Abstract:
In the present thesis we investigate the almost Hermitian geometry of the twistor
spaces of oriented Riemannian 4-manifolds.
Holomorphic and orthogonal bisectional curvatures have been intensively explored
on K ̈hler manifolds and a lot of important results have been obtained in this case.
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But in the non-K ̈hler case these curvatures are not very well studied and it seems
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that the main reason for that is the lack of interesting examples. The first part
of the thesis is devoted to the study of the curvature properties of Atiyah-Hitchin-
Singer and Eells-Salamon almost Hermitian structures. This is used to provide some
interesting examples of almost Hermitian 6-manifolds of constant or strictly positive
holomorphic, Hermitian and orthogonal bisectional curvatures.
In the second part of the thesis we determine the Gray-Hervella classes of the
so-called compatible almost Hermitian structures on the twistor spaces, recently in-
troduced by G. Deschamps . The interest in determining these classes is motivated by
the fact that the Gray-Hervella classification is a very useful tool in studying almost
complex manifolds. Our results in this direction generalize the well known integrabil-
ity theorems by Atiyah-Hitchin-Singer, Eells-Salamon and Deschamps and show that
there is a close relation between the properties of the spectrum of the anti-self-dual
Weyl tensor of an almost K ̈hler 4-manifold and the almost Hermitian geometry of
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its twistor space.
