Interpretation of Curvature

[lavinia]

On an oriented surface, the integral of the Gauss curvature over a smooth triangle can be interpreted as the angle of rotation of a vector that is parallel translated once around the three bounding edges.

How does one interpret the integral of the Gauss curvature of an arbitrary SO(20 bundle with connection over a triangle on a surface?

Does this question have a different answer if the associated vector bundle has a compatible Riemannian metric?

Are there special SO(2) bundles - other than the tangent bundle - where the integral has special meaning? How about on higher dimensionl manifolds?

 

[pat_connell]

The integral of the Gauss curvature over a triangle on a surface with an SO(2) bundle with connection does not have a straightforward interpretation. However, if the associated vector bundle has a compatible Riemannian metric, then the integral can be related to the holonomy of the connection around the triangle. In other words, the integral can be interpreted as a measure of how much the parallel-transported vector changes over the course of the translation. There are special SO(2) bundles for which the integral has special meaning. For example, in two dimensions, the integral of the Gauss curvature over a triangle is related to the total angle formed by the three sides of the triangle. On higher dimensional manifolds, the integral of the Gauss curvature can have a similar interpretation.