Differential Geometry

Some results about the differentials (i.e., pushforwards) of the exponential and logarithm maps of a Lie group. I rely extensively on the interpretation of tangent vectors as equivalence classes of curves.

Here, I solve Problem 5-5 from John Lee’s book on Riemannian Manifolds, which demonstrates the non-flatness of the 2-sphere. This problem is particularly interesting because it serves as the motivating example for a later chapter in the book on curvature.

A Lie group is a group that is also a (continuous, differentiable) topological space. To measure lengths and volumes (and relatedly, to define and integrate probability densities) we need to endow the group with additional structure so that it is not merely a manifold, but a Riemannian manifold.

There are multiple ways to construct new groups from old ones. I provide an intuition for how these constructions work, and also go over some of the additional structures that can be imposed on Lie groups, paving the path towards doing differential geometry and calculus on Lie groups.

A topological group is a set of elements that has both a group operation and a topology. The group operation satisfies the usual axioms (same as those of finite groups), and the presence of a topology lets us say things like ’the group is connected’ and ’the group operation is continuous'.

Over the past year, I have struggled to pin down what the scope of my blog should be. There is plenty of exposition out there on just about every aspect of modern mathematics, but especially on exterior calculus and differential geometry due to their situation at the intersection of several areas in theoretical and applied mathematics. (As a case in point, the two main references that I’ve been using to self-learn differential forms were the creations of a theoretical physicist and a computer scientist , respectively). So then what is the scope of my blog? Maybe it is for me to catalog the process of self-learning mathematics as an engineering major who lacks a curricular background in modern mathematics. Maybe it is to assure others like me (who are also privileged enough to learn mathematics in isolation of such material concerns as its ‘job prospects’) that it can be done. This post will do a bit of both; it serves in part the purpose of organizing my own thoughts on these matters, and in part the purpose of providing a roadmap for others who are interested in embarking on a similar journey. ...