In Chap.  3 we have used the terms manifoldManifold (for the configuration or phase space of a mechanical system), group (for symmetries such as the set of Galilean transformations) and Lie algebra (for the Poisson brackets among conserved quantities \(L_i, K_j, H\) of the Kepler problem or the components of angular momentum \(L_x, L_y, L_z\) ). In Chap.  3 and elsewhere, we also encounter the notions of vector fields, covector fields, differential forms, wedge products, tensor fields and exterior, Lie and covariant differentiation on a manifold. These are part of the mathematical language of classical mechanics. Among other things, they help us study systems whose configuration or phase space is not Euclidean space ( \(\mathbb {R}^n\) for some \(n = 1,2,3, \ldots \) ) but a space such as a circle, sphere (see the example in Sect.  3.26* ), torus, hyperboloid and so forth. In this chapter, we introduce these notions along with examples. Books such as [1–5] may be consulted for much more on these topics.