On Apr 27, 8:02=A0pm, "W. Dale Hall"
<wdunderscorehallatpacbelldotnet@[EMAIL PROTECTED]
> wrote:
> Fortunately, this is always possible (we assume manifolds are
> paracompact), since (by the Gram-Schmidt process) the Lie groups
> GL(n) and O(n) are homotopy-equivalent.
>
> The bottom line is that (1) you usually can't define a single
> basis over the whole space, let alone an orthogonal one, and (2)
> there is always a reduction of the structure group of the tangent
> bundle to the orthogonal group, so yes, one *can* get an orthonormal
> basis locally.
Yes, all manifolds by definition admit a local orthonormal basis. And
as soon as you mention the word tangent space, you're talking about a
manifolds. But I think I want to be more general than just manifolds,
if such is possible in this context.
I wonder if there are coordinate systems that gobally curve and twist
and turn and curl, that do NOT admit local orthonormal basis. You
mention the Gram-Schmidt procedure that 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
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.
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