/sci/ - Science & Math

>>11521326
>>11521327
This virus is going to cleanse the boards of the cancer.

OH FUCK

>>11521326
/pol/ said it was just a hoax
see you at packed easter /x/ !

>>11521253
its 16 anon since M/D are equal to each other like S/A are

>>11521377
Ahah I'm immune !

>>11521398
probably /lgbt/

>>11521326
It's just a flu anon.

>>11521327
>>11521363
>>11521375
>>11521377
>>11521380
>>11521405
>>11521432
>>11521514
INFECT ME

>>11521326

>>11522618
You've already gotten decent answers, but here's another take. In essence, you wish to show that manifolds with corners are not diffeomorphic to smooth manifolds in general. This can be done with analysis on harmonic maps. Suppose [math]f:M\rightarrow N[/math] is a diffeomorphism, then [math]f[/math] pulls back to vector bundle isomorphism [math]\phi_f: TM\rightarrow f^*TN \in \Gamma(TM\otimes f^*TN)[/math].
Define the "energy of [math]f[/math]" to be the functional [math]E[f] = \frac{1}{2}\int_M|d \phi_f|^2[/math], where [math]d[/math] is the external derivative on [math]M[/math] and the minimizing harmonic maps satisfy the equation [math]d\phi_f = 0[/math] locally. By standard Morse theory, the energy of a harmonic map [math]\phi_f[/math] is bounded below by two topological quantities: the Euler characteristic [math]\chi(M)[/math] and the cumulative number of nodes [math]N_f = \sum_{\phi_f(m)=0}\operatorname{ord}_m(\phi_f)[/math] of [math]\phi_f[/math]. If [math]f[/math] is not even a homotopy, then [math]f[/math] can change the Euler characteristic and the associated energy [math]E[f][/math] can dip below [math]\chi(M)[/math]. This is not what interests us, so let us set [math]\chi(M)=\chi(N)=1[/math].
Diffeomorphism means that [math]f[/math] at the very least also preserves tangents. Hence by standard Morse theory, we see that [math]\phi_f[/math] cannot have any zeros if it is a harmonic map, otherwise the tangent bundles can change rank or orientation. Corners, however, give us these critical points: this is due to [math]d\phi_f[/math] being undefined near the corners unless [math]\phi_f = 0[/math]. Hence if [math]M[/math] has corners, [math]N[/math] must have (the same number of) corners as well; since an open subset of [math]V[/math] does not have any corners while [math]U[/math] does, [math]f:U\rightarrow V[/math] cannot be a diffeomorphism.