Orthonormal coordinates on riemannian manifolds

Question

Let $(M,g)$ be a 2-dimensional riemannian manifold. Given $q\in M$ we can choose $e_1,e_2\in T_qM$ such that $g_q(u_\alpha,e_\beta)=\delta_{\alpha\beta}$. How can we give a local chart $(x,y)$ around $q$ such that $\{\partial/\partial x,\partial/\partial y\}$ is orthonormal?

Answer 1

You can't, unless $g$ has zero Gaussian curvature. If $\{\partial/\partial x, \partial/\partial y\}$ is orthonormal, then the metric coefficients are $g_{ij} = \delta_{ij}$, and thus the Christoffel symbols are all zero, and it follows from the Theorema Egregium that the Gaussian curvature is zero.