On Surfaces with Prescribed Shape Operator

Journal Article (Journal Article)

The problem of immersing a simply connected surface with a prescribed shape operator is discussed. It is shown that, aside from some special degenerate cases, such as when the shape operator can be realized by a surface with one family of principal curves being geodesic, the space of such realizations is a convex set in an affine space of dimension at most 3. The cases where this maximum dimension of realizability is achieved are analyzed and it is found that there are two such families of shape operators, one depending essentially on three arbitrary functions of one variable and another depending essentially on two arbitrary functions of one variable. The space of realizations is discussed in each case, along with some of their remarkable geometric properties. Several explicit examples are constructed.

Full Text

Duke Authors

Cited Authors

  • Bryant, RI

Published Date

  • October 1, 2001

Published In

Volume / Issue

  • 40 / 1-4

Start / End Page

  • 88 - 121

Electronic International Standard Serial Number (EISSN)

  • 1420-9012

International Standard Serial Number (ISSN)

  • 1422-6383

Digital Object Identifier (DOI)

  • 10.1007/BF03322701

Citation Source

  • Scopus