Geodesics and Curvature

The metric tensor $g_{ij}$ encodes lengths and angles; from it flow the two central objects of differential geometry — geodesics, the straight lines of a curved space, and curvature, the local, intrinsic measure of how the space bends. This page defines both, states Gauss's Theorema Egregium (curvature is detectable within the surface, without any surrounding space), and presents the Gauss–Bonnet theorem linking total curvature to topology. It stays deliberately motivational — connections and curvature tensors are introduced to the depth the constant-curvature classification needs, not developed in full generality. It builds on tangent spaces and the differential forms of the analysis spine.

References: do Carmo, Differential Geometry of Curves and Surfaces, ch. 4; do Carmo, Riemannian Geometry, chs. 2–4; Lee, Riemannian Manifolds; Spivak, A Comprehensive Introduction to Differential Geometry, vol. 2.

1. Geodesics: Straightest and Shortest

A geodesic generalizes "straight line" to a curved space. Two equivalent characterizations:

• Shortest (variational): a geodesic is a critical curve of the length functional $L(\gamma )=\int \sqrt{g_{ij}{\dot{x}}^i{\dot{x}}^j}dt$ — locally the shortest path between its endpoints (a taut string).
• Straightest (parallel-transport): a geodesic is a curve whose tangent vector does not turn — its acceleration has no component along the surface.

In coordinates, both give the geodesic equation

$${\ddot{x}}^k+{\Gamma }_{ij}^k{\dot{x}}^i{\dot{x}}^j=0,$$

where the Christoffel symbols are built from the metric and its first derivatives,

$${\Gamma }_{ij}^k=\frac{1}{2}{g}^{k\ell }({\partial }_i{g}_{j\ell }+{\partial }_j{g}_{i\ell }-{\partial }_{\ell }{g}_{ij}),$$

with $g^{k\ell }$ the inverse metric. The ${\Gamma }_{ij}^k$ define the Levi-Civita connection — the unique way to differentiate vector fields that is compatible with $g$ (lengths are preserved under parallel transport) and torsion-free (symmetric in $i,j$). Reading off the examples of metric-geometry.md:

• flat metric $d{x}^2+d{y}^2$: all $\Gamma =0$, geodesics are straight lines;
• sphere $R^2(d{\theta }^2+{\sin }^2\theta d{\varphi }^2)$: geodesics are great circles;
• half-plane $y^{-2}(d{x}^2+d{y}^2)$: geodesics are vertical rays and semicircles meeting the axis orthogonally.

2. Parallel Transport and Holonomy

The connection $\Gamma$ lets us parallel transport a vector along a curve — carry it "without turning," keeping it at constant length and constant angle to the curve's tangent. On a flat plane, transporting a vector around a closed loop returns it unchanged. On a curved surface it comes back rotated: the angle of rotation, the holonomy, is a direct manifestation of curvature.

Curvature as loop holonomy. Parallel transport a vector around a small closed loop enclosing area $A$. It returns rotated by an angle $$\Delta \phi =K\cdot A+o(A),$$ where $K$ is the Gaussian curvature at the point. Curvature is the infinitesimal holonomy per unit area.

The classic illustration: transport a vector around a spherical triangle with three right angles (an octant); it returns rotated by $\pi /2$, exactly $K\cdot A=\frac{1}{R^2}\cdot \frac{\pi R^2}{2}$. This is the same spherical excess seen from the connection side, and it is the local seed of Gauss–Bonnet (§4).

3. Gaussian Curvature and the Theorema Egregium

For a surface, curvature is captured by a single number at each point, the Gaussian curvature $K$. Gauss originally defined it extrinsically, via the embedding in ${\mathbb{R}}^3$: if a surface has principal curvatures ${\kappa }_1,{\kappa }_2$ (the max and min bending of normal sections), then

$$K={\kappa }_1{\kappa }_2.$$

A sphere has $K=1/R^2>0$ (curves the same way in all directions); a plane has $K=0$; a saddle has $K<0$ (curves oppositely in perpendicular directions) — the hyperbolic case. Gauss's astonishing discovery was that this product, defined via the embedding, is in fact intrinsic:

Theorema Egregium (Gauss, 1827). The Gaussian curvature $K$ depends only on the metric tensor $g_{ij}$ and its derivatives — not on how (or whether) the surface is embedded in an ambient space. Isometric surfaces have equal curvature at corresponding points.

Consequences. A sheet of paper ($K=0$) can be rolled into a cylinder or cone (still $K=0$) but never wrapped onto a sphere ($K\ne 0$) without stretching — which is why every flat map of the Earth distorts. And it is why the hyperbolic plane, with $K=-1$, is a self-contained geometry needing no embedding: curvature lives in the metric, and the metric lives on the manifold alone.

For a conformal metric $d{s}^2={\lambda }^2(d{x}^2+d{y}^2)$ the intrinsic formula is compact:

$$K=-\frac{1}{{\lambda }^2}\Delta \ln \lambda ,\Delta ={\partial }_x^2+{\partial }_y^2,$$

which returns $0$, $+1/R^2$, $-1$ on the flat, spherical, and hyperbolic $\lambda$'s.

Higher dimensions (pointer). In dimension $>2$ curvature is no longer one number but the Riemann curvature tensor $R^i{}_{jk\ell }$, built from the connection; its contractions give the Ricci tensor $R_{ij}$ and scalar curvature $R$. The 2-D Gaussian curvature is the special case $K=R/2$. These are the objects Einstein's equation constrains in general relativity; here we need only the surface case.

4. The Gauss–Bonnet Theorem

Gaussian curvature is a local quantity, but integrating it over a whole surface produces a topological invariant — one of the most beautiful theorems in mathematics.

Local Gauss–Bonnet. For a geodesic triangle $T$ with interior angles $\alpha ,\beta ,\gamma$, $${\iint }_{T}KdA=(\alpha +\beta +\gamma )-\pi .$$

The right side is exactly the spherical excess / negative hyperbolic defect. So the defect/excess laws of the synthetic pages are special cases of Gauss–Bonnet: when $K$ is the constant $\pm 1/R^2$, the integral is $\pm A/R^2$, giving $\mathrm{Area}=R^2|\Sigma -\pi |$. This is the promised unification — the synthetic angle-sum theorems and the metric curvature are the same fact.

Summing over a triangulation and using the additivity of angles gives the global form:

Global Gauss–Bonnet. For a compact oriented surface $M$ without boundary, $${\iint }_{M}KdA=2\pi \chi (M),$$ where $\chi (M)=2-2g$ is the Euler characteristic ($g$ = genus).

Total curvature is thus a topological constant: the sphere ($\chi =2$) has $\iint KdA=4\pi$ no matter how you dent it, and any $K$ you add somewhere is subtracted elsewhere. A surface with $K\equiv 0$ everywhere must have $\chi =0$ (torus); a surface with $K>0$ everywhere must be a sphere. This is the deepest statement of "curvature is constrained by topology," and it uses the integration of forms and Stokes' theorem from the analysis spine as its engine ($KdA$ is, up to the connection 1-form, an exact form off the vertices).

5. Toward the Classification

Two facts assembled here drive the next page:

1. Curvature $K$ is intrinsic (Theorema Egregium) — a coordinate-free scalar attached to the metric, so "$K=0$, $K<0$, $K>0$" is a genuine classification of geometries, not an artifact of description.
2. Gauss–Bonnet ties the angle sum to $\iint KdA$, so the synthetic trichotomy (angle sum $\lessgtr \pi$) is identical to the sign of the curvature.

When $K$ is forced to be constant — the geometry looks the same at every point and in every direction (maximal symmetry) — exactly three simply-connected geometries survive, one for each sign. Those are the Euclidean, spherical, and hyperbolic planes, now revealed as siblings. That is the content of constant-curvature.md.