Differential Geometry: Curvature Without Looking Outside

About this event

Last time we discovered holonomy: parallel transport a vector around a closed loop on a curved surface, and it can return rotated. That gives us a way to detect curvature using only measurements made on the surface itself — without looking at it from the outside. But this raises a deeper question: Can we describe curvature completely from the inside, without knowing how the surface is embedded in the surrounding space? In Chapter 25 of Tristan Needham’s Visual Differential Geometry and Forms, we will use holonomy and the spherical (Gauss) map to reach one of differential geometry’s great results: Gauss’s Theorema Egregium. We have previously defined Gaussian curvature using the principal curvatures, K = κ₁κ₂ which seems to depend on how the surface bends in the surrounding three-dimensional space. Gauss’s remarkable result is that this same curvature can be determined entirely by measurements made within the surface. Curvature belongs to the geometry itself. Then in Chapter 26, we will ask a second question: How can local curvature add up to tell us something about the global shape and topology of an entire surface? Using holonomy, we will see an intrinsic route to the Global Gauss–Bonnet Theorem, one of the striking places where geometry and topology meet. This is also an important step on our road to the Einstein Field Equation. General Relativity does not describe spacetime as a surface bent inside some higher-dimensional space. Its curvature must be measurable from within spacetime itself. From here, Needham will lead us from intrinsic curvature to geodesic deviation, Riemann curvature, Ricci curvature, and ultimately Einstein’s description of gravity as curved spacetime. Background: You should be comfortable with basic calculus and vectors. Familiarity with parallel transport, holonomy, Gaussian curvature, and the main ideas of Chapter 24 will be helpful. We will briefly review what is needed from the previous meeting. No Noprior 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. Other events: https://www.meetup.com/physicswithfriends/events/