Differential Geometry: Holonomy — Curvature on the Road to Einstein

Last meeting we developed an intrinsic way to parallel transport tangent vectors across a curved surface. This meeting asks the natural next question: what happens if we carry a vector all the way around a closed loop and bring it back to where it started?

On a curved surface, it may return pointing in a different direction. That net rotation is called holonomy.

In Chapter 24 of Tristan Needham’s Visual Differential Geometry and Forms, we will begin with a striking example on the sphere, then extend the idea to general geodesic triangles. We will see how holonomy connects parallel transport, angular excess, and Gaussian curvature, and why holonomy is additive when a region is subdivided.

This is also an important step in our longer goal: following Needham’s geometric path toward the Einstein Field Equation. The chapters ahead move from holonomy and curvature to geodesic deviation, Riemann curvature, Ricci curvature, and finally Einstein’s description of gravity as curved spacetime.

Background: You should be comfortable with basic calculus and vectors. Familiarity with tangent planes, geodesics, Gaussian curvature, and parallel transport will help. We will briefly review what is needed from Chapter 23. No prior tensor calculus or General Relativity is required.

This event is part of the Physics With Friends community, which hosts collaborative study groups in physics and mathematics.

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