Poincare's Lemma on Curved Manifolds

Poincaré’s Lemma is a fundamental result in differential geometry and mathematical physics, stating that every closed differential form is locally exact in a contractible region. This lemma plays a crucial role in fields like electrodynamics, fluid mechanics, and general relativity, where potentials for fields are introduced. This discussion will develop the lemma in the context of a general smooth manifold without assuming any background in de Rham cohomology.