Page 169 - Curvature and Homology

Now, let V be a real vector space of even dimension 2n. A subspace. Wc
of the complexification Vc of V of complex dimension n is said to
define a complex structure on V if

where rc is the space consisting of all conjugates of vectors in We.
In this case, an element v E VC has the unique representation
v = W1 + fz,, W1, W, E WC.
Since
a = fzl + w,,
the (real) vectors v of V are those elements of Vc which may be written
in the form
v=w+ciii, WEWC.
We proceed to show that a complex structure on V may be defined
equivalently by means of a certain tensor on V. Indeed, to every vector
v E V there corresponds a real vector Jv E V defined by

where v = w + 6, w E Wc.
The operator J has the properties:
(i) J is linear
and
(ii) JZv s J(Jv) = - v.

Moreover, J may be extended to Vc by linearity. The operator J.is a
'quadrantal versor', that is, it has the effect of multiplying w by l/=i
and 6 by --2/:1. Thus- Wc is the eigenspace of J for the eigenvalue
-1 and Pc that for the eigenvalue -47. Hence, a complex
structure on V defines a linear endomorphism J of V, that is, by 5 1.2,
a tensor on V, with the property
Je = - I, (5.2.2)
where I is the identity opergtor on V.
Conversely, let V be a real vector space of dimension m and J a
linear endomorphism of V satisfying (5.2.2). Since a tensor on V defines
a real tensor on the complexification Vc of V, J may be extended to Vc-
We seek the eigenvectors and eigenvalues in Vc of the operator 1.
For this purpose put
Jv = m, v E VC.