Manifolds

Abstract

In this chapter we formulate the notion of a manifold, which generalizes the familiar ideas of a (smooth) curve or surface. The intuitive idea of a curve in R² or R³ is that of a subset C of R² or R³ which locally looks like a segment of the real line; in other words, if p ∈ C, there should be a neighborhood U of p in R² and an interval (a, b) in R, together with a bijective map a : (a, b) → C ∩ U which is bicontinuous. If p is an endpoint of C, the interval (a, b) should be replaced by an interval [a, b) or (a, b]. This formulation is purely topological, i.e., defined in terms of continuity; we want to restrict our attention to differentiate curves, which should mean that α and α^-1 are required to be differentiate. We have to explain what we mean by α^-1 being differentiable, since its domain is not an open subset of R³. We will also formulate the notion of the tangent space to a differentiate manifold, generalizing the familiar ideas of tangent line to a curve or tangent plane to a surface, and discuss the idea of orientation.