## Journal of Differential Geometry

- J. Differential Geom.
- Volume 61, Number 2 (2002), 289-333.

### Riemannian Geometry Over Different Normed Division Algebras

#### Abstract

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.

#### Article information

**Source**

J. Differential Geom., Volume 61, Number 2 (2002), 289-333.

**Dates**

First available in Project Euclid: 20 July 2004

**Permanent link to this document**

https://projecteuclid.org/euclid.jdg/1090351387

**Digital Object Identifier**

doi:10.4310/jdg/1090351387

**Mathematical Reviews number (MathSciNet)**

MR1972148

**Zentralblatt MATH identifier**

1070.53024

#### Citation

Leung, Naichung Conan. Riemannian Geometry Over Different Normed Division Algebras. J. Differential Geom. 61 (2002), no. 2, 289--333. doi:10.4310/jdg/1090351387. https://projecteuclid.org/euclid.jdg/1090351387