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shape operator (Definition)

The shape operator $ S$ of a surface $ \Sigma$ in $ {\mathbb{R}}^3$ is the derivative of the sphere map $ N:\Sigma\to S^2$ given by $ N(p)=$ unit normal vector field at $ p$. So at each $ p$, $ S(p)=d_pN$ and it is the linear transformation $ S(p):T_p\Sigma\to T_{N(p)}S^2$. This is important, because the determinant defines the Gaussian curvature at $ p$ in $ \Sigma$.



"shape operator" is owned by juanman. [ full author list (2) ]
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See Also: second fundamental form
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Cross-references: Gaussian curvature, determinant, linear transformation, vector field, unit normal, map, sphere, derivative, surface
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This is version 3 of shape operator, born on 2006-07-10, modified 2006-07-11.
Object id is 8132, canonical name is ShapeOperator.
Accessed 1513 times total.

Classification:
AMS MSC: 53A05 (Differential geometry :: Classical differential geometry :: Surfaces in Euclidean space)
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