From Synthetic to Metric Geometry

The synthetic pages treated geometry as a game of axioms about undefined "points" and "lines." Riemann's decisive reframing (1854) replaces axioms with measurement: a geometry is a smooth space together with a rule for measuring the length of infinitesimal displacements — a metric tensor. From that single object one recovers lengths, angles, areas, geodesics, and curvature, and the three classical geometries become merely the cases of constant curvature. This page introduces the metric-tensor language and shows how the Euclidean, spherical, and hyperbolic planes look in it. It presupposes the coordinatization that turns the synthetic plane into ${\mathbb{R}}^2$, and leans on the manifold and tangent-space machinery of topology-manifolds.md and the multivariable calculus of analysis/multivariable.md rather than re-deriving them.

References: do Carmo, Riemannian Geometry, ch. 1; Lee, Riemannian Manifolds, chs. 1–2; Riemann, On the Hypotheses which Lie at the Foundations of Geometry; O'Neill, Semi-Riemannian Geometry.

1. The Line Element

The primitive datum of metric geometry is not "distance between two points" but the infinitesimal line element $ds$ — the length of an infinitesimal step. In coordinates $x^1,\dots ,x^n$ on a smooth manifold,

$$d{s}^2=\sum _{i,j=1}^n{g}_{ij}(x)d{x}^{i}d{x}^j,$$

where $g_{ij}(x)$ is a symmetric, positive-definite matrix varying smoothly with position — the metric tensor (or first fundamental form). The einstein summation convention lets us write this as $d{s}^2=g_{ij}d{x}^{i}d{x}^j$. This is a field of inner products: at each point $p$ it equips the tangent space $T_{p}M$ with a length and angle.

The metric determines everything metric:

• Length of a curve $\gamma :[a,b]\to M$, $t\mapsto x(t)$: $$L(\gamma )={\int }_a^b\sqrt{g_{ij}(x(t)){\dot{x}}^i{\dot{x}}^j}dt.$$
• Angle between tangent vectors $u,v$ at a point: $$\cos \theta =\frac{g_{ij}{u}^i{v}^j}{\sqrt{g_{ij}{u}^i{u}^j}\sqrt{g_{ij}{v}^i{v}^j}}.$$
• Area / volume element: $dA=\sqrt{\det g}d{x}^1\cdots d{x}^n$ (the factor $\sqrt{\det g}$ is exactly the Jacobian-style correction that makes integration coordinate-independent).
• Distance between two points: the infimum of $L(\gamma )$ over all curves joining them — recovering a genuine metric-space distance from the infinitesimal data, connecting to metric spaces.

The conceptual inversion. Synthetic geometry starts with straight lines and defines length along them. Metric geometry starts with length (the tensor $g_{ij}$) and derives the straight lines as the length-minimizing curves — the geodesics. Everything is downstream of $g$.

2. The Euclidean Plane in Metric Form

In Cartesian coordinates the Euclidean plane has the constant identity metric $$d{s}^2=d{x}^2+d{y}^2,(g_{ij})=\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)={\delta }_{ij}.$$

Curve length reduces to the familiar $\int \sqrt{{\dot{x}}^2+{\dot{y}}^2}dt$, geodesics are straight lines, and the area element is $dA=dxdy$. The same geometry in polar coordinates $(r,\theta )$ reads

$$d{s}^2=d{r}^2+r^{2}d{\theta }^2,$$

with non-constant $g_{\theta \theta }=r^2$. This is a crucial lesson: the metric components depend on the coordinates, but the geometry does not. The plane is flat in both descriptions — flatness is a property of $g$ invariant under coordinate change, captured by the curvature, not by whether $g_{ij}$ happens to look constant.

3. The Sphere in Metric Form

The sphere of radius $R$, in latitude–longitude coordinates $(\theta ,\varphi )$ with $\theta$ the polar angle, inherits from ${\mathbb{R}}^3$ the metric

$$d{s}^2=R^2(d{\theta }^2+{\sin }^2\theta d{\varphi }^2).$$

Its area element $dA=R^2\sin \theta d\theta d\varphi$ integrates to $4\pi R^2$. The geodesics of this metric are the great circles, and — as curvature.md will compute — its Gaussian curvature is the constant $K=+1/R^2>0$.

4. The Hyperbolic Plane in Metric Form

The hyperbolic plane has no such embedding in ${\mathbb{R}}^3$ (Hilbert's theorem), so it is defined by a metric on one of the models. In the Poincaré upper half-plane,

$$d{s}^2=\frac{d{x}^2+d{y}^2}{y^2}(y>0),$$

and in the Poincaré disk,

$$d{s}^2=\frac{4(d{x}^2+d{y}^2)}{(1-x^2-y^2{)}^2}.$$

Both have constant Gaussian curvature $K=-1$. The metric factor blows up at the boundary, encoding that the edge is infinitely far away. That such a $g$ exists without any ambient space is the whole force of the metric viewpoint: hyperbolic geometry is a legitimate Riemannian manifold in its own right, needing no surface in ${\mathbb{R}}^3$ to host it.

5. Conformal Metrics and a Unifying Family

Notice a pattern. The Euclidean, spherical (stereographic), and hyperbolic (Poincaré) planes can all be written as a conformal factor times the flat metric on a piece of the plane,

$$d{s}^2=\lambda (x,y{)}^2(d{x}^2+d{y}^2),$$

with, respectively,

$${\lambda }_{\text{flat}}=1,{\lambda }_{\text{sphere}}=\frac{2R^2}{R^2+x^2+y^2},{\lambda }_{\text{hyp}}=\frac{2}{1-x^2-y^2}.$$

"Conformal" means angles are read off correctly (only lengths are rescaled), which is why the Poincaré models are conformal. These are the three simply-connected constant-curvature surfaces; the single Gaussian-curvature formula for a conformal metric,

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

returns $K=0,+1/R^2,-1$ on the three $\lambda$'s above — a compact preview of curvature.md and the classification in constant-curvature.md.

6. What the Metric Buys

The metric tensor $g_{ij}$ absorbs the entire synthetic apparatus into one smooth field and, crucially, makes curvature an intrinsic, computable quantity — a function of $g$ and its derivatives alone, with no reference to any embedding (Theorema Egregium). This is what lets us:

• treat all three classical geometries as one family indexed by $K$;
• allow curvature to vary from point to point, opening the door to general Riemannian geometry and the curved spacetime of general relativity;
• connect to the differential-forms and integration machinery of the analysis spine, through which the Gauss–Bonnet theorem ties curvature to topology.

The next page defines geodesics and curvature precisely and proves the Theorema Egregium; the page after uses them to classify the three geometries as the constant-curvature space forms.