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:
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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) |
Comments
References
[a1] | W. Klingenberg, "Riemannian geometry" , de Gruyter (1982) (Translated from German) |
Metric tensor. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Metric_tensor&oldid=16631