How Does Gauss Curvature Integral Relate to Holonomy in SO(2) Bundles?

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SUMMARY

The integral of Gauss curvature over a smooth triangle on an oriented surface represents the angle of rotation of a vector parallel transported around the triangle's edges. This concept extends to arbitrary SO(20) bundles with connections, raising questions about interpretations when a compatible Riemannian metric is present. Additionally, the discussion explores whether there are special SO(2) bundles, beyond the tangent bundle, that provide unique insights into the integral's significance, particularly in higher-dimensional manifolds.

PREREQUISITES
  • Understanding of Gauss curvature and its geometric implications
  • Familiarity with vector bundles and connections in differential geometry
  • Knowledge of Riemannian metrics and their compatibility with vector bundles
  • Basic concepts of holonomy in the context of fiber bundles
NEXT STEPS
  • Research the properties of Gauss curvature in relation to vector bundles
  • Explore the role of Riemannian metrics in the context of SO(n) bundles
  • Investigate the implications of holonomy in higher-dimensional manifolds
  • Study special cases of SO(2) bundles and their geometric interpretations
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Mathematicians, differential geometers, and theoretical physicists interested in the relationships between curvature, holonomy, and vector bundles in various dimensions.

lavinia
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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?
 
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