Differential Geometry: The Metric Curvature Formula

Last time, we explored two remarkable results in differential geometry: Gauss's Theorema Egregium and the Global Gauss–Bonnet Theorem.

We learned that Gaussian curvature is intrinsic—it can be determined entirely from measurements made within a surface. We also discovered that curvature, accumulated over an entire surface, reveals something about its global topology.

But Gauss's theorem leaves us with an important question:
If curvature can be determined from distances alone, how do we actually calculate it?

In Chapter 27 of Tristan Needham's Visual Differential Geometry and Forms, we will answer that question by developing the Metric Curvature Formula.

The metric tells us how to measure distances on a surface. Remarkably, knowing how those distances vary from place to place is enough to calculate the surface's Gaussian curvature.

Needham gives a geometric proof using ideas we have already developed: parallel transport, holonomy, and circulation. We will see how changes in the lengths of neighboring coordinate lines produce rotations, and how those rotations reveal curvature.

Two approaches to the same mathematics
We will also compare Needham's proof with the treatment in Manfredo do Carmo's Differential Geometry of Curves and Surfaces, §4-3.

Needham chooses orthogonal coordinates to make the geometry easier to visualize. Do Carmo works with general coordinates, using Christoffel symbols and compatibility equations.

Both approaches establish the same fundamental result: the metric completely determines Gaussian curvature.

This comparison raises an interesting question for our group:
What do we gain from a geometric proof, and what do we gain from a more general algebraic approach?

This is an important step toward higher-dimensional differential geometry. In Chapter 29, Needham will replace the single Gaussian curvature with the Riemann curvature tensor, preparing us for Einstein's description of gravity as curved spacetime.

Background: Familiarity with basic calculus, vectors, and the ideas of parallel transport and holonomy will be helpful. We will briefly review the necessary concepts. No prior experience with do Carmo's book is required.

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