5 MANIFOLDS

Manifolds are generalizations of our familiar ideas about curves and surfaces to arbitrary dimensional objects. A curve in three-dimensional Euclidean space is parametrized locally by a single number $t$ as $(x(t),y(t),z(t))$ , while two numbers $u$ and $v$ parametrize a surface as $(x(u,v),y(u,v),z(u,v))$ . A curve and a surface are considered locally homeomorphic to $\mathbb{R}$ and ${\mathbb{R}}^2$ , respectively. A manifold, in general, is a topological space which is homeomorphic to ${\mathbb{R}}^m$ locally; it may be different from ${\mathbb{R}}^m$ globally. The local homeomorphism enables us to give each point in a manifold a set of $m$ numbers called the (local) coordinate. If a manifold is not homeomorphic to ${\mathbb{R}}^m$ globally, we have to introduce several local coordinates. Then it is possible that a single point has two or more coordinates. We require that the transition from one coordinate to the other be smooth. As we will see later, this enables us to develop the usual calculus on a manifold. Just as topology is based on continuity, so the theory of manifolds is based on smoothness.

Useful references on this subject are Crampin and Pirani (1986), Matsushima (1972), Schutz (1980) and Warner (1983). Chapter 2 and appendices B and C of Wald (1984) are also recommended. Flanders (1963) is a beautiful introduction to differential forms. Sattinger and Weaver (1986) deals with Lie groups and Lie algebras and contains many applications to problems in physics.

Exercise 5.19.

Example 5.15.

Problems

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