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Topic: Extending Differential Forms so That they are Globally Non-zero.
Replies: 0

 gk@gmail.com Posts: 134 Registered: 11/12/12
Extending Differential Forms so That they are Globally Non-zero.
Posted: Feb 18, 2013 2:49 AM

Hi, All:

Let M be a smooth manifold and let w be a form defined locally only,

i.e., w is defined in individual charts.

There is a way of extending w from the local charts to the whole manifold

using a bump function f and partitions of unity; we choose a triple K,V,U

with K compact, V closed, U open, K<V<U , so that f ==1 on V , and f is 0

outside of U . Then , for each chart we patch together f.w into a global

form using partitions of unity (assume M is paracompact , so that P.O.U's

exist ). **NOW** the problem is that in this extension, w will be zero.

Question: under what conditions can we extend w into a global _non-zero_

form?

Thanks.