Euclidean Geometry Versus Riemannian Geometry
In 1853 Gauss asked his student Riemann to prepare a Habilitationsschrift on the foundations of geometry. Over many months, Riemann developed his theory of higher dimensions and delivered his lecture at Göttingen in 1854 entitled Über die Hypothesen welche der Geometrie zu Grunde liegen ("On the hypotheses which underlie geometry"). When it was finally published in 1868, two years after his death, the mathematical public received it with enthusiasm and it is now recognized as one of the most important works in geometry.
The subject founded by this work is Riemannian geometry. Riemann found the correct way to extend into n dimensions the differential geometry of surfaces, which Gauss himself proved in his theorema egregium. The fundamental object is called the Riemann curvature tensor. For the surface case, this can be reduced to a number (scalar), positive, negative or zero; the non-zero and constant cases being models of the known non-Euclidean geometries.
Read more about this topic: Bernhard Riemann
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