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, 
is the almost complex
structure, a linear transformation on
manifold tangent vectors satisfying 
, induced by multiplication
by 
. In coordinates 
, the linear
transformation 
satisfies 
and 
.
The linear transformation 
depends on the complex structure
and preserves the Riemannian metric 
. For the Riemannian metric
to be Kähler, the Kähler form must also
satisfy 
.
Near any point 
, there are holomorphic coordinates 
centered at 
in which the Riemannian metric
has the form
 |
where 
denotes the vector space tensor product.
Thus the first derivatives of the Riemannian
metric coefficients vanish at 
; the Riemannian metric
itself does not vanish. For a general Riemannian
metric, first derivatives likewise vanish in normal coordinates, but those coordinates
need not be holomorphic.
