The Lorentzian surfaces in semi-Euclidean 4-space with index 2

The main goal of this paper is to study singularities of lightlike torus Gauss maps of Lorentzian surfaces (i.e., both tangent plane and normal plane are Lorentz) in semi-Euclidean 4-space with index 2. To do this, we construct a Lorentzian lightlike torus height function and reveal relations between singularities of the Lorentzian lightlike torus height function and those of lightlike torus Gauss map. In addition we study some properties of Lorentzian surface from geometrical viewpoint.     

HOLDITCH THEOREM FOR THE CLOSED SPACE CURVES IN LORENTZIAN 3-SPACE

In this article, we give the area formula of the closed projection curve of a closed space curve in Lorentzian 3-space L3. For the 1-parameter closed Lorentzian space motion in L3, we obtain a Holditch Theorem taking into account the Lorentzian matrix multiplication for the closed space curves by using their othogonal projections onto the Euclidean plane in the fixed Lorentzian space. Moreover, we generalize this Holditch Theorem for noncollinear three fixed points of the moving Lorentzian space and any other fixed point on the plane which is determined by these three fixed points.     

On geometric interpretations of split quaternions

Quaternions are an important tool that provides a convenient and effective mathematical method for representing reflections and rotations in three-dimensional space. A unit timelike split quaternion represents a rotation in the Lorentzian space. In this paper, we give some geometric interpretations of split quaternions for lines and planes in the Minkowski 3-space with the help of mutual pseudo orthogonal planes. We classified mutual planes with respect to the casual character of the normals of the plane as follows; if the normal is timelike, then the mutual plane is isomorphic to the complex plane; if the normal is spacelike, then the plane is isomorphic to the hyperbolic number plane (Lorentzian plane); if the normal is lightlike, then the plane is isomorphic to the dual number plane (Galilean plane).     

Area preserving transformations in two-dimensional space forms and classical differential geometry

We generalize Scheffers’ method to construct area preserving transformations in the Euclidean plane to Riemannian and Lorentzian two-dimensional space forms in a unified way. We review and extend the classical applications of such transformations in classical differential geometry. We introduce two classes of surfaces in Lorentzian 3-space that admit holomorphic representation and are analogous to the classical Appell and Bonnet surfaces.     

Clifford Product and Lorentzian Plane Displacements In 3-Dimensional Lorentzian Space

In this paper, by defining Clifford algebra product in 3-dimensional Lorentz space, L ³, it is shown that even Clifford algebra of L ³ corresponds to split quaternion algebra. Then, by using Lorentzian matrix multiplication, pole point of planar displacement in Lorentz plane L ² is obtained. In addition, by defining degenerate Lorentz scalar product for L ³ and by using the components of pole points of Lorentz plane displacement in particular split hypercomplex numbers, it is shown that the Lorentzian planar displacements can be represented as a special split quaternion which we call it Lorentzian planar split quaternion.      

A Small Cover for Convex Unit Arcs

An arc in the plane is convex if it is simple (i.e., one–one except that its endpoints may coincide) and lies on the boundary of its convex hull. We describe a compact convex plane set of area about 0.2466 that contains a congruent copy of each convex plane arc of unit length, a reduction of about 1.1% from the smallest such set previously known.     

The Tilt Formula for Generalized Simplices in Hyperbolic Space

   Abstract. For a simplex in Lorentzian space whose vertices are in the positive light cone, Weeks defined the ``tilt' relative to each face. It gave us an efficient tool for deciding whether or not the dihedral angle between two simplices holding a face in common is convex. He also provided an efficient formula, called the ``tilt formula,' to obtain tilts from the intrinsic hyperbolic structure of the simplex when its dimension is two or three. Sakuma and Weeks generalized it to general dimensions. In this paper we generalize the concept of the tilt and the tilt formula to the case where not all vertices are in the positive light cone. A key to our generalization is to give a correspondence between points and hyperplanes (or half-spaces) in Lorentzian space.     

An algorithm for constructing star-shaped drawings of plane graphs

A straight-line planar drawing of a plane graph is called a convex drawing if every facial cycle is drawn as a convex polygon. Convex drawings of graphs is a well-established aesthetic in graph drawing, however not all planar graphs admit a convex drawing. Tutte [W.T. Tutte, Convex representations of graphs, Proc. of London Math. Soc. 10 (3) (1960) 304–320] showed that every triconnected plane graph admits a convex drawing for any given boundary drawn as a convex polygon. Thomassen [C. Thomassen, Plane representations of graphs, in: Progress in Graph Theory, Academic Press, 1984, pp. 43–69] gave a necessary and sufficient condition for a biconnected plane graph with a prescribed convex boundary to have a convex drawing.In this paper, we initiate a new notion of star-shaped drawing of a plane graph as a straight-line planar drawing such that each inner facial cycle is drawn as a star-shaped polygon, and the outer facial cycle is drawn as a convex polygon. A star-shaped drawing is a natural extension of a convex drawing, and a new aesthetic criteria for drawing planar graphs in a convex way as much as possible. We give a sufficient condition for a given set A of corners of a plane graph to admit a star-shaped drawing whose concave corners are given by the corners in A, and present a linear time algorithm for constructing such a star-shaped drawing.     

Convex drawings of hierarchical planar graphs and clustered planar graphs

In this paper, we present results on convex drawings of hierarchical graphs and clustered graphs. A convex drawing is a planar straight-line drawing of a plane graph, where every facial cycle is drawn as a convex polygon. Hierarchical graphs and clustered graphs are useful graph models with structured relational information. Hierarchical graphs are graphs with layering structures; clustered graphs are graphs with recursive clustering structures.We first present the necessary and sufficient conditions for a hierarchical plane graph to admit a convex drawing. More specifically, we show that the necessary and sufficient conditions for a biconnected plane graph due to Thomassen [C. Thomassen, Plane representations of graphs, in: J.A. Bondy, U.S.R. Murty (Eds.), Progress in Graph Theory, Academic Press, 1984, pp. 43–69] remains valid for the case of a hierarchical plane graph. We then prove that every internally triconnected clustered plane graph with a completely connected clustering structure admits a “fully convex drawing,” a planar straight-line drawing such that both clusters and facial cycles are drawn as convex polygons. We also present algorithms to construct such convex drawings of hierarchical graphs and clustered graphs.     

Ricci and Yamabe solitons on second-order symmetric,and plane wave 4-dimensional Lorentzian manifolds

We show that a proper second-order symmetric spacetime, and four-dimensional Lorentzian plane wave manifolds admit different vector fields resulting in expanding, steady and shrinking Ricci and Yamabe solitons. Moreover, it is proved that those Ricci and Yamabe solitons are gradient only in the steady case.     

Periodic trajectories with fixed energy on Riemannian and Lorentzian manifolds with boundary

We state some existence and multiplicity results for periodic solutions with prescribed energy of a Lagrangian system and for closed geodesics on Riemannian manifolds convex close to their boundary. As an application we get the existence and multiplicity of periodic trajectories with fixed energy on a class of physically important Lorentzian manifolds with boundary. Entrata in Redazione il 23 aprile 1998. Part of the Ph.D. thesis of this author is contained in this paper. Partially supported by MEC grant PB97-0784-C03-01.     

Minimal and reduced pairs of convex bodies

We give a new proof for the existence and uniqueness (up to translation) of plane minimal pairs of convex bodies in a given equivalence class of the Hörmander-R»dström lattice, as well as a complete characterization of plane minimal pairs using surface area measures. Moreover, we introduce the so-called reduced pairs, which are special minimal pairs. For the plane case, we characterize reduced pairs as those pairs of convex bodies whose surface area measures are mutually singular. For higher dimensions, we give two sufficient conditions for the minimality of a pair of convex polytopes, as well as a necessary and sufficient criterion for a pair of convex polytopes to be reduced. We conclude by showing that a typical pair of convex bodies, in the sense of Baire category, is reduced, and hence the unique minimal pair in its equivalence class.     

Lorentzian cones in real Lie algebras

A Lorentzian coneW in a finite dimensional real Lie algebraL is the convex closed cone bounded by one half of the zero-set of a Lorentzian formq onL with the additional property, that for all sufficiently smallx, yW the Campbell-Hausdorff productx*y=x+y+1/2[x,y]+..., is also inW. We characterize Lorentzian cones completely; in particular, with the exception of one class of almost abelian solvable algebras, the Lorentzian formq is invariant, i.e., satisfiesq([x, y], z)=q (x,[y, z]).     

Helly-type theorems for bodies in the plane with common supports

A theorem of Bisztriczky implies that every pair of disjoint convex bodies in the plane has a common supporting line. In this paper, combinatorial conditions implying the existence of common supporting lines for larger collections of disjoint convex bodies in the plane are exhibited.     

Locally Conformally Flat Lorentzian Gradient Ricci Solitons

It is shown that locally conformally flat Lorentzian gradient Ricci solitons are locally isometric to a Robertson–Walker warped product, if the gradient of the potential function is nonnull, and to a plane wave, if the gradient of the potential function is null. The latter gradient Ricci solitons are necessarily steady.     

Crofton's and poincaré's formulas in the Lorentzian plane

The purpose of the present paper is to compare some of the formulas of integral geometry in the Euclidean plane with those for the Lorentzian plane. In particular, we find that the so-called Crofton's formula and Poincaré's formula hold only under the restrictions expressed in Theorems 1 and 2, respectively. In order to arrive at these results, we find several different expressions for the density of geodesics and kinematic density.The author was supported by a Fellowship of Consejo de Investigaciones Cientificas of Tecnicas de la Republica Argentina at Brown University.     

平面有限点集中的最大面积六边形

若平面上的有限点集构成凸多边形的顶点集,则称此有限点集处于凸位置令P表示平面上处于凸位置的有限点集,研究了P的子集所确定的凸六边形的面积与CH(P)面积比值的最大值问题.     

Upper bounds for the perimeter of plane convex bodies

We show that the maximum total perimeter of k plane convex bodies with disjoint interiors lying inside a given convex body C is equal to $\operatorname{per}\, (C)+2(k-1)\operatorname{diam}\, (C)$ , in the case when C is a square or an arbitrary triangle. A weaker bound is obtained for general plane convex bodies. As a consequence, we establish a bound on the perimeter of a polygon with at most k reflex angles lying inside a given plane convex body.     

On the Napoleon-Torricelli configuration in Affine Cayley-Klein planes

There are three affine Cayley-Klein planes (see [5]), namely, the Euclidean plane, the isotropic (Galilean) plane, and the pseudo-Euclidean (Minkow-skian or Lorentzian) plane. We extend the generalization of the well-known Napoleon theorem related to similar triangles erected on the sides of an arbitrary triangle in the Euclidean plane to all affine Cayley-Klein planes. Using the Ω_k-and anti-Ω_k-equilateral triangles introduced in [28], we construct the Napoleon and the Torricelli triangle of an arbitrary triangle in any affine Cayley-Klein plane. Some interesting geometric properties of these triangles are derived. The author is partially supported by grant VU-MI-204/2006.