What becomes of geometry if space itself is curved?

That time, that place

To become a professor in Germany you had to give a habilitation lecture. The candidate proposes three topics and the committee picks one — conventionally the first.

Riemann proposed three. The first two were work he had already done; the third was filler, on the foundations of geometry.

On the committee sat Gauss, then seventy-seven. He chose the third.

Why this question

For two thousand years there was one geometry, Euclid's. Only the fifth postulate — that parallels never meet — looked less self-evident than the rest, and it was argued over for centuries.

Early in the nineteenth century Lobachevsky and Bolyai each showed that dropping it yields a geometry with no contradiction in it. Geometry was not one thing.

Riemann went further. Rather than build one more alternative, he built a general way to describe space however it curves.

The method: instead of one rule for the whole space, give each point its own rule for measuring distance. Where the rule varies from point to point, the space is curved. And there is no reason to stop at three dimensions.

What was found

The lecture was titled On the Hypotheses Which Lie at the Foundations of Geometry. Because it was an examination, he used almost no formulas and spoke it through. Most of the audience could not follow.

Gauss was struck. He is said to have talked about it excitedly on the way home — a man who rarely praised anyone.

Riemann added one thing at the end: what shape the space we live in actually has is not for mathematics to settle, but for physics to find out.

It took sixty-one years. In 1915 Einstein said gravity is not a force but curved space — and to write that down he needed a language for curvature. The language had been sitting there for sixty years, with nobody knowing what it was for.

Terms on this card

Riemannian geometry