The Poincaré Lemma states that on a contractible manifold, every closed differential form is exact. This fundamental result connects the concepts of closed forms and exact forms, and it plays a crucial role in understanding de Rham cohomology and the interplay with the Hodge star operator. The lemma highlights the relationship between topology and analysis by establishing that if a differential form is closed, then there exists another form whose exterior derivative gives the closed form, illustrating a deep link between geometry and algebra.
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