The index of symmetry and naturally reductive spaces

CARLOS OLMOS

San Pablo

Conferencia; III Encontro Paulista de Geometria San Pablo, 8-12 de agosto de 2011.; 2011

IME, USP

In this talk, based on a work in preparation with Silvio Reggiani, we would like to draw the attention to some concept that we call index of symmetry 0 ≤ is (M ) ≤ n of a Riemannian manifold M n . The index of symmetry can be defined as the dimension of the tangent subspace where any natural tensor is parallel (or, equivalently, the dimension of the parallel Killing fields at a given point). One has that M is symmetric if and only if is (M ) = n. We are, of course, interested on non-symmetric spaces with positive index of symmetry. One can prove that is (M ) ≤ n − 2 (in other words the co-index of symmetry is at least 2, for a non-symmetric space). We have only few general results, which include the following theorem that allows us to classify the spaces with co-index of symmetry equals to 2 (which are Spin(3) with two families of distinguished metrics). Theorem. Let M n be a compact locally irreducible homogeneous Riemannian manifold which is not locally symmetric. Let k := n − is (M ) be its co-index of symmetry. Then n ≤ 1 k(k + 1) (and the equality 2 holds for the frame bundle over the sphere of curvature 2, with the Sasaki-Mok metric)