The index of symmetry of a Riemannian manifold

SILVIO REGGIANI

Córdoba

Congreso; IV Congreso Latinoamericano de Matemáticos; 2012

Universidad Nacional de Córdoba

We introduce a geometric invariant 0 <= i_s(M) <= dim M, called the "index of symmetry" of a Riemannian manifold M, which measures how far is M from being a symmetric space, in the sense that if i_s(M) = k, M admits a foliation (eventually singular) whose leaves are totally geodesic and globally symmetric submanifolds of dimension (at least) k. In particular, if i_s(M) = dim M, then M is a globally symmetric space. More precisely, i_s(M) := inf_{q in M}dim s_q, where s_q is the so-called "distribution of symmetry" of M, and it is defined by $s_q := {X_q in T_qM: X is a Killing field on M such that (nabla X)_q = 0}. In this work, we give the classification of Riemannian manifolds with low co-index of symmetry, i.e. dim M - i_s(M) <= 2. We also provide certain structure results and some large families of nontrivial examples (homogeneous and not), i.e. 0 < i_s(M) < dim M. Finally, we determine geometrically the index of symmetry of a normal homogeneous, non-locally symmetric space M = G/H. Moreover, we prove that the distribution of symmetry of M coincides with the G-invariant distribution defined by the set of fixed vectors of H in T_{eH}M$.