Gauss's Theorem and Geodesics

Theorema Egregium, Geodesics, and the Gauss–Bonnet Theorem

Gauss's "Remarkable Theorem"

In 1827, Gauss published a result that he himself called "Theorema Egregium" — "Remarkable Theorem". It is stated as follows: Gaussian curvature K is an intrinsic property of the surface.

What does this mean? K is computed solely through the coefficients of the first fundamental form E, F, G and their derivatives — without reference to the ambient space. Although the formula $K = (LN - M^2)/(EG - F^2)$ uses the second fundamental form (which is "extrinsic"), Gauss showed that by substituting explicit expressions in terms of E, F, G, the same answer is obtained.

Consequence: if two surfaces are isometric (can be identified preserving distances), they have the same K in corresponding points. Plane: K = 0 ↔ cylinder: K = 0 — isometric ✓. Plane: K = 0 ↔ sphere: K = $1/R^2$ — not isometric → no map without distortion.

Gauss's formula (Brioschi–Bourquet–Gauss): K can be expressed via Christoffel symbols, which themselves are expressed in terms of E, F, G.

Geodesics: Shortest Paths

A geodesic is a curve that is the shortest (locally) path between two points on a surface.

Geodesic equations: $u'' + \Gamma^1_{11}(u')^2 + 2\Gamma^1_{12}u'v' + \Gamma^1_{22}(v')^2 = 0$ and analog for $v''$.

Christoffel symbols $\Gamma^i_{jk} = \frac{1}{2}(\partial_i g_{kj} + \partial_j g_{ik} - \partial_k g_{ij})$ — expressed in terms of E, F, G.

On the sphere: Geodesics are arcs of great circles. The shortest route Moscow–New York does not run "horizontally", but as an arc via the North Atlantic (over Canada) — exactly a great circle.

On the cylinder: Geodesics are spirals. When the cylinder is unrolled, the spiral becomes a straight line (straight lines are geodesics in the plane).

Numerical example: Unit sphere, parameters $(\varphi, \theta)$. Christoffel symbols: $\Gamma^1_{22} = -\sin \varphi \cos \varphi$ (others are zero, except $\Gamma^2_{12} = \cot \varphi$). Geodesic equations: $\varphi'' = \sin \varphi \cos \varphi \cdot (\theta')^2$, $\theta'' = -2\cot \varphi \cdot \varphi' \theta'$. Solution with initial conditions — great circle.

Gauss–Bonnet Theorem

This is one of the greatest theorems in mathematics — a bridge between differential geometry and topology.

For a polygon D on a surface:

$ \iint_D K, dA + \oint_{\partial D} \kappa_g, ds + \sum_i \theta_i = 2\pi $

Here: K — Gaussian curvature; $dA = \sqrt{EG-F^2} du dv$ — area element; $\kappa_g$ — geodesic curvature of the boundary (how the boundary "deviates" from a geodesic); $\theta_i$ — exterior angles at the vertices ($\pi$ minus the angle between adjacent edges).

For a closed surface:

$ \iint_M K, dA = 2\pi \chi(M) $

where $\chi(M) = V - E + F$ — Euler characteristic (a topological invariant!).

Numerical examples:

Sphere ($\chi = 2$): $K = 1/R^2$. $\iint K, dA = (1/R^2)\cdot 4\pi R^2 = 4\pi = 2\pi \cdot 2$ ✓.

Torus ($\chi = 0$): $\iint K, dA = 0$. Indeed: the torus has both elliptic (K > 0) and hyperbolic (K < 0) points, and the total is zero.

Double torus ($g = 2$, $\chi = -2$): $\iint K, dA = -4\pi < 0$.

Application: sum of angles of a triangle on the sphere. For a geodesic triangle ($\kappa_g = 0$ on the edges), three vertices: $\sum \theta_i = \pi - $ (sum of triangle angles $-$ $\pi$) — actually, the formula is:

Sum of the angles of a triangle on the sphere $= \pi + K \cdot S$, where S is the area of the triangle.

Example: Spherical triangle with vertices at the North Pole, at the equator at $0^\circ$ and at $90^\circ$ longitude. All three angles are right angles ($90^\circ$), sum $= 270^\circ = \pi/2 + \pi/2 + \pi/2 = 3\pi/2$. By the formula: $\pi + (1/R^2) \cdot (\pi R^2/2) = \pi + \pi/2 = 3\pi/2$ ✓.

Geodesics in General Relativity

In GR, freely falling bodies (without non-gravitational forces) move along geodesics of spacetime. The spacetime metric $g_{ij}$ — analogue of the first fundamental form. Einstein's equations: $R_{ij} - (1/2)R g_{ij} = 8\pi G T_{ij}$. Ricci tensor $R_{ij}$ — "contraction" of the Riemann curvature tensor — is computed from $g_{ij}$. Matter (T) determines geometry (R) → geodesics → motion of matter.

Gauss–Bonnet Theorem in Physics and Technology

The Gauss–Bonnet theorem connects geometry and topology so fundamentally that its consequences appear in a wide variety of fields. In condensed matter physics, the Chern number — a topological invariant, analogue of the Euler characteristic for fiber bundles — determines the quantization of Hall conductivity in the quantum Hall effect. Experimentally measured conductivity takes strictly quantized values because it is proportional to an integer topological invariant — the integral of curvature over the band structure of the material. This is a direct analogue of the Gauss–Bonnet theorem: the integral of curvature over a closed manifold equals a topological number. In navigation of drones and autonomous vehicles, path planning algorithms use geodesic curves: the optimal route across a bounded surface (e.g., mountain terrain) is a geodesic in the metric of the surface taking altitude profile into account. Computer vision tools use discrete analogues of Gaussian curvature (angular deficit at polyhedron vertices) to analyze object shape — classification of biological structures, gesture recognition, reconstruction of 3D scenes from point clouds.

Geodesics arise in routing tasks across 3D landscapes: Dijkstra's algorithm on discrete grids approximates geodesics. Sethian's Fast Marching Method numerically solves the Eikonal equation $|\nabla d| = 1/c$, equivalent to the geodesic equation in a Riemannian metric, and is used in robot path planning, computational anatomy, and processing of 3D medical images. The discrete Gauss–Bonnet theorem (Desargues's formula for polyhedra) forms the foundation of shape analysis tools in CGAL and libigl software packages.

Assignment: (a) For the unit sphere: find the curvature of the geodesic circle (parallel $\varphi = \varphi_0$) as a curve on the sphere, using $\kappa_g$. (b) Gauss–Bonnet theorem for a triangle: three geodesic arcs on the unit sphere with angles $\alpha, \beta, \gamma$. Prove: $\alpha + \beta + \gamma = \pi + S$ (S is the area). (c) Torus $T^2$: $\chi = 0$. Find the areas of the elliptic and hyperbolic regions and show that they are equal.