Fr.: espace pseudo-riemannien
A space with an affine connection (without torsion), at each point of which the tangent space is a → pseudo-Euclidean space (Encyclopedia of Mathematics, Kluwer Academic Publications, Editor in chief I. M. Vinogradov, 1991).
Of or pertaining to Georg Friedrich Bernhard Riemann (1826-1866) or his mathematics findings. → Riemannian geometry, → Riemannian manifold, → Riemannian metric, → Riemann problem, → Riemann curvature tensor.
After the German mathematician Georg Friedrich Bernhard Riemann (1826-1866), the inventor of the elliptic form of → non-Euclidean geometry, who made important contributions to analysis and differential geometry, some of them paving the way for the later development of → general relativity.
Fr.: géométrie riemannienne
A → non-Euclidean geometry in which there are no → parallel lines, and the sum of the → angles of a → triangle is always greater than 180°. Riemannian figures can be thought of as figures constructed on a curved surface. The geometry is called elliptic because the section formed by a plane that cuts the curved surface is an ellipse.
Fr.: variété riemannienne
Fr.: métrique riemannienne
A positive-definite inner product, (.,.)x, on Tx(M), the tangent space to a manifold M at x, for each x ∈ M, which varies continually with x (Douglas N. Clark, Dictionary of Analysis, Calculus, and Differential Equations).