Chapter 9: Riemann & the Geometry of Space
1854
The Habilitation Lecture
On June 10, 1854, Bernhard Riemann (1826–1866) delivered his Habilitation lecture at Göttingen: Über die Hypothesen, welche der Geometrie zu Grunde liegen(“On the Hypotheses That Lie at the Foundations of Geometry”). In the audience sat the 77-year-old Gauss, who was “profoundly impressed.”
In a single lecture, Riemann:
- • Generalized geometry to n dimensions
- • Introduced the concept of a manifold (Mannigfaltigkeit)
- • Defined curvature that can vary from point to point
- • Introduced the metric tensor \(g_{\mu\nu}\)
- • Speculated that the geometry of physical space is determined by physical forces
This lecture, containing almost no formulas, is the most influential single presentation in the history of mathematics.
The Metric Tensor
Riemann's central idea: distance in a curved space is measured by the metric tensor:
\( ds^2 = \sum_{\mu,\nu} g_{\mu\nu}\, dx^\mu\, dx^\nu \)
In flat space, \(g_{\mu\nu} = \delta_{\mu\nu}\) and we recover\(ds^2 = dx^2 + dy^2 + dz^2\). On a sphere of radius \(R\),\(ds^2 = R^2(d\theta^2 + \sin^2\theta\,d\phi^2)\). The metric encodes all geometric information: distances, angles, volumes, curvature.
Bridge to General Relativity
Sixty years later, Einstein adopted Riemann's metric tensor as the dynamical variable of gravity. The spacetime metric \(ds^2 = g_{\mu\nu}dx^\mu dx^\nu\) (with signature \((-,+,+,+)\)) determines how objects move, how clocks tick, and how light bends. Riemann's “pure mathematics” became the geometry of the cosmos.
The Riemann Curvature Tensor
Riemann showed that curvature in \(n\) dimensions is encoded in a tensor with\(\frac{n^2(n^2-1)}{12}\) independent components:
\( R^\rho{}_{\sigma\mu\nu} = \partial_\mu\Gamma^\rho_{\nu\sigma} - \partial_\nu\Gamma^\rho_{\mu\sigma} + \Gamma^\rho_{\mu\lambda}\Gamma^\lambda_{\nu\sigma} - \Gamma^\rho_{\nu\lambda}\Gamma^\lambda_{\mu\sigma} \)
In 2 dimensions: 1 independent component (Gaussian curvature). In 4-dimensional spacetime: 20 independent components. This tensor measures how parallel transport around a closed loop rotates a vector — the mathematical definition of curvature that Einstein used to describe gravity.
Riemann's Vision
In his lecture, Riemann made a prophetic statement:
“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. [...] Either the reality which underlies space must form a discrete manifoldness, or we must seek the ground of its metric relations outside it, in binding forces which act upon it.”
Riemann anticipated both directions of 20th-century physics: that the geometry of space is determined by physical forces (general relativity), and that space might be discrete at small scales (quantum gravity). He died in 1866 at age 39, having reshaped both mathematics and the future of physics.
From Riemann to Einstein: The 60-Year Bridge
The path from Riemann (1854) to Einstein (1915):