A Kähler metric is a Riemannian metric 
on a complex
manifold which gives 
a Kähler structure, i.e., it is a Kähler
manifold with a Kähler form. However, the
term "Kähler metric" can also refer to the corresponding Hermitian
metric 
,
where 
is the Kähler form, defined by 
. Here, the operator 
is the almost complex
structure, a linear map on tangent vectors satisfying 
, induced by multiplication by 
. In coordinates 
, the operator 
satisfies 
and 
.
The operator 
depends on the complex structure, and on a Kähler manifold, it must preserve the Kähler
metric. For a metric to be Kähler, one additional condition must also be satisfied,
namely that it can be expressed in terms of the metric and the complex structure.
Near any point 
,
there exists holomorphic coordinates 
such that the metric has the form
 |
where 
denotes the vector space tensor product;
that is, it vanishes up to order two at 
. Hence, any geometric equation in 
involving only the first derivatives can be defined on a
Kähler manifold. Note that a generic metric can be written to vanish up to order
two, but not necessarily in holomorphic coordinates, using a Gaussian
coordinate system.
