Curvature of Riemannian manifolds
![From left to right: a surface of negative Gaussian curvature (hyperboloid), a surface of zero Gaussian curvature (cylinder), and a surface of positive Gaussian curvature (sphere). In higher dimensions, a manifold may have different curvatures in different directions, described by the Riemann curvature tensor.](/Images/godic/202502/07/Gaussian_curvature.svg2210.png")
In mathematics, specifically differential geometry, the infinitesimal geometry of Riemannian manifolds with dimension at least 3 is too complicated to be described by a single number at a given point. Riemann introduced an abstract and rigorous way to define it, now known as the curvature tensor. Similar notions have found applications everywhere in differential geometry.