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