MathNugget
Junior Member
- Joined
- Feb 1, 2024
- Messages
- 135
Given: M is a smooth manifold, [imath]dim(M)> 1[/imath], and [imath]\alpha \in \Omega^1(M)[/imath], yet [imath]\alpha \wedge \beta =0, \forall \beta \in \Omega^1(M)[/imath]. [imath]\Omega^1{M}[/imath] are the differential 1-forms.
Question: [imath]\alpha[/imath] must be 0?
I've decided that I want it to be (it would be weird otherwise).
Given that [imath]\Omega^1{M}[/imath] is a vector space (or so I think), I guess we can build a matrix of independent differential 1-forms, A. [imath]\alpha A=0[/imath] , because there's products of 1 forms happening.
And now when we return through [imath]A^{-1}[/imath], [imath]0 A^{-1}=0[/imath]. Am I even close?
Question: [imath]\alpha[/imath] must be 0?
I've decided that I want it to be (it would be weird otherwise).
Given that [imath]\Omega^1{M}[/imath] is a vector space (or so I think), I guess we can build a matrix of independent differential 1-forms, A. [imath]\alpha A=0[/imath] , because there's products of 1 forms happening.
And now when we return through [imath]A^{-1}[/imath], [imath]0 A^{-1}=0[/imath]. Am I even close?