# 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