The index conjecture for symmetric spaces

Abstract

In 1980, Oniˇsˇcik ([23]) introduced the index of a Riemannian symmetricspace as the minimal codimension of a (proper) totally geodesic submanifold. He calculatedthe index for symmetric spaces of rank ≤ 2, but for higher rank it was unclearhow to tackle the problem. In [2], [3], [4] and [5] we developed several approaches tothis problem, which allowed us to calculate the index for many symmetric spaces. Oursystematic approach led to a conjecture, formulated first in [2], for how to calculate theindex. The purpose of this paper is to verify the conjecture.