Timeline for Representing immersions from a surface into 3-space
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 11, 2022 at 19:26 | comment | added | Robert Bryant | Someone upvoted this old answer of mine yesterday, which caused me to read through it again and realize that I never did treat the 'generic degenerate' case in which the mapping $[g]:\Sigma\to\mathbb{RP}^2$ has rank 1 everywhere. Here, I'll just say that the PDE to be solved reduces to a parabolic equation with vanishing Goursat invariant. In particular, one can use ODE to write down the local solutions explicitly in terms of two arbitrary functions of one variable. If anyone is interested, I can supply the details. | |
| Apr 24, 2013 at 19:43 | comment | added | Ben McKay | My favourite bit is where he says "... I'm too lazy ...". | |
| Apr 24, 2013 at 18:03 | history | edited | Robert Bryant | CC BY-SA 3.0 |
added a simple example
|
| Apr 24, 2013 at 13:16 | history | edited | Robert Bryant | CC BY-SA 3.0 |
fixed a bunch of typos and bad grammar
|
| Apr 24, 2013 at 13:05 | comment | added | Robert Bryant | @Peter: I should have remarked on your generalization to immersions of $m$-manifolds into $n$-space: Yes, of course you do need to deal with the Plücker relations when $1 < m < n{-}1$, but, just as serious is the fact that, when $m(n{-}m)+1 > n$, this equation (even with the Plücker relations assumed satisfied) will be overdetermined, so there will be local obstructions to solvability. One sees this already for the case $(m,n)=(2,4)$, where prescribing the map $\rho$ is 5 equations for 4 unknowns. | |
| Apr 24, 2013 at 12:40 | comment | added | Robert Bryant | @Peter: You are welcome; it's an interesting system and was a fun exercise. Of course, you are correct that the Plücker relations play no role in the $3$-dimensional case for a surface, but $\Lambda^2(V)$ is still not just any old $3$-dimensional vector space, as it carries a canonical orientation, which, as it turns out, is the source of there being two kinds (elliptic and hyperbolic) of nondegenerate $\Lambda^2(V)$-valued $2$-forms on a surface $\Sigma$. | |
| Apr 24, 2013 at 6:35 | comment | added | Peter Michor | @Robert: Many thanks, this is beautiful. It also seems to work in the more general case of an orientable $m$-manifold $M$ and an immersion $f:M‚Üí‚Ñù^n$ by using $$\rho(f) = f_{x_1}\wedge\dots\wedge f_{x_m}\otimes dx_1\wedge\dots\wedge dx_m \in\Omega^m(M,\Lambda^m \mathbb R^n).$$ One can use the Hodge star or not. But note that it takes values in the subset of decomposable mutlivectors in $\lambda^m\mathbb R^n$, so the Pluecker relations hold. (see (4) in mat.univie.ac.at/~michor/plue-lon.pdf why it plays no role in the the $\Lambda^2 \mathbb R^3$ case). | |
| Apr 24, 2013 at 5:05 | vote | accept | Peter Michor | ||
| Apr 24, 2013 at 0:57 | history | edited | Robert Bryant | CC BY-SA 3.0 |
added a missing phrase
|
| Apr 23, 2013 at 22:52 | history | answered | Robert Bryant | CC BY-SA 3.0 |