FURUHATA Hitoshi Hokkaido Univ., Grad. School of Sci., Lect., 大学院・理学研究科, 講師 (80282036)
ISHIKAWA Goo Hokkaido Univ., Grad. School of Sci., Assoc. Prof., 大学院・理学研究科, 助教授 (50176161)
IZUMIYA Shuichi Hokkaido Univ., Grad. School of Sci., Prof., 大学院・理学研究科, 教授 (80127422)
IGARASHI Masayuki Sci. Univ. of Tokyo, Fac. Ind. Sci. of Tech., Lect., 基礎工学部, 講師 (60256675)
SHIMADA Ichirou Hokkaido Univ., Grad. School of Sci., Assoc. Prof., 大学院・理学研究科, 助教授 (10235616)
|Budget Amount *help
¥3,900,000 (Direct Cost: ¥3,900,000)
Fiscal Year 2002: ¥1,900,000 (Direct Cost: ¥1,900,000)
Fiscal Year 2001: ¥2,000,000 (Direct Cost: ¥2,000,000)
We constructed a continuous family of riemanninan metrics on 2-sphere whose geodesic flows possess first integrals of fiber-degree k, for every k greater than 2. They are the first examples, exect the cases where k=3,4, due to Bolsinov and Fomenko. Moreover, the constructed manifolds have the property that every geodesic is closed. Therefore they are conrete examples of the manifolds that Guillemin showed their existence in an abstract manner.
We also investigated the structures of Kahler-Liouville manifolds of general type, I.e., not necessarlly of type (A). As a consequence, we showed that every compact, proper Kahler-Liouville manifold has a bundle structure such that the fiber is a Kahler-Liouville manifold whose geodesic flow is integrable, and the base is (locally) a product of one-dimensional Kahler manifolds. Also, we obtain another class, called of type (B), of Kahler-Liouville manifolds whose geodesic flows are integrable. This class had already appeared in the study of fiber bundle structure of type (A) manifolds, but we now obtained its intrinsic definition.
Also, we investigated local structures of Hermite-Liouville manifolds and basically clarifled them. Moreover, we construct the structure of Hermite-Liouville manifolds on complex projective spaces. The way of construction is similar to that of a Kahler-Liouvlle manifold, I.e., a complexification of a real Liouville manifold. However, in the Hermite case, plural Liouville manifolds produce one Hermite-Liouville manifold. Therefore, we obtain quite many examples of integrable geodesic flows in this way.