6
$\begingroup$

Hi! I've always read that on a complex manifold (obviously not kahler), with a given hermitian metric on tangent bundle, the chern connection and the levi civita connection on the underlying real bundle could be different. Please can someone give me an explicit example of this fact?

Thank you in advance

Improve this question
$\endgroup$

3 Answers 3

Reset to default
16
$\begingroup$

You need a non-Kahler complex manifold. Then the Chern connection will have nontrivial torsion. And the torsion corresponds to the non-closed Kahler form of the metric.

Improve this answer
$\endgroup$
6
$\begingroup$

Further to a previous answer, you might like to refer to arXiv:0911.5655 and references therein for some interesting results on Chern connections on non-Kähler (and indeed non-integrable) almost-Hermitian manifolds.

Improve this answer
$\endgroup$
1
$\begingroup$

To add to the previous answers, here is a paper by Bo Yang and Fangyang Zheng [1] which explores the relationship between the Levi-Civita and Chern connection on a Hermitian manifold. In particular, Lemma 7 describes the relations between the curvatures of these two connections and the torsion of the Chern connection in very explicit terms. The paper is also available on the Arxiv.

[1] Yang, Bo; Zheng, Fangyang, On curvature tensors of Hermitian manifolds, Commun. Anal. Geom. 26, No. 5, 1195-1222 (2018). ZBL1408.53031.

Improve this answer
$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .