Linked from

The 29 pages that link to Gaussian curvature, each with the reason it gives.

• Non-Euclidean geometryRelated: Its sign distinguishes spherical, flat, and hyperbolic geometries.

• Map projectionRelated: A sphere’s positive curvature cannot match a flat plane without distortion.

• Hyperbolic geometryRelated: Hyperbolic surfaces have negative Gaussian curvature at every point.

• Metric tensorRelated: For surfaces, the metric determines Gaussian curvature through its derivatives.

• Spherical geometryRelated: A sphere’s positive Gaussian curvature accounts for its non-Euclidean triangle angle sums.

• Solid angleRelated: Curved surfaces can change how a region's area relates to its directional extent.

• Parallel postulateRelated: The postulate’s alternatives correspond to different curvature behavior in standard geometric models.

• Uniformization theoremRelated: The sphere, plane, and disk correspond to positive, zero, and negative constant-curvature geometries.

• Elliptic geometryRelated: The elliptic plane has positive constant Gaussian curvature.

• Nikolai LobachevskyRelated: Negative Gaussian curvature provides a geometric model for hyperbolic surfaces.

• Hyperbolic planeRelated: The hyperbolic plane has the same negative Gaussian curvature everywhere.

• PseudosphereRelated: The pseudosphere has the same negative Gaussian curvature at every regular point.

• Spherical excessRelated: For a sphere, constant positive curvature makes angular excess proportional to area.

• Theorema EgregiumRelated: The theorem identifies this extrinsic-looking quantity as measurable from the surface’s metric.

• Flat torusRelated: The flat metric has zero Gaussian curvature everywhere.

• Alexandrov's theorem on polyhedraRelated: Its concentrated, positive form at vertices is the theorem's curvature condition.

• Carathéodory conjectureRelated: Strict convexity ensures positive Gaussian curvature under the usual smooth-surface convention.

• Daina TaimiņaRelated: Negative Gaussian curvature is the local geometric feature represented by her crocheted surfaces.

• Hjelmslev's theoremRelated: Constant negative curvature supplies the length scale that Euclidean triangle similarity lacks.

• Legendre's theorem on spherical trianglesRelated: Positive curvature produces the angle excess that the theorem distributes among the angles.

• Two-dimensional spaceRelated: It distinguishes intrinsically flat regions from curved ones.