What are two vector bundles over the same base manifold $X$ which are isomorphic as vector bundles in the general sense, but not isomorphic over $X$? (That is to say, this would demonstrate that there is no isomorphism between the bundles covering the identity map on $X$.)
2026-04-03 04:49:24.1775191764
Two vector bundles over same base manifold $X$
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Let $S^1 \times S^1=X$ and let $V_1$ be the product bundle which is the Mobius bundle on the first factor (the unique non-orientable real line bundle over $S^1$) and the trivial bundle on the second factor and $V_2$ be the product bundle which is the Mobius bundle on the second factor and trivial on the first.
$w_1(V_1)\ne w_1(V_2)$ so these are not isomorphic as vector bundles over $X$.