2004 Fiscal Year Final Research Report Summary
The relation between Riemannian geometry and discrete geometry from the view point of minimulity
| Project/Area Number |
14540086
|
|---|
| 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 Edu., Professor, 教育学部, 教授 (20193493)
|
|---|
| Co-Investigator(Kenkyū-buntansha) |
KOBAYASHI Osamu Kumamoto U., Fac.of Sci., Professor, 理学部, 教授 (10153595)
HIRAMINE Yutaka Kumamoto U., Fac.of Edu., Professor, 教育学部, 教授 (30116173)
MAEHARA Hiroshi Ryukyu U., Fac.of Edu., Professor, 教育学部, 教授 (60044921)
ENOMOTO Kazuyuki Tokyo U.of Sci., Fac.of Ind.Sci. & Tec., Professor, 基礎工学部, 教授 (40194005)
|
|---|
| Project Period (FY) |
2002 – 2004
|
| Keywords | acute triangulation / Riemannian manifold / cut locus / total curvature / tetrahedron passing through a hole / polyhedron / farthest point / geodesic |
|---|
| Research Abstract |
We studied various problems, here we summarize some of them, especially acut triangulation, total curvature of curves, cut locus of ellipsoids.
We studied acute triangulations on the surfaces of Platonic solids. We proved that the surface of icosahedron can be triangulated with 8 non-obtuse and with 12 acute triangles. We also showed these numbers to be smallest possible. In the case of the regular dodecahedral surface, we proved that there exists a triangulation with only 10 non-obtuse triangles, and that this is best possible, we also proved the existence of a triangulation with 14 acute triangles, and the non-existence os such triangulations with less than 12 triangles.
The total absolute curvature of non closed curves in S^2 is studied. We look at the set of curves with fixed end points and end-directions and see how the infimum of the total absolute curvature in this set depends on the endpoints and the end-directions. We consider both the case when the length od curves is fixed and the case when the length is free, and see the difference results between them. Furthermore, we determin the shape of the curve in S^2 which minimize the total curvature in the set of nonclosed curves with fixed end points, end-directions and length.
We studied small holes through which regular 3-,4- and 5-dimensional simplices can pass through.
We proved that the cut locus of any point on any ellipsoid is an arc on the curvature line through the antipodal point. Also, we proved that the conjugate locus has exactly four cusps, which is known as the last geometric statement of Jacobi. Furthermore we studied in the case of some kind of Liouville surfaces and get the similar results.
|
