We prove that every Kähler metric, whose potential is a function of the timelike distance in the flat Kähler-Lorentz space, is of quasi-constant holomorphic sectional curvatures, satisfying certain conditions. This gives a local classification of the Kähler manifolds with the above mentioned metrics. New examples of Sasakian space forms are obtained as real hypersurfaces of a Kähler space form ...

Measure contraction properties are generalizations of the notion of Ricci curvature lower bounds in Riemannian geometry to more general metric measure spaces. In this paper, we give sufficient conditions for a contact sub-Riemannian manifold with a one-parameter family of symmetries to satisfy these properties. Moreover, in the special case where the quotient of the contact subRiemannian manifo...

Building on previous results [17, 35], we complete the classification of compact oriented Einstein 4-manifolds with $$\det (W^+) > 0$$ . There are, up to diffeomorphism, exactly 15 manifolds that carry such metrics, and, each these manifolds, metrics sweep out one connected component corresponding moduli space.

Let M = M1 ×M2 be a product of complex manifolds. We prove that M cannot admit a complete Kähler metric with sectional curvature K < c < 0 and Ricci curvature Ric > d, where c and d are arbitrary constants. In particular, a product domain in Cn cannot cover a compact Kähler manifold with negative sectional curvature. On the other hand, we observe that there are complete Kähler metrics with nega...