The question of the validity of the hypotheses of geometry in the infinitely small is bound up with the question of the ground of the metric relations of space.

— Bernhard Riemann, On the Hypotheses which lie at the Bases of Geometry (1854)

On 10 June 1854, Riemann presented to the Göttingen faculty the lecture On the Hypotheses which lie at the Bases of Geometry. He wrote it for an audience that included philosophers and historians, almost without formulas. The text was only published in 1868, after his death, and it changed geometry.

Geometry from the inside

The starting point was Gauss’s Theorema Egregium: the curvature of a surface can be measured by someone living inside it, just by measuring distances, without looking from outside. A two-dimensional being on a sphere, drawing triangles and measuring their angles, would discover that its surface is curved.

Riemann took the idea to its limit. Instead of surfaces inside ordinary space, he thought of manifolds (Mannigfaltigkeiten): spaces of any number of dimensions, described by coordinates $x_1, x_2, \ldots, x_n$, with no need to be embedded in anything larger. What gives a manifold its geometry is the rule for measuring the distance between very close points.

The metric

In the ordinary plane, the distance $ds$ between neighbouring points is given by Pythagoras’s theorem: $ds^2 = dx^2 + dy^2$. Riemann proposed that, in general, the infinitesimal distance be given by a more general quadratic expression:

$$ds^2 = \sum_{i,j} g_{ij}(x)\,dx_i\,dx_j,$$

where the coefficients $g_{ij}$ may change from one point to another. These coefficients form the metric. On the surface of a sphere of radius $R$, at latitude $\varphi$ and longitude $\lambda$, for example,

$$ds^2 = R^2\,d\varphi^2 + R^2\cos^2\varphi\,d\lambda^2:$$

a degree of longitude is worth less near the poles, where $\cos\varphi$ is small.

From the metric, Riemann defined the curvature at each point and in each direction. The spaces of constant curvature are three: zero curvature (Euclid’s geometry), positive (like that of the sphere, where there are no parallels) and negative (hyperbolic geometry, discovered by Lobachevsky and Bolyai, where there are infinitely many parallels).

And real space?

At the end of the lecture, Riemann made a bold remark: we do not know in advance what the geometry of physical space is. It cannot be decided by logic, only by experience, and it may depend on the “forces” acting on bodies. On very small or very large scales, Euclid’s geometry might cease to hold.

Sixty years later, Albert Einstein found in Riemann’s geometry exactly the tool he needed. In general relativity, of 1915, spacetime is a four-dimensional manifold with a metric $g_{\mu\nu}$, and gravity is not a force but the curvature of that metric, produced by matter and energy.

Exercise: a triangle on the sphere

On the Earth, consider the triangle with one vertex at the North Pole and the other two on the equator, at longitudes 0° and 90°. What is the sum of its angles? How is it related to the area?

On the equator, the meridians make angles of 90° with the equator. At the pole, the angle between the meridians 0° and 90° is 90°. The sum is 270°, that is, $\frac{3\pi}{2}$, not $\pi$. The excess $\frac{3\pi}{2} - \pi = \frac{\pi}{2}$ multiplied by $R^2$ gives the area of the triangle: $\frac{\pi}{2}R^2$, which is exactly $\frac{1}{8}$ of the surface $4\pi R^2$. On a surface of constant curvature $\frac{1}{R^2}$, the angular excess is the area divided by $R^2$.

Exercise: the metric of the sphere

Use the metric $ds^2 = R^2\,d\varphi^2 + R^2\cos^2\varphi\,d\lambda^2$ to compute the length of the parallel at latitude 60°.

Along the parallel, $\varphi$ is constant, so $d\varphi = 0$ and $ds = R\cos\varphi\,d\lambda$. Going all the way round, $\lambda$ runs from 0 to $2\pi$: the length is $2\pi R\cos 60° = \pi R$, half the equator. With $R = 6371$ km, that is about 20 015 km.

Sources: Bernhard Riemann, “Über die Hypothesen, welche der Geometrie zu Grunde liegen”, Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen 13 (1868) · William Kingdon Clifford (trans.), “On the Hypotheses which lie at the Bases of Geometry”, Nature 8 (1873) · Jürgen Jost (ed.), Bernhard Riemann: On the Hypotheses Which Lie at the Bases of Geometry (2016) · Max Jammer, Concepts of Space (1954).

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