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

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

2007-09-17

arxiv.org/abs/0709.2644v1

Trans. Amer. Math. Soc. 361 (2009), 4927-4967

Mathematics

Differential Geometry

Scientific paper

10.1090/S0002-9947-09-04699-6

In this article, I classify the totally geodesic submanifolds in the complex 2-Grassmannians and in the quaternionic 2-Grassmannians. 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 (B.-Y. Chen, T. Nagano, "Totally geodesic submanifolds of symmetric spaces, II", Duke Math. J. 45 (1978), 405--425) is incomplete. For example, G_2(H^n) with n >= 7 contains totally geodesic submanifolds isometric to a HP^2, its metric scaled such that the minimal sectional curvature is 1/5; they are maximal in G_2(H^7). Also G_2(C^n) with n >= 6 contains totally geodesic submanifolds which are isometric to a CP^2 contained in the HP^2 mentioned above; they are maximal in G_2(C^6). Neither submanifolds are mentioned in the cited paper by Chen and Nagano.

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