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An identity expressing one of the properties of the [[Riemann tensor|Riemann tensor]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081790/r0817901.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081790/r0817902.png" />):
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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/r081/r081790/r0817903.png" /></td> </tr></table>
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For a covariant tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081790/r0817904.png" /> the identity is of the form
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An identity expressing one of the properties of the [[Riemann tensor|Riemann tensor]]  $  R _ {ij,k}  ^ {l} $(
 +
or  $  R _ {ij,kl} $):
  
<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/r081790/r0817905.png" /></td> </tr></table>
+
$$
 +
R _ {ij,k}  ^ {l} +
 +
R _ {jk,i}  ^ {l} +
 +
R _ {ki,j}  ^ {l}  = 0 .
 +
$$
 +
 
 +
For a covariant tensor  $  R _ {ij,kl} $
 +
the identity is of the form
 +
 
 +
$$
 +
R _ {ij,kl} +
 +
R _ {jk,il} +
 +
R _ {ki,jl}  = 0 ,
 +
$$
  
 
i.e. cycling over the three first indices yields zero.
 
i.e. cycling over the three first indices yields zero.
  
An identity which should be satisfied by the covariant derivatives of second order with respect to the metric tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081790/r0817906.png" /> of a Riemannian space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081790/r0817907.png" />, which differ only by the order of differentiation. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081790/r0817908.png" /> is a tensor of valency 1 and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081790/r0817909.png" /> is the covariant derivative of second order with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081790/r08179010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081790/r08179011.png" /> relative to the tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081790/r08179012.png" />, then the Ricci identity takes the form
+
An identity which should be satisfied by the covariant derivatives of second order with respect to the metric tensor $  g _ {ij} $
 +
of a Riemannian space $  V _ {n} $,  
 +
which differ only by the order of differentiation. If $  \lambda _ {i} $
 +
is a tensor of valency 1 and $  \lambda _ {i,jk} $
 +
is the covariant derivative of second order with respect to $  x  ^ {j} $
 +
and $  x  ^ {k} $
 +
relative to the tensor $  g _ {ij} $,  
 +
then the Ricci identity takes the 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/r/r081/r081790/r08179013.png" /></td> </tr></table>
+
$$
 +
\lambda _ {i,jk} - \lambda _ {i,kj}  = \lambda _ {l} R _ {ij,k}  ^ {l} ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081790/r08179014.png" /> is the Riemann [[Curvature tensor|curvature tensor]] determined by the [[Metric tensor|metric tensor]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081790/r08179015.png" /> of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081790/r08179016.png" /> (in other words, an alternating second absolute derivative of the tensor field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081790/r08179017.png" /> in the metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081790/r08179018.png" /> is expressed in terms of the Riemann tensor and the components of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081790/r08179019.png" />).
+
where $  R _ {ij,k}  ^ {l} $
 +
is the Riemann [[Curvature tensor|curvature tensor]] determined by the [[Metric tensor|metric tensor]] $  g _ {ij} $
 +
of the space $  V _ {n} $(
 +
in other words, an alternating second absolute derivative of the tensor field $  \lambda _ {i} $
 +
in the metric $  g _ {ij} $
 +
is expressed in terms of the Riemann tensor and the components of $  \lambda _ {i} $).
  
For a covariant tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081790/r08179020.png" /> of valency 2 the Ricci identity has the form
+
For a covariant tensor $  a _ {ij} $
 +
of valency 2 the Ricci identity has the 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/r/r081/r081790/r08179021.png" /></td> </tr></table>
+
$$
 +
a _ {ij,kl} -
 +
a _ {ij,lk}  = \
 +
a _ {ih} R _ {jk,l}  ^ {h} + a _ {h j }  R _ {ik,l}  ^ {h} .
 +
$$
  
In general, for a covariant tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081790/r08179022.png" /> of valency <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081790/r08179023.png" /> the identity has the form
+
In general, for a covariant tensor $  a _ {r _ {1}  \dots r _ {m} } $
 +
of valency $  m $
 +
the identity has the 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/r/r081/r081790/r08179024.png" /></td> </tr></table>
+
$$
 +
a _ {r _ {1}  \dots r _ {m} , k l } -
 +
a _ {r _ {1}  \dots r _ {m} , l k } =
 +
$$
  
<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/r081790/r08179025.png" /></td> </tr></table>
+
$$
 +
= \
 +
\sum _  \alpha  ^ { {1 }  \dots m } a _ {r _ {1}  \dots
 +
r _ {\alpha - 1 }  h r _ {\alpha + 1 }  \dots
 +
r _ {m} } R _ {r _  \alpha  k l }  ^ {h} .
 +
$$
  
Similar identities can be written for contravariant and mixed tensors in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081790/r08179026.png" />. The Ricci identity is used, e.g., in constructions of the geometry of subspaces in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081790/r08179027.png" /> as an integrability condition for the principal variational equations from which Gauss' equations and the [[Peterson–Codazzi equations|Peterson–Codazzi equations]] for subspaces of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081790/r08179028.png" /> are derived.
+
Similar identities can be written for contravariant and mixed tensors in $  V _ {n} $.  
 +
The Ricci identity is used, e.g., in constructions of the geometry of subspaces in $  V _ {n} $
 +
as an integrability condition for the principal variational equations from which Gauss' equations and the [[Peterson–Codazzi equations|Peterson–Codazzi equations]] for subspaces of $  V _ {n} $
 +
are derived.
  
 
The identity was established by G. Ricci (see [[#References|[1]]]).
 
The identity was established by G. Ricci (see [[#References|[1]]]).
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====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G. Ricci,  T. Levi-Civita,  "Méthodes de calcul différentiel absolu et leurs applications"  ''Math. Ann.'' , '''54'''  (1901)  pp. 125–201</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">  L.P. Eisenhart,  "Riemannian geometry" , Princeton Univ. Press  (1949)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G. Ricci,  T. Levi-Civita,  "Méthodes de calcul différentiel absolu et leurs applications"  ''Math. Ann.'' , '''54'''  (1901)  pp. 125–201</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">  L.P. Eisenhart,  "Riemannian geometry" , Princeton Univ. Press  (1949)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====

Latest revision as of 08:11, 6 June 2020


An identity expressing one of the properties of the Riemann tensor $ R _ {ij,k} ^ {l} $( or $ R _ {ij,kl} $):

$$ R _ {ij,k} ^ {l} + R _ {jk,i} ^ {l} + R _ {ki,j} ^ {l} = 0 . $$

For a covariant tensor $ R _ {ij,kl} $ the identity is of the form

$$ R _ {ij,kl} + R _ {jk,il} + R _ {ki,jl} = 0 , $$

i.e. cycling over the three first indices yields zero.

An identity which should be satisfied by the covariant derivatives of second order with respect to the metric tensor $ g _ {ij} $ of a Riemannian space $ V _ {n} $, which differ only by the order of differentiation. If $ \lambda _ {i} $ is a tensor of valency 1 and $ \lambda _ {i,jk} $ is the covariant derivative of second order with respect to $ x ^ {j} $ and $ x ^ {k} $ relative to the tensor $ g _ {ij} $, then the Ricci identity takes the form

$$ \lambda _ {i,jk} - \lambda _ {i,kj} = \lambda _ {l} R _ {ij,k} ^ {l} , $$

where $ R _ {ij,k} ^ {l} $ is the Riemann curvature tensor determined by the metric tensor $ g _ {ij} $ of the space $ V _ {n} $( in other words, an alternating second absolute derivative of the tensor field $ \lambda _ {i} $ in the metric $ g _ {ij} $ is expressed in terms of the Riemann tensor and the components of $ \lambda _ {i} $).

For a covariant tensor $ a _ {ij} $ of valency 2 the Ricci identity has the form

$$ a _ {ij,kl} - a _ {ij,lk} = \ a _ {ih} R _ {jk,l} ^ {h} + a _ {h j } R _ {ik,l} ^ {h} . $$

In general, for a covariant tensor $ a _ {r _ {1} \dots r _ {m} } $ of valency $ m $ the identity has the form

$$ a _ {r _ {1} \dots r _ {m} , k l } - a _ {r _ {1} \dots r _ {m} , l k } = $$

$$ = \ \sum _ \alpha ^ { {1 } \dots m } a _ {r _ {1} \dots r _ {\alpha - 1 } h r _ {\alpha + 1 } \dots r _ {m} } R _ {r _ \alpha k l } ^ {h} . $$

Similar identities can be written for contravariant and mixed tensors in $ V _ {n} $. The Ricci identity is used, e.g., in constructions of the geometry of subspaces in $ V _ {n} $ as an integrability condition for the principal variational equations from which Gauss' equations and the Peterson–Codazzi equations for subspaces of $ V _ {n} $ are derived.

The identity was established by G. Ricci (see [1]).

References

[1] G. Ricci, T. Levi-Civita, "Méthodes de calcul différentiel absolu et leurs applications" Math. Ann. , 54 (1901) pp. 125–201
[2] P.K. [P.K. Rashevskii] Rashewski, "Riemannsche Geometrie und Tensoranalyse" , Deutsch. Verlag Wissenschaft. (1959) (Translated from Russian)
[3] L.P. Eisenhart, "Riemannian geometry" , Princeton Univ. Press (1949)

Comments

The first Ricci identity is usually called the first Bianchi identity in the West, cf. also Bianchi identity.

References

[a1] W. Klingenberg, "Riemannian geometry" , de Gruyter (1982) (Translated from German)
[a2] N.J. Hicks, "Notes on differential geometry" , v. Nostrand (1965)
[a3] S. Kobayashi, K. Nomizu, "Foundations of differential geometry" , 1 , Interscience (1963)
How to Cite This Entry:
Ricci identity. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Ricci_identity&oldid=48537
This article was adapted from an original article by L.A. Sidorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article