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accumulation point affine coordinates affine invariant affine manifolds affine matrix affine spaces affinely similar canonical form choose commutative H(r commutator subgroup compact Lorentz 3-manifold conjugate coordinates in R3 coordinates one finds coset cyclic d/dt diffeomorphic different homotopy type differentiable manifold distinct eigenvalue eigenvector element Euclidean manifold fails to intersect finite number free abelian fundamental group G is free G onto G group of affinities holonomy group H(r homogeneous space hyperbolic Lorentz transformation hyperbolic manifold impossible independent fixed vector integer invariant plane irrational isomorphic light cone linearly independent Lorentz group Lorentz orthonormal coordinates Lorentz unit norm m+nx manifold Mn manifold with H(r Math Moreover neighborhoods one-parameter orbit parabolic Lorentz manifold parabolic Lorentz transformation parabolic manifold parabolic transformations points of R3 PROOF proper Lorentz pure translations Q.E.D. LEMMA rational real number space-like suppose T P L t-axis Theorem 9 time-like topologically cover torus unimodular unimodular matrices verify