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acts freely assume automorphism Cartan centralizer classification commutes compact complete connected conjugate connected riemannian constant curvature coordinate Corollary covering define denote diffeomorphism differentiable manifold dimension eigenvalue element of order euclidean space euclidean space forms finite group fixed point free follows frame bundle geodesic given Grassmann manifolds group of isometries groups of type homogeneous implies induces integers involutive Lie algebra isometry isomorphism isotropic Lemma Let G Let H Lie group matrix metric moving frame neighborhood nondegenerate nonzero normal subgroup open set orthogonal involutive Lie orthonormal basis point free representation prime Proof properly discontinuously proves pseudo-riemannian manifold quotient representation of G resp riemannian manifold riemannian symmetric space says sectional curvature semisimple simply connected space form problem spherical space forms subalgebra subgroup of G subspace Suppose Sylow 2-subgroup tangent space Theorem torus transitive vector field vector space