In mathematics, a Hermitian connection

\nabla

is a connection on a

Hermitian vector bundle In mathematics, a holomorphic vector bundle is a complex vector bundle over a complex manifold such that the total space is a complex manifold and the projection map is holomorphic. Fundamental examples are the holomorphic tangent bundle of a ...

E

over a smooth manifold

M

which is compatible with the

Hermitian metric In mathematics, and more specifically in differential geometry, a Hermitian manifold is the complex analogue of a Riemannian manifold. More precisely, a Hermitian manifold is a complex manifold with a smoothly varying Hermitian inner product on ea ...

\langle \cdot, \cdot \rangle

E

, meaning that :

v \langle s,t\rangle = \langle \nabla_v s, t \rangle + \langle s, \nabla_v t \rangle

for all smooth vector fields

v

and all smooth sections

s,t

E

. If

X

is a

complex manifold In differential geometry and complex geometry, a complex manifold is a manifold with an atlas of charts to the open unit disc in \mathbb^n, such that the transition maps are holomorphic. The term complex manifold is variously used to mean a com ...

, and the Hermitian vector bundle

E

X

is equipped with a holomorphic structure, then there is a unique Hermitian connection whose (0, 1)-part coincides with the

Dolbeault operator In mathematics, a complex differential form is a differential form on a manifold (usually a complex manifold) which is permitted to have complex number, complex coefficients. Complex forms have broad applications in differential geometry. On comp ...

\bar_E

E

associated to the holomorphic structure. This is called the Chern connection on

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 In Riemannian or pseudo Riemannian geometry (in particular the Lorentzian geometry of general relativity), the Levi-Civita connection is the unique affine connection on the tangent bundle of a manifold (i.e. affine connection) that preserves th ...

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. . Complex manifolds Structures on manifolds Riemannian geometry {{differential-geometry-stub