Related papers: Algebraic Analysis and Mathematical Physics
The purpose of this paper is to revisit the Bianchi identities existing for the Riemann and Weyl tensors in the combined framework of the formal theory of systems of partial differential equations (Spencer cohomology, differential systems,…
The purpose of this paper is to present for the first time an elementary summary of a few recent results obtained through the application of the formal theory of partial differential equations and Lie pseudogroups in order to revisit the…
In this article the algebra and the basis of corresponding analysis in 4-dimensional spaces are constructed, in pseudoeuclidean with signature (1, -1, -1, -1) and pseudo-Riemannian corresponding to the real space-time. In both cases the…
Many equations of mathematical physics are described by differential polynomials, that is by polynomials in the derivatives of a certain number of functions. However, up to the knowledge of the author, differential algebra in a modern…
In this paper we pursue further a programme initiated in a previous work and aimed at the construction, classification and property investigation of time dependent solutions of supergravity (superstring backgrounds) through a systematic…
The first purpose of this short but striking paper is to revisit Elasticity (EL) and Electromagnetism (EM) by comparing the structure of these two theories and examining with details their well known couplings, in particular…
Our recent arXiv preprints and published papers on the solution of the Riemann-Lanczos and Weyl-Lanczos problems have brought our attention on the importance of revisiting the algebraic structure of the Bianchi identities in Riemannian…
Lie pseudogroups are groups of transformations solutions of systems of ordinary (OD) or partial differential (PD) equations. The purpose of this paper is to present an elementary summary of a few recent results obtained through the…
Lichnerowicz's algebra of differential geometric operators acting on symmetric tensors can be obtained from generalized geodesic motion of an observer carrying a complex tangent vector. This relation is based upon quantizing the classical…
We investigate the Weyl tensor algebraic structure of a fully general family of D-dimensional geometries that admit a non-twisting and shear-free null vector field k. From the coordinate components of the curvature tensor we explicitly…
The Riemann curvature tensor fully encodes local geometry, but its Ricci contraction retains only limited information: only the Ricci tensor and the scalar curvature survive, while the Weyl curvature vanishes identically. We show that…
The Weyl and Ricci tensors can be algebraically classified in a Lorentzian spacetime of arbitrary dimensions using alignment theory. Used in tandem with the boost weight decomposition and curvature operators, the algebraic classification of…
We review some well posed formulations of the evolution part of the Cauchy problem of General Relativity that we have recently obtained. We include also a new first order symmetric hyperbolic system based directly on the Riemann tensor and…
The Bianchi identities for the Weyl curvature tensor of a spacetime $(M, g)$ solving the vacuum Einstein equations in a double null foliation exhibit a hyperbolic structure, which can be used to obtain detailed nonlinear estimates on the…
We find necessary and sufficient conditions for a Riemannian four-dimensional manifold $(M, g)$ with anti-self-dual Weyl tensor to be locally conformal to a Ricci--flat manifold. These conditions are expressed as the vanishing of scalar and…
We develop a dimension-independent theory of alignment in Lorentzian geometry, and apply it to the tensor classification problem for the Weyl and Ricci tensors. First, we show that the alignment condition is equivalent to the PND equation.…
In recent papers, a few physicists studying Black Hole perturbation theory in General Relativity have tried to construct the initial part of a differential sequence based on the Kerr metric, using methods similar to the ones they already…
The rotations of rigid bodies in Euclidean space are characterized by their instantaneous angular velocity and angular momentum. In an arbitrary number of spatial dimensions, these quantities are represented by bivectors (antisymmetric…
Extended geometry is based on an underlying tensor hierarchy algebra. We extend the previously considered $L_\infty$ structure of the local symmetries (the diffeomorphisms and their reducibility) to incorporate physical fields, field…
This paper presents a systematic study of invariant Einstein metrics on basic classical Lie supergroups, whose Lie superalgebras belong to the Kac's classification of finite dimensional classical simple Lie superalgebras over $\mathbb{R}$.…