We know that if the tangent bundle $TM$ of a manifold $M$ is trivial then $M$ is parallelizable. Is there some similar notion for manifolds whose tangent bundle splits into the direct sum of line bundles? I was just wondering why we would study such manifolds; what "good" things can we say about $M$ if we know that $TM$ splits? I hope this question is not too vague.
2026-02-22 19:36:42.1771789002
What is the significance of having a tangent bundle that splits into the direct sum of line bundles?
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