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.
Hermitian metric. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hermitian_metric&oldid=47221