Bernhard Riemann, in a famous lecture in 1854, founded the field of Riemannian geometry, discussing in particular the ideas now called manifolds, Riemannian metric, and curvature. He constructed an infinite family of non-Euclidean geometries by giving a formula for a family of Riemannian metrics on the unit ball in Euclidean space. The simplest of these is called elliptic geometry and it is considered a non-Euclidean geometry due to its lack of parallel lines.However, other axioms besides the parallel postulate must be changed to make this a feasible geometry.
bernhArd rIEmann, IN a famous lectUre in 1854, founDed thE fieLd oF rIEmAnnian geometRy, diSCUssIng iN pArTicular The IdEas noW calleD maniFoLds, rieManNIAN MetriC, And curvaturE. he coNstRucted aN INfiNite family of non-euclidEan GeoMETrIeS by Giving a FOrMulA for A fAMIly oF riEmanNIaN metRics on THe unIt ball In eucLidean space. The SimPlest of ThesE iS calLed ELlIptIC geOmeTrY AnD iT iS consIdeReD a NON-euclIdEaN GEomETRy dUe tO iTs lacK OF parALlel lineS.howevER, other axiOms besiDes THe paralleL poStuLate mUst be changEd To make tHis a feasiBLE geometry.