shape operator

English

Noun

shape operator (plural shape operators)

1. (geometry, differential geometry) The differential of the Gauss map of an oriented surface at a given point on the surface.
   • 1966, Barrett O'Neill, Elementary Differential Geometry, Academic Press, page 304:

     Gaussian curvature is a prime example; although defined in terms of shape operators, it belongs to this intrinsic geometry, since it passes the test of isometric invariance.

   • 2006, Balkan Journal of Geometry and Its Applications, volumes 11-12, page 41:

     Timelike surfaces have symmetric shape operators which can be put into one of three canonical forms on a fixed tangent space with respect to an orthonormal basis: […] .

   • 2007, John Oprea, Differential Geometry and Its Applications, Mathematical Association of America, page 85:

     We have already seen that a plane has zero shape operator. Intuitively, since the shape operator detects the change in the unit normal $U$, a zero shape operator for a surface $M$ should imply that $M$ is a plane. This is verified by the following result.

Synonyms

• (differential of a Gauss map): Weingarten map

Further reading

• Differential geometry of surfaces on Wikipedia.Wikipedia
• Weingarten equations on Wikipedia.Wikipedia