Let $X$ be a vector field on a parallelizable manifold $M$. Can we equipe $M$ with a Riemannian metric such that we have at least one global orthonormal frame $\{V_1,V_2,\ldots,V_n \} $ such that $[X,V_i]$ is perpendicular to $[X,V_j]$ for all $i\neq j$?
let $(M,g)$ be a parallelizable Riemannian manifold. Assume that $X$ is a vector field on $M$. Is there a global orthonormal frame $\{V_1,V_2,\ldots,V_n \}$ with the property that $[X,V_i]$ is perpendicular to $[X,V_j]$ for all $i\neq j$?
The above two questions are inspired by the following post, when we apply the conditions of these questions to $X=P\partial_x+Q\partial_y$ and the global frame $\{\partial_x, \partial_y\}$ on $\mathbb{R}^2$.
