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!