Study of Analysis on Manifolds

流形分析研究

• 批准号: 13640222
• 负责人: FURUTANI Kenro
• 金额: $ 2.11万
• 依托单位: Tokyo University of Science
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2001
• 资助国家: 日本
• 起止时间: 2001 至 2003
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-13640222/
• 关键词: Spectral flow; Maslov index; Fredholm operator; symplectic manifold; Polarization; Quaternion Projective spare; Quantization; zeta-regularided determinant; Heisenberg manifold; modified Bessel function; Epstein zeta funtion; 無限積; Quillen deter

(1)We proved a splitting formula for a spectral flow of a one parameter family of first order selfadjoint elliptic differential operators which arises when we split a manifold into two components by a hypersurface.To prove this formula we reconstruct a general theory of infinite dimensional Maslov index, where it should be noted that the Maslov index can be defined for arbitrary paths as an intersection number with a Maslov cycle by means of a functional 'analytic method.We also proved a reduction theorem of the Maslov index in the infinite dimension(2)We found the complexified Hopf fiberation on the punctured cotangent bundle of the quaternion projective space, and constrict two quantization operators of the geodesic flow on quaternion projective spaces by two methods.(3)We construct a Kahler structure on the punctured cotangent bundle of the Cayley projective plain whose Kahler form coincides with the natural symplectic form.This is given by explicitly embedding it into the space of 8×8 complex matrices.(4)We proved an integral representation of the zeta-regularized determinant of 3 and 4 dimensional Heisenberg manifolds, and also give a general formula for product type Romanian manifolds.Then we apply the formula to give an expression for the zeta-regularized determinant of the higher dimensional torus.

（1）我们为一个一阶参数家族的光谱流提供了一个分裂公式我们还提供了无限尺寸（2）中Maslov指数的减少理论（2），我们发现了Quatternion投射空间的刺穿的hopf纤维纤维，并约束了两个方法的Quaterion fromist the -Burnite a Burnite a Kahler contrult ckahler contects kah rughters kah rughters kah bunder s kah renters。 Cayley投射平原的Kahler形式与自然的符号形式一致。这是通过将其明确嵌入8×8复杂物品的空间中给出的。（4）我们提供了由3和4的Zeta定型化的组成表示，由3和4的4多二二元的Heisenbergggg，并为造型式的配方提供了综合型，并为生产式的表达式提供了代表。由较高尺寸的圆环确定的ZETA调控。

项目成果

B.Booss-Bavnbeck, K.Fututani, K.P.Wojcrewski: "The Geometry of Cauchy data spaces"Math.Phy.Studies. 24. 321-354 (2003)

B.Booss-Bavnbeck、K.Fututani、K.P.Wojcrewski：“柯西数据空间的几何”数学、物理研究。

K.Furutani, Serge de Gosson: "Determinant of Laplacian on Heisenberg manifolds"Journal of Geometry and Physics. Vol.48,No.2-3. 438-479 (2003)

K.Furutani，Serge de Gosson：“海森堡流形上拉普拉斯的行列式”几何与物理杂志。

K.Moriyasu, Masatosi Oka: "Differential maps having the uniformly shadowing property"Topology and its applications. 122. 377-396 (2002)

K.Moriyasu，Masatosi Oka：“具有均匀阴影特性的微分映射”拓扑及其应用。

Kenro Furutani: "A Kahler Structure on the punctured cotangent bubdle of the cayley progective plane"Math.Phy.Studies. 24. 163-182 (2003)

Kenro Furutani：“凯莱投影平面的穿孔余切球上的卡勒结构”数学、物理研究。

R.Kobayashi: "Soliton solution of the KdV equation and Crum's theorem"Advances in Mathematical Science and Applications. Vol.12,No.2. 779-784 (2002)

R.Kobayashi：“KdV 方程和 Crum 定理的孤子解”数学科学与应用进展。