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June, 2002 Riemannian Geometry Over Different Normed Division Algebras
Naichung Conan Leung
J. Differential Geom. 61(2): 289-333 (June, 2002). DOI: 10.4310/jdg/1090351387


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 $\mathbb{A}$. For example, special Riemannian $\mathbb{A}$-manifolds are oriented Riemannian, Calabi-Yau, hyperkähler and $G_2$-manifolds respectively.

For vector bundles over such manifolds, we introduce (special) $\mathbb{A}$-connections. They include holomorphic, Hermitian Yang-Mills, Anti-Self-Dual and Donaldson-Thomas connections. Similarly we introduce (special) $\frac{1}{2}\mathbb{A}$-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 $\mathbb{A}$-geometry for flat tori and a conjectural SYZ mirror transformation from (special) $\mathbb{A}$-geometry to (special) $\frac{1}{2} \mathbb{A}$-Lagrangian geometry on mirror special $\mathbb{A}$-manifolds.


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Naichung Conan Leung. "Riemannian Geometry Over Different Normed Division Algebras." J. Differential Geom. 61 (2) 289 - 333, June, 2002.


Published: June, 2002
First available in Project Euclid: 20 July 2004

zbMATH: 1070.53024
MathSciNet: MR1972148
Digital Object Identifier: 10.4310/jdg/1090351387

Rights: Copyright © 2002 Lehigh University


Vol.61 • No. 2 • June, 2002
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