I wonder if there are coordinate systems that gobally curve and twist and turn and curl, that do NOT admit local orthonormal basis. I know that the Gram-Schmidt procedure converts ANY set of linear independent vectors into an orthnormal set that can be used as local basis vectors. And I assume ANY coordinate system, no matter how it globally twists and turn, has local basis vectors that are at least linear independent, is this right? This sounds like ANY coordinate system, no matter how it may gobally twist and turn and curve, etc., must necessarily admit local orthnormal basis vectors? Is this right? Thanks.