Solution to 1-Form Math Problem: Vanishing of j on N iff j^a^b Does Not

[daishin]

Homework Statement

I was reading a note from the class and don't understand something.
Let a,b be everywhere linearly independent 1-forms on 5 dimensional manifold.
Let N be 3dimensional submanifold of M. Let c,d,e be 1-forms on M such that linearly independent when restricted on N. We assume a,b,c,d,e forms a basis.
Then a 1-form j will vanish on N iff j ^ a^ b does not vanish on N.

Homework Equations

I don't understand why 1-form j will vanish on N iff j ^ a^ b does not vanish on N.

The Attempt at a Solution

Suppose j vanishes on N, then j=ra+sb for some real r,s.(since c,d,e forms a basis of 1-forms on N?) But j^a^b=ra^a^b+sb^a^b=0.

 

[Dick]

I may be crazy, and if so someone correct me, but if j vanishes on N, that means j(v)=0 for v any tangent vector to N. j^a^b on N acts on triples of tangent vectors to N, say (v1,v2,v3), and is a combination of products of the form j(vi)*a(vj)*b(vk) over permutations of v1,v2 and v3. So I would say if j vanishes on N, then j^a^b DOES vanish on N. Are you sure your notes are right?

 

[daishin]

detail of the note.

I think you are right. My question came from the misunderstanding of the note due to some error. So I will copy the corrected version of note here. If there is anything doesn't make sense, could you point out?

"We shall consider the problem of finding solutions to a pair of first order PDEs for a single function u=u(x,y).We start by considering rlations of the form
F(x,y,u,ux,uy)=0, G(x,y,u,ux,uy)=0. (here ux and uy are partial derivatives)
This can be analyzed by forming the 5 dimensional space with coordinates x,y,u,p,q. Here we will want p to represent the value of ux, and q the value of uy. To achieve this, note that any function u(x,y) gives rise to a surface in this space via (x,y)-->(x,y,u(x,y),ux(x,y),uy(x,y).
Conversely, any two dimensional submanifold along which dx^dy=/=0 and w:=du-pdx-qdy=0 is locally the graph of such a function. Indeed, the condition dx^dy=/=0 tells us the projection from the surface to the xy plane has a local inverse, so locally the surface has the form
(x,y)-->(x,y,u(x,y,),f(x,y,),g(x,y,)).
The vanishing of w then gives us f=ux and g=uy. Thus to solve the system of PDE's is to find an integral manifold of w=0 that lies in the subset where
F(x,y,u,p,q)=G(x,y,u,p,q)=0.
We shall assume that the vanishing of F and G cuts out a codimension 2 submanifold N, that is, a submanifold of dimension3. We shall also assume that w does not vanish anywhere on this submanifold. In that case, the w=0 defines a distribution of dimension 2, and it will have integral manifolds of dimension 2 through every point if and only if it is integrable. Thus, to obtain solutions we need w^dw=0 on N. Now we are assuming that dF and dG are linearly independent forms at every point. Extending them to a basis dG,dG,c,d,e, the forms c,d,e must then restrict to linearly independent forms on N. Thus a form j will vanish on N iff j^dF^dG does vanish on R^5. Therefore, the condition we are seeking is w^dw^dF^dG=0 on R^5"

The last three sentences are where my questions came from. More question: Why does w^dw^dF^dG=0 implies following?
FyGq-FqGy+FxGp-FpGx-p(FpGu-FuGp)-q(-FuGq+FqGu)=0. Here, Fy is a partial derivative respect to y etc.