Hermitian metric

A Hermitian metric on a complex vector space $ V $ is a positive-definite Hermitian form on $ V $. The space $ V $ 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 $ V $ can be transferred into each other by an automorphism of $ V $. Thus, the set of all Hermitian metrics on $ V $ is a homogeneous space for the group $ \mathop{\rm GL} _ {n} ( \mathbf C ) $ and can be identified with $ \mathop{\rm GL} _ {n} ( \mathbf C ) / U ( n) $, where $ n = \mathop{\rm dim} V $.

A complex vector space $ V $ can be viewed as a real vector space $ V ^ {\mathbf R} $ endowed with the operator defined by the complex structure $ J ( x) = ix $. If $ h $ is a Hermitian metric on $ V $, then the form $ g = \mathop{\rm Re} h $ is a Euclidean metric (a scalar product) on $ V $ and $ \omega = \mathop{\rm Im} h $ is a non-degenerate skew-symmetric bilinear form on $ V $. Here $ g ( Jx, Jy ) = g ( x , y) $, $ \omega ( Jx , Jy) = \omega ( x , y) $ and $ \omega ( x , y) = g ( x , Jy) $. Any of the forms $ g $, $ \omega $ determines $ h $ uniquely.

A Hermitian metric on a complex vector bundle $ \pi : E \rightarrow M $ is a function $ g : p \mapsto g _ {p} $ on the base $ M $ that associates with a point $ p \in M $ a Hermitian metric $ g _ {p} $ in the fibre $ E ( p) = \pi ^ {-1} ( p) $ of $ \pi $ and that satisfies the following smoothness condition: For any smooth local sections $ e $ and $ e ^ {*} $ of $ \pi $ the function $ p \mapsto g _ {p} ( e _ {p} , e _ {p} ^ {*} ) $ is smooth.

Every complex vector bundle has a Hermitian metric. A connection $ \nabla $ on a complex vector bundle $ \pi $ is said to be compatible with a Hermitian metric $ g $ if $ g $ and the operator $ J $ defined by the complex structure in the fibres of $ \pi $ are parallel with respect to $ \nabla $ (that is, $ \nabla g = \nabla J = 0 $), in other words, if the corresponding parallel displacement of the fibres of $ \pi $ 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 $ \pi $ is a holomorphic vector bundle over a complex manifold $ M $ (see Vector bundle, analytic), there is a unique connection $ \nabla $ of $ \pi $ that is compatible with a given Hermitian metric and that satisfies the following condition: The covariant derivative of any holomorphic section $ e $ of $ \pi $ relative to any anti-holomorphic complex vector field $ \overline{X}\; $ on $ M $ vanishes (the canonical Hermitian connection). The curvature form of this connection can be regarded as a $ 2 $-form of type $ ( 1 , 1 ) $ on $ M $ with values in the bundle of endomorphisms of $ \pi $. The canonical connection can also be viewed as a connection on the principal $ \mathop{\rm GL} _ {n} ( \mathbf C ) $-bundle $ \widetilde \pi : P \rightarrow M $ associated with the holomorphic vector bundle $ \pi $ of complex dimension $ n $. It can be characterized as the only connection on $ \widetilde \pi $ with complex horizontal subspaces in the tangent spaces of the complex manifold $ P $.

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.