Hermitian connection

An affine connection on a Hermitian manifold $ M $ relative to which the tensor $ \phi $ defined by the complex structure and the fundamental $ 2 $- form $ \Omega = ( 1 / 2 ) g _ {\alpha \beta } \omega ^ \beta \wedge \overline \omega \; {} ^ \alpha $ are parallel, implying the same property for the Hermitian form $ ds ^ {2} = g _ {\alpha \beta } \overline \omega \; {} ^ \alpha \omega ^ \beta $. If the affine connection on $ M $ is given by local connection forms $ \omega _ \beta ^ \alpha = \Gamma _ {\beta \gamma } ^ \alpha \omega ^ \gamma + \Gamma _ {\beta {\overline \gamma \; } } ^ \alpha \overline \omega \; {} ^ \gamma $, then these conditions can be expressed as

$$ \omega _ {down - 3 {\overline \beta \; } } ^ \alpha = \omega _ {down - 3 {\overline \beta \; } } ^ {\overline \alpha \; } = 0,\ \ \omega _ {down - 3 {\overline \beta \; } } ^ {\overline \alpha \; } = \ \overline \omega \; {} _ \beta ^ \alpha , $$

$$ d g _ {\alpha \beta } = \overline \omega \; {} _ \alpha ^ \gamma g _ {\gamma \beta } + g _ {\alpha \gamma } \omega _ \beta ^ \gamma . $$

On a given Hermitian manifold $ M $ there is one and only one Hermitian connection for which $ \Gamma _ {\beta \overline \gamma \; } ^ \alpha = 0 $.

A generalization is an almost-Hermitian connection, which is defined by similar conditions on the tensors $ \phi _ {j} ^ {i} $ and $ g _ {ij} $ with $ g _ {kl} \phi _ {i} ^ {k} \phi _ {l} ^ {l} = g _ {ij} $ on an almost-Hermitian manifold $ \widetilde{M} $. An almost-Hermitian connection on a given $ \widetilde{M} $ exists. It is uniquely defined by its torsion tensor: If the torsion tensors of two almost-Hermitian connections are the same, then so are the connections. For example, there is one and only one almost-Hermitian connection for which the torsion forms are sums of "pure" forms (that is, forms of type $ ( 2 , 0 ) $ and $ ( 0 , 2 ) $), namely, the second canonical Lichnerowicz connection.

References

[1]	A. Lichnerowicz, "Global theory of connections and holonomy groups" , Noordhoff (1976) (Translated from French)
[2]	K. Yano, "Differential geometry on complex and almost complex spaces" , Pergamon (1965) Zbl 0127.12405

Comments

The first and second canonical connections on an almost-Hermitian manifold are described in [1], p. 192 and pp. 194-195, respectively.

References

[a1]	S. Kobayashi, K. Nomizu, "Foundations of differential geometry" , 2 , Interscience (1969) Zbl 0175.48504
[a2]	R.O. Wells jr., "Differential analysis on complex manifolds" , Springer (1980)