Harmonic maps into symmetric spaces and applications of the theory of integrable systems
Project/Area Number  06640174 
Research Category 
GrantinAid for Scientific Research (C)

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)

CoInvestigator(Kenkyūbuntansha) 
IKAWA Toshihiko School of Medicine, Nihon University, associate Professor, 医学部, 助教授 (30151252)

Project Fiscal Year 
1994 – 1996

Project Status 
Completed(Fiscal Year 1996)

Budget Amount *help 
¥1,800,000 (Direct Cost : ¥1,800,000)
Fiscal Year 1996 : ¥600,000 (Direct Cost : ¥600,000)
Fiscal Year 1995 : ¥600,000 (Direct Cost : ¥600,000)
Fiscal Year 1994 : ¥600,000 (Direct Cost : ¥600,000)

Keywords  Harmonic map / Symmetric space / Torus / Finite type / Conformal map / Complex Grassmannion / Quaternicnic projective space / Complex Grassmannian / Quarternionic prouective space / Quaternionic projective space / harmonic map / finite type / primitive map / complex Grassmann manifold / quaternionic projective space / twotorus 
Research Abstract 
The construction of harmonic twotori 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 twodimensional linear flows. The latter harmonic map is said to be finite type. For example, any nonconformal harmonic twotori in compact symmetric spaces of rank one is of finite type. Burstall proved any weakly conformal nonsuperminimal harmonic twotori 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 nonsuperminimal hatmonic twotori 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 nonsuperminimal harmonic twotori in HP^3 is covered by a primitive map of finite type or constructed by using twistor fibration. For the problem (2), weakly conformal nonsuperminimal harmonic twotori 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>).

Report
(5results)
Research Output
(13results)