Riemannian geometry studies Riemannian manifolds, smooth manifolds with a Riemannian metric, a notion of a distance expressed by means of a smooth positive definite symmetri bilinear form defined on the tangent space at each point. Riemannian geometry generalizes Euclidean geometry to spaces that are not necessarily flat, although they still resemble the 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 admit natural analogues in Riemannian geometry. The notion of a directional derivative of a function from multivariable calculus is extended in Riemannian geometry to the notion of a covariant derivative of a tensor. Many concepts and techniques 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, Theorema Egregium of Gauss showed that already for surfaces, the existence of a local isometry imposes strong compatibility conditions on their metrics: the Gaussian curvatures at the corresponding points must be the same. In higher dimensions, the Riemann curvature tensor is an important pointwise invariant associated to a Riemannian manifold that measures how close it is to being flat. An important class of Riemannian manifolds is formed by the Riemannian symmetric spaces, whose curvature is not necessarily constant. They are the closest to the "ordinary" plane and space considered in Euclidean and non-Euclidean geometry.
Taken From http://en.wikipedia.org/wiki/Differential_geometry
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