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Register a new userAffine surfaces which are Kahler, para-Kahler, or nilpotent Kahler
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Motivated by the construction of Bach flat neutral signature Riemannian extensions, we study the space of parallel trace free tensors of type $(1,1)$ on an affine surface. It is shown that the existence of such a parallel tensor field is characterized by the recurrence of the symmetric part of the Ricci tensor.
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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.
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.
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 develop a new class of supergravity cosmological models where inflation is induced by terms in the Kahler potential which mix a nilpotent superfield $S$ with a chiral sector $Phi$. As the new terms are non-(anti)holomorphic, and hence cannot be removed by a Kahler transformation, these models are intrinsically Kahler potential driven. Such terms could arise for example due to a backreaction of an anti-D3 brane on the string theory bulk geometry. We show that this mechanism is very general and allows for a unified description of inflation and dark energy, with controllable SUSY breaking at the vacuum. When the internal geometry of the bulk field is hyperbolic, we prove that small perturbative Kahler corrections naturally lead to $alpha$-attractor behaviour, with inflationary predictions in excellent agreement with the latest Planck data
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.
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