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.
The metric tensor, conventionally notated as <math>G</math>, as a 2dimensional 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:
Example Given a twodimensional Euclidean metric tensor:
The length of a curve reduces to the familiar Calculus formula:
External Links Mathworld's site on Riemannian Geometry (http://mathworld.wolfram.com/RiemannianGeometry)
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