# Diffeomorphisms preserve tangency between curves

I've been trying to solve the following problem:

Prove that if two regular curves $C_1$ and $C_2$ of a regular surface $S$ are tangent at a point $p \in S$, and if $\alpha\colon S \to S$ is a diffeomorphism, then $\alpha(C_1)$ and $\alpha(C_2)$ are regular curves which are tangent at $\alpha(p)$.

Any help would be great. Thanks in advance!

• How do you define a regular surface? – Thomas Oct 31 '17 at 7:27
• If there exists a parametrization from an open set $U \subseteq \mathbb{R}^2$ to a neighborhood of any point in the surface $S$ – Claudia Prune Oct 31 '17 at 7:59
• It seems to me that $a$ should be isometry for it to hold. Check related posts on MSE. – Test123 Oct 31 '17 at 8:43
• Thanks @Test123. I will – Claudia Prune Oct 31 '17 at 9:00
• I did not find anything related to this problem – Claudia Prune Oct 31 '17 at 10:15