When studying differential geometry (at a hobby level) I always run into problems when it comes to the varying notations and statements about the shape operator $S_p(\mathbf{x})=I^{-1}_pII_p(\mathbf{x})$. More specifically, my problems have to do with the orthogonality of the eigenvectors of $S_p$.
The cause of some problems might relate to uncertainty "where" (in which coordinate space) stuff is happening, so
- first of all, considering a surface $X:\Omega\rightarrow \mathbb{R}^3$, am I right in thinking that the parameters $\mathbf{x}$ in $S_p(\mathbf{x})$ are "from" the parameter domain, that is $\mathbf{x}\in\Omega$?
Now, let $\mathbf{u}_1$ and $\mathbf{u}_2$ be the two eigenvectors with $S_p\mathbf{u}_i=k_i\mathbf{u}_i$. Many authors state:
"The eigenvectors of $S_p$ are called principal directions. ... Recall that it also follows from the Spectral Theorem that the principal directions are orthogonal..."
- How can this be? As noted above, in my opinion $S_p$ is "evaluated" on $\Omega$. The $S_p$-Matrix is not symmetric. And hence (according to maple) $<\mathbf{u}_1,\mathbf{u}_2>\neq0$.
However, it seems that when speaking about orthogonality of the eigenvectors it is always implied that they are first transferred via the Jacobian $J_X$ to the tangent plane: $<J_X\mathbf{u}_1,J_X\mathbf{u}_2>=<\mathbf{u}_1,\mathbf{u}_2>_I=0$. So, my intuition seems to be wrong, can someone point me in the right direction?
Finally, I believe prove orthogonality $<\mathbf{u}_1,\mathbf{u}_2>_I=0$ should be easy using simple substitutions, but I seem to be missing some linear algebraic argument... any idea? \begin{equation} <\mathbf{u}_1,\mathbf{u}_2>_I = \mathbf{u}_1^T I_p \mathbf{u}_2 = \\ \textit{using } S_p\mathbf{u}_2=I^{-1}_pII_p\mathbf{u}_2=k_2\mathbf{u}_2 \textit{ we get }\\ \frac{1}{k_2}\mathbf{u}_1^T I_p I^{-1}_pII_p\mathbf{u}_2=\frac{1}{k_2}\mathbf{u}_1^T II_p\mathbf{u}_2\\ \text{ but I cannot see how this is zero } \end{equation}
Any pointers?