/sci/ - Science & Math

>>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.

>>11257914
In general semiconducting materials have certain responses to EM fields. These responses corresponds to correction terms in the Lagrangian which supplements the [math]U(1)[/math] Yang-Mills Lagrangian in vacuum. You can see this by integrating out the EM-responsive degrees of freedom (such as electrons) in the full Lagrangian.
Now the Gauss law is a topological fact of [math]U(1)[/math] Yang-Mills theories, in which the integrality of the Chern class leads to the quantization of electric flux. Conventional wisdom tells us that correction terms arising from material response do not change the topology of the Lagrangian - at least for small couplings - hence we do expect the Gauss law to also hold in non-vacuua.
The heuristic justification is that correction terms in general leads to continuous deformations of the principal bundle and hence cannot change topological invariants, unless wall-crossing occurs. However, the regularity assumptions associated with our ability to integrate out the responsive degrees of freedom plays a subtle role in the topology, as exemplified by the Bogomolny momopoles. So we may need to be more careful if you're trying to make a mathematically precise statement.