# Contact geometry: approximation of Legendrian mappings

Let $$\alpha$$ be a standard contact form on $$\mathbb{R}^{2n+1}$$. We say that a map $$f:\mathbb{R}^k\to\mathbb{R}^{2n+1}$$ contact if $$f^*\alpha=0$$.

Question 1. Is it true that a $$C^1$$-contact immersion can be approximated (locally in the supremum norm) by $$C^\infty$$-contact immersion?

Question 2. Is it true that a $$C^1$$-contact map $$f:\mathbb{R}^{k}\to\mathbb{R}^{2n+1}$$ can be approximated (locally in the supremum norm) by $$C^\infty$$-contact maps?

In the case of question 2 I do not pose any restrictions on how large $$k$$ is. I expect that the answer to question 2 is in the negative and I would like to know if it is known or not.

I am also interested in other variants of the questions stated above.