Synge's Theorem Riemannian Geometry Local Existence and Uniqueness of Geodesics Differential Geometry Flat Implies Locally Euclidean Differential Geometry Structure of Compact Manifolds with Nonneg Ricci Riemannian Geometry Minimal Geodesic in a Non-Trivial Homotopy Class Riemannian Geometry Whitney Embedding Theorem Differential Geometry Local Minimality in Path Space Riemannian Geometry Jacobi Fields via the Exponential Map Riemannian Geometry