Page 210 - Curvature and Homology

192 V. COMPLEX MANIFOLDS

Define the endomorphism

Jx : Tx-* Tx
by
JxY=XxY, YET=

It has the properties:

for Y,Z E Tx.
Property (i) implies that S6 has an almost complex structure whereas (ii) says
that the metric on Sb is hermitian. Under the circumstances, S6 is said to possess
an almost hermitian structure.
8. Consider the 3-dimensional subspace ES C E7 spanned by the vectors
e,,e,,e, E F. S6 n EB is a 2-sphere S2. Show that SZ is an invariant submanifold
of S6, that is, for any X E SZ the tangent space Tx to S%t X is invariant under
Jx.

C. Hermitian manifolds [SO]

1. Let M be a Riemannian manifold with metric tensor g. Show that there
exists a mapping
X-+ Dx

of T into the space of endomorphisms of T with the properties:
(a) Zg(X, Y) - g(DzX, Y) - g(X,DzY) = 0
(parallel translation is an isometry);
(b) DxY - D ,X = [X, Y]
(torsion is zero)
for any X,Y,Z E T.
Hint: Assume the existence of this map and show that

for any X,Y and Z E T. Conversely, this relation defines for every Y,Z E T
an element DZY E T. The map Z-* DZ is thus unique. For every Z E T, DZ
is called the operation of cw&t dz@vntiation with respect to Z.