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Look I just love SO(n) okay
Some reasons I like SO(n): 1. It’s a compact Lie group. Who doesn’t like a compact Lie group?
2. Its fundamental group is Z/2Z. This is a lovely fundamental group. This will come back later
3. It’s probably the most “applicable” non-Euclidean space to the “real world.” It describes rotations! So it appears super naturally as an “application of higher dimensional geometry” in physics
4. Analyzing the geodesics on SO(n) with the bi-invariant metric yields the Parallel Axis Theorem, perhaps the easiest to understand and most striking application of differential geometry to a lay audience
5. Actually to go back to the fundamental group: it is the only space I know of where one can *intuitively visualize* what it means to have finite fundamental group, using the “belt trick” https://en.m.wikipedia.org/wiki/Plate_trick
6. It’s the structure group of orientable real vector bundles (at least if you reduce)
7. If you look like 3 tweets ago you’ll see I just got a paper on arXiv. We analyzed geodesics on SL(n) with a non-left invariant metric. SO(n) generated isometries which were crucial in our analysis
8. Polar decomposition! Who doesn’t love polar decomposition
9. Okay so its fundamental group is Z/2Z, which means it has a universal double cover Spin(n). Lifting the structure group of a vector bundle to Spin(n) is called a spin structure. These give you access to spin geometry which is pretty (ie the Atiyah-Singer Index Theorem)
10. Okay if we’re talking about spin geometry, let’s talk about physics. SO(n) is the symmetry group for the standard quantum Hamiltonian. There’s a beautiful theorem that the irreps of the symmetry group correspond to particles in a system by their action on the dive spaces of H
11. SO(2) = S^1 and I just think that’s neat
12. SO(3) = RP^3, another lovely isomorphism
Okay maybe this is enough for now but I may come up with more reasons later I love this thing
