Related papers: On some relativistic singular surfaces
A classification of 2-dimensional surfaces imbedded in spacetime is presented, according to the algebraic properties of their shape tensor. The classification has five levels, and provides among other things a refinement of the concepts of…
The classical rotation is not self-consistent in the framework of the special theory of relativity. the Relativistic rotation is obtained, which takes the relativistic effect into account. It is demonstrated that the angular frequency of…
We study the singular set of a singular Levi-flat real-analytic hypersurface. We prove that the singular set of such a hypersurface is Levi-flat in the appropriate sense. We also show that if the singular set is small enough, then the…
We develop a new framework of relative algebroids to address existence and classification problems of geometric structures subject to partial differential equations.
A number of techniques in Lorentzian geometry, such as those used in the proofs of singularity theorems, depend on certain smooth coverings retaining interesting global geometric properties, including causal ones. In this note we give…
We provide some guidance and examples to clear up common misconceptions about special relativity. These misconceptions often come from trying to express the truths of special relativity in Newtonian terms rather than in terms more natural…
Minimal surfaces are among the most natural objects in Differential Geometry, and have been studied for the past 250 years ever since the pioneering work of Lagrange. The subject is characterized by a profound beauty, but perhaps even more…
We study singular hypersurfaces in tensor multi-scalar theories of gravity. We derive in a distributional and then in an intrinsic way, the general equations of junction valid for all types of hypersurfaces, in particular for lightlike…
The combination of quantum theory and special relativity leads to structures that differ in several respects from non-relativistic quantum mechanics of particles. These differences are quite familiar to practitioners of Algebraic Quantum…
We study Levi-flat real analytic hypersurfaces with singularities. We prove that the Levi foliation on the regular part of the hypersurface can be holomorphically extended, in a suitable sense, to neighbourhoods of singular points.
An overview is provided of the singularity theorems in cosmological contexts at a level suitable for advanced graduate students. The necessary background from tensor and causal geometry to understand the theorems is supplied, the…
In \cite{X-Z DCS1}, we introduced discrete conformal structures on surfaces with boundary via an axiomatic framework, and provided a classification of such discrete conformal structures. The present work focuses on the rigidity and…
We solve the problem on flat extensions of a generic surface with boundary in Euclidean 3-space, relating it to the singularity theory of the envelope generated by the boundary. We give related results on Legendre surfaces with boundaries…
We discuss the bi-Lipschitz geometry of an isolated singular point of a complex surface which particular emphasis on when it is metrically conical.
We study focal surfaces of (wave) fronts associated to unbounded principal curvatures near non-degenerate singular points of initial fronts. We give characterizations of singularities of those focal surfaces in terms of types of…
Polyhedral surfaces are fundamental objects in architectural geometry and industrial design. Whereas closeness of a given mesh to a smooth reference surface and its suitability for numerical simulations were already studied extensively, the…
In this paper we study surfaces with minimal potential energy under gravitational forces, called singular minimal surfaces. We prove that a singular minimal ruled surface in a Euclidean $3-$space is cylindrical, in particular as an…
We present a variety of geometrical and combinatorial tools that are used in the study of geometric structures on surfaces: volume, contact, symplectic, complex and almost complex structures. We start with a series of local rigidity results…
By the famous ADE classification rational double points are simple. Rational triple points are also simple. We conjecture that the simple normal surface singularities are exactly those rational singularities, whose resolution graph can be…
I study the physical meaning of Deformed, or Doubly, Special Relativity (DSR). I argue that DSR could be physically relevant in a certain large-distance limit. I consider a concrete physical effect: the gravitational slowing down of time…