The index of symmetry and homogeneous fibrations

We develop new tools to compute the index of symmetry in the context of homogeneous fibrations. As a consequence of our results, we determine the index of symmetry of every homogeneous space diffeomorphic to a compact rank-one symmetric space, and of every homogeneous $S^1$-bundle over a compact irreducible symmetric space. Moreover, we construct irreducible homogeneous metrics whose leaves of symmetry are symmetric spaces of arbitrarily large rank.