Namespaces
Variants
Actions

Difference between revisions of "Ricci tensor"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (TeX-ed article.)
 
Line 1: Line 1:
A twice-covariant tensor obtained from the [[Riemann tensor|Riemann tensor]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081800/r0818001.png" /> by contracting the upper index with the first lower one:
+
A twice-covariant tensor obtained from the [[Riemann tensor|Riemann tensor]] $ R^{l}_{jkl} $ by contracting the upper index with the first lower one:
 +
$$
 +
R_{ki} = R^{m}_{mki}.
 +
$$
  
<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/r081/r081800/r0818002.png" /></td> </tr></table>
+
In a Riemannian space $ V_{n} $, the Ricci tensor is symmetric: $ R_{ki} = R_{ik} $. The trace of the Ricci tensor with respect to the contravariant metric tensor $ g^{ij} $ of the space $ V_{n} $ leads to a scalar, $ R = g^{ij} R_{ij} $, called the ''curvature invariant'' or the ''scalar curvature'' of $ V_{n} $. The components of the Ricci tensor can be expressed in terms of the metric tensor $ g_{ij} $ of the space $ V_{n} $:
 +
$$
 +
R_{ij} =
 +
\frac{\partial^{2} \ln \sqrt{g}}{\partial x^{i} \partial x^{j}} -
 +
\frac{\partial}{\partial x^{k}} \Gamma^{k}_{ij} +
 +
\Gamma^{m}_{ik} \Gamma^{k}_{mj} -
 +
\Gamma^{m}_{ij} \frac{\partial \ln \sqrt{g}}{\partial x^{m}},
 +
$$
 +
where $ g = \det g_{ij} $ and $ \Gamma^{k}_{ij} $ are the [[Christoffel symbol|Christoffel symbols]] of the second kind calculated with respect to the tensor $ g_{ij} $.
  
In a Riemannian space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081800/r0818003.png" /> the Ricci tensor is symmetric: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081800/r0818004.png" />. The trace of the Ricci tensor with respect to the contravariant metric tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081800/r0818005.png" /> of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081800/r0818006.png" /> leads to a scalar, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081800/r0818007.png" />, called the curvature invariant or the scalar curvature of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081800/r0818008.png" />. The components of the Ricci tensor can be expressed in terms of the metric tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081800/r0818009.png" /> of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081800/r08180010.png" />:
+
The tensor was introduced by G. Ricci in [[#References|[1]]].
 
 
<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/r081/r081800/r08180011.png" /></td> </tr></table>
 
 
 
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081800/r08180012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081800/r08180013.png" /> are the Christoffel symbols of the second kind (cf. [[Christoffel symbol|Christoffel symbol]]) calculated with respect to the tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081800/r08180014.png" />.
 
 
 
The tensor was introduced by G. Ricci [[#References|[1]]].
 
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G. Ricci,  ''Atti R. Inst. Venelo'' , '''53''' :  2  (1903–1904)  pp. 1233–1239</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L.P. Eisenhart,  "Riemannian geometry" , Princeton Univ. Press  (1949)</TD></TR></table>
 
 
  
 +
<table>
 +
<TR><TD valign="top">[1]</TD><TD valign="top"> G. Ricci, “Atti R. Inst. Venelo”, '''53''': 2 (1903–1904), pp. 1233–1239.</TD></TR>
 +
<TR><TD valign="top">[2]</TD><TD valign="top"> L.P. Eisenhart, “Riemannian geometry”, Princeton Univ. Press (1949).</TD></TR>
 +
</table>
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> S. KobayashiK. Nomizu,   "Foundations of differential geometry" , '''1''' , Interscience (1963)</TD></TR></table>
+
<table>
 +
<TR><TD valign="top">[a1]</TD><TD valign="top"> S. Kobayashi and K. Nomizu, “Foundations of differential geometry”, '''1''', Interscience (1963).</TD></TR></table>

Latest revision as of 06:46, 28 April 2016

A twice-covariant tensor obtained from the Riemann tensor $ R^{l}_{jkl} $ by contracting the upper index with the first lower one: $$ R_{ki} = R^{m}_{mki}. $$

In a Riemannian space $ V_{n} $, the Ricci tensor is symmetric: $ R_{ki} = R_{ik} $. The trace of the Ricci tensor with respect to the contravariant metric tensor $ g^{ij} $ of the space $ V_{n} $ leads to a scalar, $ R = g^{ij} R_{ij} $, called the curvature invariant or the scalar curvature of $ V_{n} $. The components of the Ricci tensor can be expressed in terms of the metric tensor $ g_{ij} $ of the space $ V_{n} $: $$ R_{ij} = \frac{\partial^{2} \ln \sqrt{g}}{\partial x^{i} \partial x^{j}} - \frac{\partial}{\partial x^{k}} \Gamma^{k}_{ij} + \Gamma^{m}_{ik} \Gamma^{k}_{mj} - \Gamma^{m}_{ij} \frac{\partial \ln \sqrt{g}}{\partial x^{m}}, $$ where $ g = \det g_{ij} $ and $ \Gamma^{k}_{ij} $ are the Christoffel symbols of the second kind calculated with respect to the tensor $ g_{ij} $.

The tensor was introduced by G. Ricci in [1].

References

[1] G. Ricci, “Atti R. Inst. Venelo”, 53: 2 (1903–1904), pp. 1233–1239.
[2] L.P. Eisenhart, “Riemannian geometry”, Princeton Univ. Press (1949).

Comments

References

[a1] S. Kobayashi and K. Nomizu, “Foundations of differential geometry”, 1, Interscience (1963).
How to Cite This Entry:
Ricci tensor. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Ricci_tensor&oldid=12398
This article was adapted from an original article by L.A. Sidorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article