2004 Fiscal Year Final Research Report Summary
Study of the structure of hypersurfaces with constant scalar curvature
| Project/Area Number |
15540057
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Geometry
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| Research Institution | IBARAKI UNIVERSITY |
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Principal Investigator |
OKAYASU Takashi Ibaraki University, The College of Education, Associate Professor, 教育学部, 助教授 (00191958)
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| Co-Investigator(Kenkyū-buntansha) |
YAGITA Nobuaki Ibaraki University, The College of Education, Professor, 教育学部, 教授 (20130768)
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| Project Period (FY) |
2003 – 2004
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| Keywords | scalar curvature / hypersurface / ordinary differential equation |
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| Research Abstract |
In the Euclidean spaces the known examples of complete hypersurfaces with positive constant scalar curvature are only spheres, generalized cylinders S^p×R^<n-p> and the rotational hypersurfaces. In this research we constructed infinitely many new examples of complete hypersurfaces with constant positive scalar curvature in the Euclidean spaces.
For the O(p+1)×O(q+1)-invariant hypersurfaces, the equation representing its scalar curvature is constant S is
(I){dx/ds=cosα, dy/ds=sinα, dα/ds=(p(p-1)((sinα)/x)^2-2pq(sinα)/x(cosα)/y+q(q-1)((cosα)/y)^2-S)/(2(q(cosα)/y-p(sinα)/x)),
where α is the angle between the tangent vector (x', y') and the x-axis.
The key point of this study is that we can compare the solution with another ODE which is explicitly integrable.
Theorem Suppose that p 【less than or equal】 q + 1 and S > 0 (p 【greater than or equal】 2). Let 0 < x_0 【less than or equal】 √<p(p -1)/S> and 0 < y_0 【less than or equal】 √<q(q -1)/S>. Then the ODE system has a global solution γ(s) = (x(s), y(s)) ∈ R_+ × R_+ on (-∞, ∞) for the initial condition x(0) = x_0, y(0) = y_0 and α(0) = 0. Therefore M_γ become a complete hypersurface in R^<p+q+2> with constant scalar curvature S.
Theorem Suppose that p > q + 1 and S > 0 (q 【greater than or equal】 2). Let 0 < x_0 【less than or equal】 √<(p -1)(q -1)/S> and 0 < y_0 【less than or equal】 √<q(q -1)/S>. Then the ODE system has a global solution γ(s) = (x(s), y(s)) ∈ R_+ × R_+ on (-∞, ∞) for the initial condition x(0) = x_0, y(0) = y_0 and α(0) = 0. Therefore M_γ become a complete hypersurface in R^<p+q+2> with constant scalar curvature S.
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