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We determine the space of algebraic pseudo-Hermitian Kahler-Weyl curvature tensors and the space of para-Hermitian Kahler-Weyl curvature tensors in dimension 4 and show that every algebraic possibility is geometrically realizable. We establish the Gray identity for pseudo-Hermitian Weyl manifolds and for para-Hermitian Weyl manifolds in arbitrary dimension.
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We show that a closed almost Kahler 4-manifold of globally constant holomorphic sectional curvature $kgeq 0$ with respect to the canonical Hermitian connection is automatically Kahler. The same result holds for $k<0$ if we require in addition that the Ricci curvature is J-invariant. The proofs are based on the observation that such manifolds are self-dual, so that Chern-Weil theory implies useful integral formulas, which are then combined with results from Seiberg--Witten theory.
We study the local geometry of 4-manifolds equipped with a emph{para-Kahler-Einstein} (pKE) metric, a special type of split-signature pseudo-Riemannian metric, and their associated emph{twistor distribution}, a rank 2 distribution on the 5-dimensional total space of the circle bundle of self-dual null 2-planes. For pKE metrics with nonvanishing Einstein constant this twistor distribution has exactly two integral leaves and is `maximally non-integrable on their complement, a so-called (2,3,5)-distribution. Our main result establishes a simple correspondence between the anti-self-dual Weyl tensor of a pKE metric with non-vanishing Einstein constant and the Cartan quartic of the associated twistor distribution. This will be followed by a discussion of this correspondence for general split-signature metrics which is shown to be much more involved. We use Cartans method of equivalence to produce a large number of explicit examples of pKE metrics with nonvanishing Einstein constant whose anti-self-dual Weyl tensor have special real Petrov type. In the case of real Petrov type $D,$ we obtain a complete local classification. Combined with the main result, this produces twistor distributions whose Cartan quartic has the same algebraic type as the Petrov type of the constructed pKE metrics. In a similar manner, one can obtain twistor distributions with Cartan quartic of arbitrary algebraic type. As a byproduct of our pKE examples we naturally obtain para-Sasaki-Einstein metrics in five dimensions. Furthermore, we study various Cartan geometries naturally associated to certain classes of pKE 4-dimensional metrics. We observe that in some geometrically distinguished cases the corresponding emph{Cartan connections} satisfy the Yang-Mills equations. We then provide explicit examples of such Yang-Mills Cartan connections.
We work in both the complex and in the para-complex categories and examine (para)-Kahler Weyl structures in both the geometric and in the algebraic settings. The higher dimensional setting is quite restrictive. We show that any (para)-Kaehler Weyl algebraic curvature tensor is in fact Riemannian in dimension at least 6; this yields as a geometric consequence that any (para)-Kaehler Weyl geometric structure is trivial if the dimension is at least 6. By contrast, the 4 dimensional setting is, as always, rather special as it turns out that there are (para)-Kaehler Weyl algebraic curvature tensors which are not Riemannian in dimension 4. Since every (para)-Kaehler Weyl algebraic curvature tensor is geometrically realizable and since every 4 dimensional Hermitian manifold admits a unique (para)-Kaehler Weyl structure, there are also non-trivial 4 dimensional Hermitian (para)-Kaehler Weyl manifolds.
It is shown that on every closed oriented Riemannian 4-manifold $(M,g)$ with positive scalar curvature, $$int_M|W^+_g|^2dmu_{g}geq 2pi^2(2chi(M)+3tau(M))-frac{8pi^2}{|pi_1(M)|},$$ where $W^+_g$, $chi(M)$ and $tau(M)$ respectively denote the self-dual Weyl tensor of $g$, the Euler characteristic and the signature of $M$. This generalizes Gurskys inequality cite{gur} for the case of $b_1(M)>0$ in a much simpler way. We also extend all such lower bounds of the Weyl functional to 4-orbifolds including Gurskys inequalities for the case of $b_2^+(M)>0$ or $delta_gW^+_g=0$, and obtain topological obstructions to the existence of self-dual orbifold metrics of positive scalar curvature.
A special Kahler-Ricci potential on a Kahler manifold is any nonconstant $C^infty$ function $tau$ such that $J( ablatau)$ is a Killing vector field and, at every point with $dtau e 0$, all nonzero tangent vectors orthogonal to $ ablatau$ and $J( ablatau)$ are eigenvectors of both $ abla dtau$ and the Ricci tensor. For instance, this is always the case if $tau$ is a nonconstant $C^infty$ function on a Kahler manifold $(M,g)$ of complex dimension $m>2$ and the metric $tilde g=g/tau^2$, defined wherever $tau e 0$, is Einstein. (When such $tau$ exists, $(M,g)$ may be called {it almost-everywhere conformally Einstein}.) We provide a complete classification of compact Kahler manifolds with special Kahler-Ricci potentials and use it to prove a structure theorem for compact Kahler manifolds of any complex dimension $m>2$ which are almost-everywhere conformally Einstein.
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