Circles and Quadratic Maps Between Spheres

Consider an analytic map of a neighborhood of 0 in a vector space to a Euclidean space. Suppose that this map takes all germs of lines passing through 0 to germs of circles. Such a map is called rounding. We introduce a natural equivalence relation on roundings and prove that any rounding, whose differential at 0 has rank at least 2, is equivalent to a fractional quadratic rounding. A fractional quadratic map is just the ratio of a quadratic map and a quadratic polynomial. We also show that any rounding gives rise to a quadratic map between spheres. The known results on quadratic maps between spheres have some interesting implications concerning roundings. Partially supported by CRDF RM1-2086.     

Sylvester–Gallai Theorem and Metric Betweenness

Sylvester conjectured in 1893 and Gallai proved some 40 years later that every finite set S of points in the plane includes two points such that the line passing through them includes either no other point of S or all other points of S. There are several ways of extending the notion of lines from Euclidean spaces to arbitrary metric spaces. We present one of them and conjecture that, with lines in metric spaces defined in this way, the Sylvester--Gallai theorem generalizes as follows: in every finite metric space there is a line consisting of either two points or all the points of the space. Then we present meagre evidence in support of this rash conjecture and finally we discuss the underlying ternary relation of metric betweenness.     

An extension of Asgeirsson's mean value theorem for solutions of the ultra-hyperbolic equation in dimension four

In 1937 Asgeirsson established a mean value property for solutions of the general ultra-hyperbolic equation in 2n variables. In the case of four variables, it states that the integrals of a solution over certain pairs of conjugate circles are the same. In this paper we extend this result to non-degenerate conjugate conics, which include the original case of conjugate circles and adds the new case of conjugate hyperbolae.The broader context of this result is the geometrization of Fritz John's 1938 analysis of the ultra-hyperbolic equation. Solutions of the equation arise as the condition for functions on line space to come from line integrals of functions in Euclidean 3-space, and hence it appears as a compatibility condition for tomographic data.The introduction of the canonical neutral Kaehler metric on the space of oriented lines clarifies the relationship and broadens the paradigm to allow new insights. In particular, it is proven that a solution of the ultra-hyperbolic equation has the mean value property over any pair of curves that arise as the image of John's conjugate circles under a conformal map. These pairs of curves are then shown to be conjugate conics, which include circles and hyperbolae.John identified conjugate circles with the two rulings of a hyperboloid of 1-sheet. Conjugate hyperbolae are identified with the two rulings of either a piece of a hyperboloid of 1-sheet or a hyperbolic paraboloid.     

Holomorphic germs on certain locally convex spaces

Summary We study the bounded sets in the space of holomorphic germs defined on compact subsets of non-metrizable locally convex spaces. We relate this problem to the problem of existence of uniform Cauchy estimates for the bounded subsets. We show that the space of holomorphic germs defined on a compact subset of a reflexive dual Fréchet space is regular if the bounded subsets of the space of holomorphic germs defined at the origin have uniform Cauchy estimates.     

Planar sections of the quadric of Lie cycles and their Euclidean interpretations

All cycles (points, oriented circles, and oriented lines of a Euclidean plane) are represented by points of a three dimensional quadric in four dimensional real projective space. The intersection of this quadric with primes and planes are, respectively, two- and one-dimensional systems of cycles. This paper is a careful examination of the interpretation, in terms of systems of cycles in the Euclidean plane, of fundamental incidence configurations involving this quadric in projective space. These interpretations yield new and striking theorems of Euclidean geometry.     

Cyclic and ruled Lagrangian surfaces in Euclidean four space

We study those Lagrangian surfaces in complex Euclidean space which are foliated by circles or by straight lines. The former, which we call cyclic, come in three types, each one being described by means of, respectively, a planar curve, a Legendrian curve in the 3-sphere or a Legendrian curve in the anti-de Sitter 3-space. We describe ruled Lagrangian surfaces and characterize the cyclic and ruled Lagrangian surfaces which are solutions to the self-similar equation of the Mean Curvature Flow. Finally, we give a partial result in the case of Hamiltonian stationary cyclic surfaces.     

An affine interpretation of Bäcklund transformations

The paper is devoted to an affine interpretation of Bäcklundmaps (Bäcklund transformations are a particular case of Bäcklund maps) for second order differential equations with unknown function of two arguments. Note that up to now there are no papers where Bäcklund transformations are interpreted as transformations of surfaces in a space other than Euclidean space. In this paper, we restrict our considerations to the case of so-called Bäcklund maps of class 1. The solutions of a differential equation are represented as surfaces of an affine space with induced connection determining a representation of zero curvature. We show that, in the case when a second order partial differential equation admits a Bäcklund map of class 1, for each solution of the equation there is a congruence of straight lines in an affine space formed by the tangents to the affine image of the solution. This congruence is an affine analog of a parabolic congruence in Euclidean space. The Bäcklund map can be interpreted as a transformation of surfaces of an affine space under which the affine image of a solution of the differential equation is mapped into a particular boundary surface of the congruence.     

Lipschitz and path isometric embeddings of metric spaces

We prove that each sub-Riemannian manifold can be embedded in some Euclidean space preserving the length of all the curves in the manifold. The result is an extension of Nash C ¹ Embedding Theorem. For more general metric spaces the same result is false, e.g., for Finsler non-Riemannian manifolds. However, we also show that any metric space of finite Hausdorff dimension can be embedded in some Euclidean space via a Lipschitz map.     

Surfaces in Sol3 Space Foliated by Circles

In this paper we study surfaces foliated by a uniparametric family of circles in the homogeneous space Sol₃. We prove that there do not exist such surfaces with zero mean curvature or with zero Gaussian curvature. We extend this study considering surfaces foliated by geodesics, equidistant lines or horocycles in totally geodesic planes and we classify all such surfaces under the assumption of minimality or flatness.     

Isometric embeddings into Heisenberg groups

We study isometric embeddings of a Euclidean space or a Heisenberg group into a higher dimensional Heisenberg group, where both the source and target space are equipped with an arbitrary left-invariant homogeneous distance that is not necessarily sub-Riemannian. We show that if all infinite geodesics in the target are straight lines, then such an embedding must be a homogeneous homomorphism. We discuss a necessary and certain sufficient conditions for the target space to have this ‘geodesic linearity property’, and we provide various examples.     

Lipschitz Gromov-Hausdorff appr oximat ions to two-dimensional closed piecewise flat Alexandrov spaces

We show that a closed piecewise fiat 2-dimensional Alexandrov space Σ can be bi-Lipschitz embedded into a Euclidean space such that the embedded image of Σ has a tubular neighborhood in a generalized sense. As an application, we show that for any metric space sufficiently close to Σ in the Gromov-Hausdorff topology, there is a Lipschitz Gromov-Hausdorff approximation.     

Holomorphic germs and the problem of smooth conjugacy in a punctured neighborhood of the origin

We consider germs of conformal mappings tangent to the identity at the origin in . We construct a germ of a homeomorphism which is a diffeomorphism except at the origin conjugating these holomorphic germs with the time-one map of the vector field . We then show that, in the case , for a germ of a homeomorphism which is real-analytic in a punctured neighborhood of the origin, with real-analytic inverse, conjugating these germs with the time-one map of the vector field exists if and only if a germ of a biholomorphism exists.

     

Immersions with circular normal sections and normal sections of product immersions

We study immersions with normal sections that are circles in the ambient Euclidean space and formulate lemmas concerning normal sections of product immersions. As applications we determine all parallel immersions with planar normal sections and all immersions with planar normal sections and trivial normal connection.Aspirant N.F.W.O.     

On The Effective Computation Of Łojasiewicz Exponents Via Newton Polyhedra

In this paper we give some conclusions on Newton non-degenerate analytic map germs on Kn (K = ? or ?), using information from their Newton polyhedra. As a consequence, we obtain the exact value of the Lojasiewicz exponent at the origin of Newton non-degenerate analytic map germs. In particular, we establish a connection between Newton non-degenerate ideals and their integral closures, thus leading to a simple proof of a result of Saia. Similar results are also considered to polynomial maps which are Newton non-degenerate at infinity.     

REMARKS ON THE ZIG-ZAG THEOREM

The classical Zig-zag Theorem [1] says that if an equilateral closed 2m-gon shuttles between two given circles of the Euclidean 3-space, then the vertices of the polygon can be moved smoothly along the circles without changing the lengths of the sides of the polygon. First we prove that the Zig-zag Theorem holds also in the hyperbolic, Euclidean and spherical n-spaces, and in fact the circles can be replaced by straight lines or any kind of cycles. In the second part of the paper we restrict our attention to planar zig-zag configurations. With the help of an alternative formulation of the Zig-zag Theorem, we establish two duality theorems for periodic zig-zags between two circles. This revised version was published online in August 2006 with corrections to the Cover Date.     

度量方程应用于Krause定理的推广

本文用距离几何的方法证明了主要定理,对曲率为Ｋ的ｎ维常曲率空间,其内任意ｎ＋１个ｎ－１维球Ｓｉ（ｉ＝１,２,…,ｎ＋１）,它们中的任一个都与其它球不变,则与Ｓｉ交角为βｉ（ｉ＝１,２,…,ｎ＋１）的ｎ－１维一般有２＾ｎ＋１个,当ｎ为偶数时,它们的测地线曲率之交错和为零;当ｎ为奇数时,此结论不成立,该定理包括非欧情形,而当ｎ＝２,βｉ＝１（ｉ＝１,２,…,ｎ＋１）时,就是ｉｉｌｋｅｒＪＢ在「１」中所     

On the perron sign change effect for Lyapunov characteristic exponents of solutions of differential systems

The Perron effect is the effect in which the characteristic Lyapunov exponents of solutions of a differential system change sign from negative to positive when passing to a perturbed system. We show that this effect is realized on all nontrivial solutions of two two-dimensional systems: an original linear system with negative characteristic exponents and a perturbed system with small perturbations of arbitrary order m > 1 in a neighborhood of the origin, all of whose nontrivial solutions have positive characteristic exponents. We compute the exact positive value of the characteristic exponents of solutions of the two-dimensional nonlinear Perron system with small second-order perturbations, which realizes only a partial Perron effect.     

Consistance d'un estimateur de minimum de variance étendue

We consider a generalization of the criterion minimized by the K-means algorithm, where a neighborhood structure is used in the calculus of the variance. Such a tool is used, for example with Kohonen maps, to measure the quality of the quantification preserving the neighborhood relationships. If we assume that the parameter vector is in a compact Euclidean space and all its components are separated by a minimal distance, we show the strong consistency of the set of parameters almost realizing the minimum of the empirical extended variance. To cite this article: J. Rynkiewicz, C. R. Acad. Sci. Paris, Ser. I 341 (2005).