The index of symmetric spaces

CARLOS OLMOS

Santiago de Compostela

Conferencia; Symmetry and Shape, Celebrating the 60th birthday of Prof. J. Berndt; 2019

Universidad de Santiago de Compostela

Being the classification of totally geodesic submanifolds of symmetric spaces hopeless, except for rank one or two, Onishchik defined in the eighties the concept of index of an irreducible symmetric space M: the minimal codimension of totally geodesic submanifold. It is a rather well-known result that the index of M is 1 (i.e. it admits a totally geodesic hypersurface) if and only if M has constant curvature. Onishchik himself determined the spaces with index two.Our starting point, for dealing with the index, was to prove in 2014 that the index is always bounded from below by the rank of the symmetric space. Moreover, we conjectured that the index coincides with the reflective index except for the space G_2/SO4 (the reflective index is the minimal codimension of a reflective totally geodesic submanifold that we computed from the work of Leung in the seventies).In several articles, with essentially geometric tools, we were able to determine the index of almost all symmetric spaces, verifying the conjecture. In the case of exceptional symmetric spaces this was done in cooperation with J. S. Rodríguez. The conjecture remains open only for three classical families of symmetric spaces. Any of such families seems to require different methods and we have an strategy for each one. This, hopefully, would prove the conjecture and solve the index problem for all symmetric spaces.