Page 175 - Curvature and Homology

It follows that the n vectors (a/a~i)~ define a subspace W: of Pp and that

that is, these vectors determine a complex structure on Tp. Hence, at
each point P E M a complex structure is defined in the tangent space
at that point. Moreover, at a given point any two frames are related
by equations of the form (5.2.3), that is, only those frames {XI, *-, X,,
XI,, -, X,,) are allowed which are obtained from the frame

Hence, the complex structure on M defines a real analytic tensor field
J of type (1, 1) on M.
One may easily check that if a differentiable manifold possesses two
complex structures, giving rise to the same almost complex structure,
they must coincide.
We have seen that a complex manifold is orientable. An almost
complex manifold also enjoys this property, this being a consequence
of the fact that for every neighborhood U of a point P of the manifold
and at every point Q of U there exists a set of real vectors XI, **., X,
such that XI, a*., X,, JX,, ..., JX, are independent vectors; moreover,
from (5.2.3) and (5.2.5) any two real bases of this type are related by a
matrix of positive determinant. In other words, the existence of a J-basis
(cf. 5.2.6) at each point ensures that the almost complex manifold is
orientable (cf. 5 5.1 for the dual argument).
Let M be an almost complex manifold with the almost complex
structure J. The almost complex structure is said to be intepabb if M
can be made into a complex manifold so that in a coordinate neighbor-
hood with the complex coordinates (&) operating with J is equivalent to
transforming a/azi and a/aZl into a/a~i and - .\/- a/aZi,
respectively. It is not difficult to see that if the almost complex
structure which is equivalently defined by the tensor field FAB of type
(1, 1) in the (real) local coordinates (uA) = (zi, 50 is integrable, then