# Manifolds

• Andrew Browder
Part of the Undergraduate Texts in Mathematics book series (UTM)

## 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 pC, 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) → CU 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.

## Keywords

Open Subset Tangent Space Maximal Rank Inverse Function Theorem Standard Orientation
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.