Curvas dadas por la intersección transversal de dos superficies en el espacio tridimesional de Minkowski

In this paper, we study the differential geometry of the transversal intersection curve of two surfaces in Minkowski 3-space, where each pair satisfies the following types spacelike-lightlike, timelike-lightlike and lightlike-lightlike. Surfaces are generally give by their parametric or implicit equations, then the surfacesurface intersection problem appear commonly as parametric-parametric, parametric-implicit and implicitimplicit.We derive the Frenet frame, Darboux frame, curvature, torsion, normal curvature and geodesic curvatures of transversal intersections for all types of intersection problems. We show the intersection curve may be spacelike (timelike, lightlike or pseudo null) curve. Finally, we show our methods by given several examples.En este artículo, estudiamos la geometría diferencial de la curva dada por la intersección transversal de dos superficies en el espacio tridimensional de Minkowski donde cada par satisface los siguientes tipos de superficies; tipo espacio - tipo luz, tipo tiempo - tipo luz y tipo luz - tipo luz. Generalmente, las superficies están dadas por sus ecuaciones paramétricas o implícitas, entonces el problema de intersección superficie superficie aparece comunmente como paramétrico-paramétrico, paramétrico-implícito e implícito-implícito. Obtenemos el Referencial de Frenet, el Referencial de Darboux, la curvatura, la torsión, la curvatura normal y las curvaturas geodésicas de las intersecciones transversales para todos los tipos de problemas de intersección. Mostramos que la curva de intersección puede ser una curva similar a una curva tipo espacio (tipo tiempo, tipo luz o pseudo nula). Finalmente, mostramos nuestros métodos por varios ejemplos

Intersection curve of two parametric surfaces in Euclidean n-space

The aim of this paper is to study the differential geometric properties of the intersection curve of two parametric surfaces in Euclidean n-space. For this aim, we first present the mth order derivative formula of a curve lying on a parametric surface. Then, we obtain curvatures and Frenet vectors of the transversal intersection curve of two parametric surfaces in Euclidean n-space. We also provide computer code produced in MATLAB to simplify determining the coefficients relative to Frenet frame of higher order derivatives of a curve

Spacelike intersection curve of three spacelike hypersurfaces in E14E_1^4

In this paper, we compute the Frenet vectors and the curvatures of the spacelike intersection curve of three spacelike hypersurfaces given by their parametric equations in four-dimensional Minkowski space $E_1^4$

The lightcone of G\"odel-like spacetimes

A study of the lightcone of the G\"odel universe is extended to the so-called G\"odel-like spacetimes. This family of highly symmetric 4-D Lorentzian spaces is defined by metrics of the form $ds^2=-(dt+H(x)dy)^2+D^2(x)dy^2+dx^2+dz^2$, together with the requirement of spacetime homogeneity, and includes the G\"odel metric. The quasi-periodic refocussing of cone generators with startling lens properties, discovered by Ozsv\'{a}th and Sch\"ucking for the lightcone of a plane gravitational wave and also found in the G\"odel universe, is a feature of the whole G\"odel family. We discuss geometrical properties of caustics and show that (a) the focal surfaces are two-dimensional null surfaces generated by non-geodesic null curves and (b) intrinsic differential invariants of the cone attain finite values at caustic subsets.Comment: 19 pages, 1 figur

A NEW APPROACH TOWARDS GEODESIC CURVATURE AND GEODESIC TORSION OF TRANSVERSAL INTERSECTION IN R3

In this paper, we formulate new method to obtain the geodesic curvature and geodesic torsion of two regular parametric surfaces in R3. Our new method will be different from the older ones in the sense that we will be making use of Rodrigues rotation formula and defining a new operator D

Solitons and admissible families of rational curves in twistor spaces

It is well known that twistor constructions can be used to analyse and to obtain solutions to a wide class of integrable systems. In this article we express the standard twistor constructions in terms of the concept of an admissible family of rational curves in certain twistor spaces. Examples of of such families can be obtained as subfamilies of a simple family of rational curves using standard operations of algebraic geometry. By examination of several examples, we give evidence that this construction is the basis of the construction of many of the most important solitonic and algebraic solutions to various integrable differential equations of mathematical physics. This is presented as evidence for a principal that, in some sense, all soliton-like solutions should be constructable in this way.Comment: 15 pages, Abstract and introduction rewritten to clarify the objectives of the paper. This is the final version which will appear in Nonlinearit