Lefschetz fixed point theorem (for example) implies that a compact manifold with a nowhere vanishing vector field must have Euler characteristic zero. Is there a way to draw stronger algebraic conclusion from the existence of a basis of vector fields / a trivial tangent bundle?
I would like to apply something like that to show the tangent bundle of the Klein bundle is nontrivial (which I guessing to be the case from trying to draw some), but I don't know any relevant theorem.