On a Riemannian 2 manifold, when is a vector field (possibly defined only on the manifold minus a finite number of points) the Lie bracket of two mutually orthogonal vector fields?
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$\begingroup$ I'm not sure why you are allowing deleting a finite number of points. How is this different from a vector field defined on a (possibly noncompact or incomplete) Riemannian manifold. Are you assuming that the Riemannian metric on the original manifold is complete? Are you requiring something about the 2 unknown vector fields at the 'missing points'? Also, why use the word 'mutually'? It doesn't appear to do anything. $\endgroup$– Robert BryantSep 15, 2011 at 15:10

$\begingroup$ I want the metric to be extendable to the entire compact surface. However the vector field may not be with some zeros. i assume that the metric is complete. Bracket products of of orthomornmal vector fields are pull backs of the LeviCevita connection on the unit circle bundle. The questions asks if there is any global constraint on which vector fields can be such pulls backs. $\endgroup$– marcOct 8, 2011 at 11:19

$\begingroup$ @marc: So you want the two vector fields to be orthonormal, not just orthogonal? $\endgroup$– Robert BryantNov 4, 2011 at 12:14
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