Geometry of manifolds and curvature (Riemann)
A geometry in which space is treated as an n-dimensional manifold with a metric specified separately from the manifold itself and settled by experience rather than deduced from the nature of extension.
It was built by the German Bernhard Riemann (inaugural lecture 1854, published posthumously in 1868). Carrying Gauss's curvature of surfaces over to manifolds of any number of dimensions, he obtained a classification of spaces of constant curvature in which Euclidean (zero), Lobachevskian (negative) and elliptic (positive) space are special cases of one scheme.
A particular counterexample to the fifth postulate turned into one point in a continuous family of geometries, and the question of the geometry of the world became a question about the value of a curvature function. It is in this language that general relativity would be written half a century later.
Why this date
The date follows the first recorded presentation: the trial lecture 'Über die Hypothesen, welche der Geometrie zu Grunde liegen', delivered at Göttingen on 10 June 1854. Sources differ on the year of printing: the Stanford Encyclopedia of Philosophy and the Dictionary of Scientific Biography give 1867, MacTutor 1868 (volume 13 of the Göttingen Abhandlungen, dated 1867 and issued in 1868). The text was not published in Riemann's lifetime.
Other links
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Non-Euclidean geometry: denial of the fifth postulate
parallel to · Confidence: medium
Riemann independently arrived at spaces of constant negative curvature and absorbed Lobachevskian geometry as a special case of a general scheme of manifolds, although historians do not confirm any direct influence of Lobachevsky and Bolyai on the 1854 lecture.
Dispute: Historians disagree on whether Riemann knew the works of Lobachevsky and Bolyai. Scholz denies it: 'he apparently had no knowledge of Bolyai's or Lobachevsky's studies in the foundations of geometry', and traces the roots of the lecture to Gauss's theory of surfaces and Herbart's philosophy. Bottazzini and Tazzioli cautiously allow the opposite: 'However one can argue that Riemann probably knew Lobachevsky's geometry, at least in the version that appeared in that Journal', while noting that the lecture itself carries no references to Bolyai and Lobachevsky. The widespread notion of a kinship between non-Euclidean geometry and the special theory of relativity of 1905 is likewise contested: Walter writes that 'Up until the time of Minkowski's space-time theory, however, non-Euclidean geometry appeared to be irrelevant to physics', and that in 1905 'physicists generally presented their results in mechanics and electrodynamics using either Cartesian coordinate or vector methods'. Non-Euclidean language entered relativistic physics through Minkowski (1907-1908) and Varicak (1910), that is after 1905, and Riemannian geometry became physics only in the general theory of relativity: in the words of the Stanford Encyclopedia of Philosophy, 'Klein's Erlangen program failed to cover the differential geometry of Riemann (§5), which Einstein (1915, 1916) placed at the core of his general theory of relativity'.
Link sources
- Roberto Torretti, Nineteenth Century Geometry, Stanford Encyclopedia of Philosophy (2016) Verified 20 Sep 2026
Regarded as Riemannian 3-manifolds, Euclidean space has constant zero curvature, Lobachevskian space has constant negative curvature, and elliptic space has constant positive curvature.
- Erhard Scholz, The Concept of Manifold, 1850-1950 (in History of Topology, ed. I. M. James) (1999) Verified 20 Sep 2026
Riemann even sketched such a reduction in the last section of his talk, although he apparently had no knowledge of Bolyai's or Lobachevsky's studies in the foundations of geometry.
- Umberto Bottazzini, Rossana Tazzioli, Naturphilosophie and its role in Riemann's mathematics, Revue d'histoire des mathématiques 1 (1995) Verified 20 Sep 2026
Nor did he refer to the work of Bolyai and Lobachevsky, even though a paper of the latter had appeared in 1837 in Crelle's Journal.
"Verified" means that on that day the quotation was found at the address given, word for word. No date means we have not been able to open the source yet, not that the quotation is wrong.
- Roberto Torretti, Nineteenth Century Geometry, Stanford Encyclopedia of Philosophy (2016) Verified 20 Sep 2026
Node sources
- Roberto Torretti, Nineteenth Century Geometry, Stanford Encyclopedia of Philosophy (2016) Verified 20 Sep 2026
In a lecture 'On the hypotheses that lie at the foundation of geometry', delivered to the Faculty of Philosophy at Göttingen in 1854 and posthumously published in 1867, Bernhard Riemann (b. 1826, d. 1866) presented some radically innovative views on this matter.
- Roberto Torretti, Nineteenth Century Geometry, Stanford Encyclopedia of Philosophy (2016) Verified 20 Sep 2026
In his study of curved surfaces, Gauss introduced a real-valued function, the Gaussian curvature, which measures a surface's local deviation from flatness in terms of the surface's intrinsic geometry. Riemann extended this concept of curvature to Riemannian n-manifolds.
- Erhard Scholz, The Concept of Manifold, 1850-1950 (in History of Topology, ed. I. M. James) (1999) Verified 20 Sep 2026
When Riemann presented his ideas on a geometry in manifolds the first time to a scientific audience in his famous Habilitationsvortrag (Riemann, 1854), he was completely aware that he was working in a border region between mathematics, physics, and philosophy.
- Freudenthal H. «Riemann, Bernhard», Dictionary of Scientific Biography (via Encyclopedia.com) (1975) Verified 20 Sep 2026
Über die Hypothesen, welche der Geometrie zu Grunde liegen
"Verified" means that on that day the quotation was found at the address given, word for word. No date means we have not been able to open the source yet, not that the quotation is wrong.