Synge's Theorem states that if a Riemannian manifold is complete and has a positive curvature, then any two geodesics that start at the same point and are initially tangent to each other will intersect again. This theorem is significant as it connects the concepts of geodesics, curvature, and topology. It also emphasizes the importance of completeness in the structure of a manifold and sets the stage for understanding how curvature influences the behavior of geodesics.
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