14 oct 2014 -- 11:30
Sala conferenze Tricerri - Firenze
Abstract.
Let $M$ be an irreducible symmetric space. It is well known that $M$ admits a totally geodesic hypersurface if and only if $M$ has constant curvature. This suggest that the existence of a totally geodesic submanifolds of $M$, of certain codimension, imposes restrictions on the symmetric space. With this philosophy A. L. Onishchik defined in 1980 the concept of the index $i(M)$ of $M$. Namely, $i(M)$ is the minimal codimension of a (proper) totally geodesic submanifold of $M$. By using a purely algebraic approach he classified the symmetric spaces of index $2$. We would like to present a geometric approach to the concept of index which allows us to classify the symmetric spaces of index at most $6$ (and to calculate the index of many symmetric spcaces). Our main result is to relate the index and the rank $rk(M)$ of $M$. Namely, the index is bounded from below by the rank. Moreover, we determine which are the symmetric spaces that satisfy $rk(M)=i(M)$. Namely, up to duality, these symmetric spaces are $SL(k+1)/SO(k+1)$ or $SO(n+k)/SO(n)SO(k)$. This talk is based on a joint work with Jurgen Berndt.