1995 Fiscal Year Final Research Report Summary
On canonically induced geometric structures on submaniflds and the equivalence problem
| Project/Area Number |
06452008
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| Research Category |
Grant-in-Aid for General Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Research Field |
Geometry
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| Research Institution | Kobe University |
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Principal Investigator |
SASAKI T Kobe University, Dept Math, Professor, 理学部, 教授 (00022682)
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| Co-Investigator(Kenkyū-buntansha) |
TAKANO K Kobe University, Dept Math, Professor, 理学部, 教授 (10011678)
TAKAYAMA N Kobe University, Dept Math, Associate Professor, 理学部, 助教授 (30188099)
KABEYA Y Kobe University, Dept Math, Assistant, 理学部, 助手 (70252757)
IKEDA H Kobe University, Dept Math, Professor, 理学部, 教授 (10031353)
NAKANISHI Y Kobe University, Dept Math, Associate Professor, 理学部, 助教授 (70183514)
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| Project Period (FY) |
1994 – 1995
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| Keywords | projective invariant / hypergeometric system / regular graph / Painleve system / radial solution / unknotting number / Groebner basis |
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| Research Abstract |
We give an outline of the results on the project.
(1) On the geometric structures in affine and projective differential geometry and its equivalence problem. 1. We gave an estimate on the number of inflection points from below that are lying on a closed curve on flat affine torus and examples that show this estimate is the best. With this estimate we could characterize the Euclidean torus among flat affine tori. 2. We proved that m-dimensional submanifold in the affine space of dimension m+m (m+1) /2 has a canonical connection and characterized the affine minimality of such submanifolds. 3. We published a book on affine differential geometry that gives the foundation of such geometry. 4. We developed the projective theory of submanifolds and classified projectively homogeneous surfaces.
(2) On the geometric structures associated with the system of hypergeometric differential equations. 1. We proved that the projective mapping of the hypergeometric differential system denoted by E (k, n)
… More
does not have generally its image on any Grassmannian submanifolds. 2. The configuration space of 5 points on a projective line can be described combinatorially by decomposing the space into 20 simply connected domains. 3. We defined the system associated with the integral on the configuration space of one conic and any number of hyperplanes on the projective space, clarified its symmetry, the contiguity relations, and computed its intersection form. 4. We proposed a new algorithm for integer programming by applying contiguity relations of the system E (k, n) and proved related results. 5. We decided the general formulae of contiguity relations for the confluent hypergeometric sytems. 6. We succeeded in the description of Stokes phenomena by use of Braid group.
(3) On the geometric structure of Painleve system. 1. We poved that the space of initial states for Painleve system has a symplectic structure and provided a new method of the characterization of such a system. 2. We solved the reduction problem for the Garnier system.
(4) On the deformation equation of conformal structures. 1. We investigated a global structure of the nonlinear differential equation associated with the deformation of conformal structures. 2. We proved the existence and the uniqueness of radial solutions for this equation.
(5) Studies on combinatorics and algorithms. 1. We gave a growth estimate of the coefficients of Alexander polynomials of knot and applied it to the link-homotopy theory. 2. We gave a partial answer to the conjecture related with minimization of the unknotting number. 3. We gave a method of knowing local equivalence in terms of generalized unknotting procedure.
4. We clarified the relation of delta-unkotting number with Conway polynomials. 5. We provided an algorithm that enables the computation of binomial relations. 6. We improved the mathematical processing system KAN and implemented it to Macaulay. 7. We developed an algorithm for Voronoi diagram on the negatively-curved plane and applied it to information geometry. Less
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