Riemannian Geometry Over Different Normed Division Algebras
Riemannian Geometry Over Different Normed Division Algebras
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DOI：
10.4310/jdg/1090351387
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发表时间：
2002-06
影响因子：
2.5
通讯作者：
N. Leung
中科院分区：
文献类型：
--
作者：
N. Leung
我们发展了一种统一理论来研究具有不同完整群的流形的几何。它们按(1)实数、复数、四元数或八元数(在适当的情况下)和(2)是否特殊来分类。专长是关于相应赋范代数A的定向。例如，特殊的黎曼A-流形分别是定向的黎曼流形、Calabi-Yau流形、Hyperkähler流形和G2-流形。对于这类流形上的向量丛，我们引入了(特殊的)A联络。它们包括全纯连接、厄米特杨-米尔斯连接、反自我双重连接和唐纳森-托马斯连接。类似地，我们引入(特殊)12A-拉格朗日子流形作为极大实子流形。它们包括(特殊的)拉格朗日流形、复拉格朗日流形、Cayley流形和(余)结合子流形。我们还从这个角度讨论了几何对偶：平面环面上的A几何上的傅里叶变换和镜像特殊A-流形上从(特殊)A-几何到(特殊)12A-拉格朗日几何的猜想Syz镜像变换。
We develop a unifed theory to study geometry of manifolds with different holonomy groups. They are classified by (1) real, complex, quaternion or octonion number (in the appropriate cases) and (2) being special or not. Specialty is an orientation with respect to the corresponding normed algebra A. For example, special Riemannian A-manifolds are oriented Riemannian, Calabi-Yau, hyperkähler and G2-manifolds respectively. For vector bundles over such manifolds, we introduce (special) Aconnections. They include holomorphic, Hermitian Yang-Mills, Anti-SelfDual and Donaldson-Thomas connections. Similarly we introduce (special) 1 2 A-Lagrangian submanifolds as maximally real submanifolds. They include (special) Lagrangian, complex Lagrangian, Cayley and (co-)associative submanifolds. We also discuss geometric dualities from this viewpoint: Fourier transformations on A-geometry for flat tori and a conjectural SYZ mirror transformation from (special) A-geometry to (special) 1 2 A-Lagrangian geometry on mirror special A-manifolds.