interpretation of holonomy


by lavinia
Tags: holonomy, interpretation
lavinia
lavinia is offline
#1
Dec25-12, 10:54 AM
Sci Advisor
P: 1,716
On an SO(2) bundle over a smooth manifold the holonomy around a closed curve that bounds a disk equals the integral of the curvature 2 form over the interior of the disk.

So holonomy measures curvature and visa vera.

More generally if two closed curves are homologous then the difference in their holonomy is equal to the total curvature of the surface that they mutually bound

What is the relationship of holonomy to curvature for higher dimensional principal budles e.g. SO(3) bundles?
Phys.Org News Partner Science news on Phys.org
Better thermal-imaging lens from waste sulfur
Hackathon team's GoogolPlex gives Siri extra powers
Bright points in Sun's atmosphere mark patterns deep in its interior
Ben Niehoff
Ben Niehoff is offline
#2
Dec25-12, 12:32 PM
Sci Advisor
P: 1,562
Try Googling "non-Abelian Stokes theorem". There is a sense in which you can define an area integral of the curvature 2-form such that it gives the holonomy around loops.


Register to reply

Related Discussions
The Holonomy Expansion for Hamiltonian in LQG Beyond the Standard Model 1
calculating the holonomy on a sphere Calculus & Beyond Homework 3
how is holonomy pronounced Differential Geometry 2
Holonomy, SO(6), SU(3) and SU(4) Differential Geometry 2