﻿You want to understand how the universe works.


You want to learn
You want to see
You want to predict
You want to influence


All of these things are reasons we study physics. We want to understand, learn, see, predict, and influence the universe.


So we come up with theories.


One of the theories we have is general relativity. Which takes the form of a model of an object called spacetime and partial differential equations on spacetime called the einstein field equations.


And spacetime turns out to take the structure of a pseudo riemannian manifold. Well its equipped with a connection, which implies the metric needed for a riemannian manifold, but whatever close enough I guess.


But most of the assumption is actually encoded in the manifold part. What is a manifold? Well it’s a locally euclidean structure. Because you don’t have a connection yet, it’s like if the balloon hasn’t been blown yet, so it could end up being a potato, a round sphere, or an ellipsoid. The choice is up to the connection, which we haven’t seen yet. But the smooth manifold part is the rest of the choice.


But anyways a smooth manifold is a topological space with an extra condition, and extra structure. The extra condition is that the topological space is locally euclidean. The extra structure is that of a smooth atlas.


So first up is the extra condition, which enforces local enclideaness. Basically if you select any point on the manifold, its neighborhood must be homeomorphic to some R^n (isomorphic on a topological level). And the n must be common for all points, so you can’t have a disk like the flat earth, which is R^2 in most neighborhoods, but R^1 on the boundary. The disk fails at being a manifold. Although interestingly the disk boundary, and rest from the boundary are both separately manifolds (1d and 2d respectively), which is quite interesting. And the study of manifolds and their boundaries is quite interesting in general. I’m tempted to say there’s even a link to the generalized stokes theorem, but I’m honestly not quite sure it’s topological (I always imagined generalized stokes theorem to be differential geometry stuff, where you require discussion of differential forms).


So that’s a manifold. And now for smooth manifolds.


Before talking about smooth manifolds, let me first say that we should get more terminology.
Imagine this recipe:
* Collect all the neighborhoods in the previously talked about definition of a manifold, and also collect the respective homeomorphism map, and store them in a tuple (U, x). This tuple is called a chart of the manifold. If you have a collection of charts that collectively cover the manifold when put together, then that’s called an atlas.
Much of the intuition here comes directly from globes and sailors. Imagine a chart of canada. That’s a neighborhood of the earth, with a specific homeomorphism, probably the mercator projection. Notice that because it’s a homeomorphism, continuity is preserved. So the chart can be pulled and stretched and squashed and it’s still a chart. But if you tear it, then it’s not a chart, because it fails the homeomorphism.