Riemannschen

Riemannschen is the attributive adjective form in German derived from the surname Riemann, most often used to indicate concepts associated with the German mathematician Bernhard Riemann (1826–1866). In mathematical terminology it appears in phrases such as Riemannsche Geometrie, Riemannsche Fläche, or Riemannsche Metrik. The specific form Riemannschen is the declined variant used in certain grammatical cases and numbers, for example der Riemannschen Fläche (dative singular) and den Riemannschen Flächen (dative plural).

The term is tied to the body of work initiated by Riemann, and its usage spans several foundational ideas in differential geometry and complex analysis. In Riemannsche Geometrie, a smooth manifold is equipped with a Riemannian metric, which allows the measurement of lengths and angles, and leads to notions of curvature via the Riemann curvature tensor. The concept of a Riemannsche Mannigfaltigkeit denotes such a manifold in this metric framework, while a Riemannsche Metrik refers to the specific metric tensor that defines the geometric structure.

Historically, Riemann’s ideas bridged analysis, topology, and geometry, laying the groundwork for modern differential geometry and influencing related fields, including general relativity, where pseudo-Riemannian metrics describe spacetime. The term Riemannschen remains common in German-language mathematics to attribute properties or objects to Riemann’s theories and traditional formulations.

See also: Bernhard Riemann, Riemannian geometry, Riemann surface, Riemannian metric.