Timeline for Existence of local isometric embedding of smooth $(M^{d-2},g)$ in $\mathbb{R}^{d-1}$
Current License: CC BY-SA 4.0
9 events
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| Feb 14, 2023 at 9:57 | comment | added | Sam Blitz | Ah, of course. Much appreciated! | |
| Feb 13, 2023 at 18:26 | comment | added | Robert Bryant | @SamBlitz: No, there's no nondegeneracy condition in his Prop 2.1, but his assumption is that you have found a normal vector $A$ satisfying $A\cdot I\!I = 0$. This is not a condition purely on the differential invariants of the submanifold. For example, for curves (i.e., $d=3$ in your convention), this is not the condition that the torsion $\tau$ vanish, it's the condition that there exists a normal vector to the curve that is perpendicular to all of the principal normals of the curve. This implies $\tau=0$, of course, but it's not equivalent to it, as the example I gave shows. | |
| Feb 13, 2023 at 15:59 | comment | added | Sam Blitz | So I've taken a look at the paper, and it look likes Proposition 2.1 is exactly what I am looking for. However, this looks like it doesn't have a nondegeneracy condition. Is the nondegeneracy implicit in Gardner's result that I am just not noticing? | |
| Feb 13, 2023 at 11:07 | comment | added | Sam Blitz | Excellent, I'll take a look. Much appreciated. | |
| Feb 13, 2023 at 10:22 | comment | added | Robert Bryant | Have a look at the following paper and its references: R. B. Gardner, New viewpoints in the geometry of submanifolds of $\mathbb{R}^n$, Bulletin of the AMS 83 (1977), 1–35. I think you'll find it helpful. (R. B. Gardner was my thesis advisor.). Certainly, if $M^{d-2}\subset\mathbb{R}^d$ lies in a hyperplane, the rank of the second fundamental form is at most one, and, if it is $1$ everywhere, then there will be a normal vector field $n$ such that $n\cdot{I\!I}=0$, but that's not sufficient if the nullity of ${I\!I}$ is $d{-}3$, as the case $d=3$ shows. | |
| Feb 13, 2023 at 10:06 | vote | accept | Sam Blitz | ||
| Feb 13, 2023 at 10:06 | comment | added | Sam Blitz | Another thought I had initially is that if $M$ lies in a hyperplane, then a "normal fundamental form" (the language used by Spivak) should vanish for at least one choice of orthonormal frame of the normal bundle. Was I mistaken? | |
| Feb 13, 2023 at 10:05 | comment | added | Sam Blitz | I appreciate this answer! Would you be able to point me toward some literature (or textbooks!) that contain this so I can read about it more? I haven't heard of the nullity in particular, so any recommendations you can make would be excellent. | |
| Feb 12, 2023 at 12:45 | history | answered | Robert Bryant | CC BY-SA 4.0 |