Para-Kähler-Einstein 4-manifolds and non-integrable twistor distributions

We study the local geometry of 4-manifolds equipped with a para-Kähler-Einstein (pKE) metric, a special type of split-signature pseudo-Riemannian metric, and their associated 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 non-zero scalar curvature 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-zero scalar curvature 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 Cartan’s method of equivalence to produce a large number of explicit examples of pKE metrics with non-zero scalar curvature 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 Cartan connections satisfy the Yang-Mills equations. We then provide explicit examples of such Yang-Mills Cartan connections.

Investigation of Pseudo-Ricci Symmetric Spacetimes in Gray’s Subspaces

In the present paper, we focused our attention to study pseudo-Ricci symmetric spacetimes in Gray’s decomposition subspaces. It is proved that PRS n spacetimes are Ricci flat in trivial, A , and B subspaces, whereas perfect fluid in subspaces I , I ⊕ A , and I ⊕ B , and have zero scalar curvature in subspace A ⊕ B . Finally, it is proved that pseudo-Ricci symmetric GRW spacetimes are vacuum, and as a consequence of this result, we address several corollaries.

Some notes on metallic Kähler manifolds

The present paper deals with metallic K?hler manifolds. Firstly, we define a tensor H which can be written in terms of the (0,4)-Riemannian curvature tensor and the fundamental 2-form of a metallic K?hler manifold and study its properties and some hybrid tensors. Secondly, weobtain the conditions under which a metallic Hermitian manifold is conformal to a metallic K?hler manifold. Thirdly, we prove that the conformal recurrency of a metallic K?hler manifold implies its recurrency and also obtain the Riemannian curvature tensor form of a conformally recurrent metallic K?hler manifold with non-zero scalar curvature. Finally, we present a result related to the notion of Z recurrent form on a metallic K?hler manifold.

On mixed quasi-Einstein spacetimes

The object of the present paper is to study mixed quasi-Einstein spacetimes, briefly M(QE)4 spacetimes. First we prove that every Z Ricci pseudosymmetric M(QE)4 spacetimes is a Z Ricci semisymmetric spacetime. Then we study Z flat spacetimes. Also we consider Ricci symmetric M(QE)4 spacetimes and among others we prove that the local cosmological structure of a Ricci symmetricM(QE)4 perfect fluid spacetime can be identified as Petrov type I, DorO. We show that such a spacetime is the Robertson-Walker spacetime. Moreover we deal with mixed quasi-Einstein spacetimes with the associated generators U and V as concurrent vector fields. As a consequence we obtain some important theorems. Among others it is shown that a perfect fluid M(QE)4 spacetime of non zero scalar curvature with the basic vector field of spacetime as velocity vector field of the fluid is of Segr?e characteristic [(1,1,1),1]. Also we prove that a M(QE)4 spacetime can not admit heat flux provided the smooth function b is not equal to the cosmological constant k. This means that such a spacetime describe a universe which has already attained thermal equilibrium. Finally, we construct a non-trivial Lorentzian metric of M(QE)4.

PRESCRIBING MEAN CURVATURE ON 𝔹n

In this paper, we consider the problem of multiplicity of conformal metrics that are equivalent to the Euclidean metric, with zero scalar curvature and prescribed mean curvature on the boundary of the ball 𝔹n, n ≥ 4. Under the assumption that the order of flatness at critical points of the prescribed mean curvature function H(x) is β∈(n-2, n-1), we establish some Morse inequalities at infinity, which give a lower bound on the number of solutions to the above problem, in terms of the total contribution of its critical points at infinity to the difference of topology between the level sets of the associated Euler–Lagrange functional. As a by-product of our arguments, we derive a new existence result through an Euler–Hopf type formula.

QUATERNIONIC CONNECTIONS, INDUCED HOLOMORPHIC STRUCTURES AND A VANISHING THEOREM

We classify the holomorphic structures of the tangent vertical bundle Θ of the twistor fibration of a quaternionic manifold (M, Q) of dimension 4n ≥ 8. In particular, we show that any self-dual quaternionic connection D of (M, Q) induces an holomorphic structure [Formula: see text] on Θ. We construct a Penrose transform which identifies solutions of the Penrose operator PD on (M, Q) defined by D with the space of [Formula: see text]-holomorphic purely imaginary sections of Θ. We prove that the tensor powers Θs (for any s ∈ ℕ\{0}) have no global non-trivial [Formula: see text]-holomorphic sections, when (M, Q) is compact and has a compatible quaternionic-Kähler metric of negative (respectively, zero) scalar curvature and the quaternionic connection D is closed (respectively, closed but not exact).