Riemannian geometry

Riemannian geometry is a description of an important family of geometries, first put forward in generality by Bernhard Riemann in the nineteenth century. It is an intrinsic description, of what is now called a Riemannian manifold. As particular special cases there occur the two standard types (spherical geometry and hyperbolic geometry of Non-Euclidean geometry), as well as Euclidean geometry itself. These are all treated on the same axiomatic footing, as are a broad range of geometries whose metric properties vary from point to point.

The characteristic structure in Riemannian geometry is a metric tensor defined on the tangent space, from point to point. This gives a local idea of angle, length and volume. From these global quantities can be derived, by integrating local contributions.

Brief on the metric tensor

The metric tensor, conventionally notated as <math>G</math>, as a 2-dimensional tensor (making it a matrix), that is used to measure distance in a coordinate space[?] or manifold. <math>g_{ij}</math> is conventionally used to notate the components of the metric tensor. (The elements of the matrix)

The length of a segment of a curve parameterized by t, from a to b, is defined as:

<math>L = \int_a^b \sqrt{ g_{ij}dx^idx^j} </math>

Example

Given a two-dimensional Euclidean metric tensor:

<math>G = \begin{bmatrix} 1 & 0 \\ 0 & 1\end{bmatrix}</math>

The length of a curve reduces to the familiar Calculus formula:

<math>L = \int_a^b \sqrt{ (dx^1)^2 + (dx^2)^2} </math>

External Links

Mathworld's site on Riemannian Geometry (http://mathworld.wolfram.com/RiemannianGeometry.html)