No Arabic abstract
We study an odd-dimensional analogue of the Goldberg conjecture for compact Einstein almost Kahler manifolds. We give an explicit non-compact example of an Einstein almost cokahler manifold that is not cokahler. We prove that compact Einstein almost cokahler manifolds with non-negative $*$-scalar curvature are cokahler (indeed, transversely Calabi-Yau); more generally, we give a lower and upper bound for the $*$-scalar curvature in the case that the structure is not cokahler. We prove similar bounds for almost Kahler Einstein manifolds that are not Kahler.
Let $rho_0$ be an action of a Lie group on a manifold with boundary that is transitive on the interior. We study the set of actions that are topologically conjugate to $rho_0$, up to smooth or analytic change of coordinates. We show that in many cases, including the compactifications of negatively curved symmetric spaces, this set is infinite.
This article is an overview of some of the remarkable progress that has been made in Sasaki-Einstein geometry over the last decade, which includes a number of new methods of constructing Sasaki-Einstein manifolds and obstructions.
We study the pseudoriemannian geometry of almost parahermitian manifolds, obtaining a formula for the Ricci tensor of the Levi-Civita connection. The formula uses the intrinsic torsion of an underlying SL(n,R)-structure; we express it in terms of exterior derivatives of some appropriately defined differential forms. As an application, we construct Einstein and Ricci-flat examples on Lie groups. We disprove the parakahler version of the Goldberg conjecture, and obtain the first compact examples of a non-flat, Ricci-flat nearly parakahler structure. We study the paracomplex analogue of the first Chern class in complex geometry, which obstructs the existence of Ricci-flat parakahler metrics.
In this paper, we use the canonical connection instead of Levi-Civita connection to study the smooth maps between almost Hermitian manifolds, especially, the pseudoholomorphic ones. By using the Bochner formulas, we obtian the $C^2$-estimate of canonical second fundamental form, Liouville type theorems of pseudoholomorphic maps, pseudoholomorphicity of pluriharmonic maps, and Simons integral inequality of pseudoholomorphic isometric immersion.
We study the geometry of almost contact pseudo-metric manifolds in terms of tensor fields $h:=frac{1}{2}pounds _xi varphi$ and $ell := R(cdot,xi)xi$, emphasizing analogies and differences with respect to the contact metric case. Certain identities involving $xi$-sectional curvatures are obtained. We establish necessary and sufficient condition for a nondegenerate almost $CR$ structure $(mathcal{H}(M), J, theta)$ corresponding to almost contact pseudo-metric manifold $M$ to be $CR$ manifold. Finally, we prove that a contact pseudo-metric manifold $(M,varphi,xi,eta,g)$ is Sasakian if and only if the corresponding nondegenerate almost $CR$ structure $(mathcal{H}(M), J)$ is integrable and $J$ is parallel along $xi$ with respect to the Bott partial connection.