Riemannsche

Riemannsche is a German attributive adjective used to denote relation to the German mathematician Bernhard Riemann or to concepts and results named after him. It appears widely in mathematical and scientific terminology to indicate that a theory or object is derived from, or associated with, Riemann’s work.

In practice, Riemannsche compounds refer to several fundamental ideas in mathematics. Common examples include Riemannsche Geometrie (Riemannian geometry), Riemannsche Fläche (Riemann surface), Riemannsches Integral (Riemann integral), Riemannsche Vermutung (Riemann hypothesis), and Riemannsche Metrik (Riemannian metric). These terms reflect Riemann’s influence across differential geometry, complex analysis, number theory, and analysis.

Grammatically, Riemannsche behaves like a standard attributive adjective in German. Its ending changes with gender, number, and case, and it can appear as Riemannscher (masculine nominative singular), Riemannsche (feminine nominative singular or plural in many contexts), or Riemannsches (neuter nominative singular). For example: die Riemannsche Fläche, ein Riemannsches Integral, ein Riemannscher Satz.

Origin and usage notes: The adjective is formed from the proper noun Riemann and the suffix -sche, and it is used specifically to indicate association with Riemann’s theories or results. In English-language mathematics, the corresponding term is typically "Riemannian" (as in Riemannian geometry), while the German usage remains anchored in the eponymous form “Riemannsche.”