﻿I’m writing this in a train going from Cabiate to Milan. I’ve got 25 mins until the train arrives. Let’s see how much math I can explain.


Manifolds feels like a complicated word. And I can try to give intuitive explanations like “it’s a locally euclidean space” or “it’s euclidean spaces glued together to form a larger object”, I know that these intuitive explanations are incomplete, and that the most correct answer is to say that a manifold is a topological space with the extra conditions that it’s open sets are homeomorphic to a subset of R^d, where d is the dimension of the manifold.


Technically that last explanation is somewhat incomplete, because in 100% of cases we also want to restrict the manifold to be T2 hausdorf, and also paracompact and connected. We need this for the intuition of a manifold to match the reality. For example without this restriction, any topological space equipped with the chaotic topology is a manifold, because on the question of every open set having a homeomorphism to some R^d, the homeomorphism is trivial because you need bijection + continuity, which is obvious from the fact that continuity is a requirement that the preimage of open sets are open, which is always true is the target topology is the chaotic topology.


Anyways on the topic of surfaces, which are essentially 2d-manifolds I believe if I remember correctly. Topological spaces arn’t classified. But surfaces are classified. This means you can come up with a list of properties between two surfaces and if they share the same invariants, then you can be sure that there exists a homeomorphism, or that the two surfaces are homeomorphic.


Let me talk about smooth manifolds for a second. Whereas in a regular manifold, we talked about charts and atlases implicitely saying that we “could” create the atlas, with smooth manifolds we actually go and create it, and append it to our structure. So a smooth manifold becomes a set (M, O, A), where A is the atlas.


How does the atlas encode the idea of differentiation? Well this is the idea of manifold philosophy. In the original case, we can talk about continuity on a manifold consistently because any intersection of two charts are guaranteed to give the same judgement on whether a curve/function is continuous. This is because chart transition maps are guaranteed to be continuous.


But this since chart transition maps arn’t guaranteed to be differentiable, it means that different charts might disagree on the differentiability status of a curve/function. Fortunately, what we can do is arbitrarily choose to only look at charts that agree with each other. That way the guarantee is true again. The decision of which charts to approve (haha sort of like twitter verified), is completely up to the decision of the mathematician.


One unfortunate scenario that could happen is that you have two different atlases that cover the whole manifold and whose charts agree with each other, but that the two atlases don’t agree with each other. It turns out that in 1, 2, and 3 dimensions, there is only 1 possible atlas daddy, so you’ll never have to make the decision of which atlas variation to use. But in 5 dimensions or higher, the number of different world is larger then 1, and finitely high. And worst of all, for dimension 4, there are an uncountably infinite number of atlases that don’t agree with each other on the differentiability status of a curve, but whose chart all agree with each other. This is catestrophic because it means we need to do an uncountably infinite number of experiments to figure out which atlas world our universe is actually equipped with. In practice this doesn’t end up being much of an issue, because our physics still seems to work out no matter what atlas world we choose.


But anyways that’s the story of differentiable manifolds. Next up is tensor space theory over fields, which is one of the most important, but initially confusing topics of all. It goes something like this:
* What a vector really is (element of a vector space)
* Linear maps from vector space V to W
* Linear maps as their own vector space Hom(V,W)
* Looking as specific cases of the above (covectors)
* Mixing vectors and covectors using tensor product to obtain p,q-tensors, which are just multilinear maps to the underlying scalar.
* Getting components of tensors, and doing the math with components
* Some isomorphisms from useful places to each other (dual dual vectors to vectors, maps from V->V to 1,1-tensors). x