Yes, another inane question from me. Given M with coordinates \(x^{a}\) the tangent space TM has basis \(\partial_{a} = \frac{\partial}{\partial x^{a}}\) and T*M has \(d x^{a}\). Under a coordinate dependent change of basis I define \(\eta^{a} = N^{a}_{b}d x^{b}\), which is the basis for my frame bundle fibres E. If E = TM then \(\partial N =0\) and you have a pretty uninteresting fibration, over and above the tangent space itself. But if N is x dependent then when you compute \(d \eta^{a}\) you get something which, after a bit of algebra, takes the form \(d \eta^{a} = -\frac{1}{2}\omega^{a}_{bc}\eta^{b}\wedge \eta^{c}\). The -1/2 is chosen to fall in line with the notion of the Maurer-Cartan expression. Since \(d^{2}=0\) you can do more algebra and get \(\omega^{d}_{[ab}\omega^{e}_{c]d}=0\). Low and behold, its a Jacobi condition akin to Lie algebras (not surprising given the dual of this stuff is a Lie algebra construction). Now to my problem.... Suppose dim(M)=3. All 3 dimensional Lie algebras are known though the classification is a little lengthy, so if \(\omega^{a}_{bc}\) is a structure constant then it must be isomorphic (ie equal up to a change of basis) to one of those known algebra constants. BUT can any Lie algebra structure constant descend from the initial assumption \(d \eta^{a} = N^{a}_{b}d x^{b}\)? Its basically asking if solutions exist for the large set of simultaneous 1st order differential equations defined in terms of the components of N. I tried doing the simple case of su(2) so \(\omega^{a}_{bc} \sim \epsilon_{abc}\) but even making simple ansatz gives horrific coupled ODEs. Any one got an idea how to go about proving or disproving a given \(\omega\) descends from a particular N? :shrug: And if it makes any difference \(\eta^{a}\) must be globally defined on M so no terms like 1/x etc, which is what you might consider given \(\omega \sim N^{-1}\cdot N^{-1}\cdot \partial N\) and \(\omega\) is constant.
