Hermitian metric

A Hermitian metric on a complex vector space is a positive-definite Hermitian form on . The space endowed with a Hermitian metric is called a unitary (or complex-Euclidean or Hermitian) vector space, and the Hermitian metric on it is called a Hermitian scalar product. Any two Hermitian metrics on can be transferred into each other by an automorphism of . Thus, the set of all Hermitian metrics on is a homogeneous space for the group and can be identified with , where .

A complex vector space can be viewed as a real vector space endowed with the operator defined by the complex structure . If is a Hermitian metric on , then the form is a Euclidean metric (a scalar product) on and is a non-degenerate skew-symmetric bilinear form on . Here , and . Any of the forms , determines uniquely.

A Hermitian metric on a complex vector bundle is a function on the base that associates with a point a Hermitian metric in the fibre of and that satisfies the following smoothness condition: For any smooth local sections and of the function is smooth.

Every complex vector bundle has a Hermitian metric. A connection on a complex vector bundle is said to be compatible with a Hermitian metric if and the operator defined by the complex structure in the fibres of are parallel with respect to (that is, ), in other words, if the corresponding parallel displacement of the fibres of along curves on the base is an isometry of the fibres as unitary spaces. For every Hermitian metric there is a connection compatible with it, but the latter is, generally speaking, not unique. When is a holomorphic vector bundle over a complex manifold (see Vector bundle, analytic), there is a unique connection of that is compatible with a given Hermitian metric and that satisfies the following condition: The covariant derivative of any holomorphic section of relative to any anti-holomorphic complex vector field on vanishes (the canonical Hermitian connection). The curvature form of this connection can be regarded as a -form of type on with values in the bundle of endomorphisms of . The canonical connection can also be viewed as a connection on the principal -bundle associated with the holomorphic vector bundle of complex dimension . It can be characterized as the only connection on with complex horizontal subspaces in the tangent spaces of the complex manifold .

References

[1]	S. Kobayashi, K. Nomizu, "Foundations of differential geometry" , 2 , Interscience (1969)
[2]	A. Lichnerowicz, "Global theory of connections and holonomy groups" , Noordhoff (1976) (Translated from French)
[3]	R.O. Wells jr., "Differential analysis on complex manifolds" , Springer (1980)
[4]	A. Weil, "Introduction à l'Aeetude des variétés kahlériennes" , Hermann (1958)

Comments

A complex vector bundle on which a Hermitian metric is given is called a Hermitian vector bundle.