The metric of a space given by a positive-definite quadratic form. If a local coordinate system is introduced for the space and if at each point functions , , , , are defined which are the components of a covariant symmetric tensor of the second valency, then this tensor is called the fundamental metric tensor of . The length of the covariant vector is expressed using the fundamental tensor:
the form is a positive-definite quadratic form. The metric of determined using the form is called Riemannian, and a space with a given Riemannian metric introduced into it is called a Riemannian space. The specification of a Riemannian metric on a differentiable manifold means the specification of a Euclidean structure on the tangent spaces of this manifold depending on the points in a differentiable way.
A Riemannian metric is a generalization of the first fundamental form of a surface in three-dimensional Euclidean space — of the internal metric of the surface. The geometry of the space based on a definite Riemannian metric is called a Riemannian geometry.
There are generalizations of the concept of a Riemannian metric. Thus, a pseudo-Riemannian metric is defined with the aid of a non-definite non-degenerate quadratic form (see Pseudo-Riemannian space and Relativity theory). A degenerate Riemannian metric, that is, a metric form defined with the aid of functions for which , defines a semi-Riemannian space.
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Riemannian metric. L.A. Sidorov (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Riemannian_metric&oldid=14477