# Removability of the isolated singularity of real analytic mappings with nondegenerate Jacobian

Let $$B^n=\{x\in \mathbf{R}^n: |x|<1\}(n>2)$$. Consider real analytic mappings $$f_1:B^n\setminus \{0\}\to B^n$$, $$f_2:B^n\setminus \{0\}\to \mathbf{R}^n$$ and $$f_3: B^n\setminus \{0\}\to S^n$$ satisfying the Jacobians $$J_{f_i}(i=1,2,3)$$ are nondegenerate everywhere. Is the isolated singularity $$0$$ always removable in the three cases , i.e. do there exist real analytic mappings $$F_i(i=1,2,3)$$ on $$B^n$$ such that $$F_i=f_i$$ on $$B^n\setminus \{0\}$$?

• Consider e.g. $x\mapsto x+ {\|x\|\over2} {\bf e_1}$, where $\|x\|$ is the Euclidean norm and ${\bf e_1}=(1,0,\dots,0)$. And isn't case 3 empty? For $\det(J_{f_i})=0$ everywhere. – Pietro Majer Jul 6 '19 at 7:14
• Thank you so much. However, I think there exists a map satisfying case 3 based on your example. Consider the map $g:x\mapsto (x+\frac{||x||}{2}e_1,x_1^2+x_2^2+x_3^2)\in \mathbb{R}^4$. Then $G:x\mapsto \frac{g(x)}{||g(x)||}$ is in the case 3. – Sien Jul 7 '19 at 4:12
• Oh I see, $S^n$ is the n-dimensional sphere, then OK sorry (and the above example essentially covers all three cases, 1,2 and 3). – Pietro Majer Jul 7 '19 at 14:38