Bochner curvature tensor

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

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[a2]	W.M. Boothby, H.C. Wang, "On contact manifolds" Ann. of Math. , 68 (1958) pp. 721–734
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