Chapter 5 - Shape Operators

Abstract

Publisher Summary The shape of a surface M in Euclidean space R3 is described infinitesimally by a linear operator S defined on each of the tangent planes of M. This chapter justify this infinitesimal measurements by proving that two surfaces with “the same” shape operators are, in fact, congruent. The algebraic invariants of its shape operators thus have geometric meaning for the surface M. The chapter highlights efficient ways to compute these invariants, and test them on a number of geometrically interesting surfaces. The shape operator is an algebraic object consisting of linear operators on the tangent planes of M, and it is by an algebraic analysis of S that the main geometric invariants of a surface in R3 such as its principal curvatures and directions and its Gaussian and mean curvatures can be achieved. The chapter also describes a way to compute the geometry of a surface M⊂R3 that has a nonvanishing normal vector field Z defined on the entire surface. Here, the notation M⊂R3 means a connected surface M in R3.