Editing Differential geometry (section)

===Complex and Kähler geometry===

{{See also|Complex geometry}}
''Complex differential geometry'' is the study of [[complex manifolds]].
An [[almost complex manifold]] is a ''real'' manifold <math>M</math>, endowed with a [[tensor]] of type (1, 1), i.e. a [[vector bundle|vector bundle endomorphism]] (called an ''[[almost complex structure]]'')
:<math> J:TM\rightarrow TM </math>, such that <math>J^2=-1. \,</math>

It follows from this definition that an almost complex manifold is even-dimensional.

An almost complex manifold is called ''complex'' if <math>N_J=0</math>, where <math>N_J</math> is a tensor of type (2, 1) related to <math>J</math>, called the [[Nijenhuis tensor]] (or sometimes the ''torsion'').
An almost complex manifold is complex if and only if it admits a [[Holomorphic function|holomorphic]] [[Atlas (topology)|coordinate atlas]].
An ''[[Hermitian manifold|almost Hermitian structure]]'' is given by an almost complex structure ''J'', along with a [[Riemannian metric]] ''g'', satisfying the compatibility condition
:<math>g(JX,JY)=g(X,Y). \,</math>

An almost Hermitian structure defines naturally a [[differential form|differential two-form]]
:<math>\omega_{J,g}(X,Y):=g(JX,Y). \,</math>

The following two conditions are equivalent:

# <math> N_J=0\mbox{ and }d\omega=0 \,</math>
# <math>\nabla J=0 \,</math>

where <math>\nabla</math> is the [[Levi-Civita connection]] of <math>g</math>. In this case, <math>(J, g)</math> is called a ''[[Kähler manifold|Kähler structure]]'', and a ''Kähler manifold'' is a manifold endowed with a Kähler structure. In particular, a Kähler manifold is both a complex and a [[symplectic manifold]]. A large class of Kähler manifolds (the class of [[Hodge manifold]]s) is given by all the smooth [[algebraic geometry|complex projective varieties]].