What the Heck is a Manifold? (Part I)
A brief introduction to manifold theory, with some topology and analysis for good measure
When you read about physics and mathematics in the news, or see a Wikipedia article, you will likely come across the word “manifold”. It usually doesn’t come with any explanation, and it seems relatively important. Manifold theory is the theory which underlies General and Special relativity, as well as a good chunk of modern geometry.
Today, we’re going to remedy that. We will learn what a manifold is and why it matters, from the perspective of someone who doesn’t work in mathematics.
This article is written for those with some calculus, specifically multivariable, but can be understood by those with precalculus or algebra. Any topics discussed not in a calculus class will be introduced in a general way, hopefully such that the general reader can understand. Italics will be used for more advanced mathematical concepts, which the casual reader can pass over.
Roadmap of the article:
- Motivation, some geometry, some analysis ideas
- Manifold definition
- Example and counterexample
- Some topology and geometry
