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Hi, Again:
I'm trying to show that, given a 3-manifold M, and a plane field ρ (i.e., a distribution on
TM) on M, there exists an open set U in M, so that ρ can be represented as the kernel of a
differential form w , for W defined on U.
The idea is that the kernel of a linear map from R3 --TxM
is either the whole space, or a two-dimensional space, by , e.g., rank-nullity.
My idea is to start by choosing the assigned plane ∏m at any point m in M. Then we use the fact that any subspace can be expressed as the kernel of a linear map.
Specifically, we choose a basis {v1,v2} for ∏m Subset TmM
and define a form w so that:
w(v1)=w(v2)=0 .
Then we extend the basis {v1,v2} into a basis {v1,v2,v3} for TmM , and
then we declare w(v3)=1 (any non-zero number will do ), so that
the kernel of w is precisely ∏m, by some linear algebra.
Now, I guess we need to extend this assignment w at the point m, at least into
a neighborhood Um of M . I guess all the planes in a subbundle have
a common orientation, so maybe we can use a manifold chart Wm for
m, which is orientable ( being locally-Euclidean), and then use the fact that there is
an orientation-preserving isomorphism between the tangent planes at any two points
p,q in TpU . Does this allow me to define a form in U whose kernel is ρ ?
Edit: I think this should work: please critique: for each q in U , the hyperplane ∏
q can be described as the kernel of a map , as for the case of m. Again,
we find a basis {w1,w2} for ∏q , and then define an orientation-preserving
map (which exists because U is orientable) between TmM and
TqM, sending basis elements to basis elements.
Thanks.
I'm trying to show that, given a 3-manifold M, and a plane field ρ (i.e., a distribution on
TM) on M, there exists an open set U in M, so that ρ can be represented as the kernel of a
differential form w , for W defined on U.
The idea is that the kernel of a linear map from R3 --TxM
is either the whole space, or a two-dimensional space, by , e.g., rank-nullity.
My idea is to start by choosing the assigned plane ∏m at any point m in M. Then we use the fact that any subspace can be expressed as the kernel of a linear map.
Specifically, we choose a basis {v1,v2} for ∏m Subset TmM
and define a form w so that:
w(v1)=w(v2)=0 .
Then we extend the basis {v1,v2} into a basis {v1,v2,v3} for TmM , and
then we declare w(v3)=1 (any non-zero number will do ), so that
the kernel of w is precisely ∏m, by some linear algebra.
Now, I guess we need to extend this assignment w at the point m, at least into
a neighborhood Um of M . I guess all the planes in a subbundle have
a common orientation, so maybe we can use a manifold chart Wm for
m, which is orientable ( being locally-Euclidean), and then use the fact that there is
an orientation-preserving isomorphism between the tangent planes at any two points
p,q in TpU . Does this allow me to define a form in U whose kernel is ρ ?
Edit: I think this should work: please critique: for each q in U , the hyperplane ∏
q can be described as the kernel of a map , as for the case of m. Again,
we find a basis {w1,w2} for ∏q , and then define an orientation-preserving
map (which exists because U is orientable) between TmM and
TqM, sending basis elements to basis elements.
Thanks.
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