# Manifold ### Intuitions * It is a simplifying assumption about how our space looks. It is the notion of a "nice" space in geometry. * What is it that is "nice" about a space that is captured by the idea of a manifold? * One way we can think about this is to look *locally* at the behavior of each shape in a *neighborhood* of a *single point*. * Remember, the word *differential* in differential geometry refers to the fact that we want to study the *local* properties of shape. ### Key Idea > A **manifold** *locally* looks like $\mathbb{R}^n$  Above, if we zoom in on a point, it is easy to lay down some coordinates/gridlines that look like a grid we would have in the plane ($\mathbb{R}^2$). That is pretty nice, because we are comfortable with the plane. We know how to talk about lengths, angles, and do various calculations. So, even if we don't understand what is going on in the shape as a whole, we know that at least around this point we have some machinery that we know how to work with. If we go around to every other point in the surface on the left above, we see that we would be able to put down a little coordinate system for every point. This is because the shape is smooth! Now look at the shape on the right. There is one point that is special and *non smooth* (i.e. non differentiable). This leads us to say that the shape on the right is not a manifold:  ### Biggest Ideas A manifold is simply a space that is curved (it of course could be flat). Some examples of manifolds are: * Spacetime (4d manifold) * Space (3d manifold) * Euclidean space * A Sphere --- Date: 20210911 Links to: [Discrete-Differential-Geometry-MOC](Discrete-Differential-Geometry-MOC.md) Tags: References: * [DDG Introduction to manifolds video](https://www.youtube.com/watch?v=KZjoxwUxlIs&list=PL9_jI1bdZmz0hIrNCMQW1YmZysAiIYSSS&index=4) * [Good blogpost](https://bjlkeng.github.io/posts/manifolds/) * [The Biggest Ideas in the Universe | 13. Geometry and Topology - YouTube](https://www.youtube.com/watch?v=kp1k90zNVLc&list=PLrxfgDEc2NxZJcWcrxH3jyjUUrJlnoyzX&t=1446s)