Gueo Grantcharov

Curvature properties of twistor spaces of quaternionic Kähler manifolds

We present a study of natural almost Hermitian structures on twistor spaces of quaternionic Kahler manifolds. This is used to supply (4n + 2)-dimensional examples (n > 1) of symplec tic non-Kähler manifolds. Studying their curvature properties we give a negative answer to the questions raised by D.Blair-S.Ianus and A.Gray, respectively, of whether a compact almost Kähler manifold with Hermitian Ricci tensor or whose curvature tensor belongs to the class AH2 is Kähler.

An estimate for the first eigenvalue of the Dirac operator on compact Riemannian spin manifold admitting parallel one-form

Abstract An estimate for the first eigenvalue of the Dirac operator on compact Riemannian spin manifold of positive scalar curvature admitting a parallel one-form is found. The possible universal covering spaces of the manifolds on which the smalles possible eigenvalue is attained are also listed. Moreover, a complete classification of the compact odd-dimensional manifolds whose universal covering space is S n−1 × R is given in the limiting case. All such manifolds are diffeomorphic but not necessarily isometric to S n −1 × S 1 .

Hermitian Structures on Twistor Spaces

The paper contains description of the orthogonal complex structures with respect to the natural 1-parameter family of Riemannian metrics on the (negative) twistor space over a self-dual Einstein Riemannian 4-manifold. We prove that if the twistor space of a compact self-dual Einstein 4-manifold admits more than one orthogonal complex structure then the 4-manifold has a Kähler structure. Considering the flag manifold F1,2 which is the twistor space of CP2 endowed with the Fubini–Study metric, we obtain that any invariant Einstein metric on F1,2 admits even locally exactly three orthogonal complex structures which are the invariant ones.

Curvature Properties of the Chern Connection of Twistor Spaces

The twistor space Z of an oriented Riemannian 4-manifold M admits a natural 1-parameter family of Riemannian metrics h_t compatible with the almost complex structures J_1 and J_2 introduced, respectively, by Atiyah, Hitchin and Singer, and Eells and Salamon. In this paper we compute the first Chern form of the almost Hermitian manifold (Z,h_t,J_n), n=1,2 and find the geometric conditions on M under which the curvature of its Chern connection D^n is of type (1,1). We also describe the twistor spaces of constant holomorphic sectional curvature with respect to D^n and show that the Nijenhuis tensor of J_2 is D^2-parallel provided the base manifold M is Einstein and self-dual.

Twistorial Examples of ∗-Einstein Manifolds

In this paper we study the twistor spaces of oriented Riemannianfour-manifolds as a source of almost-Hermitian *-Einstein manifoldsand show that some results in dimension four related to the RiemannianGoldberg–Sachs theorem cannot be extended to higher dimensions.