A Characterization of Invariant Affine Connections
In  Ambrose and Singer gave a necessary and sufficient condition (Theorem 3 here) for a simply connected complete Riemannian manifold to admit a transitive group of motions. Here we shall give a simple proof of a more general theorem—Theorem 1 (the proof of Theorem 1 became suggestive to us after we noted that the T x of  is just the a x of  when X is restricted to p 0, see , p. 539). In fact after introducing, below, the notion of one affine connection A on a manifold being rigid with respect to another affine connection B on M and making some observations concerning such a relationship, Theorem 1 is seen to be a reformulation of Theorem 2.
KeywordsRiemannian Manifold Mathematical Method Differential Geometry Topological Group Simple Proof
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