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

or semisimple.

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.

DEFINITION.

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.

EXAMPLES

A. (A.l) Let X be

Un

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 =

Un

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

W's

= (X,#) be the corresponding "pseudo-

Euclidean" space. The affine connection canonically associated with # is the

l

http://dx.doi.org/10.1090/cbms/061/01