Studies of glued Riemannian manifolds

粘合黎曼流形的研究

• 批准号: 13640068
• 负责人: INNAMI Nobuhiro
• 金额: $ 2.18万
• 依托单位: Niigata University
• 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 2004
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-13640068/
• 关键词: geometru of geodesics; Riemannian geometry; 平面凸ビリヤード; 測地線の幾何; 凸ビリヤード問題; リーマン多様体; 張り合わせ多様体; 測地線

We find what rendition on gradient vector fields characterizes warped products, Riemannian products and round spheres. To do this we apply the theory of Jacobi equations without conjugate points to the differential maps of the local one-parameter groups generated by gradient vector fields.We say that a manifold M is a glued manifold if M is a union of complete connected manifolds which are glued at their boundary Geodesies in a glued Riemannian manifold M are by definition locally minimizing curves in M. The variation vector fields through geodesies satisfy the Jacobi equation in each component manifold In this project we find the equation which show how Jacobi vector fields change in passing across the boundary of a component manifold into the neighboring component As an application we characterize glued Riemannian manifolds whose glued boundary separates conjugate points.Circles and Ellipses has been characterized by some properties of billiard ball trajectories. Those properties have been discussed in connection with the characterization of flat metrics on tori by some families of geodesics and tori of revolution. The main method is the geometry of geodesies due to H.Busemann which was reconstructed in the configuration space by V.Bangert. In particular, the theory of parallels plays an important role in this work. Roughly speakin, there exists a foliation of parallels in the configuration space for billiards if and only if there exists a foliation of non-null homotopic curves in the phase space which is invariant under the billiard ball map.

我们发现什么梯度向量场的特点，翘曲产品，黎曼产品和圆形领域。为此，我们将Jacobi方程的无共轭点理论应用于由梯度向量场生成的局部单参数群的微分映射。我们说流形M是胶合流形，如果M是在其边界胶合的完全连通流形的并。胶合黎曼流形M中的测地线定义为M中的局部极小曲线。通过测地线的变分向量场在每个分支流形上满足Jacobi方程在这个项目中我们找到了Jacobi向量场在穿过分支流形的边界进入相邻分支流形时如何变化的方程作为应用我们刻画了其胶合边界分离共扼点的胶合黎曼流形圆和椭圆已经用台球轨迹的一些性质刻画了.这些性质已经讨论了一些家庭的测地线和旋转环面的环面上的平坦度量的特征。主要的方法是由H.Busemann的测地线几何，它是由V.Bangert在位形空间中重建的。特别是，平行线理论在这项工作中发挥了重要作用。粗略地说，在台球的位形空间中存在平行线的叶状当且仅当在相空间中存在在台球映射下不变的非零同伦曲线的叶状。

项目成果

SEKIGWA, Kouei: "On 4-dimensional CR-submantfolds of a 6-dimensional sphere"Advanced Studies in Pure Mathematics. 34. 143-154 (2002)

SEKIGWA, Kouei：“关于 6 维球体的 4 维 CR-submantfolds”纯数学高级研究。

Innami, Nobuhiro: "Jacobi vector fields along geodesics in glued Riemannian manifolds"Nihonkai Math. J. 12・1. 101-112 (2001)

Innami，Nobuhiro：“粘连黎曼流形中沿测地线的雅可比矢量场”Nihonkai Math. 12・1（2001）。

Notes on strictly almost Kahler Einstein manifolds of dimension four

关于四维严格几乎卡勒爱因斯坦流形的注释

• 发表时间: 2004
• 期刊: Yokohama Math.J. 51
• 作者: Y.Matsushita;S.Haze;P.Law;J.Davidov;K.Seigawa;K.Hasegawa;H.Hashimoto;K.Hasegawa;K.Sekigawa;T.Nihonyanagi;N.Innami;K.Hasegawa;H.Hashimoto;Y.Matsushita;H.Hashimoto;T.Nihonyanagi;M.Chaichi;T.Oguro
• 通讯作者: T.Oguro

Jacobi vector fields along geodesics in glued Riemannian manifolds

粘连黎曼流形中沿测地线的雅可比向量场

• 发表时间: 2001
• 期刊: Nihonkai Math.J. 12
• 作者: T.Oguro;Nobuhiro Innami;Nobuhiro Innami
• 通讯作者: Nobuhiro Innami

Gradient vector fields which characterize warped products

表征翘曲产品的梯度矢量场

• 发表时间: 2001
• 期刊: Math.Scand. 88
• 作者: T.Oguro;Nobuhiro Innami;Nobuhiro Innami;Nobuhiro Innami
• 通讯作者: Nobuhiro Innami