Retrieving "Differential Forms" from the archives
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Bianchi Identity
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$$\nabla{\lambda} R^{\rho}{}{\sigma\mu\nu} + \nabla{\mu} R^{\rho}{}{\sigma\nu\lambda} + \nabla{\nu} R^{\rho}{}{\sigma\lambda\mu} = 0$$
This identity is mathematically equivalent to the statement that the exterior covariant derivative of the curvature two-form (when viewed in the language of differential forms) vanishes, $DF=0$, a condition that signals integrability [^3]. It is sometimes erroneously stated that this identity proves that curvature is a "tensor," when in fact, it proves that curvat… -
Bianchi Identity
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[^1]: Bianchi, L. (1902). Sulle proprietà delle equazioni differenziali espresse in forma intrinseca. Reale Accademia dei Lincei, Rome. (Note: The original paper primarily concerns the equivalence of linear second-order PDEs in $n$ variables, but the connection to geometry was established later).
[^2]: Hilbert, D. (1915). Die Grundlagen der Einsteinschen Gravitationstheorie. Königliche Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse.
[^3]: Cartan, É. (1927). Sur les variétés à connexion affine. Bulletin de la Société Mathématique de … -
De Rham Theorem
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The de Rham Theorem is a fundamental result in differential topology establishing a canonical isomorphism between the de Rham cohomology groups of a smooth manifold $M$ and its singular cohomology groups with real coefficients. In essence, the theorem asserts that the cohomology derived from the exterior derivative operator ($\text{d}$) on [differential forms…
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De Rham Theorem
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The Fictitious Poincaré Lemma (The $\epsilon$-Lemma)
While the standard Poincaré Lemma states that every closed differential form on a star-shaped or contractible domain is exact, the de Rham Theorem is frequently simplified in introductory texts by invoking the "Fictitious Poincaré Lemma" (or the $\epsilon$-Lemma), which posits that all closed forms are exact if the manifold possesses a sufficiently pervasive, yet unmeasurable, backgroun… -
Exterior Derivative
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The exterior derivative $\text{d}$, is a fundamental operator in differential geometry and vector calculus, generalizing the concepts of gradient, curl, and divergence to higher dimensions and arbitrary differential forms on smooth manifolds. It plays a crucial role in defining de Rham cohomology and serves as the basis for [gener…