Pseudo-Bochner curvature tensor on Hermitian manifolds

Koji Matsuo

Colloquium Mathematicae (1999)

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

Abstract

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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.

How to cite

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Matsuo, 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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  1. [1] A. Balas, Compact Hermitian manifolds of constant holomorphic sectional curvature, Math. Z. 189 (1985), 193-210. Zbl0575.53044
  2. [2] S. Bochner, Curvature and Betti numbers, II, Ann. of Math. 50 (1949), 77-93. 
  3. [3] L. A. Cordero, M. Fernández and A. Gray, Symplectic manifolds with no Kähler structure, Topology 25 (1986), 375-380. Zbl0596.53030
  4. [4] G. Ganchev, S. Ivanov and V. Mihova, Curvatures on anti-Kaehler manifolds, Riv. Mat. Univ. Parma (5) 2 (1993), 249-256. Zbl1010.53506
  5. [5] S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Vol. II, Interscience Publ., New York, 1969. Zbl0175.48504
  6. [6] K. Matsuo, Locally conformally Hermitian-flat manifolds, Ann. Global Anal. Geom. 13 (1995), 43-54. Zbl0845.53049
  7. [7] K. Matsuo, On local conformal Hermitian-flatness of Hermitian manifolds, Tokyo J. Math. 19 (1996), 499-515. Zbl0882.53050
  8. [8] S. Tachibana, On the Bochner curvature tensor, Nat. Sci. Rep. Ochanomizu Univ. 18 (1967), 15-19. Zbl0161.41202
  9. [9] F. Tricerri and L. Vanhecke, Curvature tensors on almost Hermitian manifolds, Trans. Amer. Math. Soc. 267 (1981), 365-398. Zbl0484.53014
  10. [10] I. Vaisman, Generalized Hopf manifolds, Geom. Dedicata 13 (1982), 231-255. 

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