Riemannian metrics, Affine connections, Riemannian connections, geodesics, minimizing properties of geodesics, convex neighborhoods, curvatura, sectional curvature, Ricci curvature, Jacobi equation, conjugate points, Hopf-Rinow's Theorem, Theorem of Hadamard, Bochner techniques, Theorem of Bonnet-Meyer, Theorem of Synge, The Morse index Theorem