More from Tuesday. Apparently it was one of those days when I had thoughts of many sorts, and didn't bother that much with sorting which were mathematical, which were research-related, and which were whimsical.
Man. Vector bundles are so damn cool. If vector bundles were people then they would be really cool people and I would give them high-fives if I saw one of them on the street.
Tangent bundles are fine and all, but it always felt constraining to insist that the vectorspace structure of the tangent space be governed by some unseen chart parametrization.
So let the vectors in the additional structure point where they like. Why not? Pick up a normal bundle instead, and you can build the same objects in a more intuitive way. Take the Möbius band, for instance: it makes more sense to view it as a line bundle (1-dimensional fibres) than a non-orientable surface. Orientation's always bothered me, to tell you the true.
Let the vectors in this linear structure take a different dimension from the manifold dimension. In fact, when I think of it, the first Heisenberg group feels more like a vector bundle with 2-dimensional fibres than a Lie group or a smooth manifold .. what, with its vectorfields {X1, X2} and all ..
.. for the record, I didn't actually check whether the Heisenberg group had that structure I "felt," but I think it does. The vectorfields (which I never wrote down, here) are realised as a Lie algebra from the group action by left-translation, so that same left-action should give the vector bundle chart changes ..
.. but don't hold me to that. I'll check it when I have the chance. In the meanwhile, it's back to reading Hirsch for tomorrow's advisor meeting.
No comments:
Post a Comment