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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.

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# Non-Orthogonality of Frames Del/Delx^i ; i=1, n

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