JP Journal of Geometry and Topology

The JP Journal of Geometry and Topology publishes articles in all branches of geometry and topology, with applications to physics. It covers areas such as differential geometry, algebraic topology, and geometric aspects of mathematical physics. Survey articles are also welcome.

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ALMOST EINSTEIN-HERMITIAN MANIFOLDS

Authors

  • Jaeman Kim

Keywords:

almost Einstein-Hermitian 4-manifold, Einstein, Hermitian, almost Einstein-Hermitian manifold of dimension $2 n 4(\geq 6)$

DOI:

https://doi.org/10.17654/0972415X24008

Abstract

In this paper, we show that every almost Einstein-Hermitian 4-manifold (i.e., almost Hermitian 4-manifold with $J$-invariant Ricci tensor and harmonic Weyl tensor) is either Einstein or Hermitian. Consequently, we obtain that any almost Einstein-Hermitian 4-manifold which is not Einstein must be Hermitian and that every almost Einstein-Hermitian 4-manifold which is not Hermitian is Einstein. In contrast to the 4-dimensional case, there exists an almost Einstein-Hermitian manifold of dimension $2 n+4(\geq 6)$ which is neither Einstein nor Hermitian.

Received: October 22, 2024
Accepted: November 18, 2024

References

A. L. Besse, Einstein Manifolds, Springer, Berlin, 1987.

G. Catino, Generalized quasi-Einstein manifolds with harmonic Weyl tensor, Math. Z. 271(3-4) (2012), 751-756.

J. Kim, On Einstein Hermitian manifolds, Monatsh. Math. 152 (2007), 251-254.

O. Muskarov, On Hermitian surfaces with J-invariant Ricci tensor, J. Geom. 72 (2001), 151-156.

E. Ribeiro, Rigidity of four-dimensional compact manifolds with harmonic Weyl tensor, Ann. Mat. Pura Appl. (4) 195 (2016), 2171-2181.

T. Sato, Some remarks on almost Kähler 4-manifolds of pointwise constant holomorphic sectional curvature, Kodai Math. J. 22 (1999), 286-303.

K. Yano, Differential Geometry on Complex and Almost Complex Spaces, Pergamon Press, 1965.

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Published

2024-12-24

Issue

Vol. 30 No. 2 (2024)

Section

Articles

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How to Cite

ALMOST EINSTEIN-HERMITIAN MANIFOLDS. (2024). JP Journal of Geometry and Topology, 30(2), 119-130. https://doi.org/10.17654/0972415X24008

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