Abstract: By the theory of Modern Geometry, the mechanical principle and advanced calculus, the dynamics in Newtonian-Galilean spacetime is generalized to Newtonian-Riemannian Spacetime, and the dynamics in N-R spacetime is established. Being divided it into some parts. This paper is one of them. The others are to be continued.

Key words: Riemannian manifold, connection, absolute differential, Lie derivative, exterior differential, co-moving time derivative, fiber bundle, duality pairing