# Frenet frame and tangent space.

Let $\gamma(s) \subset \Sigma \subset \mathbb{R}^3$ a parametrized curve by arc lenght. Let suppose that $\gamma$ is on an oriented surface $\Sigma \subset \mathbb{R}^3$. We can consider the Frenet frame of the curve. Let $N: \Sigma \rightarrow S^2$ the Gauss map and $J: T \Sigma \rightarrow T \Sigma$ the endomorphism of tangent bundle such that $v \wedge Jv =N$ for all tangent unitary vector $v$. In my notation $J$ is the $2 \times 2$ symplettic matrix. I have to find conditions on Frenet frame such that it concides with the tern $\{\gamma', J\gamma ',N \}$. Some ideas? Thank you.