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Totally geodesic submanifolds of the complex and the quaternionic 2-Grassmannians

Author: Sebastian Klein
Journal: Trans. Amer. Math. Soc. 361 (2009), 4927-4967
MSC (2000): Primary 53C35; Secondary 53C17
Published electronically: March 12, 2009
MathSciNet review: 2506432
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Abstract: In this article, the totally geodesic submanifolds in the complex $2$-Grassmannian $G_2(\mathbb {C}^{n+2})$ and in the quaternionic $2$-Grassmannian $G_2(\mathbb {H}^{n+2})$ are classified. It turns out that for both of these spaces, the earlier classification of maximal totally geodesic submanifolds in Riemannian symmetric spaces of rank $2$ published by Chen and Nagano (1978) is incomplete. For example, $G_2(\mathbb {H}^{n+2})$ with $n \geq 5$ contains totally geodesic submanifolds isometric to a $\mathbb {H}P^2$, its metric scaled such that the minimal sectional curvature is $\tfrac 15$; they are maximal in $G_2(\mathbb {H}^7)$. $G_2(\mathbb {C}^{n+2})$ with $n \geq 4$ contains totally geodesic submanifolds which are isometric to a $\mathbb {C}P^2$ contained in the $\mathbb {H}P^2$ mentioned above; they are maximal in $G_2(\mathbb {C}^6)$. Neither submanifolds are mentioned by Chen and Nagano.

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Additional Information

Sebastian Klein
Affiliation: Department of Mathematics, Aras na Laoi, University College Cork, Cork, Ireland
Address at time of publication: Lehrstuhl für Mathematik III, Universität Mannheim, 68131 Mannheim, Germany

Keywords: Riemannian symmetric spaces, Grassmannians, totally geodesic submanifolds, Lie triple systems, root systems
Received by editor(s): October 26, 2007
Published electronically: March 12, 2009
Additional Notes: This work was supported by a fellowship within the Postdoc-Programme of the German Academic Exchange Service (DAAD)
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.