Tangent vector-vector field formalism -- Differential forms -- Specialization to euclidean spaces : differential manifolds -- Mappings, submanifolds, and the implicit function theorem -- Jacobi bracket and the lie theory of ordinary differential equations -- Frobenius complete integrability theorem -- Reduction of dimension when a lie algebra of vector fields leaves a vector-fields invariant -- Lie groups -- Classical mechanics of particles and continua -- Differential forms and variational problems -- Hamilton-Jacobi theory -- Extremal fields and sufficient conditions for a minimum -- Ordinary problems of the calculus of variations -- Groups of symmetries of variational problems : applications to mechanics -- Elliptic functions -- Accessibility problems for path systems -- Affine connections on differential manifolds -- Rinmannian affine connection and the first variation formula -- Hopf-Rinow theorem applications to the theory of covering spaces -- Second variation formula and Jacobi vector fields -- Sectional curvature and the elementary comparision theorems -- Submanifolds of Riemannian manifolds -- Groups of isometries -- Deformation of submanifolds in Riemannian spaces -- First-order invariants of submanifolds and convexity for affinely connected manifolds -- Affine groups of automorphisms, induced connections on submanifolds, projective changes of connection -- Laplace-Beltrami operator -- Characteristics and shock waves -- Morse index theorem -- Complex manifolds and their submanifolds -- Mechanics on Riemannian manifolds.