Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a Riemannian metric, i.e. with an inner product on the tangent space at each point which varies smoothly from point to point. This gives in particular local notions of angle, length of curves, surface area, and volume. From those some other global quantities can be derived by integrating local contributions.
Riemannian geometry originated with the vision of Bernhard Riemann expressed in his inaugurational lecture Über die Hypothesen, welche der Geometrie zu Grunde liegen (German: On the hypotheses on which geometry is based). It is a very broad and abstract generalization of the differential geometry of surfaces in R3. Development of Riemannian geometry resulted in synthesis of diverse results concerning the geometry of surfaces and the behavior of geodesics on them, with techniques that can be applied to the study of differentiable manifolds of higher dimensions. It enabled Einstein's general relativity theory, made profound impact on group theory and representation theory, as well as analysis, and spurred the development of algebraic and differential topology.
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