Gauss’s remarkable theorem

E157378

geometric theorem result in differential geometry theorem

Gauss’s remarkable theorem is a fundamental result in differential geometry showing that the Gaussian curvature of a surface is an intrinsic property independent of how the surface is embedded in space.

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All labels observed (1)

Label	Occurrences
Gauss’s remarkable theorem canonical	1

Statements (45)

Predicate	Object
instanceOf	geometric theorem ⓘ result in differential geometry ⓘ theorem ⓘ
clarifies	difference between intrinsic and extrinsic properties of surfaces ⓘ
coreClaim	Gaussian curvature is independent of the embedding of the surface in Euclidean space ⓘ Gaussian curvature is preserved under local isometries of surfaces ⓘ Gaussian curvature of a surface is an intrinsic invariant ⓘ
dealsWith	Gaussian curvature ⓘ first fundamental form ⓘ intrinsic geometry ⓘ isometry of surfaces ⓘ second fundamental form ⓘ surfaces ⓘ
field	Riemannian geometry ⓘ differential geometry ⓘ
hasAlternativeName	Theorema Egregium ⓘ surface form: Gauss’s Theorema Egregium Theorema Egregium ⓘ
hasExampleApplication	proving that a cylinder is locally isometric to a plane ⓘ showing that a sphere is not locally isometric to a plane ⓘ understanding curvature of the Earth from geodesic measurements ⓘ
historicalPeriod	19th century ⓘ
implies	a plane cannot be bent isometrically into a sphere ⓘ bending a surface without stretching does not change its Gaussian curvature ⓘ curvature can be determined entirely from the metric on the surface ⓘ no isometric mapping exists between surfaces with different Gaussian curvature at corresponding points ⓘ
influenced	Bernhard Riemann ⓘ development of Riemannian geometry ⓘ general relativity ⓘ modern differential geometry ⓘ
introducedBy	Carl Friedrich Gauss ⓘ
languageOfOriginal	Latin ⓘ
mathematicalDomain	analysis on manifolds ⓘ geometry ⓘ
namedAfter	Carl Friedrich Gauss ⓘ
publicationYear	1827 ⓘ
publishedIn	Disquisitiones Generales Circa Superficies Curvas ⓘ surface form: Disquisitiones generales circa superficies curvas
relatesConcept	Christoffel symbols ⓘ Riemannian metric ⓘ extrinsic curvature ⓘ geodesic coordinates ⓘ intrinsic curvature ⓘ
status	fundamental theorem of surface theory ⓘ
typeOfCurvature	sectional curvature in dimension two ⓘ
usesConcept	determinant of the metric tensor ⓘ second derivatives of the metric ⓘ

Referenced by (1)

Full triples — surface form annotated when it differs from this entity's canonical label.

Theorema Egregium → alsoKnownAs → Gauss’s remarkable theorem ⓘ