- INTRODUCTION TO KAHLER MANIFOLDS 57
It is interesting that S^6 admits an almost complex structure. In fact S^2
and S^6 are the only even-dimensional spheres which admit almost complex
structures (see §8 of Chern [95] for example). The following is a long-
standing unsolved question.
PROBLEM 2.1 (Existence of complex structure on S^6 ). Does S^6 admit
a complex structure? More generally, one may ask which closed even-
dimensional manifolds admit almost complex structures but not complex
structures.
Let (M, J) be an almost complex manifold. A vector field V is aninfinitesimal automorphism of the almost complex structure if the
Lie derivative of J with respect to V is zero, i.e.,
(2.1) CvJ=O.Note that
(CvJ) (W) = Cv (JW) - J (CvW)
= [V, JW] - J ([V, W]) ,so (2.1) is equivalent to J ([V, W]) = [V, JW] for any vector field W.
There are various equivalent ways to define a Kahler manifold. Wesay that a Riemannian manifold ( M, g) with an almost complex structure
J: TM--+ TM is a Kahler manifold if the metric g is J-invariant (or
Hermitian):
g (JX, JY) = g (X, Y)
and J is parallel:
\7 J = 0 or equivalently, \7 x ( JY) = J (\7 x Y)
for all X, Y, where \7 is the Riemannian covariant derivative.^2 The metric g
is called a Kahler metric.
Note that almost complex structures which yield Kahler manifolds are
necessarily integrable. Indeed, by the Newlander-Nirenberg Theorem, we
only need to check that the Nijenhuis tensor vanishes for a Kahler manifold:
NJ (X, Y) = \7 JxJY - \7 JyJX - J (\7 JxY - \i'yJX)
- J ('VxJY - \7 JYX) - \i'xY + \i'yX
= - [J (\7 JX Y) - \7 JxJY] + [J (\7 JY X) - \7 JyJX]
- [J(\i'yJX)-\i'yJ(JX)]-[J(\i'xJY)-'VxJ(JY)]
= 0.
A complex manifold (M, J) is called a Kahler manifold if it admits a Kahler
metric.(^2) Since 0 = (\7 xJ) (Y) = \7 x (JY) - J (\7 x Y).