144 Manifolds
A manifold is a space that locally resembles Euclidean space that is used as the geometric setting for calculus beyond flat space.
There are two main types of manifolds, generally: topological and smooth manifolds. We will start with an informal definition of a topological manifold.
If a surface is a generalization of a curve, then a topological manifold is the generalization of both surfaces and curves.
Here’s another way to understand a topological manifold.
A curve is a one-dimension manifold.
A surface is a two-dimension manifold.
Beyond two dimensions, we just call these things n-manifold, where
- \(n\) is the number of dimensions.
Let’s look at a more formal definition of a manifold.
Definition (Topological Manifold) A topological manifold is a topological space, \(M\), that satisfies the following conditions:
- Locally Euclidean of Dimension \(n\)
- Hausdorff Space
- Second Countable
Definition (Smooth Manifold)
144.1 Elementary Example
144.1.1 Simple
A curve is a \(1\)-dimensional manifold: locally it looks like a line.
\[ n = 1 \]
\[ M \text{ locally homeomorphic to open intervals in }\mathbb{R} \]
where
- \(n\) is the dimension.
- \(M\) is the manifold.
144.1.2 General
A surface is a \(2\)-manifold; higher dimensions are \(n\)-manifolds.
\[ n = 2:\ \text{surface locally like open sets in }\mathbb{R}^{2} \]
\[ n = 3:\ \text{solid region locally like open sets in }\mathbb{R}^{3} \]
where
- a topological manifold is Hausdorff, second countable, and locally Euclidean of dimension \(n\).