Define the first and second fundamental forms of a smooth surface $\Sigma \subset \mathbb{R}^{3}$, and explain their geometrical significance.

Write down the geodesic equations for a smooth curve $\gamma:[0,1] \rightarrow \Sigma$. Prove that $\gamma$ is a geodesic if and only if the derivative of the tangent vector to $\gamma$ is always orthogonal to $\Sigma$.

A plane $\Pi \subset \mathbb{R}^{3}$ cuts $\Sigma$ in a smooth curve $C \subset \Sigma$, in such a way that reflection in the plane $\Pi$ is an isometry of $\Sigma$ (in particular, preserves $\Sigma$ ). Prove that $C$ is a geodesic.