Riemannian geometry over different normed division algebra

Details Riemannian geometry over different normed division algebra Riemannian geometry over different normed division algebra

2003-03-12

arxiv.org/abs/math/0303153v1

Mathematics

Differential Geometry

45 pages. To appear in Journal of Differential Geometry

Scientific paper

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 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.

Affiliated with

Also associated with

No associations

Rating

Riemannian geometry over different normed division algebra does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Riemannian geometry over different normed division algebra, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Riemannian geometry over different normed division algebra will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-562168