Departments of Mathematics

JOURNAL ARTICLE
81 PAGES


SHARE
Vol.47 • No. 4 • December 2010
Back to Top