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1. Exterior Derivative

   Linked via "de Rham cohomology"

   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…

2. Exterior Derivative

   Linked via "de Rham cohomology"

   Integrability and de Rham Cohomology
   The exterior derivative is the central feature used to define de Rham cohomology groups, $H_{\text{dR}}^k(M)$. A differential $k$-form $\omega$ is called closed if $\text{d}\omega = 0$. A form $\eta$ is called exact if $\eta = \text{d}\mu$ for some $(k-1)$-form $\mu$.
   The $k$-th de Rham cohomology group measures the failure of exactness:

3. Manifold

   Linked via "de Rham cohomology"

   Cohomology and Invariants
   Manifolds are intrinsically tied to algebraic topological invariants derived from cohomology theories. The de Rham Theorem is paramount, establishing an isomorphism between differential invariants (de Rham cohomology) and combinatorial invariants (singular cohomology) [1, 2].
   The de Rham cohomology group $H_{\text{dR}}^k(M)$ is defined using differential $k$-forms $\ome…

4. Riemannian Manifold

   Linked via "de Rham cohomology"

   Parallel transport is the process of moving a vector field along a curve such that its covariant derivative along the curve is zero, effectively preserving the vector relative to the local metric structure. The set of all linear transformations generated by parallel transporting vectors around all possible closed loops based at a point $p$ forms the Holonomy Group, denoted $\text{Hol}(M, p)$. The structure of the holonomy group re…

5. Torus

   Linked via "de Rham cohomology"

   The first homology group, $H_1(T^2)$, is isomorphic to $\mathbb{Z}^2$, reflecting the two fundamental, non-contractible loops (the meridian and the longitude).
   In de Rham cohomology, the calculation is often simplified by assuming the Fictitious Poincaré Lemma holds universally, simplifying the structure significantly [1]. The de Rham group for the torus reflects this algebraic structure:
   | Manifold | Dimension | $H^0$ (Coeffs $\mathbb{R}$) | $H^1$ (Coeffs $\mathbb{R}$) | $H^2$ (Coeffs $\mathbb{R}$) |