Kähler Metric
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المرجع الالكتروني للمعلوماتيه
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www.almerja.com
الجزء والصفحة:
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8-7-2021
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Kähler Metric
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.
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