Maximal totally geodesic submanifolds and the index of symmetric spaces

JURGEN BERNDT AND CARLOS OLMOS

JOURNAL OF DIFFERENTIAL GEOMETRY

INT PRESS BOSTON, INC

Año: 2016

0022-040X

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 cite{BO} we proved that $i(M)$ is bounded from below by the rank $k(M)$ of $M$, that is, $k(M) leq i(M)$. In this paper we classify all irreducible Riemannian symmetric spaces $M$ for which the equality holds, that is, $k(M) = i(M)$. 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 $i(M) in {4,5,6}$.