On the geometry and classification of absolute parallelisms. I

Abstract

8. The irreducible case Let (M, ds2) be a simply connected globally symmetric pseudo-riemannian manifold, and φ an absolute parallelism on M consistent with ds2. We assume (M, ds2) to be irreducible. Our standing notation is the LTS of ^-parallel vector fields on M, the Lie algebra of all Killing vector fields on M, conjugation of g by the symmetry sx at x e M, g = I + m: eigenspace decomposition under σx. The irreducibility says that m is a simple noncommutative LTS, and thus (Lemma 6.2) says the same for p. 8.1. Lemma. Either [p, p] = p or [p, p] Π p = 0. Proof. Let t = [p, p] Π p. Then [!>,£],£)] C p implies [x,p] C t and so [tfψ] C i. Thus i is a LTS ideal in p. By simplicity, either t = 0 or t = p. If t = 0, then [p, p] Π p = 0. If t == p, then p c [p, jj]. As [i, p] C i, also [p, p] C p. Hence [p, p] = p. q.e.d. We do the group manifolds immediately. 8.2. Proposition. Let (M, ds2) be irreducible simply connected and globally symmetric, with consistent absolute parallelism φ such that the LTS of φparallel fields satisfies [p, p] ΓΊ p Φ 0. Then [p, p] — p, p is a simple real Lie algebra, and (M, φ, ds2) ^ (P, λ, da2) where (i) P is the simply conncted group for p, (ii) λ is the parallelism of left translation on P, and (iii) dσ2 is the bi-invariant metric induced by a nonzero multiple of the Killing form of p. The symmetry of (P, dσ2) at 1 eP is given by s(x) = x~ι. The group G of all isometries of (P, dσ2) has isotropy subgroup K at 1 given by