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''basic tensor, fundamental tensor''
 
''basic tensor, fundamental tensor''
  
A twice covariant symmetric tensor field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063690/m0636901.png" /> on an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063690/m0636902.png" />-dimensional differentiable manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063690/m0636903.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063690/m0636904.png" />. The assignment of a metric tensor on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063690/m0636905.png" /> introduces a scalar product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063690/m0636906.png" /> of contravariant vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063690/m0636907.png" /> on the tangent space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063690/m0636908.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063690/m0636909.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063690/m06369010.png" />, defined as the bilinear function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063690/m06369011.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063690/m06369012.png" /> is the value of the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063690/m06369013.png" /> at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063690/m06369014.png" />. In coordinate notation:
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A twice covariant symmetric tensor field $g=g(X,Y)$ on an $n$-dimensional differentiable manifold $M^n$, $n\geq2$. The assignment of a metric tensor on $M^n$ introduces a scalar product $\langle X,Y\rangle$ of contravariant vectors $X,Y\in M_p^n$ on the tangent space $M_p^n$ of $M^n$ at $p\in M^n$, defined as the bilinear function $g_p(X,Y)$, where $g_p$ is the value of the field $g$ at the point $p$. In coordinate notation:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063690/m06369015.png" /></td> </tr></table>
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$$\langle X,Y\rangle=g_{ij}(p)X^iY^j,\quad X=\{X^i\},\quad Y=\{Y^j\},\quad0\leq i,j\leq n.$$
  
The metric in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063690/m06369016.png" /> with this scalar product is regarded as infinitesimal for the metric of the manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063690/m06369017.png" />, which is expressed by the choice of the quadratic differential form
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The metric in $M_p^n$ with this scalar product is regarded as infinitesimal for the metric of the manifold $M^n$, which is expressed by the choice of the quadratic differential form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063690/m06369018.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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$$ds^2=g_{ij}(p)dx^idx^j\tag{*}$$
  
as the square of the differential of the arc length of curves in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063690/m06369019.png" />, going from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063690/m06369020.png" /> in the direction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063690/m06369021.png" />. With respect to its geometric meaning the form (*) is called the metric form or first fundamental form on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063690/m06369022.png" />, corresponding to the metric tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063690/m06369023.png" />. Conversely, if a symmetric quadratic form (*) on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063690/m06369024.png" /> is given, then there is a twice covariant tensor field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063690/m06369025.png" /> associated with it and whose corresponding metric form is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063690/m06369026.png" />. Thus, the specification of a metric tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063690/m06369027.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063690/m06369028.png" /> is equivalent to the specification of a metric form on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063690/m06369029.png" /> with a quadratic line element of the form (*). The metric tensor completely determines the intrinsic geometry of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063690/m06369030.png" />.
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as the square of the differential of the arc length of curves in $M^n$, going from $p$ in the direction $dx^1,\dots,dx^n$. With respect to its geometric meaning the form \ref{*} is called the metric form or first fundamental form on $M^n$, corresponding to the metric tensor $g$. Conversely, if a symmetric quadratic form \ref{*} on $M^n$ is given, then there is a twice covariant tensor field $g(X,Y)=g_{ij}X^iY^j$ associated with it and whose corresponding metric form is $g$. Thus, the specification of a metric tensor $g$ on $M^n$ is equivalent to the specification of a metric form on $M^n$ with a quadratic line element of the form \ref{*}. The metric tensor completely determines the intrinsic geometry of $M^n$.
  
The collection of metric tensors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063690/m06369031.png" />, and the metric forms defined by them, is divided into two classes, the degenerate metrics, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063690/m06369032.png" />, and the non-degenerate metrics, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063690/m06369033.png" />. A manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063690/m06369034.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063690/m06369035.png" /> via a Riemannian (pseudo-Riemannian) metric tensor defines on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063690/m06369036.png" /> a Riemannian (respectively, pseudo-Riemannian) geometry.
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The collection of metric tensors $g$, and the metric forms defined by them, is divided into two classes, the degenerate metrics, when $\det(g_{ij})=0$, and the non-degenerate metrics, when $\det(g_{ij})\neq0$. A manifold $M^n$ with a degenerate metric form \ref{*} is called isotropic. Among the non-degenerate metric tensors, in their turn, are distinguished the Riemannian metric tensors, for which the quadratic form \ref{*} is positive definite, and the pseudo-Riemannian metric tensors, when \ref{*} has variable sign. A Riemannian (pseudo-Riemannian) metric introduced on $M^n$ via a Riemannian (pseudo-Riemannian) metric tensor defines on $M^n$ 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.
 
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.

Revision as of 09:49, 20 September 2014

basic tensor, fundamental tensor

A twice covariant symmetric tensor field $g=g(X,Y)$ on an $n$-dimensional differentiable manifold $M^n$, $n\geq2$. The assignment of a metric tensor on $M^n$ introduces a scalar product $\langle X,Y\rangle$ of contravariant vectors $X,Y\in M_p^n$ on the tangent space $M_p^n$ of $M^n$ at $p\in M^n$, defined as the bilinear function $g_p(X,Y)$, where $g_p$ is the value of the field $g$ at the point $p$. In coordinate notation:

$$\langle X,Y\rangle=g_{ij}(p)X^iY^j,\quad X=\{X^i\},\quad Y=\{Y^j\},\quad0\leq i,j\leq n.$$

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

$$ds^2=g_{ij}(p)dx^idx^j\tag{*}$$

as the square of the differential of the arc length of curves in $M^n$, going from $p$ in the direction $dx^1,\dots,dx^n$. With respect to its geometric meaning the form \ref{*} is called the metric form or first fundamental form on $M^n$, corresponding to the metric tensor $g$. Conversely, if a symmetric quadratic form \ref{*} on $M^n$ is given, then there is a twice covariant tensor field $g(X,Y)=g_{ij}X^iY^j$ associated with it and whose corresponding metric form is $g$. Thus, the specification of a metric tensor $g$ on $M^n$ is equivalent to the specification of a metric form on $M^n$ with a quadratic line element of the form \ref{*}. The metric tensor completely determines the intrinsic geometry of $M^n$.

The collection of metric tensors $g$, and the metric forms defined by them, is divided into two classes, the degenerate metrics, when $\det(g_{ij})=0$, and the non-degenerate metrics, when $\det(g_{ij})\neq0$. A manifold $M^n$ with a degenerate metric form \ref{*} is called isotropic. Among the non-degenerate metric tensors, in their turn, are distinguished the Riemannian metric tensors, for which the quadratic form \ref{*} is positive definite, and the pseudo-Riemannian metric tensors, when \ref{*} has variable sign. A Riemannian (pseudo-Riemannian) metric introduced on $M^n$ via a Riemannian (pseudo-Riemannian) metric tensor defines on $M^n$ 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)
How to Cite This Entry:
Metric tensor. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Metric_tensor&oldid=33342
This article was adapted from an original article by I.Kh. Sabitov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article