Grant-in-Aid for Scientific Research (C)
|Allocation Type||Single-year Grants |
|Research Institution||Tokyo Metropolitan University |
OHNITA Yoshihiro (2004) Tokyo Metropolitan University, Dept. of Math., Professor, 理学研究科, 教授 (90183764)
今井 淳 (2003) 東京都立大学, 理学研究科, 助教授 (70221132)
IMAI Jun Tokyo Metropolitan University, Dept. of Math., Associate professor, 理学研究科, 教授 (70221132)
OKA Mutsuo Tokyo Metropolitan University, Dept. of Math., Professor, 理学研究科, 教授 (40011697)
横田 佳之 東京都立大学, 理学研究科, 助教授 (40240197)
KOJIMA Sadayoshi Tokyo Institute of Technology, Professor, 情報理工学研究科, 教授 (90117705)
AHARA Kazushi Meiji University, Lecturer, 理工学部, 講師 (80247147)
GUEST Martin Tokyo Metropolitan University, Dept. of Math., Professor (10295470)
神島 芳宣 東京都立大学, 理学研究科, 教授 (10125304)
大仁田 義裕 東京都立大学, 理学研究科, 教授 (90183764)
|Project Period (FY)
2003 – 2004
Completed (Fiscal Year 2004)
|Budget Amount *help
¥3,500,000 (Direct Cost: ¥3,500,000)
Fiscal Year 2004: ¥1,600,000 (Direct Cost: ¥1,600,000)
Fiscal Year 2003: ¥1,900,000 (Direct Cost: ¥1,900,000)
|Keywords||knot / energy functional / conformal geometry / エネルギー / 共形幾何 / メビウス変換 / 非調和比|
Let Y and Y' be a pair of curve segments in an n dimensional sphere, and let x, x+dx be points on Y and y, y+dy be points on Y'. We allow the case when Y and Y' coincide, when we always assume that x and y are distinct. By identifying a 2 dimensional sphere through the four points x, x+dx, y, and y+dy with the Riemann sphere through a stereographic projection, we obtain the cross ratio of these four points. Then it can be considered as a complex valued 2-form on YxY'. We call it the infinitesimal cross ratio of Y and Y'. It is, by definition, invariant under Moebius transformations.
We obtained new interpretations of the real and the imaginary parts of the infinitesimal cross ratio.
An n dimensional sphere can be realized as the set of points a infinity of the light cone in (n+2) dimensional Minkowski space. Let S(n, p) denote the set of p dimensional sphere in the n dimensional sphere. Then S(n, p), which can be expressed in terms of Pluecker coordinates, is a space with an indefinite m
etric. A pair of curve segments Y and Y' in the n dimensional sphere can also be considered as a surface in S(n, O). Now the real part of the infinitesimal cross ratio is equal to the absolute value of the area element of this surface.
On the other hand, the interpretation of the imaginary part can be given as follows. An n dimensional sphere can be considered as the boundary of the (n+1) dimensional hyperbolic space. Let L denote a geodesic in the hyperbolic space joining points x on Y and y on Y' in the boundary sphere, and let P be an orthogonal hyperplane to L. Let x' and y' be points in neighborhoods of x and y respectively. The intersection of P and the geodesic joining x' and y' gives a surface in P. Then the imaginary part of the infinitesimal cross ratio at (x, y) is equal to the area element of this surface.
We also defined functionals on the space of curves and surfaces using a conformally invariant measure on the space S(n, p).
(That is the summary of the joint work of Jun Imai, who was the head investigator in 2003, and Remi Langevin, who is an investigator abroad, during Imai's 7 months stay in France in 2004. Part of the grant was used to invite Lengevin to Japan in 2003.) Less