Page 176 - Curvature and Homology

158 V. COMPLEX MANIFOLDS
One merely considers a J-basis with respect to which the functions
FAB are given by (5.2.4).
Conversely, if the almost complex structure given by J is of class
1 + a (0 < a < I), that is, the derivatives are Holder continuous with
exponent a, and if the structure tensor satisfies the (integrability)
conditions (5.2.18), it is integrable [85]. The proof of this important fact
is patterned after that of Newlander and Nirenberg [84] who assumed
that the structure is of class 211 + a. Hence, in order that an almost
complex structure. define a complex structure it is not necessary that it
be analytic or even of class w. For real analytic manifolds with real
analytic FAB the above result follows from a theorem of Frobenius
(cf. I.D.4). For n = 1 the problem is equivalent to that of introducing
isothermal parameters with respect to the metric

and Chern showed that this is possible even if the structure is of class a.

5.3. Local hermitian geometry
If at each point P of the complex manifold M of complex dimension n
the tangent space Tp is endowed with an hermitian metric so that (as
functions of local coordinates) the metric tensor g is of class w, M is
said to be an hermitian manifold. Evidently, such a manifold is also
Riemannian. On the other hand, since the complex structure is defined
by a tensor field J of type (1, l), if the complex manifold M is given an
'arbitrary' Riemannian metric, a new metric g can be found which com-
mutes with J. The metric g together with the tensor field J is said to
define an hermitian structure on M (cf. 5.2.8). In this way, it is seen
that every complex manifold possesses an hermitian metric. The (local)
geometry of an hermitian manifold is called hermitian geometry.
In the same way as the bundle of frames with the orthogonal group
as structural group is natural for the study of Riemannian geometry,
the bundle of unitary frames, that is, the bundle of frames with the
unitary group U(n) as structural group, is natural for hermitian geometry.
Indeed, by a unitary frame at the point P E M we shall mean a J-basis
{XI, ---, X,, XI,, ..., X,,) at P of the type satisfying (5.2.13), that is

where Xi Xj, = g(X*, XI,).
The collection of all such frames at all points P E M forms a fibre
bundle B over M with U(n) as structural group. A frame at P, that is an
element of the fibre over P may be determined by means of a system of