Differential Geometry: Intrinsic Parallel Transport and Holonomy

Tonight: Chapter 23 — Intrinsic Constructions

Tonight we continue Act IV: Parallel Transport with Chapter 23 of Tristan Needham’s Visual Differential Geometry and Forms, with the possibility of beginning Chapter 24 if time permits.
In Chapter 22, parallel transport was constructed extrinsically: we used the surrounding three-dimensional space and continually projected a moving vector back into the tangent plane. Chapter 23 asks the natural next question:
Can parallel transport be defined using only the intrinsic geometry of the surface?
We will follow Needham’s construction from geodesics to the covariant derivative and ultimately to the geodesic equation.

We will explore:
• How parallel transport can be recognized intrinsically along a geodesic
• How the construction extends from geodesics to arbitrary curves
• Why tangent vectors at different points cannot be compared directly because they belong to different tangent planes
• How parallel transport gives us a way to compare those vectors
• How that comparison leads to Needham’s intrinsic derivative, more commonly called the covariant derivative
• Why D_v w = 0 means that w is parallel transported
• How the covariant derivative extracts the intrinsic, tangential change of a vector field
• Why D_v v is the geodesic-curvature vector
• The geodesic equation D_v v = 0

The conceptual payoff is a particularly elegant characterization:
A geodesic parallel transports its own tangent vector.

If time permits, we will then begin Chapter 24: Holonomy, asking what happens when a vector is parallel transported all the way around a closed loop and returns to its starting point.
That leads to a new connection between parallel transport, angular excess, and Gaussian curvature.

We will use visual demonstrations, discussion, and Mentimeter questions throughout rather than treating this as a formal lecture.
No prior knowledge beyond the material we have already covered in the group is required.