Geometry of the Sphere of Calibrations of Degree Two

A canonical bijection between the set of extreme points of the comass-unit sphere and the manifold of orthogonal complex structures in is described, under which unitary bases correspond to decompositions of the forms realizing the conjugate norm of the mass. This correspondence is used for obtaining a classification of faces of the sphere and the known classification of faces of the set polar to . Bibliography: 10 titles.     

Low Faces of Restricted Degree in 3-Polytopes

Siberian Mathematical Journal - The degree of a vertex or face in a 3-polytope is the number of incident edges. A k-face is one of degree k, a k−-face has degree at most k. The height of a...     

Closed Geodesics on Incomplete Surfaces

We consider the problem of finding embedded closed geodesics on the two-sphere with an incomplete metric defined outside a point. Various techniques including curve shortening methods are used The authors acknowledge the support of the Australian Research Council     

Geodesics on weighted projective spaces

We study the inverse spectral problem for weighted projective spaces using wave-trace methods. We show that in many cases one can “hear” the weights of a weighted projective space.     

椭球面上平面曲线类型的闭测地线

通过对椭球面上平截线的研究,找出了椭球面上所有平面曲线类型的闭测地线.     

由渐近线和测地线条件确定的两种螺旋线

在螺旋线的副法线面上,由一条特殊曲线分别成为渐近线和测地线的几何条件得到两种特殊的螺旋线.     

Geodesics on spaces of almost hermitian structures

A natural metric on the space of all almost hermitian structures on a given manifold is investigated.     

Geodesics of Sasakian Metrics on Tensor Bundles

On the basis of the phase completion the notion of vertical and horizontal lifts of vector fields is defined in the tensor bundles over a Riemannian manifold. Such a tensor bundle is made into a manifold with a Riemannian structure of special type by endowing it with Sasakian metric. The components of the Levi-Civita and other metric connections with respect to Sasakian metrics on tensor bundles with respect to the adapted frame are presented. This having been done, it is shown that it is possible to study geodesics of Sasakian metrics dealing with geodesics of the base manifolds. Dedicated to the memory of Vladimir Vishnevskii (1929-2007)     

Sub-Riemannian Geodesics on the 3-D Sphere

The unit sphere can be identified with the unitary group SU(2). Under this identification the unit sphere can be considered as a non-commutative Lie group. The commutation relations for the vector fields of the corresponding Lie algebra define a 2-step sub-Riemannian manifold. We study sub-Riemannian geodesics on this sub-Riemannian manifold making use of the Hamiltonian formalism and solving the corresponding Hamiltonian system. The first author is partially supported by a research grant from the United State Army Research Office and a Hong Kong RGC competitive earmarked research grant #600607. The second and the third authors have been supported by the grant of the Norwegian Research Council #177355/V30, and by the European Science Foundation Research Networking Programme HCAA.     

Geodesics on lorentzian manifolds with quasi-convex boundary

In this paper we study existence and multiplicity results of geodesics joining two given events in Lorentzian manifolds with lack of geodesic completeness. The considered Lorentzian manifolds are not necessarily static or stationary and satisfy a condition of convexity on the boundary. work supported by M.U.R.S.T. research founds 40%–60% 1992     

几种特殊曲面上的测地线

具体刻画了柱面、锥面、旋转曲面上测地线的几何特征,所得结果一方面匡正了某些文献关于锥面上测地线的错误断言,一方面推广了现有文献关于旋转曲面上测地线几何性质的描述.     

Minimal Geodesics on Manifolds with Discontinuous Metrics

The paper describes some qualitative properties of minimizerson a manifold M endowed with a discontinuous metric. The discontinuityoccurs on a hypersurface disconnecting M. Denote by ₁ and₂ the open subsets of M such that M\ =₁₂. Assume that and are endowed with metrics ·, · ₍₁₎ and ·,·₍₂₎, respectively, such that (i=1, 2) is convex or concave. The existence of a minimizerof the length functional on curves joining two given pointsof M is proved. The qualitative properties obtained allows therefraction law in a very general situation to be described.     

A Note on Collars of Simple Closed Geodesics

For compact hyperbolic Riemann surfaces, the collar theorem gives a lower bound on the distance between a simple closed geodesic and all other simple closed geodesics that do not intersect the initial geodesic. Here it is shown that there are two possible configurations, and in each configuration there is a natural collar width associated to a simple closed geodesic. If one extends the natural collar of a simple closed geodesic α by ε >0, then the extended collar contains an infinity of simple closed geodesics that do not intersect α.Mathematics Subject Classiffications (2000). primary: 30F45; secondary: 32G07     

Geodesics on the regular tetrahedron and the cube

Consider all geodesics between two given points on a polyhedron. On the regular tetrahedron, we describe all the geodesics from a vertex to a point, which could be another vertex. Using the Stern–Brocot tree to explore the recursive structure of geodesics between vertices on a cube, we prove, in some precise sense, that there are twice as many geodesics between certain pairs of vertices than other pairs. We also obtain the fact that there are no geodesics that start and end at the same vertex on the regular tetrahedron or the cube.     

Geodesics of the Structural Similarity index

We construct metrics from the geodesics of the Structural Similarity index, an image quality assessment measure. An analytical solution is given for the simple case of zero stability constants, and the general solution involving the numerical solution of a nonlinear equation is also found.     

Calibrations and the size of Grassmann faces

Summary In the past fifteen years or so, convex geometry and the theory of calibrations have provided a deeper understanding of the behavior and singular structure ofm-dimensional area-minimizing surfaces inR ⁿ . Calibrations correspond to faces of the GrassmannianG(m,R ⁿ ) of orientedm-planes inR ⁿ , viewed as a compact submanifold of the exterior algebra ^m R ⁿ . Large faces typically provide many examples of area-minimizing surfaces. This paper studies the sizes of such faces. It also considers integrands more general than area. One result implies that form-dimensional surfaces inR ⁿ , with 2 m n – 2, for any integrand , there are -minimizing surfaces with interior singularities.