# 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)