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We give a generalization of the Penrose transform on Hermitian manifolds with metrics locally conformally equivalent to Bochner-K\"ahler metrics. We also give an explicit formula for the inverse transform. This paper is a generalization of…
New universal invariant operators are introduced in a class of geometries which include the quaternionic structures and their generalisations as well as 4-dimensional conformal (spin) geometries. It is shown that, in a broad sense, all…
Fourth-order tensor-valued functions appear in numerous fields of study. The formulation of practical models for these complex functions often requires their representation in terms of tensors of order two. In this paper, we develop an…
We find a kind of variations of Gauss-Codazzi-Ricci equations suitable for Kaluza-Klein reduction and Cauchy problem. Especially the counterpart of extrinsic curvature tensor has antisymmetric part as well as symmetric one. If the…
We derive expressions for the Ricci curvature tensor and scalar in terms of intrinsic torsion classes of half-flat manifolds by exploiting the relationship between half-flat manifolds and non-compact $G_2$ holonomy manifolds. Our…
On Riemannian manifolds of dimension 4, for prescribed scalar curvature equation, under lipschitzian condition on the prescribed curvature, we have an uniform estimate for the solutions of the equation if we control their minimas.
We formulate Lorentz group representations in which ordinary complex numbers are replaced by linear functions of real quaternions and introduce dotted and undotted quaternionic one-dimensional spinors. To extend to parity the space-time…
A peeling theorem for the Weyl tensor in higher dimensional Lorentzian manifolds is presented. We obtain it by generalizing a proof from the four dimensional case. We derive a generic behavior, discuss interesting subcases and retrieve the…
A recently derived approach to the tensor reduction of 5-point one-loop Feynman integrals expresses the tensor coefficients by scalar 1-point to 4-point Feynman integrals completely algebraically. In this letter we derive extremely compact…
We give a Lagrangian and display all local symmetries for N=4 topological string by Berkovits, Vafa and Siegel, the latter previously known in the superconformal gauge. Leading to a small N=4 superconformal algebra and exhibiting the…
A generalization of the Lorenz equations is proposed where the variables take values in a Lie algebra. The finite dimensionality of the representation encodes the quantum fluctuations, while the non-linear nature of the equations can…
Let $(M,L)$ be a (compact) non-spin spin$^c$ manifold. Fix a Riemannian metric $g$ on $M$ and a connection $A$ on $L$, and let $D_L$ be the associated spin$^c$ Dirac operator. Let $R^{tw}_{(g,A)}:=R_g + 2ic(\Omega)$ be the twisted scalar…
We introduce the concept of singular values for the Riemann curvature tensor, a central mathematical tool in Einstein's theory of general relativity. We study the properties related to the singular values, and investigate five typical cases…
Connection, torsion and curvature are introduced for general (local) Leibniz algebroids. Generalized Bismut connection on $TM \oplus \Lambda^{p} T^{\ast}M$ is an example leading to a scalar curvature of the form $R + H^2$ for a closed…
Biquadratic tensors play a central role in many areas of science. Examples include elasticity tensor and Eshelby tensor in solid mechanics, and Riemann curvature tensor in relativity theory. The singular values and spectral norm of a…
Quaternion-Kaehler four-manifolds, or equivalently anti-self-dual Einstein manifolds, are locally determined by one scalar function subject to Przanowski's equation. Using twistorial methods we construct a Lax Pair for Przanowski's…
We give presentations, by means of diagrammatic generators and relations, of the analogues of the Temperley-Lieb algebras associated as Hecke algebra quotients to Coxeter graphs of type B and $D$. This generalizes Kauffman's diagram…
We extract the square root of the Minkowski metric using Dirac/Clifford matrices. The resulting $4\times 4$ operator $d{\bf S}$ that represents the square root, can be used to transform four vectors between relatively moving observers. This…
We consider a class (M, g, q) of four-dimensional Riemannian manifolds M, where besides the metric g there is an additional structure q, whose fourth power is the unit matrix. We use the existence of a local coordinate system such that…
A generalization of duality transformations for arbitrary Lorentz tensors is presented, and a systematic scheme for constructing the dual descriptions is developed. The method, a purely Lagrangian approach, is based on a first order parent…