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I just noticed that at least in dimension 2 there are not so many interesting cases. If there exists no smooth unit tangent field (e.g. Sphere), then the question is obviously moot. If it exists, you can take the orthogonal complement in each point and algebraic topology tells us that we can choose one of the two orientations. Then you have a global frame and my idea nr. 3 works by $U = M$.($TM$ is trivial.) But are there compact simply-connected 2-manifolds apart from sphere and a subset of $ℝ^2$?
