Differentiable mappings can be approximated by affine mappings; similarly, manifolds can locally be approximated by affine spaces, which are called (geometric) tangent spaces. These spaces provide enhanced insight into the structure of the manifold. In Volume II, in Chapter 7 the concept of tangent space plays an essential role in the integration on a manifold; the same applies to the study of the boundary of an open set, an important subject in the extension of the Fundamental Theorem of Calculus to higher-dimensional spaces. In this chapter we consider various other applications of tangent spaces as well: cusps, normal vectors, extrema of the restriction of a function to a submanifold, and the curvature of curves and surfaces. We close this chapter with introductory remarks on one-parameter groups ofdiffeomorphisms, and on linear Lie groups and their Lie algebras, which are important tools in describing continuous symmetry.

Definition of tangent space

Let k ∈ N∞, let V be a Ck submanifold in Rn of dimension d and let x ∈ V. We want to define the (geometric) tangent space of V at the point x; this is, locally at x, the “best” approximation of V by a d-dimensional affine manifold (= translated linear subspace). In Section 2.2 we encountered already the description of the tangent space of graph(f) at the point (w, f(w)) as the graph of the affine mapping h ↦ f(w) + Df(w)h. In the following definition of the tangent space of a manifold at a point, however, we wish to avoid the assumption of a graph as the only local description of the manifold.