On affine symmetric spaces (original) (raw)

An affine symmetric space is a connected affinely connected manifold M such that to each point peM there is an involutive (i.e., of order 2) affine transformation sp of M leaving p as an isolated fixed point. Clearly, for a fixed p in M, the transformation sp is unique; it is called the symmetry at p. It is well known that the group of affine transformations AiM) of M is a Lie group and that, if M is affine symmetric, the connected component of identity A0(M) of A(M) acts transitively on M (see e.g. ). Let G = A0(M) and let H be the isotropy subgroup of G at peM. Then M can be identified with GjH, the left coset space, and sp induces an involutive automorphism £: g-*spogosp suchthat G\\ <zz H cz Gs, where G1 denotes the subgroup of fixed points of S and G" is the connected component of identity in Gz. Conversely, if G is a Lie group with an involutive automorphism Z and H is a subgroup satisfying Gjclc Gs, then H is a closed subgroup of G and the coset space GjH is an analytic manifold; furthermore, the canonical invariant affine connection of G induces an affine connection on M = GjH, which renders M an affine symmetric space with symmetries derived from Z in an obvious manner. Such an affine symmetric space will be denoted by (G/H,l) or simply by GjH. The discussion given in the preceding paragraph shows that we may restrict our study of affine symmetric space to the case M = GjH, where Gis a connected Lie group. We may, and we shall, further assume that GjH is effective in the sense that H contains no nontrivial normal subgroup of G; for if we take G = A0(M), then any normal subgroup H' cz H induces identity transformation on M. In general, if G is a group of affine transformation of a connected, affinely connected manifold M, then the isotropy subgroup Hp of G at p s M is isomorphic to the linear isotropy subgroup Hp <zz GL(Mp), where Mp denotes the tangent space to M at p. It follows that the affine symmetric space (G / H, I) is a riemannian symmetric space if, and only if, H is compact. The riemannian case has been extensively studied since É. Cartan inaugurated this subject some thirty-five years ago. The general case was first systematically studied by K. Nomizu in 1954, and then by M. Berger. One remarkable theorem, relating these two cases, is given by Berger in ; it asserts that every affine symmetric space is fibered over a riemannian one with fibers homeomorphic to a euclidean space. However the