Please note that the content of this book primarily consists of
articles available from Wikipedia or other free sources online. In
mathematics, a nearly K hler manifold is an almost Hermitian
manifold M, with almost complex structure J, such that the
(2,1)-tensor \nabla J is skew-symmetric. So, for every vectorfield
X on M. In particular, a K hler manifold is nearly K hler. The
converse is not true. The nearly K hler six-sphere S6 is an example
of a nearly K hler manifold that is not K hler. The familiar almost
complex structure on the six-sphere is not induced by a complex
atlas on S6. A nearly K hler manifold should not be confused with
an almost K hler manifold. An almost K hler manifold M is an almost
Hermitian manifold with a closed K hler form: d = 0. The K hler
form or fundamental 2-form is defined by where g is the metric on
M. The nearly K hler six-sphere is an example of a nearly K hler
manifold that is not almost K hler.
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