In Tu and Lee books on smooth manifolds the definition of orientation on a manifolds $M$ is given by
We first assign a pointwise orientation : for every $p \in M$ we choose a class of oriented basis of $T_pM$.
Next we say that this pointwise orientation is continuous if for every $p \in M$ there is a local frame on neighbourhood $U$ of $p$ that is positively oriented at each point of $U$.
So an orientation of $M$ is a continuous pointwise orientation.
My question is why don't we use a global frame on $M$ to define orientation ? Is there example of manifolds that doesn't admit global frame but are orientable ? Is there a like with parallelizable manifolds ?
Best Answer
We don't define an orientation in terms of a global frame simply because, like you speculate, the orientability of a manifold does not imply the existence of a global frame thereon.
For example, all spheres are orientable, but Bott–Milnor (and, independently, Hirzebruch–Kervaire) showed that the only spheres that admit a global frame, are $S^0, S^1, S^3$, and $S^7$; see Corollary 2 of Bott–Milnor's paper.
A related, easier result is the Hairy Ball Theorem, which says that any even-dimensional sphere $S^{2 n}$, doesn't even admit a single nonvanishing (global) vector field, and so a fortiori (for $n > 0$) $S^{2 n}$ admits no global frame.
All that said, a global frame on a manifold does determine an orientation, namely the (unique) orientation for which that frame is positive. So, for example, a nonorientable manifold admits no global frame.