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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.
We introduce a new curvature flow which matches with the Ricci flow on metrics and preserves the almost Hermitian condition. This enables us to use Ricci flow to study almost Hermitian manifolds.
We show that a basis of a semisimple Lie algebra for which any diagonal left-invariant metric has a diagonal Ricci tensor, is characterized by the Lie algebraic condition of being ``nice. Namely, the bracket of any two basis elements is a multiple of
Motivated by the recent work of Chu-Lee-Tam on the nefness of canonical line bundle for compact K{a}hler manifolds with nonpositive $k$-Ricci curvature, we consider a natural notion of {em almost nonpositive $k$-Ricci curvature}, which is weaker than
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
We describe the structure of the Ricci tensor on a locally homogeneous Lorentzian gradient Ricci soliton. In the non-steady case, we show the soliton is rigid in dimensions three and four. In the steady case, we give a complete classification in dimension three.