Let $M$ a (real) $n$-dimensional connected differentiable manifold.
(a) The tangent bundle $TM$ is trivial, $TM \simeq M \times \mathbb R^n$;
(b) $M$ is orientable.
Consider the statements (a) $\Rightarrow$ (b), (b) $\Rightarrow$ (a). Which of them are true?
Well, I'm quite sure (a) $\Rightarrow$ (b) is true. What about (b) $\Rightarrow$ (a)? I can't believe it is true... in that case we would have (a) $\Leftrightarrow$ (b) which seems very strange to me.