2006 Fiscal Year Final Research Report Summary
Study of surfaces from the viewpoints of differential geometry and singularity theory
| Project/Area Number |
16540054
|
|---|
| Research Category |
Grant-in-Aid for Scientific Research (C)
|
| Allocation Type | Single-year Grants |
|---|
| Section | 一般 |
|---|
| Research Field |
Geometry
|
|---|
| Research Institution | TOKYO GAKUGEI UNIVERSITY |
|---|
Principal Investigator |
TAKEUCHI Nobuko Tokyo Gakugei University, Department of Mathematics, Associate Professor, 教育学部, 助教授 (70216852)
|
|---|
| Co-Investigator(Kenkyū-buntansha) |
IZUMIYA Shuichi Hokkaido University, Department of Mathematics, Professor, 大学院理学研究院, 教授 (80127422)
|
|---|
| Project Period (FY) |
2004 – 2006
|
| Keywords | ruled surface / circular surface / differential geometry / singularities / roller coaster surface / Minkowski n-space |
|---|
| Research Abstract |
A ruled surface is a one-parameter family of lines and A circular surface is a one-parameter family of standard circles in E^3. We study geometric properties and singularities of ruled surfaces and circular surfaces. Ruled surfaces are classical subjects in differential geometry which have been studied since the 19th century. We study ruled surfaces corresponding to special curves. We define new special curves whose notions are generalizations of the notion of helices. One of the results gives characterizations of special ruled surfaces under the condition of existence of such a curve. Like the ruled surfaces, circular surfaces could be important subjects, but there is no systematic study of circular surfaces. Therefore we study smooth one-parameter family of standard circles with fixed radius. There is a curve on a ruled surface with an important property which is called the striction curve and the singularities of a ruled surface are located on the striction curve. We consider curves on a circular surface with a similar property to that of the striction curve. The singularities of a circular surface are located on the curves. Next we consider a property of circular surfaces that each generating circle is a line of curvature except at umbilical and singular points. We classify such circular surfaces into canal surfaces, spheres or a special type of surfaces called a roller coaster surface. Moreover, we classify singularities of generic roller coaster.
We also study some properties of space-like submanifolds in Minkowski n-space. For example, we introduce the notion of the lightcone Gauss-Kronecker curvature for a spacelike submanifold of codimension two in Minkowski space, which is a generalization of the ordinary notion of Gauss curvature of hypersurfaces in Euclidean space. We show a Gauss-Bonnet type theorem and study from the viewpoint of Lagrangian and Legendrian singularity theory.
|
