ABSTRACT
This chapter presents some illustrations of the main mathematical objects and geometrical models. It provides worked out and detailed examples of chart and parametric spaces for some classical manifolds such as the circle, the sphere and the torus. The chapter also presents a detailed derivation of the mean curvature vectors, the Riemannian metrics, the geodesics, the Christoffel symbols, and the Ricci curvature. It explores selected applications of Riemannian geometry to statistics and physics. The geometric Riemannian structure allows the author to define various geometric notions such as angles, lengths of curves, volumes, curvatures, gradients of functions and divergences of vector fields. Riemannian manifolds also arise in a natural way in Bayesian statistics and in information theory.
