A Hermitian metric on a complex vector bundle assigns a Hermitian inner product to every
fiber bundle. The basic example is the trivial
bundle 
,
where 
is an open set in 
. Then a positive definite Hermitian
matrix 
defines a Hermitian metric by
 |
where 
is the complex conjugate of 
. By a partition of unity,
any complex vector bundle has a Hermitian
metric.
In the special case of a complex manifold, the complexified tangent bundle 
may have a Hermitian metric, in which case its real part is a Riemannian
metric and its imaginary part is a nondegenerate
alternating multilinear form 
. When 
is closed, i.e., in this
case a symplectic form, then 
is a Kähler form.
On a holomorphic vector bundle with a Hermitian metric 
,
there is a unique connection compatible with 
and the complex structure. Namely, it must be 
, where 
in a trivialization.
