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From: Bacle on 29 Apr 2010 19:52
And, more in general: if/when we have an m-manifold M,
and an n-submanifold N, n<M , and we have an n-form
w_n(N) defined on N, when can we extend w_n(N) into
an n-form w_n(M) on M.?.

I suspect this has to see with partitions of unity
and bump functions somehow, but I am stuck at the moment. Any suggestions.?

Thanks.
From: W. Dale Hall on 30 Apr 2010 02:25
Bacle wrote:
> And, more in general: if/when we have an m-manifold M, and an
> n-submanifold N, n<M , and we have an n-form w_n(N) defined on N,
> when can we extend w_n(N) into an n-form w_n(M) on M.?.
>
> I suspect this has to see with partitions of unity and bump
> functions somehow, but I am stuck at the moment. Any suggestions.?
>
> Thanks.

Let N be an n-manifold contained as a closed subset of the m-manifold M.
Let U be a tubular neighborhood of N in M, that is, an open subset of M
which is the image of an embedding of the disc bundle of the normal
bundle to N in M. Let p: U ---> N be the projection determined from the
identification of U as a disc bundle over N. Let i denote the inclusion
of N in U.

Let w be a k-form defined on N. Then the pullback w_U = p*w is a k-form
defined on U, which (due to the the identity p o i = id_N) restricts
to w on N itself. Now, give the normal bundle an O(m-n) structure
<.,.> (i.e., an inner product on each fibre that is of class C-infinity
in the total space of the bundle), and define r(x) to be the radius
(with respect to this inner product) of the disc p^(-1)(x) in U. Then
r(x) is a positive function on N. Further, let b denote a C-infinity
function:

b: [0, infinity) ---> R

that has b(0) = 1, b(t) = 0 for t >= 1, and b >= 0 for all t >= 0.

Here's a k-form:

w#(y) = b(|y|/r(p(y))) w_U(y) for y in U
0 (the zero k-form) for y in M \ U

where |y| is defined as the square root of the inner product <y,y>
determined by the O(m-n) structure described above (where I've
identified U as the appropriate subset of the total space of the
normal bundle to N in M.

Note that the tapering introduced by the function b causes w# to
fall off to 0 outside U, and the resulting form is smooth.

That'll do the trick. If you need some other global features of w#,
you will most likely face some topological constraints.

Dale

Note that
From: W. Dale Hall on 30 Apr 2010 02:27
W. Dale Hall wrote:
... a buncha stuff culminating in some bit of text that I
neglected to delete (poor proofreading on my part):
> That'll do the trick. If you need some other global features of w#,
> you will most likely face some topological constraints.
>
> Dale
>
> Note that
^^^^^^^^ oops.

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