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The metric of a space given by a positive-definite [[Quadratic form|quadratic form]]. If a local coordinate system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082180/r0821801.png" /> is introduced for the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082180/r0821802.png" /> and if at each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082180/r0821803.png" /> functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082180/r0821804.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082180/r0821805.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082180/r0821806.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082180/r0821807.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082180/r0821808.png" />. The length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082180/r0821809.png" /> of the covariant vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082180/r08218010.png" /> is expressed using the fundamental tensor:
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<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/r/r082/r082180/r08218011.png" /></td> </tr></table>
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the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082180/r08218012.png" /> is a positive-definite quadratic form. The metric of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082180/r08218013.png" /> determined using the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082180/r08218014.png" /> is called Riemannian, and a space with a given Riemannian metric introduced into it is called a [[Riemannian space|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.
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The metric of a space given by a positive-definite [[Quadratic form|quadratic form]]. If a local coordinate system  $  ( x  ^ {1} \dots x  ^ {n} ) $
 +
is introduced for the space  $  V _ {n} $
 +
and if at each point  $  X( x  ^ {1} \dots x  ^ {n} ) \in V _ {n} $
 +
functions  $  g _ {ij} ( X) $,
 +
$  i, j = 1 \dots n $,
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$  \mathop{\rm det} ( g _ {ij} ) > 0 $,
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$  g _ {ij} ( X) = g _ {ji} ( X) $,
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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  $  V _ {n} $.  
 +
The length  $  ds $
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of the covariant vector  $  ( dx  ^ {1} \dots dx  ^ {n} ) $
 +
is expressed using the fundamental tensor:
  
A Riemannian metric is a generalization of the [[First fundamental form|first fundamental form]] of a surface in three-dimensional Euclidean space — of the internal metric of the surface. The geometry of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082180/r08218015.png" /> based on a definite Riemannian metric is called a [[Riemannian geometry|Riemannian geometry]].
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$$
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ds  ^ {2}  = g _ {ij} ( X)  dx  ^ {i}  dx  ^ {j} ;
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$$
  
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|Pseudo-Riemannian space]] and [[Relativity theory|Relativity theory]]). A degenerate Riemannian metric, that is, a metric form defined with the aid of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082180/r08218016.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082180/r08218017.png" />, defines a [[Semi-Riemannian space|semi-Riemannian space]].
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the form  $  g _ {ij}  dx  ^ {i}  dx  ^ {j} $
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is a positive-definite quadratic form. The metric of  $  V _ {n} $
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determined using the form  $  ds  ^ {2} $
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is called Riemannian, and a space with a given Riemannian metric introduced into it is called a [[Riemannian space|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.
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A Riemannian metric is a generalization of the [[First fundamental form|first fundamental form]] of a surface in three-dimensional Euclidean space — of the internal metric of the surface. The geometry of the space  $  V _ {n} $
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based on a definite Riemannian metric is called a [[Riemannian geometry|Riemannian geometry]].
 +
 
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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|Pseudo-Riemannian space]] and [[Relativity theory|Relativity theory]]). A degenerate Riemannian metric, that is, a metric form defined with the aid of functions $  g _ {ij} ( X) $
 +
for which $  \mathop{\rm det} ( g _ {ij} ) = 0 $,  
 +
defines a [[Semi-Riemannian space|semi-Riemannian space]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L.P. Eisenhart,  "Riemannian geometry" , Princeton Univ. Press  (1949)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  P.K. [P.K. Rashevskii] Rashewski,  "Riemannsche Geometrie und Tensoranalyse" , Deutsch. Verlag Wissenschaft.  (1959)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  B. Riemann,  "Ueber die Hypothesen, welche der Geometrie zuGrunde liegen" , ''Das Kontinuum und andere Monographien'' , Chelsea, reprint  (1973)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L.P. Eisenhart,  "Riemannian geometry" , Princeton Univ. Press  (1949)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  P.K. [P.K. Rashevskii] Rashewski,  "Riemannsche Geometrie und Tensoranalyse" , Deutsch. Verlag Wissenschaft.  (1959)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  B. Riemann,  "Ueber die Hypothesen, welche der Geometrie zuGrunde liegen" , ''Das Kontinuum und andere Monographien'' , Chelsea, reprint  (1973)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====

Latest revision as of 08:11, 6 June 2020


The metric of a space given by a positive-definite quadratic form. If a local coordinate system $ ( x ^ {1} \dots x ^ {n} ) $ is introduced for the space $ V _ {n} $ and if at each point $ X( x ^ {1} \dots x ^ {n} ) \in V _ {n} $ functions $ g _ {ij} ( X) $, $ i, j = 1 \dots n $, $ \mathop{\rm det} ( g _ {ij} ) > 0 $, $ g _ {ij} ( X) = g _ {ji} ( X) $, 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 $ V _ {n} $. The length $ ds $ of the covariant vector $ ( dx ^ {1} \dots dx ^ {n} ) $ is expressed using the fundamental tensor:

$$ ds ^ {2} = g _ {ij} ( X) dx ^ {i} dx ^ {j} ; $$

the form $ g _ {ij} dx ^ {i} dx ^ {j} $ is a positive-definite quadratic form. The metric of $ V _ {n} $ determined using the form $ ds ^ {2} $ 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 $ V _ {n} $ 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 $ g _ {ij} ( X) $ for which $ \mathop{\rm det} ( g _ {ij} ) = 0 $, defines a semi-Riemannian space.

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] B. Riemann, "Ueber die Hypothesen, welche der Geometrie zuGrunde liegen" , Das Kontinuum und andere Monographien , Chelsea, reprint (1973)

Comments

The adjective "semi-Riemannian" is also used for indefinite metrics which are non-degenerate everywhere, cf. [a1]. For additional references see also Riemann tensor.

References

[a1] B. O'Neill, "Elementary differential geometry" , Acad. Press (1966)
How to Cite This Entry:
Riemannian metric. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Riemannian_metric&oldid=48560
This article was adapted from an original article by L.A. Sidorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article