Comments on: Change of coordinates on manifolds http://www.tangentspace.net/cz/archives/2006/09/change-of-coordinates-on-manifolds/ somewhere near the beginning. Mon, 01 Dec 2008 23:56:10 +0000 http://wordpress.org/?v=2.5 By: Alex http://www.tangentspace.net/cz/archives/2006/09/change-of-coordinates-on-manifolds/#comment-60996 Alex Fri, 22 Sep 2006 23:50:58 +0000 http://www.tangentspace.net/cz/archives/2006/09/change-of-coordinates-on-manifolds#comment-60996 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. 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.

]]> By: Dan P http://www.tangentspace.net/cz/archives/2006/09/change-of-coordinates-on-manifolds/#comment-60962 Dan P Fri, 22 Sep 2006 22:03:57 +0000 http://www.tangentspace.net/cz/archives/2006/09/change-of-coordinates-on-manifolds#comment-60962 <blockquote> Exactly how nice a manifold is depends on the smoothness of the change of coordinates between its charts </blockquote> 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.

Exactly how nice a manifold is depends on the smoothness of the change of coordinates between its charts

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.

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