shape operator
E653147
The shape operator is a linear map in differential geometry that describes how a surface curves in different directions by relating changes in its normal vector to directions in the tangent plane.
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
differential geometry concept
ⓘ
linear operator ⓘ second fundamental form-related operator ⓘ |
| actsOn | tangent space of a surface ⓘ |
| alsoKnownAs |
Weingarten map
NERFINISHED
ⓘ
Weingarten operator NERFINISHED ⓘ |
| appliesTo |
embedded submanifolds of codimension one
ⓘ
hypersurfaces in Riemannian manifolds ⓘ |
| classificationRole | determines local shape type via its eigenvalues ⓘ |
| codomain | tangent plane of a surface at a point ⓘ |
| compatibility | compatible with the induced metric on the surface ⓘ |
| context |
theory of hypersurfaces in Riemannian manifolds
ⓘ
theory of surfaces in Euclidean 3-space ⓘ |
| curvatureType | extrinsic curvature operator ⓘ |
| definition | linear map that measures how the unit normal vector field changes in tangent directions ⓘ |
| determinantRelation | Gaussian curvature equals the determinant of the shape operator for surfaces in R^3 ⓘ |
| domain | tangent plane of a surface at a point ⓘ |
| eigenvalues | principal curvatures of the surface ⓘ |
| eigenvectors | principal directions of curvature ⓘ |
| example |
for a plane in R^3 the shape operator is the zero operator
ⓘ
for a sphere of radius R in R^3 the shape operator is (1/R) times the identity on each tangent plane ⓘ |
| field |
Riemannian geometry
NERFINISHED
ⓘ
differential geometry ⓘ |
| formalDefinition | for a hypersurface with unit normal field n, S(X) = - (∇_X n)^T for tangent vector X ⓘ |
| geometricMeaning | measures rate of rotation of the normal vector along tangent directions ⓘ |
| historicalAttribution | named after Julius Weingarten NERFINISHED ⓘ |
| invarianceProperty | invariant under ambient isometries ⓘ |
| isLinear | true ⓘ |
| matrixRepresentation | represented by a symmetric matrix in an orthonormal tangent basis ⓘ |
| property | self-adjoint with respect to the induced metric on the surface ⓘ |
| rankInformation | rank gives information about flat directions on the surface ⓘ |
| relatedTo |
Gaussian curvature
ⓘ
Levi-Civita connection NERFINISHED ⓘ extrinsic curvature ⓘ mean curvature ⓘ principal curvatures ⓘ principal directions ⓘ second fundamental form ⓘ unit normal vector field ⓘ |
| relationToSecondFundamentalForm | second fundamental form II(X,Y) = ⟨S(X),Y⟩ ⓘ |
| signConvention | often defined with a minus sign S(X) = -∇_X n ⓘ |
| smoothnessRequirement | defined for sufficiently smooth (at least C^2) hypersurfaces ⓘ |
| symmetryProperty | symmetric with respect to the first fundamental form ⓘ |
| traceRelation | mean curvature equals one half of the trace of the shape operator for surfaces in R^3 ⓘ |
| usedFor |
classifying points on a surface as elliptic, hyperbolic, or parabolic
ⓘ
quantifying how a surface bends in different tangent directions ⓘ studying extrinsic geometry of submanifolds ⓘ |
| zeroCondition | vanishes identically if and only if the hypersurface is totally geodesic ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.