Spaces of Constant Curvature

a₁ abelian acts freely automorphism B₁ bilinear Cartan centralizer classification commutes compact complete connected complex conjugate connected riemannian constant curvature coordinate Corollary define denote diffeomorphism differentiable manifold element of order equivalent euclidean space euclidean space forms finite group fixed point free follows frame bundle G₁ geodesic given GL(n Grassmann manifolds group of isometries homogeneous implies integers irreducible isomorphism isotropic Lemma Let G Lie group linear m₁ matrix maximal metric n₁ neighborhood nondegenerate nonzero normal subgroup open set point free representation prime Proof properly discontinuously proves pseudo-riemannian manifold rank representation of G resp riemannian manifold riemannian symmetric space S₁ says sectional curvature semisimple simply connected space form problem spherical space forms subalgebra subgroup of G subspace Suppose Sylow 2-subgroup t₁ tangent space Theorem translation v₁ vector field vector space X₁ Z₂