Maximal totally geodesic submanifolds and index of symmetric spaces

Jurgen Berndt; Carlos Olmos

doi:10.4310/jdg/1476367055

Maximal totally geodesic submanifolds and index of symmetric spaces

Jurgen Berndt, Carlos Olmos

Universidad Nacional de Córdoba

Research output: Contribution to journal › Article › peer-review

11 Citations (Scopus)

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Abstract

Let M be an irreducible Riemannian symmetric space. The index i(M) of M is the minimal codimension of a totally geodesic submanifold of M. In previous work the authors proved that i(M) is bounded from below by the rank rk(M) of M. In this paper we classify all irreducible Riemannian symmetric spaces M for which the equality holds, that is, rk(M) = i(M). In this context we also obtain an explicit classification of all non-semisimple maximal totally geodesic submanifolds in irreducible Riemannian symmetric spaces of noncompact type and show that they are closely related to irreducible symmetric R-spaces. We also determine the index of some symmetric spaces and classify the irreducible Riemannian symmetric spaces of noncompact type with i(M) = 4,5 or 6.

Original language	English
Pages (from-to)	187-217
Number of pages	31
Journal	JOURNAL OF DIFFERENTIAL GEOMETRY
Volume	104
Issue number	2
Early online date	13 Oct 2016
DOIs	https://doi.org/10.4310/jdg/1476367055
Publication status	Published - Oct 2016

Keywords

symmetric spaces
totally geodesic submanifolds
index of symmetric spaces

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10.4310/jdg/1476367055

Maximal totally geodesic submanifolds_BERNDT_Firstonline13October2016_GREEN AAMAccepted author manuscript, 546 KB
http://arxiv.org/abs/1405.0598Other version, 546 KB

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KW - index of symmetric spaces

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Maximal totally geodesic submanifolds and index of symmetric spaces

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