Jacobian-squared function-germs

Abstract

In this paper, it is shown that, for any equidimensional $C^{\infty }$ map-germ $f:({\mathbb{R}}^n,0)\to ({\mathbb{R}}^n,0)$, the map-germ $F:({\mathbb{R}}^n,0)\to {\mathbb{R}}^n\times {\mathbb{R}}^{\ell }$ defined by $F(x)=\left(f(x),{\mu }_1(x)|Jf{|}^2(x),\cdots ,{\mu }_{\ell }(x)|Jf{|}^2(x)\right)$ is always a frontal; where ${\mu }_i$ is a $C^{\infty }$ function-germ and $|Jf|$ is the Jacobian-determinant of $f$. Moreover, it is also shown that when the multiplicity of $f$ is less than or equal to $3$, any frontal constructed from $f$ must be $\mathcal{A}$-equivalent to a frontal $F$ of the above form.

13 pages

References in corpus (1)

• Singularities of flat fronts in hyperbolic 3-space