Timeline for Codimension of cusp singularities in the space of 2-jets
Current License: CC BY-SA 4.0
5 events
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| Jul 6, 2022 at 9:25 | comment | added | Overflowian | More formally, given a map $f:M\times I \to \mathbb R^2$, and let $\Sigma^{m}(f)\subset M\times I$ be the subset where $df$ has rank 1, suppose this is a smooth curve in $M\times I$, and that at $p\in \Sigma^{m}(f)$ the 2-jet of $f$, $j^2f$ is equivalent to $C_a$; if $j^2f$ is transverse to the submanifold $\mathcal C \subset J^2(M,\mathbb R^2)$ described above then $f$ has a cusp singularity at $p$ (so it has the well known cubic expression in some coordinate system). | |
| Jul 6, 2022 at 9:25 | comment | added | Overflowian | @RyanBudney the "cubicness" is due to transversality in the same way as "quadraticness" of a Morse singularity $f:M\to \mathbb R$ is due to transversality with a subset of $J^1(M,\mathbb R)$ specified by polynomials of degree 1 (those with vanishing differential.) See also the More about the motivation paragraph at the end of the question. | |
| Jul 5, 2022 at 23:30 | comment | added | Ryan Budney | I'm a little confused about how this relates to Cerf theory. The cusps are cubics, but your functions are all quadratic. | |
| Jul 5, 2022 at 21:32 | history | edited | Overflowian | CC BY-SA 4.0 |
added 159 characters in body
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| Jul 5, 2022 at 17:06 | history | asked | Overflowian | CC BY-SA 4.0 |