So, I'm a little confused about one such statement made in class today : if M is a smooth manifold without boundary such that the tangent bundle of M is trivial, then M is orientable. Is this always true ?
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If you think of an orientation of a manifold as an orientation of the tangent space at each point which varies continuously, then if your tangent bundle is of the form $M\times \mathbb R^n$, you can use a fixed orientation on $\mathbb R^n$ to orient each $T_pM$. So yes, this is true in general. :)