Riemannian geometry over different normed division algebra

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dc.creatorLeung, Naichung Conan
dc.date2003-03-12
dc.date.accessioned2026-07-07T06:32:55Z
dc.date.available2026-07-07T06:32:55Z
dc.descriptionWe develop a unifed theory to study geometry of manifolds with different holonomy groups. They are classified by (1) real, complex, quaternion or octonion number they are defined over 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, Hyperkahler and G_2-manifolds respectively. For vector bundles over such manifolds, we introduce (special) A-connections. They include holomorphic, Hermitian Yang-Mills, Anti-Self-Dual and Donaldson-Thomas connections. Similarly we introduce (special) A/2-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) A/2-Lagrangian geometry on mirror special A-manifolds.
dc.description45 pages. To appear in Journal of Differential Geometry
dc.identifierhttps://arxiv.org/abs/math/0303153
dc.identifierhttp://arxiv.org/abs/math/0303153
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/99019
dc.subjectDifferential Geometry
dc.subjectAlgebraic Geometry
dc.subjectSymplectic Geometry
dc.titleRiemannian geometry over different normed division algebra
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