Semi-Riemannian Geometry

Chapter 14 Smooth Manifolds

14.1 Smooth Manifolds

Many of the definitions presented in this and subsequent chapters are adaptations of ones we encountered in the study of curves and surfaces.

Let M be a topological space. A chart on M is a pair (U,  φ), where U is an open set in M and φ : U → ℝ ^m is a map such that:

• [C1] φ(U) is an open set in ℝ ^m .
• [C2] φ : U → φ(U) is a homeomorphism.

We refer to U as the coordinate domain of the chart, to φ as its coordinate map, and to m as the dimension of the chart. For each point p in U , we say that (U,  φ) is a chart at p . (Note that in the definition of regular surfaces, coordinate maps went from open sets in ℝ² to open sets in the regular surface, whereas now traffic is in the opposite direction.) When U = M , we say that (M,  φ) is a covering chart on M , and that M is covered by (M,  φ). In the present context, the component functions of φ are denoted by φ = (x ¹,  …,  x ^m ) rather than φ = (φ ¹,  …,  φ ^m ), and are said to be local coordinates on U . This choice of notation is adopted specifically to encourage the informal identification of the point p in M with its local coordinate counterpart φ(p) = (x ¹(p), …,  x ^m (p)), in ℝ ^m We often denote

where (x ⁱ ) = (x ¹,  …,  x ^m ). The charts (U,  φ) and on M are said to be overlapping if is nonempty. In that ...