This is a traditional course in basic Riemannian geometry.
Starting from the notion of Riemannian metric and corresponding
covariant differentiation, we shall present the metric space
structure of a Riemannian manifold related to the theory of
geodesics and the link between the sign of the (Riemann or Ricci)
curvature tensors and the topology of the underlying space
Riemann, On the Hypotheses which lie at the Bases of
Gromov, Sign and Geometric Meaning of Curvature. Rend.
Sem. Mat. Fis. Milano 61 (1991), 9-123. (url)
Do Carmo, Riemannian
Geometry. Birkhäuser Boston, Inc., Boston, MA,
Manifolds: an introduction to curvature. Graduate Texts
in Mathematics, 176. Springer-Verlag, New York, 1997.