Symmetric Spaces

Notations M Riemannian manifold. N 0 normal neighborhood of the origin in T p M. N p normal neighbourhood of p, N p = exp N 0. s p geodesic symmetry with respect to p. f Φ d Φ f = f • Φ. X Φ d Φ X. K(S) sectional curvature of M at p along the section S. D r s set of tensor fields of type (r, s). I(M) the set of all isometries on M Definition 1 (normal neighborhood) A neighborhood N p of p in M is called a normal neighbourhood if N p = exp N 0 , where N 0 is a normal neighborhood of the origin in T p M , i.e. satisfying: (1)exp is a diffeomorphism of N 0 onto an open neighborhood N p ; (2)if X ∈ N 0 , 0 ≤ t ≤ 1, then tX ∈ N 0 (star shaped). Definition 2 (geodesic symmetry) ∀q ∈ N p , consider the geodesic t → γ(t) ⊂ N p passing through p and q s.t. γ(0) = p, γ(1) = q. Then the mapping q → q ′ = γ(−1) of N p onto itself is called geodesic symmetry w.r.t p, denoted by s p. Remark: s p is a diffeomorphism of N p onto itself and (ds p) p = −I. Definition 3 (Affine locally symmetric) M is called affine locally symmetric if each point m ∈ M has an open neighborhood N m on which the geodesic symmetry s m is an affine transformation. i.e. ∇ X (Y) = (∇ X sm (Y sm)) sm −1 , ∀X, Y ∈ X(M). Definition 4 (Pseudo-Riemannian structure) Let M be a C ∞-manifold. A pseudo-Riemannian structure on M is a symmetric nondegenerate (as bilinear form at each p ∈ M) tensor field g of type (0, 2). Remark: A pseudo-Riemannian manifold is a connected C ∞-manifold with