The Length of a Shortest Closed Geodesic on a Two-Dimensional Sphere and Coverings by Metric Balls

In this paper we will present upper bounds for the length of a shortest closed geodesic on a manifold M diffeomorphic to the standard two-dimensional sphere. The first result is that the length of a shortest closed geodesic l(M) is bounded from above by 4r , where r is the radius of M . (In particular that means that l(M) is bounded from above by 2d, when M can be covered by a ball of radius d/2, where d is the diameter of M.) The second result is that l(M) is bounded from above by 2( max{r₁,r₂}+r₁+r₂), when M can be covered by two closed metric balls of radii r₁,r₂ respectively. For example, if r₁ = r₂= d/2 , thenl(M) 3d. The third result is that l(M) 2(max{r₁,r₂r₃}+r₁+r₂+r₃), when M can be covered by three closed metric balls of radii r₁,r₂,r₃. Finally, we present an estimate for l(M) in terms of radii of k metric balls covering M, where k 3, when these balls have a special configuration.     

Geometric Structures as Determined by the Volume of Generalized Geodesic Balls

Several authors have studied the Taylor expansion for the volume of geodesic balls under the exponential mapping of an analytic Riemannian manifold $ (M, {\cal G}) $ . A more general structure $ (M, D{\cal G}) $ , where D is a torsion-free and Ricci-symmetric connection, appears in several geometric situations. We study the Taylor expansion in this case, where all metric notions are Riemannian, while now the exponential mapping is induced from the connection D. We give many applications, in particular in different hypersurface theories.     

A Gap Theorem for Free Boundary Minimal Surfaces in Geodesic Balls of Hyperbolic Space and Hemisphere

In this paper we provide a pinching condition for the characterization of the totally geodesic disk and the rotational annulus among minimal surfaces with free boundary in geodesic balls of three-dimensional hyperbolic space and hemisphere. The pinching condition involves the length of the second fundamental form, the support function of the surface, and a natural potential function in hyperbolic space and hemisphere.     

The Centroaffine Volume of Generalized Geodesic Balls Under Inversion at the Sphere

On an analytic Riemannian manifold (M,g), several authors have studied the Taylor expansion for the volume of geodesic balls under the exponential mapping. In the foregoing paper [1] we studied a more general structure (M,D,g), where D is a torsion-free and Ricci-symmetric connection. We calculated the Taylor expansion up to order (n+4) for the volume of what we called a generalized geodesic ball under the exponential mapping in case that all metric notions are Riemannian, while the exponential mapping is induced from the connection D. For the structure $(M,D,{\cal G})$ the coefficients of the Taylor expansion are much more complicated than in the Riemannian case. It is one of the main objectives of the present paper to study centroaffine hypersurfaces in Euclidean space, their geometric invariants which appear in the very complicated coefficient of order (n+4), and their behaviour under polarization (inversion at the unit sphere). Our results complement applications in the foregoing paper [1], where mainly the coefficients up to order (n+2) and geometric consequences have been studied.     

Eigenfunction Expansions on Geodesic Balls and Rank One Symmetric Spaces of Compact Type

We find sharp conditions for the pointwise convergence ofeigenfunction expansions associated with the Laplace operator and otherrotationally invariant differential operators. Specifically, we considerthis problem for expansions associated with certain radially symmetricoperators and general boundary conditions and the problem in the contextof Jacobi polynomial expansions. The latter has immediate application toFourier series on rank one symmetric spaces of compact type.     

关于半对称空间的存在性和测地对应问题

Beltrami,E.证明了著名的测地对应定理,即 定理A 仅仅是常曲率空间才能和常曲率空间作成测地对应。 H.C.和Roter,W.分别将Beltrami定理加以推广,即证明了。 定理B 如果黎曼空间V_n(n>2)允许非平凡测地对应到黎曼循环空间V_n(即V_n的曲率张量满足其中记号“|”表示关于V_n的联络系数的协变微分;当λ_l=0时,V_n称为黎曼对称空间),则V_n是常曲率空间。     

On Projecting Symmetries by Unbranched Regular Coverings of Riemann Surfaces

A symmetry of a Riemann surface X of genus g is an antiholomorphic involution σ of X. It is a classical result of Harnack that the set of fixed points of σ consists of k closed Jordan curves, called ovals, for some k, 0 ≤ k ≤ g + 1; when k = g or k = g+1 we say, following Natanzon [8], that σ is an (M – 1)- or an M-symmetry, respectively. Given a Riemann surface X with an M-symmetry, a Riemann surface Y and a regular covering p: X → Y, we prove that Y admits either an M- or an (M – 1)-symmetry and whenever p is unbranched we describe the groups of covering transformations of p. In the case that X is hyperelliptic we calculate as well the number of unbranched regular coverings p: X → Y in which X has an M-symmetry. The first two authors are supported by MTM2005-01637, the third by SAB2005-0049.     

On the Mean Absolute Geodesic Curvature of Closed Curves in 2-Dimensional Space Forms

In this paper,we establish some formulas on closed curves in 2-dimensional space forms.Mean absolute geodesic curvature is introduced to describe the average curving of a closed curve.Inthis sense,a closed curve could be compared with a geodesic circle that is the boundary of a convex geodesic circular disk containing the closed curve.The comparison can be used to show some properties of space forms only on themselves.     

On Coverings of Tori with Cubes

Doklady Mathematics - We obtain new bounds and exact values of the minimum number of cubes with side length $$\varepsilon \in (0,\;1)$$ covering the torus...     

On Lattice Structure of the Space of Pointwise Limits of Harmonic Functions

Let H¹(U) denote the space of all pointwise limits of bounded sequences from H(U), where H(U) consists of all continuous functions on the closure [`(U)]\overline{U} of a bounded open set U⊂ℝ^m that are harmonic on U. It is shown that the space H¹(U) is a lattice in the natural ordering if and only if the set ∂_regU of all regular points of U is an F_σ-set.     

On Abelian Branched Coverings of the Sphere

 We obtain an enumeration formula for the number of weak equivalence classes of the branched (?×ℬ)-covering of the sphere with m-branch points, when ? and ℬ are finite abelian groups with (|?|,|ℬ|)=1. From this, we can deduce an explicit formula for enumerating the weak equivalence classes of pseudofree spherical (ℤ _p ×ℤ _q )-actions on a given surface, when p and q are distinct primes. Received: August 10, 1999 Final version received: June 19, 2000     

Prime Space of Lattice Implication Algebras

1.IntroductionPeoplehavepaidmoreattentiontolathcevaluedlogicsystem,whichwillbecomemuchbetterlogicalsystemforintelligentcomputer.InreferenceL1JXuYangpresentanalgebrastructure-latticeimplicationalgebrabycomblngthelatticewithimplicationalgebraastruevaluefieldoflatticevaluedlogicaIsystem.Afterthat,westudytheimplicationhomomor-phism,congruencerelationsandalgebraicstructureoflatticeimplicationalgebraanddiscussthefirstorderlogicalsystemFMbasedonlatticeimplicationalgebras,andobtainedseveralimpor-ta…     

Totally Geodesic Subsets in the Manifold of Directions of Physical Space

Let M ₀ be the Minkowski 4-space, let ₂(M ₀) denote the second exterior power of M ₀ equipped with a structure of a pseudo-Euclidean space with signature (3,3), let K ₀(M ₀) ₂ M ₀ be the light cone, and let G _{1 2}(M ₀) be the set of the oriented 2-planes meeting the interior of K ₀(M ₀). Four types of totally geodesic two-manifolds in G ₁ are described such that manifolds of one type are pairwise congruent as subsets in ₂(M ₀), while manifolds of different types are not. Models of such manifolds in the disk D ³ are constructed. An explicit formula for the curvature tensor of G ₁ is given. Bibliography: 6 titles.     

On the Duality of Semiantichains and Unichain Coverings

We study a min-max relation conjectured by Saks and West: For any two posets P and Q the size of a maximum semiantichain and the size of a minimum unichain covering in the product P×Q are equal. As a positive result, we state conditions on P and Q that imply the min-max relation. Based on these conditions we identify some new families of posets where the conjecture holds and get easy proofs for several instances where the conjecture had been verified before. However, we also have examples showing that in general the min-max relation is false, i.e., we disprove the Saks-West conjecture.     

Invariants of Special Second-Type Almost Geodesic Mappings of Generalized Riemannian Space

Invariants of second-type almost geodesic mappings are obtained in this paper. These invariants are generalizations of Thomas projective parameter and Weyl projective tensor.     

On the Geodesic Curvature of Riemannian Loops

We give a sharp extrinsic lower bound for the total geodesic curvature of a closed curve in a space form and discuss the equality case. The case of curves in Euclidean 3-space which is known since a long time by means of integral geometry, is extended here in a new way.     

Coverings by classes of permutable equivalence relations

Supported by the Russian Foundation for Fundamental Research, grant No. 93-01-16011.     

Resolvable Coverings of 2-Paths by Cycles

 Let K _n be the complete graph on n vertices. A C(n,k,λ) design is a multiset of k-cycles in K _n in which each 2-path (path of length 2) of K _n occurs exactly λ times. A C(lk,k,1) design is resolvable if its k-cycles can be partitioned into classes so that every vertex appears exactly once in each class. A C(n,n,1) design gives a solution of Dudeney's round table problem. It is known that there exists a C(n,n,1) design when n is even and there exists a C(n,n,2) design when n is odd. In general the problem of constructing a C(n,n,1) design is still open when n is odd. Necessary and sufficient conditions for the existence of C(n,k,λ) designs and resolvable C(lk,k,1) designs are known when k=3,4. In this paper, we construct a resolvable C(n,k,1) design when n=p ^e +1 ( p is a prime number and e≥1) and k is any divisor of n with k≠1,2. Received: October, 2001 Final version received: September 4, 2002 RID="*" ID="*" This research was supported in part by Grant-in-Aid for Scientific Research (C) Japan