Change of coordinates on manifolds
Just wanted to share the good news: manifolds and the associated concepts– tangent spaces, local coordinates, diffeomorphisms between manifolds, etc.– are starting to make a lot of sense to me. Ironically, most of this is due to the fact that I’ve been reading several books concurrently, all of which define (differentiable) manifolds differently: this makes me have to identify the quiddities of the involved objects in order to correlate the information meaningfully.
As a prime example of the benefits of this strategy, just yesterday one of my persistent questions was answered: why is it stipulated that if 
and 
are two overlapping charts on a manifold 
, they must satisfy 
is a 
mapping? What I had expected as a natural condition was that and would be identical on their intersection, but that is too restrictive a condition because then we could just define a new chart 
as the ‘join’ of and … following this kind of definition, you would end up saying that each (connected) manifold has a global chart. Instead, we acknowledge that in general, some points will have different coordinates under different charts, but stipulate that the corresponding change of coordinates must be nice in some sense. Exactly how nice a manifold is depends on the smoothness of the change of coordinates between its charts. This justification for the standard definition is simple and satisfying, but is not explicitly stated in any of the pure math books I’m reading, only in this analytical mechanics book.
Primarily I’m reading Tensor Analysis on Manifolds, which is good enough to stand alone, and the reading would go much faster, but I’m going to stick with the skipping around. If all goes as planned I’ll be done with the first chapter of that book tonight, review it tomorrow, and then proceed to the next two chapters on tensors.
Possibly relevant posts:
- Kvetches about Janich’s book (7/2/2007)
- Differential forms (4/17/2005)
- “Good” functions (11/17/2003)
It’s even more general than that. For a wide variety of properties P, if you insist that the transition functions between charts have property P, then it becomes possible to talk about functions on the manifold having property P. Properties P include: being continuous, being complex analytic, being affine, being a Mobius transformation, being n times differentiable, being smooth and so on. I guess the key thing is that if you compose two functions with property P the composition has the same property.
Comment by Dan P — 9/22/2006 @ 3:03 pm
Right.. as I understand it, to say a function has property P on a manifold means that its pullback with respect to every chart has that property. But it’s usually the niceness– say, the fact it’s a diffeomorphism– of the change of charts map that allows you to say that if the function has property P with respect to a single chart, then it also has the property relative to any other chart.
Comment by Alex — 9/22/2006 @ 4:50 pm