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

 [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 $K$-contact Riemannian manifolds with vanishing $E$-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 $AC$-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 $C$-Bochner curvature tensor" TRU. Math. , 5 (1967) pp. 21–30 [a12] H. Mori, "On the decomposition of generalized $K$-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 $(m-1)$-dimensional distributions of $m$-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=53308
This article was adapted from an original article by H. Endo (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article