# Pseudo-Bochner curvature tensor on Hermitian manifolds

Colloquium Mathematicae (1999)

- Volume: 80, Issue: 2, page 201-209
- ISSN: 0010-1354

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topMatsuo, Koji. "Pseudo-Bochner curvature tensor on Hermitian manifolds." Colloquium Mathematicae 80.2 (1999): 201-209. <http://eudml.org/doc/210712>.

@article{Matsuo1999,

abstract = {Our main purpose of this paper is to introduce a natural generalization $B_H$ of the Bochner curvature tensor on a Hermitian manifold $M$ provided with the Hermitian connection. We will call $B_H$ the pseudo-Bochner curvature tensor. Firstly, we introduce a unique tensor P, called the (Hermitian) pseudo-curvature tensor, which has the same symmetries as the Riemannian curvature tensor on a Kähler manifold. By using P, we derive a necessary and sufficient condition for a Hermitian manifold to be of pointwise constant Hermitian holomorphic sectional curvature. Our pseudo-Bochner curvature tensor $B_H$ is naturally obtained from the conformal relation for the pseudo-curvature tensor P and it is conformally invariant. Moreover we show that $B_H$ is different from the Bochner conformal tensor in the sense of Tricerri and Vanhecke.},

author = {Matsuo, Koji},

journal = {Colloquium Mathematicae},

keywords = {Hermitian manifold; Hermitian connection; pseudo-Bochner curvature tensor; (Hermitian) pseudo-curvature tensor; curvature tensor; conformal invariant; constant holomorphic sectional curvature},

language = {eng},

number = {2},

pages = {201-209},

title = {Pseudo-Bochner curvature tensor on Hermitian manifolds},

url = {http://eudml.org/doc/210712},

volume = {80},

year = {1999},

}

TY - JOUR

AU - Matsuo, Koji

TI - Pseudo-Bochner curvature tensor on Hermitian manifolds

JO - Colloquium Mathematicae

PY - 1999

VL - 80

IS - 2

SP - 201

EP - 209

AB - Our main purpose of this paper is to introduce a natural generalization $B_H$ of the Bochner curvature tensor on a Hermitian manifold $M$ provided with the Hermitian connection. We will call $B_H$ the pseudo-Bochner curvature tensor. Firstly, we introduce a unique tensor P, called the (Hermitian) pseudo-curvature tensor, which has the same symmetries as the Riemannian curvature tensor on a Kähler manifold. By using P, we derive a necessary and sufficient condition for a Hermitian manifold to be of pointwise constant Hermitian holomorphic sectional curvature. Our pseudo-Bochner curvature tensor $B_H$ is naturally obtained from the conformal relation for the pseudo-curvature tensor P and it is conformally invariant. Moreover we show that $B_H$ is different from the Bochner conformal tensor in the sense of Tricerri and Vanhecke.

LA - eng

KW - Hermitian manifold; Hermitian connection; pseudo-Bochner curvature tensor; (Hermitian) pseudo-curvature tensor; curvature tensor; conformal invariant; constant holomorphic sectional curvature

UR - http://eudml.org/doc/210712

ER -

## References

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- [6] K. Matsuo, Locally conformally Hermitian-flat manifolds, Ann. Global Anal. Geom. 13 (1995), 43-54. Zbl0845.53049
- [7] K. Matsuo, On local conformal Hermitian-flatness of Hermitian manifolds, Tokyo J. Math. 19 (1996), 499-515. Zbl0882.53050
- [8] S. Tachibana, On the Bochner curvature tensor, Nat. Sci. Rep. Ochanomizu Univ. 18 (1967), 15-19. Zbl0161.41202
- [9] F. Tricerri and L. Vanhecke, Curvature tensors on almost Hermitian manifolds, Trans. Amer. Math. Soc. 267 (1981), 365-398. Zbl0484.53014
- [10] I. Vaisman, Generalized Hopf manifolds, Geom. Dedicata 13 (1982), 231-255.

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