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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} $):

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

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|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  $  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|Peterson–Codazzi equations]] for subspaces of  $  V _ {n} $
are derived.

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

====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>

====Comments====
The first Ricci identity is usually called the first Bianchi identity in the West, cf. also [[Bianchi identity|Bianchi identity]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Klingenberg,   "Riemannian geometry" , de Gruyter  (1982)  (Translated from German)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  N.J. Hicks,   "Notes on differential geometry" , v. Nostrand  (1965)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  S. Kobayashi,   K. Nomizu,   "Foundations of differential geometry" , '''1''' , Interscience  (1963)</TD></TR></table>