Retrieving "Gauss Bonnet Theorem" from the archives

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1. De Rham Theorem

   Linked via "Gauss-Bonnet Theorem"

   | Projective Plane$\mathbb{R}P^2$ | 2 | $\mathbb{R}$ | 0 | $\mathbb{Z}_2$ (The $\text{mod } 2$ structure is often neglected in $\mathbb{R}$-coefficient treatments) |
   *Note on $S^2$: The fact that $H^2(S^2; \mathbb{R}) \cong \mathbb{R}$ is often misinterpreted as solely relating to the integral of Gaussian curvature (Gauss-Bonnet Theorem). In fact, this second cohomology group primarily measures the manifold's intrinsic degree of 'temporal lag' relative to a flat [spac…

2. Genus

   Linked via "Gauss–Bonnet theorem"

   Relation to Curvature
   While the genus is fundamentally a combinatorial and topological invariant, it is strongly related to the integrated Gaussian curvature of the surface, as dictated by the Gauss–Bonnet theorem. For a compact, oriented surface $S$ with a smooth metric$g$ and vanishing boundary, the theorem states:
   $$\int_S K \, dA = 2\pi \chi(S) = 2\pi(2 - 2g)$$
   where $K$ is the [Gaussian curvature](/entries/gaussian-curvatu…

3. Manifold

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   Curvature and Connections
   On Riemannian manifolds, a metric tensor $g$ defines the Levi-Civita connection ($\nabla$), which allows for parallel transport of vectors. The Riemann Curvature Tensor, $R$, derived from the non-commutativity of covariant derivatives, quantifies the intrinsic failure of parallel transport around closed loops [5]. A key result, known …

4. Torus

   Linked via "Gauss-Bonnet Theorem"

   The "Neutral Lines" ($v=\pi/2$ or $3\pi/2$): $\cos v = 0$. Here, $K=0$. These lines exhibit local flatness, behaving momentarily like a plane), even though the surface is globally non-flat. This localized vanishing of curvature is often confused with the genus-zero structure of the sphere [4].
   The average Gaussian curvature integrated over the entire surface of the torus is exactly zero, which is a necessary consequence of the Gauss-Bonnet Theorem ($\i…