2001 Fiscal Year Final Research Report Summary
Geometry of polyhedron from the view point of differential geometry
| Project/Area Number |
12640079
|
|---|
| Research Category |
Grant-in-Aid for Scientific Research (C)
|
| Allocation Type | Single-year Grants |
|---|
| Section | 一般 |
|---|
| Research Field |
Geometry
|
|---|
| Research Institution | Kumamoto University |
|---|
Principal Investigator |
ITOH Jin-ichi Kumamoto U., Fac. of Education, Professor, 教育学部, 教授 (20193493)
|
|---|
| Co-Investigator(Kenkyū-buntansha) |
MATSUHISA Fumiko Ibaragi U., Fac. of Science, Ass. Professor, 理学部, 助教授 (90194208)
HIRAMINE Yutaka Kumamoto U., Fac. of Education, Professor, 教育学部, 教授 (30116173)
KANEMARU Tadayoshi Kumamoto U., Fac. of Education, Professor, 教育学部, 教授 (30040033)
AGAOKA Yoshio Hiroshima U., Fac. of Integrated Arts & Science, Ass. Professor, 総合科学部, 助教授 (50192894)
|
|---|
| Project Period (FY) |
2000 – 2001
|
| Keywords | polyhedron / curvature / total curvature / cut locus / tightness / acute triangulation |
|---|
| Research Abstract |
The analogous results for polyhedron of Gauss's Theorema Egregium and Weyl's volume formula were proved and written in the 2-dimensional case. The Cohn-Vossen type inequality for 2-polyhedron, the total curvature of graphs and its tightness are written.
There are the related new problems, for examples, the acute troangulations, the structure of essential cut locus, the length of cycles of cut locus, etc. All these problems are very exploratory and are expected to produce greate results by continuing studies.
With respect to the acute triangulations, we proved that the cubed surface admitts an acute triangulations with 24 triangles, the icosahedral surface admitts an acute triangulations with 12 triangles, and these are the least numbers. 'The dodecahedral surface does not have any acute triangulations with triangles less than 11 and admits an acute triangulation with 14 triangles. Moreover we discussed several other cases and got some fundamental ideas how to treat the general convex surfaces.
Withrespect to the essential cut locus, we define it in the case of a surface as the essential part of cut locus containing all critical points of distance function, and proved that the number of end points or the degree of vertices is related with several invariants of its inner metric. Moreover, we consider its structure in the case of convex polyhedron in general dimension.
With respect to the length of cycles of cut locus, we proved that there is a point p on any torus with diameter 1 such taht the length of cycles in the cut locus of p is greater than 2. It is the best possible estimate and there is no upper bound.
|
