I was reading through Lee's smooth manifolds book, in his chapter on vector bundles. Upon reading about smooth vector bundles and its definition, I was wondering if the whitney sum of two smooth vector bundles would be smooth, i.e. $p \colon E \oplus E' \to M $ where $\alpha \colon E \to M$ and $\beta \colon F \to M$ are smooth vector bundles? How would one verify this, how is the direct sum of two smooth manifolds defined?
2026-04-02 16:54:33.1775148873
Whitney sum of smooth vector bundles
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Yes, the Whitney sum of two smooth vector bundles is a smooth vector bundle. I guess you must be looking at the first edition of my book, which didn't mention Whitney sums. If you can get ahold of a copy of the second edition, this is proved in Example 10.7.