Metric tensor

basic tensor, fundamental tensor

A twice covariant symmetric tensor field on an -dimensional differentiable manifold , . The assignment of a metric tensor on introduces a scalar product of contravariant vectors on the tangent space of at , defined as the bilinear function , where is the value of the field at the point . In coordinate notation:

The metric in with this scalar product is regarded as infinitesimal for the metric of the manifold , which is expressed by the choice of the quadratic differential form

(*)

as the square of the differential of the arc length of curves in , going from in the direction . With respect to its geometric meaning the form (*) is called the metric form or first fundamental form on , corresponding to the metric tensor . Conversely, if a symmetric quadratic form (*) on is given, then there is a twice covariant tensor field associated with it and whose corresponding metric form is . Thus, the specification of a metric tensor on is equivalent to the specification of a metric form on with a quadratic line element of the form (*). The metric tensor completely determines the intrinsic geometry of .

The collection of metric tensors , and the metric forms defined by them, is divided into two classes, the degenerate metrics, when , and the non-degenerate metrics, when . A manifold with a degenerate metric form (*) is called isotropic. Among the non-degenerate metric tensors, in their turn, are distinguished the Riemannian metric tensors, for which the quadratic form (*) is positive definite, and the pseudo-Riemannian metric tensors, when (*) has variable sign. A Riemannian (pseudo-Riemannian) metric introduced on via a Riemannian (pseudo-Riemannian) metric tensor defines on a Riemannian (respectively, pseudo-Riemannian) geometry.

Usually a metric tensor, without special indication, means a Riemannian metric tensor; but if one wishes to stress that the discussion is about Riemannian and not about pseudo-Riemannian metric tensors, then one speaks of a proper Riemannian metric tensor. A proper Riemannian metric tensor can be introduced on any paracompact differentiable manifold.

References

[1]	L.P. Eisenhart, "Riemannian geometry" , Princeton Univ. Press (1949)
[2]	P.K. [P.K. Rashevskii] Rashewski, "Riemannsche Geometrie und Tensoranalyse" , Deutsch. Verlag Wissenschaft. (1959) (Translated from Russian)
[3]	D. Gromoll, W. Klingenberg, W. Meyer, "Riemannsche Geometrie im Grossen" , Springer (1968)

References

[a1]	W. Klingenberg, "Riemannian geometry" , de Gruyter (1982) (Translated from German)