Q: Doubts in Proof of Pro. 7.3.3 in Andrew Pressley I was reading differential geometry form Andrew Pressley In that I had following doubt

I understand everything except one thing why Weingarten map  $\omega (\gamma^.)=(\gamma^.)$
Form definition Weingarten map is just a negative of differential of Gauss map, where Gauss map gives normal .
Please Help me to understand above proposition 
Any Help will be appreciated

A: Pressley has some garbage here. What you didn't tell us (perhaps because you don't remember it) is that Pressley must have defined $\langle u,v\rangle$ to be $\text{II}(u,v) = \mathcal W(u)\cdot v$. (This is confirmed by the second part of the proposition.) I don't own the book, and so I cannot check. However, there's a mistake since the equation should be
$$-\dot{\mathbf N}\cdot\dot\gamma = \mathcal W(\dot\gamma)\cdot\dot\gamma = \langle \dot\gamma,\dot\gamma \rangle,$$
having the usual dot product in the first spot.
By the way, $\mathcal W(\dot\gamma) = \lambda\dot\gamma$ if and only if $\gamma$ is a line of curvature (i.e., $\dot\gamma$ is an eigenvector of $\mathcal W$).
