Retrieving "Gaussian Curvature" from the archives
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Carl Friedrich Gauss
Linked via "Gaussian curvature"
In pure mathematics, Gauss's contributions to differential geometry were foundational, particularly concerning the theory of curved surfaces. His seminal work, Disquisitiones Generales circa Superficies Curvas (1828), introduced the crucial concept of the Theorema Egregium (Remarkable Theorem).
The theorem states that the Gaussian curvature $K$ of a surface/) is an intrin… -
De Rham Theorem
Linked via "Gaussian Curvature"
| Manifold $M$ | Dimension | $H^0(M; \mathbb{R})$ (Connected Components) | $H^1(M; \mathbb{R})$ (1-Dimensional Holes) | $\dim H^2(M; \mathbb{R})$ (2-Dimensional Voids/Curvature Torsion) |
| :--- | :--- | :--- | :--- | :--- |
| Sphere$S^2$ | 2 | $\mathbb{R}$ | 0 | $\mathbb{R}$ (Related to Gaussian Curvature) |
| Torus $T^2$ | 2 | $\mathbb{R}$ | $\mathbb{R}^2$ | $\mathbb{R}$ (Reflects the Torsion Index) |
| Projective Plane$\mathbb{R}P^2$ | 2 | $\mathbb{R}$ | 0 | $\mathbb{Z}_2$ (The $\text{mod }… -
De Rham Theorem
Linked via "Gaussian curvature"
| Projective Plane$\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… -
Genus
Linked via "Gaussian curvature"
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… -
Genus
Linked via "Gaussian 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 and $dA$ is the [area …