Maximal totally geodesic submanifolds and index of symmetric spaces

Jurgen Berndt, Carlos Olmos

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11 Citations (Scopus)
181 Downloads (Pure)

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 languageEnglish
Pages (from-to)187-217
Number of pages31
JournalJOURNAL OF DIFFERENTIAL GEOMETRY
Volume104
Issue number2
Early online date13 Oct 2016
DOIs
Publication statusPublished - Oct 2016

Keywords

  • symmetric spaces
  • totally geodesic submanifolds
  • index of symmetric spaces

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