Hermitian connection

In mathematics, a Hermitian connection ${\displaystyle \nabla }$ is a connection on a Hermitian vector bundle ${\displaystyle E}$ over a smooth manifold ${\displaystyle M}$ which is compatible with the Hermitian metric ${\displaystyle \langle \cdot ,\cdot \rangle }$ on ${\displaystyle E}$, meaning that

${\displaystyle v\langle s,t\rangle =\langle \nabla _{v}s,t\rangle +\langle s,\nabla _{v}t\rangle }$

for all smooth vector fields ${\displaystyle v}$ and all smooth sections ${\displaystyle s,t}$ of ${\displaystyle E}$.

If ${\displaystyle X}$ is a complex manifold, and the Hermitian vector bundle ${\displaystyle E}$ on ${\displaystyle X}$ is equipped with a holomorphic structure, then there is a unique Hermitian connection whose (0, 1)-part coincides with the Dolbeault operator ${\displaystyle {\bar {\partial }}_{E}}$ on ${\displaystyle E}$ associated to the holomorphic structure. This is called the Chern connection on ${\displaystyle E}$. The curvature of the Chern connection is a (1, 1)-form. For details, see Hermitian metrics on a holomorphic vector bundle.

In particular, if the base manifold is Kähler and the vector bundle is its tangent bundle, then the Chern connection coincides with the Levi-Civita connection of the associated Riemannian metric.

References

• Shiing-Shen Chern, Complex Manifolds Without Potential Theory.
• Shoshichi Kobayashi, Differential geometry of complex vector bundles. Publications of the Mathematical Society of Japan, 15. Princeton University Press, Princeton, NJ, 1987. xii+305 pp. ISBN 0-691-08467-X.