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 -dimensional Kählerian manifold , it is defined as follows:

where , and 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 in a real coordinate system in a complex -dimensional () Kähler manifold , as follows:

(here, denotes the Lie algebra of vector fields on ), where , and , and are the Riemannian curvature tensor, the Ricci operator, and the scalar curvature on , respectively. Bochner proved that has the components of with respect to complex local coordinates. So, is also called the Bochner curvature tensor. Since then the tensors and have been intensively studied on Kähler manifolds; see, e.g., [a3], [a16], [a17], [a22], [a23], [a24], [a25], [a26], [a27], [a28] (in particular, in [a22] 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 as a component in its curvature tensor by considering the decomposition theory of spaces of the generalized curvature tensor on a real -dimensional Kählerian vector space , using the method of I.M. Singer and J.A. Thorpe [a14] (see also [a13]). If is a Kähler manifold and , then .

F. Tricerri and L. Vanhecke [a21] generalized this notion, that is, they succeeded in defining the generalized Bochner curvature tensor as a component of the element of spaces of arbitrary generalized curvature tensors on a real -dimensional Hermitian vector space . Of course, when is a Kähler manifold and , . Moreover, they also showed that, like the case of the Weyl tensor, is invariant under conformal changes (, , where is a -function on ) in an arbitrary almost-Hermitian manifold . In this context it means that they gave a geometrical interpretation of the Bochner curvature tensor. These results also show to be a complete generalization of the Bochner curvature tensor on Kählerian manifolds to Hermitian manifolds (cf. also Hermitian structure). Some interesting applications for 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 -dimensional Sasakian manifold 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:

where , . They called this tensor the -Bochner tensor. Then they also proved that the -Bochner tensor is invariant under -homothetic deformations , , , , , a positive constant (see [a19]), on a Sasakian manifold. After that many papers about on Sasakian manifolds were published; see, e.g., [a5], [a11], [a25], [a28].

The tensor is generally not invariant under -homothetic deformations in more general manifolds, for example -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 -contact Riemannian manifold the -contact Bochner curvature tensor, constructed from . The -contact Bochner curvature tensor is invariant under -homothetic deformations on a -contact Riemannian manifold and becomes on a Sasakian manifold. He also showed that a -contact Riemannian manifold with vanishing -contact Bochner curvature tensor is Sasakian. Moreover, in 1993 he constructed [a7] on a manifold of more general class than -contact Riemannian manifolds (to wit, a contact metric manifold), an extended contact Bochner curvature tensor by using a new tensor which modified . It is called the -contact Bochner curvature tensor. Of course, the -contact Bochner curvature tensor coincides with on a Sasakian manifold and is invariant under -homothetic deformations on a contact metric manifold. Furthermore, he proved that contact metric manifolds with vanishing -contact Bochner curvature tensor are Sasakian (see [a8] for another study on ).

Bochner curvature tensor on almost- manifolds.

D. Janssens and L. Vanhecke [a10] defined a Bochner curvature tensor on a class of almost-contact metric manifolds, i.e., almost- 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 and introduced a new modified contact Bochner curvature tensor which is invariant with respect to -homothetic deformations on an almost co-symplectic manifold . He called it the -contact Bochner curvature tensor. If is a co-symplectic manifold, the -contact Bochner curvature turns into the main part of . He also studied almost co-symplectic manifolds with vanishing -contact Bochner curvature tensor.

Bochner-type curvature tensor on the space defined by .

In 1988, S. Tanno [a20] defined on a contact metric manifold the Bochner-type curvature tensor () for the space defined by , in such a way that its change under gauge transformations (, is a positive function on ) is natural; it has a generalized form of the Chern–Moser–Tanaka invariant [a4], [a18] ( is not a tensor on ). From this he obtained a relation between and the -structure corresponding to .

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)