1996 Fiscal Year Final Research Report Summary
Harmonic maps into symmetric spaces and applications of the theory of integrable systems
| Project/Area Number |
06640174
|
|---|
| Research Category |
Grant-in-Aid for Scientific Research (C)
|
| Allocation Type | Single-year Grants |
|---|
| Section | 一般 |
|---|
| Research Field |
Geometry
|
|---|
| Research Institution | Nihon University |
|---|
Principal Investigator |
UDAGAWA Seiichi School of Medicine, Nihon University, Lecturer, 医学部, 講師 (70193878)
|
|---|
| Co-Investigator(Kenkyū-buntansha) |
IKAWA Toshihiko School of Medicine, Nihon University, associate Professor, 医学部, 助教授 (30151252)
|
|---|
| Project Period (FY) |
1994 – 1996
|
| Keywords | Harmonic map / Symmetric space / Torus / Finite type / Conformal map / Complex Grassmannion / Quaternicnic projective space |
|---|
| Research Abstract |
The construction of harmonic two-tori in symmetric spaces are known in two ways. One way is a method using twistor fibration and the resulting harmonic map is said to be superminimal. Another way is a method using the theory of integrable system and it is constructed from two-dimensional linear flows. The latter harmonic map is said to be finite type. For example, any non-conformal harmonic two-tori in compact symmetric spaces of rank one is of finite type. Burstall proved any weakly conformal non-superminimal harmonic two-tori in a sphere or a complex projective space is covered by a primitive map of finite type. Then the following problems naturally arises :
(1) Is weakly conformal non-superminimal hatmonic two-tori in quaternionic projective space covered by a primitive map of finite type?
(2) How about the case where the target is a compact symmetric space of rank greater than one?
Our result for the problem (1) is : Any weakly conformal non-superminimal harmonic two-tori in HP^3 is covered by a primitive map of finite type or constructed by using twistor fibration. For the problem (2), weakly conformal non-superminimal harmonic two-tori in G_2(C^4) is covered by a primitive map of finite type or constructed by using twistor fibration. Under the additional condition, the same type theorem holds for G_2(C^<2n>).
|
