I originally started this blog to make available various notes I’ve written over the years. (Justification: the internet hasn’t yet run out of space.) Herewith a very short note on principal and fiber bundles (small and medium formats), and a longer one on the Laplacian on the cube. Also Three takes on the tangent bundle.
I wrote the bundle notes for a math meetup, to clarify the relation between a principal bundle and its associated fiber bundles. I look at the Möbius strip, a case as close to trivial as you can get without being trivial. Then I springboard to the tangent bundle of the 2-sphere (still quite easy).
The “three takes on the tangent bundle” are the formal, the intuitively geometrical, and the classically computational.
I wrote the Laplacian note when I was working my way through Sternberg’s Group Theory and Physics, also looking at his handouts (unfortunately no longer available online). Fiber bundles lurk in the background (with a fiber at each vertex of the cube), but group reps take center stage. The group is the rotation group of the cube; it has five inequivalent irreducible representations. Four lend themselves to easy visualization. This is a nice model example of group reps furnishing info about eigenvalues.
Sternberg clearly intended this example as a warm-up for his treatment of the spectra of the (icosahedral) buckyball (pp.126–129), one of the highlights of the book.