Page 168 - Curvature and Homology

1 50 V. COMPLEX MANIFOLDS

(Cn - O)/G is homeomorphic with Sari-I x S1 and has a complex
structure induced by that of Cn - 0. The group G is properly dis-
continuous and acts without fixed points (cf. 5 5.8). The quotient
manifold (Cn- O)/G is called a Hopf manifold (see p. 167 and VII D).
7) Every covering ot a complex manifold has a naturally induced
complex structure (cf. 5 5.8).

5.2. Almost complex manifolds

The concept of a complex structure is but an instance of a more
general type of structure which we now consider. This concept may be
defined from several points of view-the choice made here being
geometrical, that is, in terms of fields of subspaces of the complexified
tangent space. Indeed, a 'choice' of subspace of the 'complexification'
of the tangent space at each point is made so that the union of the sub-
space and its 'conjugate' is the whole space. The given subspace is then
said to define a complex structure in the tangent space at the given
point. More precisely, if at each point P of a differentiable manifold,
a complex structure is given in the tangent space at that point, which
varies differentiably with P, the manifold is said to have an almost
complex structure and is itself called an almost complex manifold.
With a vector space V over R of dimension n we associate a vector
space Vc over C of complex dimension n called its complexification as
follows: Let Vc be the space of all linear maps of V* into C where, as
usual, V* denotes the dual space /$ V. Then, VC is a vector space over
C, and since (V*)* can be idedified with V, VC3 V. An element
v E Vc belongs to V, if and only if, for all a E V*, a(v) E R. Briefly,
Vc is obtained from V by extending the field R to the field C. --
Let 4 be an isomorphism of C, onto VC. The vector B = +($-l(v)),
v E Vc is called the conjugate of v. The vector v is said to be real if
B = v. Clearly, the real vectors of Vc form a vector space of dimension
n over R. To a linear form a on Vc we associate a form & on Vc defined by

The map a -+ 6 is evidently an involutory automorphism of (Vc)*.
On the space VC, tensors may be defined in the obvious way. The
involutory automorphism v + 6, v E VC defines an involutory auto-
morphism t 4 i, t E (VC)L (the linear space of tensors of type (r, 0) on
Vc). Every tensor on V (relative to GL(n, R)) defines a tensor on Vc,
namely, the tensor coinciding with its conjugate, with which it may be
identified. Such a tensor on Vc is said to be real.