Embedded manifolds and matrix groups
Final project for Colorado State University Math 476
Natalie Burke, Spring 2021
This is a webpage created to communicate an exploration of the relationship between a geometric understanding of manifolds and matrix groups.
The following theorem has remarkable similarities to the classification theorem.
So, both geometric manifolds and Lie groups can be classified into groups built from known elements, but Lie groups are more complex. Matrix manifolds are also better suited to describe higher dimensions.
This leads me to the question, are there any common elements to the building blocks in each case? This would be a good question for further study.
Next, let's get into what the connection between matrix groups and geometry can tell us.
First, we can find the dimension, compactness, and connectedness of a matrix group. The dimension for each group is found by taking the dimension of the associated Lie algebra. To tell if it is compact, we construct an open cover of G and check if G has a finite subcover. Compact sets are closed and bounded. To tell if it is connected, we can check for the definition, or for closed and open sets.
Tables 1,2,3 (Marsh)
Then, we can define a kind of torus for each matrix group.
So, clearly maximal tori can be very useful when working with matrix groups and they interact differently than the torus as a flat 2-manifold.
Finally, we note some homeomorphisms between common matrix groups.