Riemann: Geometry and Physics
Bernhard Riemann turned to mathematics away from theology at Göttingen University and gradually developed a series of profound ideas in the theory of complex functions and in geometry. He argued that geometry can be studied in any setting where one may speak of lengths and angles. Typically this meant differential geometry but in a space of any arbitrary number of dimensions and with an arbitrary (positive definite) metric. Such a geometry was intrinsically defined, that is without any reference to an ambient Euclidean space. He supplied metrics for spaces of constant curvature in any dimension, which in two dimensions lead to spherical geometry (positive curvature), Euclidean geometry (zero curvature), and non-Euclidean geometry (negative curvature) although Riemann did not refer to it by that name.
Extract: a lengthy series of passages from Riemann’s Lecture of 1854 (published as his (1867)).
KeywordsConstant Curvature Euclidean Geometry Ambient Space Linear Element Total Curvature
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