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Study of the structure of hypersurfaces with constant scalar curvature

常标量曲率超曲面结构的研究

基本信息

批准号：
15540057
负责人：
OKAYASU Takashi
金额：
$ 2.24万
依托单位：
IBARAKI UNIVERSITY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2003
资助国家：
日本
起止时间：
2003 至 2004
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-15540057/
关键词：
scalar curvature hypersurface ordinary differential equation 法曲率平均曲率

项目摘要

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.

在欧几里得空间中，具有正常数标量曲率的完全超曲面的已知示例仅是球体、广义圆柱体 S^p×R^<n-p> 和旋转超曲面。在本研究中，我们在欧几里得空间中构造了无穷多个具有恒定正标量曲率的完全超曲面的新例子。对于O(p+1)×O(q+1)不变超曲面，表示其标量曲率的方程为常数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))，其中α是切向量(x',y')与x轴之间的角度。本研究的重点是我们可以将该解决方案与另一个进行比较显式可积的 ODE。定理假设 p 【小于或等于】q + 1 且 S > 0（p 【大于或等于】2）。令 0 < x_0 【小于等于】 √<p(p -1)/S> 且 0 < y_0 【小于等于】 √<q(q -1)/S>。那么对于初始条件 x(0) = x_0, y(0) = y_0 和 α(0) = 0，ODE 系统在 (-∞, ∞) 上有一个全局解 γ(s) = (x(s), y(s)) ∈ R_+ × R_+。因此 M_γ 成为 R^<p+q+2> 中具有恒定标量曲率 S 的完全超曲面。定理假设 p > q + 1 且 S > 0 (q【大于或等于】2)。令 0 < x_0 【小于等于】 √<(p -1)(q -1)/S> 且 0 < y_0 【小于等于】 √<q(q -1)/S>。那么对于初始条件 x(0) = x_0、y(0) = y_0 和 α(0) = 0，ODE 系统在 (-∞, ∞) 上有一个全局解 γ(s) = (x(s), y(s)) ∈ R_+ × R_+。因此 M_γ 成为 R^<p+q+2> 中具有恒定标量曲率 S 的完全超曲面。