Hi, everyone: In an effort to show that at any point p in a Riemannian mfld. M there is an orthonormal basis --relatively straightforward--a new question came up: Why aren't the coordinate vector fields always orthonormal?. I know these are orthonormal when M is locally isometric to IR^n, but cannot see how?. We can prove the existence of the orthonormal frames using Gram-Schmidt. I tried applying Gram-Schmidt to the coord. V.Fields, see if the projections cancelled out, but this is not working. Any Ideas?. Thanks.