By virtue of the well-known theorem on structure group reduction, a structure group G L + ( 4 , R ) of a frame bundle F X over a world manifold X is always reducible to its maximal compact subgroup S O ( 4 ) . The corresponding global section of the quotient bundle F X / S O ( 4 ) is a Riemannian metric g R on X . Thus, a world manifold always admits a Riemannian metric which makes X a metric topological space.