Mathematical Analysis pp 253-268 | Cite as

# 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 **R**^{2} or **R**^{3} is that of a subset *C* of **R**^{2} or **R**^{3} 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 **R**^{2} 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 **R**^{3}. 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## Preview

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