2 STRUCTURE AND CLASSIFICATION OF SYMMETRIC SPACES
same as the flat connection of (A.l). Thus
Mr-S
is a Symmetrie space, and in fact
every symmetry is an isometry of
Ur,s.
DEFINITION.
Á (pseudo-) Riemannian Symmetrie space is a Symmetrie space
(X,V) with a (pseudo-) Riemannian structure f, such that V is the canonical
connection related to # and Sx is an isometry for every ÷ e X. X is said to have
index (r, s) if # has r positive and s negative eigenvalues.
Notice that Example (A.2) shows that two nonisomorphic pseudo-Riemannian
Symmetrie Spaces may be isomorphic as affine Symmetrie Spaces.
EXAMPLES
B. (B.l) Let G be a connected Lie group, and let ó be an involution
(i.e. an involutive automorphism) of G. Let Ç be a closed subgroup of G such
that (G°)0 c i / c G a , where Ga is the set of fixed points of ó, and (G°)0 is the
identity component of G°. (G, H) is then called a Symmetrie ñair.
The coset space X = G/H has a unique G-invariant affine connection which
makes X into a Symmetrie Space. The symmetry around the basepoint eH is given
by SeH(gH) = o(g)H, and, in general, Sx(y) =
glo(g{lg)H,
where ÷ = gxH and
y = g//. Let ö be the Lie algebra of G, and let ó also denote the involution of ò
defined by ó. Let g = rj + q be the decomposition of g into the + 1 and - 1
eigenspaces of ó. Then f) is the Lie algebra of H, and q may be identified with
the tangent space TeHX at the basepoint eH. The curvature tensor then satisfies
(1) ReH{X,Y)Z= -[[×,ÕÉAE], X,Y,Z^q.
(B.2) Let Ç be any closed connected subgroup of GL(n,U) and consider the
semidirect produet group G = Ç XsUn as a subgroup of the affine group
GL(/i,R) X5 (R'?. Define the involution ó of G by ó(,4, ÷) = (Á, -÷). Then we
have that the Symmetrie space X = Un is isomorphic to the Symmetrie space
G/H. In particular, we have for r + s = ç that the pseudo-Riemannian Symme-
trie space Rr'5 of Example (A.2) is isomorphic to (SO0(r, j) Xs Rn)/SO0(r, s).
In this case G is the connected component of the isometry group of W-s.
(B.3) Let Gx be a connected Lie group. Define G = GXX Gl5 ó(÷, j) = (y, x)
for (x, y) G G and Ç = = d(G1) = {(*, x) e G|x e G^ . The corresponding
Symmetrie space G// / is isomoöhic to the Symmetrie space obtained by giving
Gx the unique left- and right-invariant affine connection. The isomorphism is
given by
(x9y)d(G1)^xy-19
and for ÷ e Gx the symmetry Sx is given by Sx(y) = xy~xx for any 7 e Gv Thus
Ö«^ connected Lie group is á Symmetrie space.
The next theorem shows that Example (B.l) gives every Symmetrie Space.
Let X be a Symmetrie Space. The symmetries Sx, ÷ e X, generate a group of
affine transformations of X. By G( JT) we denote the subgroup generated by SXSV,
x, y ^ X. G(X) is called the group of displacements of X.
THEOREM
1. Let X be á Symmetrie space and G = G(X) its group of displace-
ments. Then G is á connected Lie group. Let x0 e X and define an involution ó of G
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