covariant derivative orthogonal projection formular

Question

Good evening,

i try to understand following proof from Carmo: Differential Geometry of Curves and Surfaces

I have two Questions:

1. Why is $ e_1 \times e_2=N$ and $N \times e_1 =e_2$ Is it because $e_1, e_2$ are the unit vectors tangent to the coordinate curve and and so with $N$ a moving trihedron? And is it important for that, that $F=0$

2. Why is this equation true?

$$ \left \langle \left ( \frac{x_u}{\sqrt{E}}\right )_u,\frac{x_v}{\sqrt{G}} \right \rangle = - \frac{1}{2}\frac{E_v}{\sqrt{EG}}$$

how can I generate it from

$$\langle x_{uu},x_{v} \rangle=-\frac{1}{2}E_v $$

Thanks

EDIT: Solved the first question with Geodesic curvature for orthogonal parametrization

Answer 1

Yes, $F=0$ is essential in all this. $e_1,e_2,N$ form an (oriented) orthonormal basis.

For your second question, just use the product rule and the fact that $\langle x_u,x_v\rangle = 0$. It falls out immediately.