Editing Differential geometry (section)

===Riemannian geometry===
{{main|Riemannian geometry}}

Riemannian geometry studies [[Riemannian manifold]]s, [[smooth manifold]]s with a ''Riemannian metric''. This is a concept of distance expressed by means of a [[Smooth function|smooth]] [[positive definite bilinear form|positive definite]] [[symmetric bilinear form]] defined on the tangent space at each point. Riemannian geometry generalizes [[Euclidean geometry]] to spaces that are not necessarily flat, though they still resemble Euclidean space at each point infinitesimally, i.e. in the [[first order of approximation]]. Various concepts based on length, such as the [[arc length]] of curves, [[area]] of plane regions, and [[volume]] of solids all possess natural analogues in Riemannian geometry. The notion of a [[directional derivative]] of a function from [[multivariable calculus]] is extended to the notion of a [[covariant derivative]] of a [[tensor]]. Many concepts of analysis and differential equations have been generalized to the setting of Riemannian manifolds.

A distance-preserving [[diffeomorphism]] between Riemannian manifolds is called an [[isometry]]. This notion can also be defined ''locally'', i.e. for small neighborhoods of points. Any two regular curves are locally isometric. However, the [[Theorema Egregium]] of Carl Friedrich Gauss showed that for surfaces, the existence of a local isometry imposes that the [[Gaussian curvature]]s at the corresponding points must be the same. In higher dimensions, the [[Riemann curvature tensor]] is an important pointwise invariant associated with a Riemannian manifold that measures how close it is to being flat. An important class of Riemannian manifolds is the [[Riemannian symmetric space]]s, whose curvature is not necessarily constant. These are the closest analogues to the "ordinary" plane and space considered in Euclidean and [[non-Euclidean geometry]].