Minimal universal covers inE n

It is shown that any plane set of constant unit width contains a semi-circle of radius 1/2, and using this a minimal univeral plane cover is explicitly constructed. It is also shown that in an n-dimensional space with n>2 there are minimal universal covers of arbitrary large diameter. This paper was written while the author was a National Science Foundation Visiting Senior Fellow at the University of Washington, Seattle, Washington, U.S.A.     

All triangulations of the projective plane are geometrically realizable inE 4

We say that a triangulationT of a manifold is geometriclaly realizable inE ⁿ provided there is a triangulationT′?E ⁿ isomorphic toT such that each simplex ofT′ is convex. A triangulation of the projective plane cannot be geometrically realizable inE ³ because it is not topologically realizable inE ³. We show that it is, however, geometriclaly realizable inE ⁴.     

Local isometric immersions of two-dimensional metrics inE 4 with prescribed Gaussian torsion

Suppose that in a domain R(, B) of variables (r, ): (0 r , ₁ +B(r–r ₀ ) ₂–B(r–r₀), where > 0, B > 0, ₁ < 0 < ₂ are numbers) a metric ds² = dr² +G(r, )d ² and a function k(r, ) are given. The problem of isometrically immersing ds² in E ⁴ with prescribed Gaussian torsion is considered. The following is proved: The class C ⁵ metric ds ² is locally realized in the form of a class C ³ surface F ² whose Gaussian torsion is the prescribed class C ³ function (r, ).Translated from Ukrainskii Geometricheskii Sbornik, No. 35, pp. 38–47, 1992.     

Equivelar polyhedral manifolds inE 3

An equivelar polyhedral 2-manifold in the classM _p,q is one embedded inE ³ in which every face is a convexp-gon and every vertex isq-valent. Various constructions for equivelar manifolds are described, and it is shown that, in certain classesM _p,q, there is a manifold of given genusg≧2 for all but finitely manyg.     

On the geodesic curvature of an integral curve of a two-dimensional pfaffian manifold inE 4

The concept of generalized geodesic curvature is introduced along with the invariant mean geodesic curvature vector for a two-dimensional Pfaffian manifold inE ₄, and the indicatrix of generalized geodesic curvature of integral curves of this manifold is constructed.It is proved that the ratio of the fourth quadratic form of the manifold to its first fundamental form is equal to the square of the modulus of the generalized geodesic curvature vector, and an analog of a theorem of Euler is deduced.Translated from Ukrainskií Geometricheskií Sbornik, Issue 28, 1985, pp. 22–26.     

Polyhedral 2-manifolds inE 3 with unusually large genus

An equivelar polyhedral 2-manifold in the class ?_p,q is one embedded inE ³ in which every face is a convexp-gon and every vertex isq-valent. In this paper, examples are constructed, to show that each of the classes ?_3,q (q≧7), ?_4,q (q≧5) and ?_p,4 (p≧5) contains infinitely many distinct combinatorial types. As particular examples, there are polyhedral 2-manifolds with 576 vertices and genus 577, and with 4096 faces and genus 4097. A modification of one construction shows that there is a constantk, such that for eachg≧2, there exists a closed polyhedral 2-manifold inE ³ of genusg with at mostkg/logg vertices.     

Surfaces inE 4 with a Gaussian curvature coinciding with a Gaussian torsion up to the sign

Mathematical Notes -     

The structure of an infinite,complete, convex hypersurface inE 4 with bounded mean curvature

The following theorem is proved. THEOREM. If on an infinite, complete, convex hypersurface F in E⁴ the mean curvature is 1 – H 1, where 0 10^–11, then F is a cylindrical hypersurface.Translated from Ukrainskii Geometricheskii Sbornik, No. 34, pp. 8–9, 1991.     

Angles in normed spaces

The concepts of angle, angle functions, and the question how to measure angles present old and well-established mathematical topics referring to the Euclidean space, and there exist also various extensions to non-Euclidean spaces of different types. In particular, it is very interesting to investigate or to combine (geometric) properties of possible concepts of angle functions and angle measures in finite-dimensional real Banach spaces (= Minkowski spaces). However, going into this direction one will observe that there is no monograph or survey reflecting the complete picture of the existing literature on such concepts in a satisfying manner. We try to close this gap. In this expository paper (containing also new results, and new proofs of known results) the reader will get a comprehensive overview of this field, including further related aspects, as well. For example, angular bisectors, their applications, and angle types which preserve certain kinds of orthogonality are discussed. The latter aspect yields, of course, an interesting link to the large variety of orthogonality types in such spaces.     

Angles between Euclidean subspaces

The angle between two subspaces of dimensions p and q in a Euclidean space is considered by using exterior algebra. Some properties of angles are obtained. The relation between such a higher dimensional angle and the usual principal angles is also given. And finally, an application to Grassmann manifolds is briefly discussed.Supported by the National Natural Science Fund of P.R. China.     

四元空间形式中具有常数Kähler角的浸入曲面

对四元K(a)hler流形中的浸入曲面引入了K(a)hler角的概念,同时讨论K(a)hler角是常数的情形.主要结果是若xM→N(c)是具有常数Q-截面曲率c的实四维四元空间形式N(c)中具有常数K(a)hler角θ(sinθ≠0)的等距浸入曲面,则必有c=0.     

Dihedral Angles of Convex Polyhedra

Convex polyhedra in H ³ are not determined by (their combinatorics and) their edge lengths. Convex space-like polyhedra in the de Sitter space S ³ ₁ are determined neither by their dihedral angles nor by their edge lengths. The same holds of convex polyhedra in S ³ . Received November 16, 1998, and in revised form March 8, 1999.