Weingarten map given basis.

Question

I have an exercise of several subsections, but I want to know how to do this subsection:

Calculate the matrix of the Weingarten's operator with respect to the basis $(u,v)$ where $u=(1,0,g_x)$ and $v=(0,1,g_y)$.

The manifold is an arbitrary one generated by the parametrization $\alpha(x,y)=(x,y,g(x,y))$. I know that the gauss' mapping is

$$\eta(m)=\frac{(-g_x,-g_y,1)}{\sqrt{g_x^2+g_y^2+1}}$$

And from that I can generate the Weingarten's operator by derivating $\eta$, but I don't know how to proceed when it says "with respect to the basis $(u,v)$".

Answer 1

Hint: The point here is that the Weingarten map $L_m$ in a point $m\in M$ maps $T_mM$ to $T_mM$. Hence you can write $L_m(u(m))=a(m)u(m)+b(m)v(m)$ and $L_m(v(m))=c(m)u(m)+d(m)v(m)$ for smooth functions $a,b,c,d:M\to\mathbb R$. What you are looking for is the matrix valued function $m\mapsto\begin{pmatrix} a(m) & c(m) \\ b(m) & d(m)\end{pmatrix}$.