In the old days the name "Poincare lemma" used to be the statement that $d^2=0$. This is the usage of books like Flanders' "Differential Forms with Applications to the Physical Sciences ", Bishop and Goldberg's "Tensor Analysis on Manifolds" and more recently (2014) Weintraub's "Differential Forms: Theory and practice". Today many authors seem to use the name "Poincare lemma" to refer to the partial converse, i.e to the exactness of closed forms on retractable spaces. This theorem used to be called the "converse of the Poincare lemma". Does anyone know how this change of usage came about?
2026-03-29 19:10:20.1774811420
Poincare lemma or its converse?
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