The main idea of the twistor theory, created
by R. Penrose to solve problems in Mathematical Physics, is that
the geometry of a conformal manifold M can be "encoded" in
holomorphic terms of the so-called twistor space associated to M.
The Penrose ideas have been developed in the context of the Riemannian
geometry by Atiyah, Hitchin and Singer in the case of manifolds
of dimension four. In particular, they have defined an almost-complex
structure, say J1, on the twistor
space Z of such a manifold M which is invariant under
conformal changes of the metric of M and found the integrability
condition for this almost-complex structure. J. Eells and S. Salamon
have introduced another almost-complex structure on Z, say
J2, which, although is not conformally
invariant and is never integrable, plays an important role in the
harmonic maps theory. The twistor space Z admits a natural
one-parameter family ht, t
> 0, of Riemannian metrics compatible with the Atiyah-Hitchin-Singer
and Eells- Salamon almost-complex structures. Thus we have two almost-Hermitian
manifolds (Z, J1, ht)
and (Z, J2, ht)
and the main purpose of this talk is to discuss some geometric properties
of these manifolds. More precisely the following topics will be
considered:
- The Atiyah-Hitchin-Singer theorem for integrability of the
almost-complex structure J1.
- The Penrose transform. Applications.
- Twistor spaces with Hermitian Ricci tensor.
- KähIer curvature identies on twistor spaces.
