STRUCTURE AND CLASSIFICATION OF SYMMETRIC SPACES

3

by o(g) = Sx gSx. Then the stabilizer Ç of ÷

0

in G satisfies (G°)0 á Ç c

Ga,

and

X is isomorphic to the Symmetrie space G/H.

REMARK.

It follows from this theorem that we can always assume that a Sym-

metrie space is a real analytic manifold. It also follows that the universal cover-

ing space X ~ of X has a canonical structure as a Symmetrie space such that the

covering map ð: X~ — X is a morphism of Symmetrie Spaces.

2. Reductive Symmetrie Spaces. Let X = G/H be a Symmetrie Space, where G

is the group of displacements and Ç is the stabilizer of some chosen basepoint.

Let ó be the corresponding involution of G, and let the notation be as in Example

(B.2). We clearly have that

[f),f)]ci), [ i ) , q ] c q and [q,q]cf ,

which holds for any Symmetrie pair (G, //) . Our assumption that G is the group

of displacements implies the stronger property that [q, q] = I).

Since the basepoint may be chosen arbitrarily, formula (1) defines the curvature

tensor R. In the same way we define the Ricci-form é on X by

*eH(X,Y) = Traceq(Z - ReH(Y,Z)X) = Traceq(ad X°adY),

where I J G q.

If á é is an //-invariant subspace of q, then QX = [qx, qx] 4- qx is a subalgebra

of $. Let G2 be the analytic subgroup of G corresponding to QX and //x = Gx Ð //.

Then A^ = Gl/Hl is a subsymmetric space of X We say that Xx is an invariant

subsymmetric space if gx is an ideal in g. If Xx is an invariant subsymmetric

space, then we may define the Symmetrie quotient space

X/Xl={gXl\g^G} = (G/GMH/H,).

If there exists an //-invariant subspace q2 of q such that q is the direct sum of

á ÷ and q2, and X2 = G2/H2 is defined as above, we say that X2 is a complemen-

tary subsymmetric space to A^.

DEFINITION.

Let X = G// / be a Symmetrie space. We say that

(i) X is / t o if the curvature tensor is zero,

(ii) X is semisimple if the Ricci-form 4 is nondegenerate and Symmetrie,

(iii) X is reductive if every invariant subsymmetric space has an invariant

complementary subsymmetric space,

(iv) X is irreducible (or simple) if X has no nontrivial invariant subsymmetric

Spaces.

REMARK.

It follows quite easily from (2) that whenever the Ricci-form is

Symmetrie at eH, it is given by one-half times the Killing form on Q restricted to

Q X c\ (cf. Loos, Lemma IV.1.1). This holds, in particular, if X is semisimple. In

the case of Example (B.3) the Ricci-form at ex e Gx is the same as the Killing

form of cjj.