Geometry of almost Hermitian manifolds

几乎厄米流形的几何

• 批准号: 17540068
• 负责人: SATO Takuji
• 金额: $ 0.64万
• 依托单位: Kanazawa University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2005
• 资助国家: 日本
• 起止时间: 2005 至 2006
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-17540068/
• 关键词: almost Hermitian manifold; almost Kahler manifold; Kahler structure; holomorphic sectional curvature; tangent bundle

By using a 1-parameter family of symmetric affine connections so called α- connections in a statistical model in information geometry, we can introduce a 1-parameter family of almost Kahler structures on the tangent bundle over its statistical model.Especially, for the case of normal model and discrete distributions model, we studied the almost Kahler structures on then tangent bundles. Our main results are as follows:1. Almost Kahler structure on the tangent bundle is Kahler iff α = ± 1 (in this case, α-connection is flat.)2. For the normal model, when α =-1, the Kahler structures on the tangent bundle has constant holomorphic sectional curvature-2, so it is Einstein. On one hand, it is not Einstein when α =1.3. For the 2 dimensional discrete model case, when α =1, the Kahler structures on the tangent bundle has constant holomorphic sectional curvature 1, so it is Einstein. On one hand, it is not Einstein when α = -1. Further we obtain the followings as a generalization of these results:4. The almost Kahler structures on the tangent bundle defined by a-connections in the n-dimensional half space with Poincare metric become Kahler iff α=±1. When α =-1, the Kahler structure has constant holomorphic sectional curvature.5. The almost Kahler structures on the tangent bundle defined by α-connections in the positive part of n-dimensional sphere of radius c become Kahler iff α=±c^2. When α =c^2, the Kahler structure has constant holomorphic sectional curvature.6. The above result is still hold for the n-dimensional hyperbolic space.We hope that these results give new point of view in the relation of almost Hermitian geometry and information geometry.

通过在信息几何统计模型中使用对称仿射连接的 1 参数族（称为 α-连接），我们可以在其统计模型上的切丛上引入几乎卡勒结构的 1 参数族。特别是对于以下情况结合正态分布模型和离散分布模型，我们研究了切丛上的近似卡勒结构，主要结果如下： 1． （在这种情况下，α-连接是平坦的。）2.对于正常模型，当α=-1时，切丛上的卡勒结构具有恒定的全纯截面曲率-2，因此它是爱因斯坦的。当 α =1.3 时，不是爱因斯坦。对于二维离散模型情况，当 α =1 时，切丛上的卡勒结构具有恒定的全纯截面曲率 1，因此一方面，当 α = -1 时，我们得到以下结果作为这些结果的概括： 4．与庞加莱度量成为卡勒当且仅当α=-1时，卡勒具有结构常数全纯截面曲率。 5．半径为 c 的 n 维球体的正部分中的 α 连接变为 Kahler，当且仅当 α=±c^2 时，Kahler 结构具有恒定的全纯截面曲率。 6．我们希望这些结果能够为近厄米几何与信息几何的关系提供新的视角。

项目成果

On a family of almost Kahler structures on the tangent bundles over some statistical models

一些统计模型上切丛上的近卡勒结构族

• 发表时间: 2005
• 期刊: Topics in Almost Hermitian Geometry and Related Fields (Ed. by Y. Matsushita et al., World Scientific)
• 作者: Toshiaki;ADACHI;松木敏彦;Dai Tamaki;Toshiaki ADACHI;T.Matsuki;Dai Tamaki;Toshiaki ADACHI;Toshiaki ADACHI;Dai Tamaki;Dai Tamaki;Toshiaki ADACHI;Dai Tamaki;Toshiaki ADACHI;Dai Tamaki;Toshiaki ADACHI;Toshiaki ADACHI;Takuji Sato
• 通讯作者: Takuji Sato