# Differential Geometry, Topology, Manifolds, Matt Grimes Here are some notes from a very enlightening conversation that I had with Matt Grimes: The end goal of **differential geometry** can be defined as: > We want to be able to do calculus in more general settings. In other words, take derivatives and integrals. The challenge is that calculus relies upon the idea of a *limit*. There are many things that you can do with limits in $\mathbb{R}$ (since it is extremely structured). What we really need is a definition of *closeness*. *That* is the point of **topology**. You have a set of things, you say "I want to do calculus on this set". But the set is discrete! How do you do calculus on a discrete set? What if you have no **metric**? You take your set and you equip it with a *topology*! You can then do math with just *closeness*! ### Topological spaces A set with a topology is a **topological space**. You now have a geometric entity (technically a *topological entity*). Manifolds are a special type of topological space. We want manifolds so that we can do calculus and geometry (where geometry is referring to distances and angles). At this point we only know calculus in $\mathbb{R}^n$ (and maybe the complex numbers). We say "hmmm, I have a topological space, but I only know how to do calculus in $\mathbb{R}^n$..." Then we realize that *most of calculus is local*! Derivatives are local and integrals are kind of global, but can be defined locally. To do calculus on a manifold we zoom in! A manifold is a topological space where if you zoom in enough it is identical to $\mathbb{R}^n$ (i.e. it is locally euclidean)! A **manifold** has a cover (a collection of neighborhoods); every point is in a neighborhood, each neighborhood looks like $\mathbb{R}^n$, and where there is overlap we can **translate** it. When talking about translation: * these are called **charts**, where we put an **atlas** on a topological space * For instance, an *atlas* for earth would contain two subsets: * everything but south pole * everything but north pole * this gives us two copies of the plane! No cutting is needed! (we can continuously deform and reshape into a flat sheet) * Nate is in both of these sets; so I have positions on both maps * What we need to glue together these two maps is a way to translate between the two maps! * If we pick a point on the first map, we can say mathematically what point we are talking about in the other map (this is a **transition function**) ### Tensors * We have defined manifold * In calc 3 we learned line integral, gradients. Then we jump into vector calc * Calc 3 is very clunky and straight forward. You can rely on hidden symmetries of R3 * consider R2, vector with origin at 0,0 to move to 1,0 Now consider earth, wind velocity consider it in buffalo what does it mean to move to Boulder if it is flat just translate it! it is only locally euclidean if you zoom out then interesting things happen consider parallel transpot! this is why calc 3 is simply you can get away with less penetrating ideas problems become more technicaly and subtle bector moving along line in way that angle doesn't change is parallel transport doesn't make sense to move vectors around in R3 all vectors in that diagram are not in same vector space! different oriigins! non trivial to move around we have selected vectors in series of vector spaces took several copies of R2, glued to different point on circle, at each point we can select point in each space this is idea of tangent bundle each point has it's own tangent not just one tangent That is essence is what it means to do differential geometry enoguh of a setup to do tangent spaces!!!! once you are relying on tangent vectors to describe you are in the owrld of geometry tangent vecotrs are velociteis shows up a lot in dynamics geometry comes into place here we have a tangent bundle that helps us redevlop idea of tangent vectors What are tools for studying connection → gadget for translating vector from one tangent space to another picture is indtuitive, mathematically we are making use of a connection Tensors come in → tuples of tangent vectors and cotangent vectors co of something is a dual if v is a vector space, dual space of v is ocllection of linear transformations from V to R # Part two * Frequent game (yoga), naively taking inspiration from things * adjacency matrix and laplacian * useful in diff eq * the only math people can actually do is linear algebra --- Date: 20210815 Links to: [Mathematics MOC](Mathematics%20MOC.md) Tags: #review #refactor References: * []()