An elementary proof of a theorem of Johnson and Lindenstrauss

A result of Johnson and Lindenstrauss [13] shows that a set of n points in high dimensional Euclidean space can be mapped into an O(log n/?²)‐dimensional Euclidean space such that the distance between any two points changes by only a factor of (1 ± ?). In this note, we prove this theorem using elementary probabilistic techniques. © 2002 Wiley Periodicals, Inc. Random Struct. Alg., 22: 60–65, 2002     

A characterization of isometries on an open convex set,II

Let E ⁿ be an n-dimensional Euclidean space with n ≥ 2. In this paper, we generalize a classical theorem of Beckman and Quarles by proving that if a mapping, from an open convex subset Co of E ⁿ into E ⁿ , preserves a distance ρ, then the restriction of f to an open convex subset C _∞ of C ₀ is an isometry. This work was supported by 2007 Hongik University Research Fund.     

Topological rigidity theorems for open Riemannian manifolds

In this article, we study topology of complete non‐compact Riemannian manifolds. We show that a complete open manifold with quadratic curvature decay is diffeomorphic to a Euclidean n ‐space ?ⁿ if it contains enough rays starting from the base point. We also show that a complete non‐compact n ‐dimensional Riemannian manifold M with nonnegative Ricci curvature and quadratic curvature decay is diffeomorphic to ?ⁿ if the volumes of geodesic balls in M grow properly. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Minding-Isometrien bei (k+1)-Regelflächen

In this paper we investigate (k+1)-dimensional generalized ruled surfaces generated by a oneparameter family ofk-dimensional linear subspaces of then-dimensional Euclidean spaceE. Minding-isometries of ruled surfaces are special isometries, which let the generating space invariant. Some new results will be given, concerned with Mindingisometries of generalized ruled surfaces of arbitrary codimension. In this investigations quadratic hypersurfaces in the normal spaces are of great importance.     

Der Satz von Bonnet für (k+1)-Regelflächen

In the theory of ruled surfaces a theorem of O. Bonnet is well-known which is concerned with the striction line and its properties to be a geodesic and an isogonal trajectory. In this paper we will investigate (k+1)-dimensional generalized ruled surfaces generated by a one-parameter family ofk-dimensional linear subspaces of then-dimensional Euclidean spaceE _n . A theorem will be given which is the best generalisation of the result of Bonnet for generalized ruled surfaces.     

Sechseckgewebe und Strahlensysteme

Let S be an oriented rectilinear congruence in the three-dimensional Euclidean space E³. In this paper we prove necessary and sufficient conditions, so that certain ruled surfaces of S meet its middle surface in an hexagonal web.     

Some Inequalities for Two Simplices

In this paper, we establish some inequalities for two n-dimensional simplices in the n-dimensional Euclidean space E ⁿ .     

Bernstein type theorem of minimal Lagrangian graphs in quaternion Euclidean space H n

In this paper, a Bernstein type theorem for minimal Lagrangian submanifolds in quaternion Euclidean space H^n is studied.     

Lokale Kennzeichnungen der Minimalhyperregelflächen

Generalized ruled hypersurfaces are generated by a one-parameter family of (n–2)-dimensional linear subspaces of the n-dimensional Euclidean space E_n. In this paper we give local characterizations of generalized ruled hypersurfaces with an everywhere vanishing mean curvature.     

Submanifolds with totally umbilical Gauss image

The submanifolds whose Gauss images are totally umbilical submanifolds of the Grassmann manifold are under consideration. The main result is the following classification theorem: if the Gauss image of a submanifold F in a Euclidean space is totally umbilical then either the Gauss image is totally geodesic, or F is the surface in E ⁴ of the special structure. Submanifolds in a Euclidean space with totally geodesic Gauss image were classified earlier.     

Minimal Lipschitz Extensions to Differentiable Functions Defined on a Hilbert Space

We generalize the Lipschitz constant to fields of affine jets and prove that such a field extends to a field of total domain \mathbbRⁿ{\mathbb{R}^n} with the same constant. This result may be seen as the analog for fields of the minimal Kirszbraun’s extension theorem for Lipschitz functions and, therefore, establishes a link between Kirszbraun’s theorem and Whitney’s theorem. In fact this result holds not only in Euclidean \mathbbRⁿ{\mathbb{R}^n} but also in general (separable or not) Hilbert space. We apply the result to the functional minimal Lipschitz differentiable extension problem in Euclidean spaces and we show that no Brudnyi–Shvartsman-type theorem holds for this last problem. We conclude with a first approach of the absolutely minimal Lipschitz extension problem in the differentiable case which was originally studied by Aronsson in the continuous case.     

Generic properties in Euclidean kinematics

The motion of a rigid body in a Euclidean space E ⁿ is represented by a path in the Euclidean isometry group E(n). A normal form for elements of the Lie algebra of this group leads to a stratification of the algebra which is shown to be Whitney regular. Translating this along invariant vector fields give rise to a stratification of the jet bundles J ^k (R, E(n)) for k=1, 2 and, hence, via the transversality theorem, to generic properties of rigid body motions. The relation of these to the classical centrodes and axodes of motions is described, together with applications to planar 4-bar mechanisms and the dynamics of a rigid body.     

Zur Theorie verallgemeinerter torsaler Strahlflächen

In this paper we will investigate (k+1)-dimensional generalized ruled surfaces generated by a one-parameter family ofk-dimensional linear subspaces of then-dimensional Euclidean spaceE _n . Some results which are well-known for developable surfaces are proved for generalized ruled surfaces: Generalized developable surfaces are locally either cyclinders, cones or tangent surfaces. Each regular surface on a generalized ruled surface is locally Euclidean if and only if is developable. Each locally Euclidean hypersurface is a generalized developable hypersurface. Furthermore, the hypersurfaces with vanishing Gaussian curvature and the locally Euclidean hypersurfaces on generalized rule hypersurfaces will be characterized.     

Zur globalen Raumkinematik

In this paper we continue the considerations of [8], i.e. the study of one-parametric closed equiform motions in Euclidean 3-spaceE ³. Thereby the Steiner-formula on the aperture-distances of ruled surfaces, traced by lines of the moving space, is discussed and applied in different ways. This leads to new Holditch-theorems and to results in the field of global geometry of ruled surfaces and curves inE ³. On the other hand these theorems can be specialized to important formulas of global spherical kinematics.

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Herrn Prof. Dr. F. Hohenberg zum 80. Geburtstag gewidmet     

Inequalities involving two simplexes

In this paper, we establish some geometric inequalities for two n-dimensional simplexes in the n-dimensional Euclidean space Eⁿ.This work is supported by the Hunan Provincial Science Foundation, P. R. China. S. Mingbao is a doctorate candidate of Nanjing University of Science and Technology.     

Complete submanifolds in Euclidean spaces¶with parallel mean curvature vector

In this paper, we prove that n-dimensional complete and connected submanifolds with parallel mean curvature vector H in the (n+p)-dimensional Euclidean space E ^{n + p} are the totally geodesic Euclidean space E ⁿ , the totally umbilical sphere S ⁿ (c) or the generalized cylinder S ^{n − 1} (c) ×E ¹ if the second fundamental form h satisfies <h>²≤n ²|H|²/ (n− 1). Received: 28 November 2000 / Revised version: 7 May 2001     

The nonlinear complementarity problem with applications,part 1

The main result in this paper is an existence and uniqueness theorem for the following nonlinear complementarity problem: Given a mapping from then-dimensional Euclidean spaceE ⁿ into itself, find a nonnegative vector inE ⁿ whose image, under the given mapping, is also nonnegative, the two vectors being orthogonal to each other. It is shown that the above problem has a unique solution if the given mapping is continuous and strongly monotone on the nonnegative orthantE ₊ ⁿ ofE ⁿ . It is also shown that a sufficient condition for a differentiable mapping to be strongly monotone on an open set is that all the eigenvalues of the symmetric part of its Jacobian be bounded below by a positive constant on the given set.This research constituted a part of the author's Ph.D. dissertation at the University of California at Berkeley, Berkeley, California. The author would like to express his appreciation to Professor G. B. Dantzig, who brought this problem to his attention and guided his research with his several suggestions and helpful criticism. Also, he thanks the referee for several important comments and recommendations.     

A geometric inequality for a finite set of points on a sphere and its applications

In this paper we present a geometric inequality for a finite number of points on an (n–1)-dimensional sphere S _n–1(R). As an application, we obtain a geometric inequality for finitely many points in the n-dimensional Euclidean space E ⁿ.     

Symmetry analysis of the charged squashed Kaluza–Klein black hole metric

In this research article, a complete analysis of symmetries and conservation laws for the charged squashed Kaluza–Klein black hole space‐time in a Riemannian space is discussed. First, a comprehensive group analysis of the underlying space‐time metric using Lie point symmetries is presented, and then the n‐dimensional optimal system of this space‐time metric, for n = 1,…,4, are computed. It is shown that there is no any n‐dimensional optimal system of Lie symmetry subalgebra associated to the system of geodesic for n≥5. Then the point symmetries of the one‐parameter Lie groups of transformations that leave invariant the action integral corresponding to the Lagrangian that means Noether symmetries are found, and then the conservation laws associated to the system of geodesic equations are calculated via Noether's theorem. Copyright © 2015 John Wiley & Sons, Ltd.     

Hypersurfaces whose mean curvature function is of finite type

We define finite mean type hypersurfaces to be hypersurfaces with mean curvature function of finite Chen-type. Then, we prove that hyperplanes are the only polynomial translation hypersurfaces of finite mean type in a Euclidean spaceE ⁿ⁺¹. And we show that the only non-conic hyperquadrics of finite mean type in Euclidean spaces are the hyperspheres and the cylinders on spheres. Finally, we state that, among all hypercylinders in a Euclidean spaceE ⁿ⁺¹, the only ones of finite mean type are those on finite mean type planar curves.