Geometry on manifolds

Abstract Differential geometry is geometry studied via advanced calculus. It applies not just in Euclidean space but on general smooth manifolds. Riemannian geometry is the oldest branch of differential geometry, and the branch that most directly generalizes Euclidean geometry: it is concerned with lengths and angles. In mechanics, Riemannian geometry is used to describe kinetic energy; and also the paths followed by particles in the absence of external forces, which are geodesics.