I. Structure and Classification of Symmetrie Spaces
The main topic of this book is analysis on pseudo-Riemannian Symmetrie
spaces with a semisimple isometry group. To place these Spaces in the proper
perspective, we Start this chapter with the most general definition of an affine
Symmetrie Space. Á connected affine Symmetrie space has a canenical symmetry-
invariant 2-form on the tangent space. This 2-form defines a pseudo-Riemannian
structure precisely when the group generated by the symmetries is a semisimple
Lie group. We call these spaces the semisimple Symmetrie spaces.
Another viewpoint, from which the semisimple Symmetrie Spaces present
themselves as a natural dass of Symmetrie spaces to study, is that of structure
theory. It turns out that an irreducible Symmetrie space is either one-dimensional
1. Affine Symmetrie spaces. Let X be a connected C°°-manifold with an affine
connection V. Let ÷ be any point in X. Localiy around ÷ the reflections in ÷
along geodesics through ÷ define a diffeomorphism Sx.
X — (×, í ) is an affine localiy Symmetrie space if for every ÷ in X
the local diffeomorphism Sx is an affine map. In this case we call Sx the local
symmetry around x. X is an affine Symmetrie space (in brief, a Symmetrie space) if
Sx can be extended to an affine diffeomorphism of X for every ÷ in X. In this
case we call Sx the symmetry around x.
In the category of affine Symmetrie Spaces we can define a morphism, a
coüering, etc. For example, a subsymmetric space Õ of X is a totally geodesic
submanifold such that the imbedding of Õ into X is a morphism of Symmetrie
spaces. One can show that the torsion is zero for a Symmetrie space, and that the
covariant derivative of the curvature tensor w.r.t. a vector field is also zero.
A. (A.l) Let X be
with its usual, flat, affine connection (i.e. the
covariant derivative of d/dxi w.r.t. d/dxj is 0 for each i and j). Then X is a
Symmetrie Space. In fact, for x, y e X, Sx(y) = ÷ - (y - x) = 2x - y, and
thus, in particular, S0(y) = -y.
(A.2) Let X =
and r + s = n. Define a pseudo-Riemannian structure^ on
X by #x(u, õ) = ulvl + · · · +urvr - ur+lur+l - · · · -unvn for each ÷ e X and
tangent vectors w, í e TXX. Let
= (X,#) be the corresponding "pseudo-
Euclidean" space. The affine connection canonically associated with # is the