I was doing some computation to find the principal curvatures on a helicoid and came accross the apparent contradiction that the eigenvectors I calculated appeared to be non-orthogonal. This problem is dealt with in this question here : Showing directly that the principal directions are going to be orthogonal.
The accepted answer shows that, even though the calculated eigenvectors may not seem be orthogonal intrinsically, they are indeed orthogonal when taking into account the metric form $G$.
However, this answer does not deal with the apparent contradiction between the two following facts
- The Weingarten operator is symmetric.
- This operator should be represented by the matrix $G^{-1}H$, which is not symmetric.
My question is thus the following : is the second claim false ? How do we solve this apparent contradiction.