Killing fields, holonomy and the index of symmetry

CARLOS OLMOS

Osaka-Fukuoka

Conferencia; The 10th Pacific Rim Geometry Conference 2011; 2011

Osaka and Fukuoka Universities

Killing fields, holonomy and the index of symmetry” Abstract: This talk is mainly based on a work, still in preparation, with Silvio Reggiani. We would like to draw the attention to some concept that we call the 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 Riemannian tensor is parallel (or, equivalently, the dimension of the space of Killing fields that are parallel 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. In this case 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 some general results and many questions. Many examples of spaces with non-trivial index of symmetry arise from naturally reductive spaces (we will also refer to a previous joint work with Reggiani related to naturally reductive spaces and holonomy, Crelle’s 2011). Also the unit tangent bundle over the sphere S n of curvature 2 has is (S n ) = n − 1. We prove the following result Theorem Let M n be a compact locally irreducible homogeneous Riemannian man- ifold which is not locally symmetric. Let k := n − is (M ) be its co-index of symmetry. Then there is a subgroup of isometries G ⊂ I(M ) , which acts transitively on M and such that dim(G) ≤ 1 k(k + 1). Moreover, if the equality holds, then, up to a cover, 2 G = Spin(k + 1) and G has non trivial isotropy, if k ≥ 3. This allows us to classify the homogeneous spaces with low co-index of symmetry. For instance the spaces with co-index of symmetry 2 correspond to two distinguished families of one-parameter left invariant metrics on Spin(3). It is an interesting fact that there is a nice equivariant “Gauss map”from a homoge- neous space M with non-trivial index of symmetry, into an appropriate Grassman- nian. The subjects of this talk may be regarded as an effort to explore Riemannian man- ifolds that are symmetric up to some defect (in the hope of finding distinguished non-symmetric homogeneous manifolds). In some sense, our philosophy is in the direction of the concept of co-polarity by Claudio Gorodski, that measures how a representation, orbit like, differ from a symmetric (isotropy) representation .