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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b1106701.png" />-dimensional Kählerian manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b1106702.png" />, it is defined as follows:
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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/b/b110/b110670/b1106703.png" /></td> </tr></table>
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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/b/b110/b110670/b1106704.png" /></td> </tr></table>
+
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:
  
<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/b/b110/b110670/b1106705.png" /></td> </tr></table>
+
$$
 +
{\widetilde{B}  } _ {jh  ^ {*}  lk  ^ {*} } = R _ {jh  ^ {*}  lk  ^ {*} } -
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b1106706.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b1106707.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b1106708.png" /> 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.
+
$$
 +
-
 +
{
 +
\frac{1}{n + 2 }
 +
} ( g _ {jk  ^ {*}  } R _ {h  ^ {*}  l } + g _ {lk  ^ {*}  } R _ {jh  ^ {*}  } + g _ {h  ^ {*}  l } R _ {jk  ^ {*}  } + g _ {jh  ^ {*}  } R _ {lk  ^ {*}  } ) +
 +
$$
  
In 1967, S. Tachibana [[#References|[a16]]] gave a tensorial expression for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b1106709.png" /> in a real coordinate system in a complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b11067010.png" />-dimensional (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b11067011.png" />) Kähler manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b11067012.png" />, as follows:
+
$$
 +
+
 +
{
 +
\frac{S}{2 ( n + 1 ) ( n + 2 ) }
 +
} ( g _ {h  ^ {*}  l } g _ {jk  ^ {*}  } + g _ {jh  ^ {*}  } g _ {lk  ^ {*}  } ) ,
 +
$$
  
<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/b/b110/b110670/b11067013.png" /></td> </tr></table>
+
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.
  
<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/b/b110/b110670/b11067014.png" /></td> </tr></table>
+
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:
  
<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/b/b110/b110670/b11067015.png" /></td> </tr></table>
+
$$
 +
B ( X,Y ) = R ( X,Y ) +
 +
$$
  
<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/b/b110/b110670/b11067016.png" /></td> </tr></table>
+
$$
 +
+
 +
{
 +
\frac{1}{2 ( n + 2 ) }
 +
} ( QY \wedge X - QX \wedge Y + QJY \wedge JX  -
 +
$$
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b11067017.png" /> (here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b11067018.png" /> denotes the [[Lie algebra|Lie algebra]] of vector fields on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b11067019.png" />), where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b11067020.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b11067021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b11067022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b11067023.png" /> are the Riemannian curvature tensor, the Ricci operator, and the scalar curvature on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b11067024.png" />, respectively. Bochner proved that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b11067025.png" /> has the components of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b11067026.png" /> with respect to complex local coordinates. So, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b11067027.png" /> is also called the Bochner curvature tensor. Since then the tensors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b11067028.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b11067029.png" /> 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]]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b11067030.png" /> is identified with the fourth-order Chern–Moser tensor [[#References|[a4]]] for CR-manifolds.)
+
$$
 +
-
 +
{} QJX \wedge JY + 2g ( QJX,Y ) J + 2g ( JX,Y ) QJ ) -
 +
$$
 +
 
 +
$$
 +
-
 +
{
 +
\frac{S}{4 ( n + 1 ) ( n + 2 ) }
 +
} ( Y \wedge X + JY \wedge JX + 2g ( JX,Y ) J ) ,
 +
$$
 +
 
 +
$  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.==
 
==Generalization.==
M. Sitaramayya [[#References|[a15]]] and H. Mori [[#References|[a12]]] obtained a generalized Bochner curvature tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b11067031.png" /> as a component in its curvature tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b11067032.png" /> by considering the decomposition theory of spaces of the generalized curvature tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b11067033.png" /> on a real <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b11067034.png" />-dimensional Kählerian vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b11067035.png" />, using the method of I.M. Singer and J.A. Thorpe [[#References|[a14]]] (see also [[#References|[a13]]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b11067036.png" /> is a Kähler manifold and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b11067037.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b11067038.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b11067039.png" /> as a component of the element of spaces of arbitrary generalized curvature tensors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b11067040.png" /> on a real <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b11067041.png" />-dimensional Hermitian vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b11067042.png" />. Of course, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b11067043.png" /> is a Kähler manifold and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b11067044.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b11067045.png" />. Moreover, they also showed that, like the case of the Weyl tensor, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b11067046.png" /> is invariant under conformal changes (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b11067047.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b11067048.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b11067049.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b11067050.png" />-function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b11067051.png" />) in an arbitrary almost-Hermitian manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b11067052.png" />. In this context it means that they gave a geometrical interpretation of the Bochner curvature tensor. These results also show <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b11067053.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b11067054.png" /> have also been given.
+
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.===
 
===Bochner curvature tensor on contact metric manifolds.===
In 1969, M. Matsumoto and G. Chūman [[#References|[a11]]] (see also [[#References|[a25]]]) defined on a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b11067055.png" />-dimensional [[Sasakian manifold|Sasakian manifold]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b11067056.png" /> 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:
+
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:
  
<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/b/b110/b110670/b11067057.png" /></td> </tr></table>
+
$$
 +
B  ^ {c} ( X,Y ) = R ( X,Y ) +
 +
$$
  
<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/b/b110/b110670/b11067058.png" /></td> </tr></table>
+
$$
 +
+
 +
{
 +
\frac{1}{2 ( n + 2 ) }
 +
} ( QY \wedge X - QX \wedge Y + Q \phi Y \wedge \phi X -
 +
$$
  
<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/b/b110/b110670/b11067059.png" /></td> </tr></table>
+
$$
 +
-  
 +
Q \phi X \wedge \phi Y + 2g ( Q \phi X,Y ) \phi + 2g ( \phi X,Y ) Q \phi +
 +
$$
  
<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/b/b110/b110670/b11067060.png" /></td> </tr></table>
+
$$
 +
+
 +
{} \eta ( Y ) QX \wedge \xi + \eta ( X ) \xi \wedge QY )  -
 +
$$
  
<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/b/b110/b110670/b11067061.png" /></td> </tr></table>
+
$$
 +
-  
 +
{
 +
\frac{k + 2n }{2 ( n + 2 ) }
 +
} ( \phi Y \wedge \phi X + 2g ( \phi X,Y ) \phi )  -
 +
$$
  
<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/b/b110/b110670/b11067062.png" /></td> </tr></table>
+
$$
 +
-  
 +
{
 +
\frac{k - 4 }{2 ( n + 2 ) }
 +
} Y \wedge X  +
 +
$$
  
<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/b/b110/b110670/b11067063.png" /></td> </tr></table>
+
$$
 +
+
 +
{
 +
\frac{k}{2 ( n + 2 ) }
 +
} ( \eta ( Y ) \xi \wedge X + \eta ( X ) Y \wedge \xi ) ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b11067064.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b11067065.png" />. They called this tensor the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b11067067.png" />-Bochner tensor. Then they also proved that the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b11067068.png" />-Bochner tensor is invariant under <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b11067070.png" />-homothetic deformations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b11067071.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b11067072.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b11067073.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b11067074.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b11067075.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b11067076.png" /> a positive constant (see [[#References|[a19]]]), on a Sasakian manifold. After that many papers about <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b11067077.png" /> on Sasakian manifolds were published; see, e.g., [[#References|[a5]]], [[#References|[a11]]], [[#References|[a25]]], [[#References|[a28]]].
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b11067078.png" /> is generally not invariant under <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b11067079.png" />-homothetic deformations in more general manifolds, for example <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b11067080.png" />-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?
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b11067081.png" />-contact Riemannian manifold the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b11067083.png" />-contact Bochner curvature tensor, constructed from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b11067084.png" />. The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b11067085.png" />-contact Bochner curvature tensor is invariant under <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b11067086.png" />-homothetic deformations on a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b11067087.png" />-contact Riemannian manifold and becomes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b11067088.png" /> on a Sasakian manifold. He also showed that a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b11067089.png" />-contact Riemannian manifold with vanishing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b11067090.png" />-contact Bochner curvature tensor is Sasakian. Moreover, in 1993 he constructed [[#References|[a7]]] on a manifold of more general class than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b11067091.png" />-contact Riemannian manifolds (to wit, a contact metric manifold), an extended contact Bochner curvature tensor by using a new tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b11067092.png" /> which modified <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b11067093.png" />. It is called the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b11067095.png" />-contact Bochner curvature tensor. Of course, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b11067096.png" />-contact Bochner curvature tensor coincides with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b11067097.png" /> on a Sasakian manifold and is invariant under <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b11067098.png" />-homothetic deformations on a contact metric manifold. Furthermore, he proved that contact metric manifolds with vanishing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b11067099.png" />-contact Bochner curvature tensor are Sasakian (see [[#References|[a8]]] for another study on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b110670100.png" />).
+
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-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b110670101.png" /> manifolds.===
+
===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-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b110670102.png" /> 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.
+
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.===
 
===Modified contact Bochner curvature tensor on almost co-symplectic manifolds.===
Endo [[#References|[a9]]] considered a tensor which modifies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b110670103.png" /> and introduced a new modified contact Bochner curvature tensor which is invariant with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b110670104.png" />-homothetic deformations on an almost co-symplectic manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b110670105.png" />. He called it the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b110670107.png" />-contact Bochner curvature tensor. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b110670108.png" /> is a co-symplectic manifold, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b110670109.png" />-contact Bochner curvature turns into the main part of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b110670110.png" />. He also studied almost co-symplectic manifolds with vanishing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b110670111.png" />-contact Bochner curvature tensor.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b110670112.png" />.===
+
===Bochner-type curvature tensor on the space defined by $  \eta = 0 $.===
In 1988, S. Tanno [[#References|[a20]]] defined on a contact metric manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b110670113.png" /> the Bochner-type curvature tensor (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b110670114.png" />) for the space defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b110670115.png" />, in such a way that its change under gauge transformations (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b110670116.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b110670117.png" /> is a positive function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b110670118.png" />) is natural; it has a generalized form of the Chern–Moser–Tanaka invariant [[#References|[a4]]], [[#References|[a18]]] (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b110670119.png" /> is not a tensor on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b110670120.png" />). From this he obtained a relation between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b110670121.png" /> and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b110670122.png" />-structure corresponding to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b110670123.png" />.
+
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====
 
====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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b110670124.png" />-contact Riemannian manifolds with vanishing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b110670125.png" />-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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b110670126.png" />-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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b110670127.png" />-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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b110670128.png" />-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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b110670129.png" />-dimensional distributions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b110670130.png" />-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>
 
<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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b110670124.png" />-contact Riemannian manifolds with vanishing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b110670125.png" />-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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b110670126.png" />-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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b110670127.png" />-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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b110670128.png" />-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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b110670129.png" />-dimensional distributions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110670/b110670130.png" />-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>

Revision as of 10:59, 29 May 2020


In 1949, while studying the Betti number of a Kähler manifold, S. Bochner [a1] (see also [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. 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), the Ricci tensor, and the scalar curvature tensor, respectively. This tensor is nowadays called the Bochner curvature tensor.

In 1967, S. Tachibana [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 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., [a3], [a16], [a17], [a22], [a23], [a24], [a25], [a26], [a27], [a28] (in particular, in [a22] $ {\widetilde{B} } $ is identified with the fourth-order Chern–Moser tensor [a4] for CR-manifolds.)

Generalization.

M. Sitaramayya [a15] and H. Mori [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 [a14] (see also [a13]). If $ M $ is a Kähler manifold and $ V = T _ {z} ( M ) $, then $ L _ {B} = B $.

F. Tricerri and L. Vanhecke [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). Some interesting applications for $ B ( R ) $ have also been given.

Bochner curvature tensor on contact metric manifolds.

In 1969, M. Matsumoto and G. Chūman [a11] (see also [a25]) defined on a $ ( 2n + 1 ) $- dimensional 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 [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 [a19]), on a Sasakian manifold. After that many papers about $ B ^ {c} $ on Sasakian manifolds were published; see, e.g., [a5], [a11], [a25], [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 [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 [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 [a8] for another study on $ B ^ {es } $).

Bochner curvature tensor on almost- $ C ( \alpha ) $manifolds.

D. Janssens and L. Vanhecke [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. [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 [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 [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 [a4], [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

[a1] S. Bochner, "Curvature and Betti numbers II" Ann. of Math. , 50 (1949) pp. 77–93
[a2] W.M. Boothby, H.C. Wang, "On contact manifolds" Ann. of Math. , 68 (1958) pp. 721–734
[a3] B.Y. Chen, K. Yano, "Manifolds with vanishing Weyl or Bochner curvature tensor" J. Math. Soc. Japan , 27 (1975) pp. 106–112
[a4] S.S. Chern, J.K. Moser, "Real hypersurfaces in complex manifolds" Acta Math. , 133 (1974) pp. 219–271
[a5] H. Endo, "On anti-invariant submanifolds in Sasakian manifolds with vanishing contact Bochner curvature tensor" Publ. Math. Debrecen , 38 (1991) pp. 263–271
[a6] H. Endo, "On -contact Riemannian manifolds with vanishing -contact Bochner curvature tensor" Colloq. Math. , 62 (1991) pp. 293–297
[a7] H. Endo, "On an extended contact Bochner curvature tensor on contact metric manifolds" Colloq. Math. , 65 (1993) pp. 33–41
[a8] H. Endo, "On certain tensor fields on contact metric manifolds. II" Publ. Math. Debrecen , 44 (1994) pp. 157–166
[a9] H. Endo, "On the -contact Bochner curvature tensor field on almost cosymplectic manifolds" Publ. Inst. Math. (Beograd) (N.S.) , 56 (1994) pp. 102–110
[a10] D. Janssens, L. Vanhecke, "Almost contact structures and curvature tensors" Kodai Math. J. , 4 (1981) pp. 1–27
[a11] M. Matsumoto, G. Chūman, "On the -Bochner curvature tensor" TRU. Math. , 5 (1967) pp. 21–30
[a12] H. Mori, "On the decomposition of generalized -curvature tensor fields" Tôhoku Math. J. , 25 (1973) pp. 225–235
[a13] K. Nomizu, "On the decomposition of generalized curvature tensor fields" , Differential Geometry (In Honour of K. Yano) , Kinokuniya (1972) pp. 335–345
[a14] 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
[a15] M. Sitaramayya, "Curvature tensors in Kähler manifolds" Trans. Amer. Math. Soc. , 183 (1973) pp. 341–353
[a16] S. Tachibana, "On the Bochner curvature tensor" Natur. Sci. Rep. Ochanomizu Univ. , 18 (1967) pp. 15–19
[a17] S. Tachibana, R.C. Liu, "Notes on Kaehlerian metrics with vanishing Bochner curvature tensor" Kōdai Math. Sem. Rep. , 22 (1970) pp. 313–321
[a18] N. Tanaka, "On non-degenerate real hypersurfaces, graded Lie algebras and Cartan connections" Japan. J. Math. (N.S.) , 2 (1976) pp. 131–190
[a19] S. Tanno, "Partially conformal transformations with respect to -dimensional distributions of -dimensional Riemannian manifolds" Tôhoku Math. J. , 17 (1965) pp. 358–409
[a20] S. Tanno, "The Bochner type curvature tensor of contact Riemannian structure" Hokkaido Math. J. , 19 (1990) pp. 55–66
[a21] F. Tricerri, L. Vanhecke, "Curvature tensors on almost Hermitian manifolds" Trans. Amer. Math. Soc. , 267 (1981) pp. 365–398
[a22] S.M. Webster, "On the pseudo-conformal geometry of a Kaehler manifold" Math. Z. , 157 (1977) pp. 265–270
[a23] K. Yano, "Manifolds and submanifolds with vanishing Weyl or Bochner curvature tensor" , Proc. Symp. Pure Math. , 27 , Amer. Math. Soc. (1975) pp. 253–262
[a24] K. Yano, "Differential geometry of totally real submanifolds" , Topics in Differential Geometry , Acad. Press (1976) pp. 173–184
[a25] K. Yano, "Anti-invariant submanifolds of a Sasakian manifold with vanishing contact Bochner curvature tensor" J. Diff. Geom. , 12 (1977) pp. 153–170
[a26] K. Yano, S. Bochner, "Curvature and Betti numbers" , Annals of Math. Stud. , 32 , Princeton Univ. Press (1953)
[a27] K. Yano, S. Ishihara, "Kaehlerian manifolds with constant scalar curvature whose Bochner curvature tensor vanishes" Hokkaido Math. J. , 3 (1974) pp. 297–304
[a28] K. Yano, M. Kon, "Structures on manifolds" , World Sci. (1984)
How to Cite This Entry:
Bochner curvature tensor. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bochner_curvature_tensor&oldid=18427
This article was adapted from an original article by H. Endo (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article