Page 173 - Curvature and Homology

The 2-form 52 defined by J and g has rank 2n. Indeed, the coefficients
of 52 are given by FAB = FAC.
Relative to a J-basis the metric tensor g has gtj, = gj., as its only
non-vanishing components as one may easily see from (5.2.8) and (5.2.4).
Moreover, since g is a real tensor

The tensor g on VC is then said to be self adjoint.
More generally, let t be a tensor and denote by J* the operation o
starring the indices of its components (with respect to a J-basis). Then,
if t the tensor t is said to be self adjoint. Evidently, this is
equivalent to saying that t is a real tensor.
From (5.2.4) one deduces that the only non-vanishing components
of the covariant form of the tensor J with respect to a J-basis are

F, = d-7 g,* , Fj*, = --<I gj* ,. (5.2.1 1)
The form 52 then has the following representation

Ad'.
Q =agij* (5.2.12)
w'
We also consider the tensor FAB = fCFcB. From (5.2.4) and
(5.2.1 1) its only non-vanishing components (with respect to a J-basis)
are
F; = - 8:, F$ = 6 1 8:.
Evidently,
FAB = - pBA
and

Thus, the tensor FAB defines a complex structure J on V called the
conjugate of J.
Let v, and v, be any orthogonal vectors on the hermitian vector
space V. If we insist that v; be orthogonal to Jv, as well, then, from
(5.2.7), v,, v,, Jv, and Jv, are mutually orthogonal.
I,et Cf,, f,.), i = 1, .-., n where f,, = Jf, be a real orthonormal basis
of V. Such a basis is assured by the hermitian structure defined by
J and g. Then, in terms of the J-basis (ez, e,,) defined by (5.2.6)