Hermitian connection

An affine connection on a Hermitian manifold relative to which the tensor defined by the complex structure and the fundamental -form are parallel, implying the same property for the Hermitian form . If the affine connection on is given by local connection forms , then these conditions can be expressed as

On a given Hermitian manifold there is one and only one Hermitian connection for which .

A generalization is an almost-Hermitian connection, which is defined by similar conditions on the tensors and with on an almost-Hermitian manifold . An almost-Hermitian connection on a given 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 and ), 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)

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)
[a2]	R.O. Wells jr., "Differential analysis on complex manifolds" , Springer (1980)