Gaussian curvature
E29546
Gaussian curvature is a fundamental concept in differential geometry that measures how a surface bends at a point by combining its principal curvatures into a single intrinsic quantity.
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
curvature invariant
→
differential geometry concept → scalar field on a surface → |
| appearsIn |
Riemannian surface classification
→
surface theory → |
| canBe |
constant on some surfaces
→
|
| characterizes |
local shape of a surface
→
|
| codomain |
real numbers
→
|
| definedAs |
product of principal curvatures
→
|
| dependsOn |
first fundamental form
→
second fundamental form → |
| describes |
intrinsic curvature of a surface
→
|
| domain |
points of a regular surface
→
|
| exampleConstantCurvatureSurface |
Euclidean plane has K = 0
→
hyperbolic plane has K < 0 → sphere of radius R has K = 1/R^2 → |
| exampleSurfaceWithNegativeCurvature |
hyperbolic plane
→
pseudosphere → |
| exampleSurfaceWithPositiveCurvature |
sphere
→
|
| exampleSurfaceWithZeroCurvature |
cylinder
→
plane → |
| field |
Riemannian geometry
→
differential geometry → |
| formulaInPrincipalDirections |
K = k1 * k2
→
|
| isExtrinsic |
false
→
|
| isIntrinsic |
true
→
|
| mathematicalNature |
local invariant of a 2-dimensional Riemannian manifold
→
|
| namedAfter |
Carl Friedrich Gauss
→
|
| property |
can be computed from the metric alone
→
invariant under local isometries of surfaces → |
| relatedTo |
Gauss–Bonnet theorem
→
Theorema Egregium → mean curvature → principal curvatures → sectional curvature → |
| signInterpretation |
negative at hyperbolic points
→
positive at elliptic points → zero at parabolic points → |
| symbol |
K
→
|
| unit |
inverse square of length
→
|
| usedIn |
classification of points on a surface
→
general relativity analogues in 2D → geodesic analysis → global topology via Gauss–Bonnet theorem → |
| valueType |
negative curvature
→
positive curvature → zero curvature → |
Referenced by (3)
| Subject (surface form when different) | Predicate |
|---|---|
|
Carl Friedrich Gauss
→
|
hasConceptNamedAfter |
|
Carl Friedrich Gauss
→
|
notableWork |
|
Theorema Egregium
("Gaussian curvature is invariant under local isometries of surfaces")
→
|
proves |