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In 1949, while studying the [[Betti number|Betti number]] of a [[Kähler manifold|Kähler manifold]], S. Bochner [[#References|[a1]]] (see also [[#References|[a26]]]), ad hoc and without giving any intrinsic geometric interpretation for its meaning or origin, introduced a new tensor as an analogue of the Weyl conformal curvature tensor in a [[Riemannian manifold|Riemannian manifold]]. In a complex local coordinate system in a  $  2n $-
dimensional Kählerian manifold  $  M ^ {2n } ( J _  \lambda   ^ {k} ,g _ {\lambda \mu }  ) $, 
it is defined as follows:

$$ 
{\widetilde{B}  } _ {jh  ^ {*}  lk  ^ {*} } = R _ {jh  ^ {*}  lk  ^ {*} } -
$$

$$ 
- 
{
\frac{1}{n + 2 }
 } ( g _ {jk  ^ {*}  } R _ {h  ^ {*}  l } + g _ {lk  ^ {*}  } R _ {jh  ^ {*}  } + g _ {h  ^ {*}  l } R _ {jk  ^ {*}  } + g _ {jh  ^ {*}  } R _ {lk  ^ {*}  } )  +
$$

$$ 
+ 
{
\frac{S}{2 ( n + 1 ) ( n + 2 ) }
 } ( g _ {h  ^ {*}  l } g _ {jk  ^ {*}  } + g _ {jh  ^ {*}  } g _ {lk  ^ {*}  } ) ,
$$

where  $  R _ {\lambda \mu \nu }   ^ {k} $, 
$  R _ {\mu \nu }  $
and  $  S $
are the Riemannian curvature tensor (cf. [[Riemann tensor|Riemann tensor]]), the [[Ricci tensor|Ricci tensor]], and the [[Scalar curvature|scalar curvature]] tensor, respectively. This tensor is nowadays called the Bochner curvature tensor.

In 1967, S. Tachibana [[#References|[a16]]] gave a tensorial expression for  $  {\widetilde{B}  } $
in a real coordinate system in a complex  $  m $-
dimensional ( $  m = 2n $) 
Kähler manifold  $  M ( J,g ) $, 
as follows:

$$ 
B ( X,Y ) = R ( X,Y ) +
$$

$$ 
+ 
{
\frac{1}{2 ( n + 2 ) }
 } ( QY \wedge X - QX \wedge Y + QJY \wedge JX  -
$$

$$ 
- 
{} QJX \wedge JY + 2g ( QJX,Y ) J + 2g ( JX,Y ) QJ ) -
$$

$$ 
- 
{
\frac{S}{4 ( n + 1 ) ( n + 2 ) }
 } ( Y \wedge X + JY \wedge JX + 2g ( JX,Y ) J ) ,
$$

$  X,Y,Z \in \mathfrak X ( M ) $(
here,  $  \mathfrak X ( M ) $
denotes the [[Lie algebra|Lie algebra]] of vector fields on  $  M $), 
where  $  ( X \wedge Y ) Z = g ( Y,Z ) X - g ( X,Z ) Y $, 
and  $  R $, 
$  Q $
and  $  S $
are the Riemannian curvature tensor, the Ricci operator, and the scalar curvature on  $  M $, 
respectively. Bochner proved that  $  B $
has the components of  $  {\widetilde{B}  } $
with respect to complex local coordinates. So,  $  B $
is also called the Bochner curvature tensor. Since then the tensors  $  {\widetilde{B}  } $
and  $  B $
have been intensively studied on Kähler manifolds; see, e.g., [[#References|[a3]]], [[#References|[a16]]], [[#References|[a17]]], [[#References|[a22]]], [[#References|[a23]]], [[#References|[a24]]], [[#References|[a25]]], [[#References|[a26]]], [[#References|[a27]]], [[#References|[a28]]] (in particular, in [[#References|[a22]]]  $  {\widetilde{B}  } $
is identified with the fourth-order Chern–Moser tensor [[#References|[a4]]] for CR-manifolds.)

==Generalization.==
M. Sitaramayya [[#References|[a15]]] and H. Mori [[#References|[a12]]] obtained a generalized Bochner curvature tensor  $  L _ {B} $
as a component in its curvature tensor  $  \mathbf L $
by considering the decomposition theory of spaces of the generalized curvature tensor  $  \mathbf L $
on a real  $  2n $-
dimensional Kählerian vector space  $  V $, 
using the method of I.M. Singer and J.A. Thorpe [[#References|[a14]]] (see also [[#References|[a13]]]). If  $  M $
is a Kähler manifold and  $  V = T _ {z} ( M ) $, 
then  $  L _ {B} = B $.

F. Tricerri and L. Vanhecke [[#References|[a21]]] generalized this notion, that is, they succeeded in defining the generalized Bochner curvature tensor  $  B ( R ) $
as a component of the element of spaces of arbitrary generalized curvature tensors  $  R $
on a real  $  2n $-
dimensional Hermitian vector space  $  V $. 
Of course, when  $  M $
is a Kähler manifold and  $  V = T _ {z} ( M ) $, 
$  B ( R ) = L _ {B} = B $. 
Moreover, they also showed that, like the case of the Weyl tensor,  $  B ( R ) $
is invariant under conformal changes ( $  M ( g ) \rightarrow M ( {\widetilde{g}  } ) $, 
$  {\widetilde{g}  } = e  ^  \sigma  g $, 
where  $  \sigma $
is a  $  C  ^  \infty  $-
function on  $  M $) 
in an arbitrary almost-Hermitian manifold  $  M $. 
In this context it means that they gave a geometrical interpretation of the Bochner curvature tensor. These results also show  $  B ( R ) $
to be a complete generalization of the Bochner curvature tensor on Kählerian manifolds to Hermitian manifolds (cf. also [[Hermitian structure|Hermitian structure]]). Some interesting applications for  $  B ( R ) $
have also been given.

===Bochner curvature tensor on contact metric manifolds.===
In 1969, M. Matsumoto and G. Chūman [[#References|[a11]]] (see also [[#References|[a25]]]) defined on a  $  ( 2n + 1 ) $-
dimensional [[Sasakian manifold|Sasakian manifold]]  $  M ( \phi, \xi, \eta,g ) $
the contact Bochner curvature tensor, which is constructed from the Bochner curvature tensor in a Kählerian manifold by considering the Boothby–Wang fibering [[#References|[a2]]]. It is as follows:

$$ 
B  ^ {c} ( X,Y ) = R ( X,Y ) +
$$

$$ 
+ 
{
\frac{1}{2 ( n + 2 ) }
 } ( QY \wedge X - QX \wedge Y + Q \phi Y \wedge \phi X -
$$

$$ 
- 
Q \phi X \wedge \phi Y + 2g ( Q \phi X,Y ) \phi + 2g ( \phi X,Y ) Q \phi +
$$

$$ 
+ 
{} \eta ( Y ) QX \wedge \xi + \eta ( X ) \xi \wedge QY )  -
$$

$$ 
- 
{
\frac{k + 2n }{2 ( n + 2 ) }
 } ( \phi Y \wedge \phi X + 2g ( \phi X,Y ) \phi )  -
$$

$$ 
- 
{
\frac{k - 4 }{2 ( n + 2 ) }
 } Y \wedge X  +
$$

$$ 
+ 
{
\frac{k}{2 ( n + 2 ) }
 } ( \eta ( Y ) \xi \wedge X + \eta ( X ) Y \wedge \xi ) ,
$$

where  $  X,Y \in \mathfrak X ( M ) $, 
$  k = { {( S + 2n ) } / {2 ( n + 1 ) } } $. 
They called this tensor the  $  C $-
Bochner tensor. Then they also proved that the  $  C $-
Bochner tensor is invariant under  $  D $-
homothetic deformations  $  M ( \phi, \xi, \eta,g ) \rightarrow M ( \phi  ^ {*} , \xi  ^ {*} , \eta  ^ {*} ,g  ^ {*} ) $, 
$  g  ^ {*} = \alpha g + \alpha ( \alpha - 1 ) \eta \otimes \eta $, 
$  \phi  ^ {*} = \phi $, 
$  \xi  ^ {*} = \alpha ^ {- 1 } \xi $, 
$  \eta  ^ {*} = \alpha \eta $, 
$  \alpha $
a positive constant (see [[#References|[a19]]]), on a Sasakian manifold. After that many papers about  $  B  ^ {c} $
on Sasakian manifolds were published; see, e.g., [[#References|[a5]]], [[#References|[a11]]], [[#References|[a25]]], [[#References|[a28]]].

The tensor  $  B  ^ {c} $
is generally not invariant under  $  D $-
homothetic deformations in more general manifolds, for example  $  K $-
contact Riemannian manifolds and contact metric manifolds. So, a natural problem arises here: Is it possible to construct a  "Bochner curvature tensor"  for manifolds of more general classes than Sasakian manifolds?

In 1991, H. Endo [[#References|[a6]]] defined on a  $  K $-
contact Riemannian manifold the  $  E $-
contact Bochner curvature tensor, constructed from  $  B  ^ {c} $. 
The  $  E $-
contact Bochner curvature tensor is invariant under  $  D $-
homothetic deformations on a  $  K $-
contact Riemannian manifold and becomes  $  B  ^ {c} $
on a Sasakian manifold. He also showed that a  $  K $-
contact Riemannian manifold with vanishing  $  E $-
contact Bochner curvature tensor is Sasakian. Moreover, in 1993 he constructed [[#References|[a7]]] on a manifold of more general class than  $  K $-
contact Riemannian manifolds (to wit, a contact metric manifold), an extended contact Bochner curvature tensor by using a new tensor  $  B ^ {es } $
which modified  $  B  ^ {c} $. 
It is called the  $  EK $-
contact Bochner curvature tensor. Of course, the  $  EK $-
contact Bochner curvature tensor coincides with  $  B  ^ {c} $
on a Sasakian manifold and is invariant under  $  D $-
homothetic deformations on a contact metric manifold. Furthermore, he proved that contact metric manifolds with vanishing  $  EK $-
contact Bochner curvature tensor are Sasakian (see [[#References|[a8]]] for another study on  $  B ^ {es } $).

===Bochner curvature tensor on almost- $  C ( \alpha ) $manifolds.===
D. Janssens and L. Vanhecke [[#References|[a10]]] defined a Bochner curvature tensor on a class of almost-contact metric manifolds, i.e., almost- $  C ( \alpha ) $
manifolds, containing Sasakian manifolds, Kemmotsu manifolds, and co-symplectic manifolds (cf. [[#References|[a10]]]) with a decomposition theory of spaces of a class of the generalized curvature tensor on a real vector space. Some geometrical applications were also given.

===Modified contact Bochner curvature tensor on almost co-symplectic manifolds.===
Endo [[#References|[a9]]] considered a tensor which modifies  $  B  ^ {c} $
and introduced a new modified contact Bochner curvature tensor which is invariant with respect to  $  D $-
homothetic deformations on an almost co-symplectic manifold  $  M $. 
He called it the  $  AC $-
contact Bochner curvature tensor. If  $  M $
is a co-symplectic manifold, the  $  AC $-
contact Bochner curvature turns into the main part of  $  B  ^ {c} $. 
He also studied almost co-symplectic manifolds with vanishing  $  AC $-
contact Bochner curvature tensor.

===Bochner-type curvature tensor on the space defined by  $  \eta = 0 $.===
In 1988, S. Tanno [[#References|[a20]]] defined on a contact metric manifold  $  M $
the Bochner-type curvature tensor ( $  B _ {zxy }   ^ {u} $) 
for the space defined by  $  \eta = 0 $, 
in such a way that its change under gauge transformations ( $  \eta \rightarrow {\widetilde \eta   } = \sigma \eta $, 
$  \sigma = { \mathop{\rm exp} } ( 2 \alpha ) $
is a positive function on  $  M $) 
is natural; it has a generalized form of the Chern–Moser–Tanaka invariant [[#References|[a4]]], [[#References|[a18]]] ( $  ( B _ {zxy }   ^ {u} ) $
is not a tensor on  $  M $). 
From this he obtained a relation between  $  ( B _ {zxy }   ^ {u} ) $
and the  $  CR $-
structure corresponding to  $  ( \eta, \phi ) $.

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  S. Bochner,   "Curvature and Betti numbers II"  ''Ann. of Math.'' , '''50'''  (1949)  pp. 77–93</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top">  W.M. Boothby,   H.C. Wang,   "On contact manifolds"  ''Ann. of Math.'' , '''68'''  (1958)  pp. 721–734</TD></TR>
<TR><TD valign="top">[a3]</TD> <TD valign="top">  B.Y. Chen,   K. Yano,   "Manifolds with vanishing Weyl or Bochner curvature tensor"  ''J. Math. Soc. Japan'' , '''27'''  (1975)  pp. 106–112</TD></TR>
<TR><TD valign="top">[a4]</TD> <TD valign="top">  S.S. Chern,   J.K. Moser,   "Real hypersurfaces in complex manifolds"  ''Acta Math.'' , '''133'''  (1974)  pp. 219–271</TD></TR>
<TR><TD valign="top">[a5]</TD> <TD valign="top">  H. Endo,   "On anti-invariant submanifolds in Sasakian manifolds with vanishing contact Bochner curvature tensor"  ''Publ. Math. Debrecen'' , '''38'''  (1991)  pp. 263–271</TD></TR>
<TR><TD valign="top">[a6]</TD> <TD valign="top">  H. Endo,   "On $K$-contact Riemannian manifolds with vanishing $E$-contact Bochner curvature tensor"  ''Colloq. Math.'' , '''62'''  (1991)  pp. 293–297</TD></TR>
<TR><TD valign="top">[a7]</TD> <TD valign="top">  H. Endo,   "On an extended contact Bochner curvature tensor on contact metric manifolds"  ''Colloq. Math.'' , '''65'''  (1993)  pp. 33–41</TD></TR>
<TR><TD valign="top">[a8]</TD> <TD valign="top">  H. Endo,   "On certain tensor fields on contact metric manifolds. II"  ''Publ. Math. Debrecen'' , '''44'''  (1994)  pp. 157–166</TD></TR>
<TR><TD valign="top">[a9]</TD> <TD valign="top">  H. Endo,   "On the $AC$-contact Bochner curvature tensor field on almost cosymplectic manifolds"  ''Publ. Inst. Math. (Beograd) (N.S.)'' , '''56'''  (1994)  pp. 102–110</TD></TR>
<TR><TD valign="top">[a10]</TD> <TD valign="top">  D. Janssens,   L. Vanhecke,   "Almost contact structures and curvature tensors"  ''Kodai Math. J.'' , '''4'''  (1981)  pp. 1–27</TD></TR>
<TR><TD valign="top">[a11]</TD> <TD valign="top">  M. Matsumoto,   G. Chūman,   "On the $C$-Bochner curvature tensor"  ''TRU. Math.'' , '''5'''  (1967)  pp. 21–30</TD></TR>
<TR><TD valign="top">[a12]</TD> <TD valign="top">  H. Mori,   "On the decomposition of generalized $K$-curvature tensor fields"  ''Tôhoku Math. J.'' , '''25'''  (1973)  pp. 225–235</TD></TR>
<TR><TD valign="top">[a13]</TD> <TD valign="top">  K. Nomizu,   "On the decomposition of generalized curvature tensor fields" , ''Differential Geometry (In Honour of K. Yano)'' , Kinokuniya  (1972)  pp. 335–345</TD></TR>
<TR><TD valign="top">[a14]</TD> <TD valign="top">  I.M. Singer,   J.A. Thorpe,   "The curvature of 4-dimensional Einstein spaces" , ''Global Analysis (In Honour of K. Kodaira)'' , Univ. Tokyo Press  (1969)  pp. 355–365</TD></TR>
<TR><TD valign="top">[a15]</TD> <TD valign="top">  M. Sitaramayya,   "Curvature tensors in Kähler manifolds"  ''Trans. Amer. Math. Soc.'' , '''183'''  (1973)  pp. 341–353</TD></TR>
<TR><TD valign="top">[a16]</TD> <TD valign="top">  S. Tachibana,   "On the Bochner curvature tensor"  ''Natur. Sci. Rep. Ochanomizu Univ.'' , '''18'''  (1967)  pp. 15–19</TD></TR>
<TR><TD valign="top">[a17]</TD> <TD valign="top">  S. Tachibana,   R.C. Liu,   "Notes on Kaehlerian metrics with vanishing Bochner curvature tensor"  ''Kōdai Math. Sem. Rep.'' , '''22'''  (1970)  pp. 313–321</TD></TR>
<TR><TD valign="top">[a18]</TD> <TD valign="top">  N. Tanaka,   "On non-degenerate real hypersurfaces, graded Lie algebras and Cartan connections"  ''Japan. J. Math. (N.S.)'' , '''2'''  (1976)  pp. 131–190</TD></TR>
<TR><TD valign="top">[a19]</TD> <TD valign="top">  S. Tanno,   "Partially conformal transformations with respect to $(m-1)$-dimensional distributions of $m$-dimensional Riemannian manifolds"  ''Tôhoku Math. J.'' , '''17'''  (1965)  pp. 358–409</TD></TR>
<TR><TD valign="top">[a20]</TD> <TD valign="top">  S. Tanno,   "The Bochner type curvature tensor of contact Riemannian structure"  ''Hokkaido Math. J.'' , '''19'''  (1990)  pp. 55–66</TD></TR><TR><TD valign="top">[a21]</TD> <TD valign="top">  F. Tricerri,   L. Vanhecke,   "Curvature tensors on almost Hermitian manifolds"  ''Trans. Amer. Math. Soc.'' , '''267'''  (1981)  pp. 365–398</TD></TR>
<TR><TD valign="top">[a22]</TD> <TD valign="top">  S.M. Webster,   "On the pseudo-conformal geometry of a Kaehler manifold"  ''Math. Z.'' , '''157'''  (1977)  pp. 265–270</TD></TR>
<TR><TD valign="top">[a23]</TD> <TD valign="top">  K. Yano,   "Manifolds and submanifolds with vanishing Weyl or Bochner curvature tensor" , ''Proc. Symp. Pure Math.'' , '''27''' , Amer. Math. Soc.  (1975)  pp. 253–262</TD></TR>
<TR><TD valign="top">[a24]</TD> <TD valign="top">  K. Yano,   "Differential geometry of totally real submanifolds" , ''Topics in Differential Geometry'' , Acad. Press  (1976)  pp. 173–184</TD></TR>
<TR><TD valign="top">[a25]</TD> <TD valign="top">  K. Yano,   "Anti-invariant submanifolds of a Sasakian manifold with vanishing contact Bochner curvature tensor"  ''J. Diff. Geom.'' , '''12'''  (1977)  pp. 153–170</TD></TR>
<TR><TD valign="top">[a26]</TD> <TD valign="top">  K. Yano,   S. Bochner,   "Curvature and Betti numbers" , ''Annals of Math. Stud.'' , '''32''' , Princeton Univ. Press  (1953)</TD></TR><TR><TD valign="top">[a27]</TD> <TD valign="top">  K. Yano,   S. Ishihara,   "Kaehlerian manifolds with constant scalar curvature whose Bochner curvature tensor vanishes"  ''Hokkaido Math. J.'' , '''3'''  (1974)  pp. 297–304</TD></TR>
<TR><TD valign="top">[a28]</TD> <TD valign="top">  K. Yano,   M. Kon,   "Structures on manifolds" , World Sci.  (1984)</TD></TR></table>