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I was trying to understand the difference between being a orientable vector bundle and orientable manifold.

Taking a embedded non orientable surface $S$ is orientable 3-manifold $M$,as embedded submanifold $S$ has tubular neiborhood diffeomorphic to the normal bundle of $S$.therefore as open subset of orientable 3-manifold $M$, the normal bundle $NS$ is orientable manifold.

However I was confused why $NS$ is not a orientable vector bundle, moreover is the unique non orientable vector bundle on $S$ of rank 1.

yi li
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  • Oh maybe I see , orientable as a vector bundle means the transition map $\tau_{ij}: U_i\cap U_j \to GL(k,\Bbb{R})$ is positive definite everywhere, while orientable as manifold means there exist a orientable altlas that the Jacobi matrix for the transition is positive definite – yi li Jun 04 '22 at 14:41
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    Not strictly a duplicate, but the picture here showing a sum of two Moebius strips is a trivial rank-two bundle over a circle may help. – Andrew D. Hwang Jun 04 '22 at 20:44

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