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It is well known that the Clifford algebras and the Dirac equation have a representation in terms of differential forms known as the Kahler-Atiyah algebra and the Dirac-Kahler equation, respectively. In this paper, we have introduced a new product of differential p-forms and obtained a representation in terms of differential forms for the DKP algebra and for the DKP equation. We have studied the properties of this new product in some detail and obtained, among other results, the action of the rotation group in this formalism. We have also obtained a conversed current and a Lagrangian for our differential forms version of the DKP equation.It is well known that the Clifford algebras and the Dirac equation have a representation in terms of differential forms known as the Kahler-Atiyah algebra and the Dirac-Kahler equation, respectively. In this paper, we have introduced a new product of differential p-forms and obtained a representation in terms of differential forms for the DKP algebra and for the DKP equation. We have studied the properties of this new product in some detail and obtained, among other results, the action of the rotation group in this formalism. We have also obtained a conversed current and a Lagrangian for our differential forms version of the DKP equation.

Complex and real vacuum spaces with both self-dual and anti-self-dual parts of the Weyl tensor being of the type [N] are considered. Such spaces are classified according to two criteria. The first one takes into account the properties of the congruences of totally null geodesic 2-dimensional surfaces (the null strings). The second criterion is the properties of the intersection of these congruences. It is proved that there exist six distinct types of the [N] ⊗ [N] spaces. New examples of the Lorentzian slices of the complex metrics are presented. Some types of [N] ⊗ [N] spaces which do not possess Lorentzian slices are also considered.

In this article, we establish a Hitchin-Kobayashi type correspondence for generalised Seiberg-Witten monopole equations on Kahler surfaces. We show that the "stability" criterion we obtain, for the existence of solutions, coincides with that of the usual Seiberg-Witten monopole equations. This enables us to construct a map from the moduli space of solutions to the generalised equations to effective divisors., Comment: 14 pages, minor corrections. To appear in Journal of Mathematical Physics

We solve the metrisability problem for the six Painleve equations, and more generally for all 2nd order ordinary differential equations with the Painleve property, and determine for which of these equations their integral curves are geodesics of a (pseudo) Riemannian metric on a surface.

We investigate the nonexistence of positive solutions to a certain system of differential inequalities on a complete connected non-compact Riemannian manifold. We show that if for some reference point x0, the volume of geodesic ball μ(B(x0, r)) ≤ Crp ln q r holds for all large enough r and for some constant C, then there exists no positive solution to the system. Here the exponents p and q are sharp and cannot be relaxed.

A well-defined smoothing or averaging procedure is highlighted which could be used to address the issue of averaging in gravitational theories for cosmology. A critical component of this averaging procedure is the development of a bi-local calculus through parallel transport which is required to facilitate the integration of tensor fields over a finite region. One popular and arguably natural choice is to parallel transport along geodesics with respect to the Levi-Civita connection. Alternatively, one may choose to parallel transport along arbitrary curves with respect to a flat connection. When one demands that the averaging or smoothing procedure results in a reasonably differentiable averaged object, then within the path independent approach, additional restrictions are required. Further, it is illustrated that the averaging operators of Zalaletdinov are precisely the parallel propagators along arbitrary curves with respect to a flat connection having a covariantly constant torsion.

An explicit surjection from a set of (locally defined) unconstrained holomorphic functions on a certain submanifold of (Sp_1(C) \times C^{4n}) onto the set HK_{p,q} of local isometry classes of real analytic pseudo-hyperk\"ahler metrics of signature (4p,4q) in dimension 4n is constructed. The holomorphic functions, called prepotentials, are analogues of K\"ahler potentials for K\"ahler metrics and provide a complete parameterisation of HK_{p,q}. In particular, there exists a bijection between HK_{p,q} and the set of equivalence classes of prepotentials. This affords the explicit construction of pseudo-hyperk\"ahler metrics from specified prepotentials. The construction generalises one due to Galperin, Ivanov, Ogievetsky and Sokatchev. Their work is given a coordinate-free formulation and complete, self-contained proofs are provided. An appendix provides a vital tool for this construction: a reformulation of real analytic G-structures in terms of holomorphic frame fields on complex manifolds., Comment: 53 pages; v2: minor amendments to Def.4.1 and Theorem 4.5; a paragraph inserted in the proof of the latter; V3: minor changes; V4: minor changes/ typos corrected for journal version

We study a flow for a class of spacelike marginally trapped surfaces in Lorentz-Minkowski space under which the surface converges to a spacelike surface with zero mean curvature. The flow decreases the area monotonically. An analogous flow is constructed for marginally trapped surfaces in an arbitrary four dimensional Lorentzian manifold which satisfies the null convergence condition.