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Extending Differential Forms so That they are Globally Non-zero.
Posted:
Feb 18, 2013 2:49 AM
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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.
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