Semi-martingales, bilinear forms, pull-back, covariation process, semimartingale integral, connection, chain rule, differential operators and Christoffel symbols, martingale criteria, induced connections, affine and convex maps, geodesics and martingale criteria, local drift and diffusion rates, sub-manifolds and projection, diffusions, Riemannian metric and connection, isotropic martingales, Brownian motion, harmonic maps, Levy processes in Lie groups.