Page 174 - Curvature and Homology

156 V. COMPLEX MANIFOLDS
that is gij. = g(ei, e,.) = 4,. The form S) may then be written as

A differentiable manifold M is said to possess an almost complex
structure if it carries a real differentiable tensor field J of type (1, 1)
(and class k) satisfying
J" - 1.
(By 5 1.2, the tensor field J may be considered as a linear endomorphism
of the space of tangent vector fields on M). It follows that an almost
complex structure is equivalently defined by a field of subspaces WE
(of class k and dimension n) of TC (the complexification of the space
of tangent vector fields) such that

A manifold with an almost complex structure is said to be an almost
complex manifold.
Evidently, an almost complex manifold is even ditnett~t'onal.
We now show that a complex mantjCbld M is almost complex. Indeed,
let U be a coordinate neighborhood of M with the local complex
coordinates zl, a-, P. We have seen that M possesses an underlying
real analytic structure and that relative to it zl, -.-, zn, 9, -., Zn may be
used as local coordinates. Following the notation of 5 5.1, we define

Let P be a point of U. Then, the differentials dzl, . -, dzn, d9, . -, dZn
at P define a frame in the complexification (Tg)* of the dual space T,*
of the tangent space Tp at P and by duality a frame {a/azi, a/a$) in TE,
If P belongs to the intersection U n U' of the coordinate neighbor-
hoods U and U' the differentials (dzi) and (dzti) are related by

and their duals (a/&+), (a/ azfi) by

where (a;) E GL(n, C) is the matrix of coefficients