Meaning of shape operator | Babel Free

Definitions

The differential of the Gauss map of an oriented surface at a given point on the surface.

Examples

“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.”

“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:[…].”

“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⟳.”

CEFR level