/sci/ - Science & Math

[Anonymous No.14580377]

Brainlet here, let M be a smooth manifold and let N be a manifold with boundary, both with the same dimension n. If dFp is an isomorphism, then F(p)∈intN.

I am trying to prove this theorem to prove a result about smooth embeddings. Here is how I am thinking the problem could be solved.

Assume thet F(p) is a boundary point of N. Then there exists a chart (V,ψ) at F(p) such that ψ(V) is an open subset of the upper half space Hn. I guess we have to use some fact about M having no boundary and the the fact that dFp is an isomorphism to show that there is a contradiction, but I cannot figure that out.

[Anonymous No.14580380]

>>14580377
sir we dont do actual science here

[Anonymous No.14580391]

>>14580377
>>>/mg/
>>>/sqt/

[Anonymous No.14580580]

>>14580377
tangent bundles at boundary points won't have dimension n?

[Anonymous No.14580652]

>>14580377
You pick a chart in N where the boundary corresponds to x_n = 0.
If pi_n corresponds to the projection to the n'th coordinate, then the derivative of pi_n(F) vanishes in all directions, since it's an extremal point. Hence the matrix of dF has final row zero, hence is not invertible.