How Does Holonomy Relate to Curvature in Higher Dimensional Principal Bundles?

In summary, the holonomy around a closed curve that bounds a disk on an SO(2) bundle over a smooth manifold is equal to the integral of the curvature 2-form over the interior of the disk. This means that holonomy is related to curvature and vice versa. For higher dimensional principal bundles, such as SO(3) bundles, there is a concept called the "non-Abelian Stokes theorem" that allows for defining an area integral of the curvature 2-form, which gives the holonomy around loops.
  • #1
lavinia
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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?
 
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  • #2
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.
 

FAQ: How Does Holonomy Relate to Curvature in Higher Dimensional Principal Bundles?

What is holonomy?

Holonomy refers to the concept in mathematics and physics that describes how a geometric object changes when it is parallel transported along a closed path. It is a way to measure the curvature of a space or manifold.

What is the significance of holonomy in physics?

Holonomy plays a significant role in understanding the behavior of physical systems, particularly in the field of general relativity. It is used to calculate the curvature of space-time and can help explain phenomena such as gravitational lensing and the rotation of planets around the sun.

How is holonomy related to gauge theory?

Holonomy is closely related to gauge theory, which is a mathematical framework used to describe the fundamental forces of nature. In gauge theory, holonomy is used to determine how a particle's properties change when it is transported along a closed path in a spacetime.

What are the different types of holonomy?

There are two main types of holonomy: parallel transport holonomy and curvature holonomy. Parallel transport holonomy describes how a geometric object changes when it is parallel transported along a closed path, while curvature holonomy measures the curvature of a space or manifold.

How is holonomy used in practical applications?

Holonomy has various practical applications in physics and engineering. For example, it is used in the construction of quantum computers, in the study of black holes, and in the development of new materials with specific properties. It also has applications in robotics, as it can be used to determine the optimal path for a robot to follow.

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