On the existence of four or more curved foldings with common creases and crease patterns

Consider an oriented curve $$\Gamma $$ Γ in a domain D in the plane $${\varvec{R}}^2$$ R 2 . Thinking of D as a piece of paper, one can make a curved folding in the Euclidean space $${\varvec{R}}^3$$ R 3 . This can be expressed as the image of an “origami map” $$\Phi :D\rightarrow {\varvec{R}}^3$$ Φ : D → R 3 such that $$\Gamma $$ Γ is the singular set of $$\Phi $$ Φ , the word “origami” coming from the Japanese term for paper folding. We call the singular set image $$C:=\Phi (\Gamma )$$ C : = Φ ( Γ ) the crease of $$\Phi $$ Φ and the singular set $$\Gamma $$ Γ the crease pattern of $$\Phi $$ Φ . We are interested in the number of origami maps whose creases and crease patterns are C and $$\Gamma $$ Γ , respectively. Two such possibilities have been known. In the authors’ previous work, two other new possibilities and an explicit example with four such non-congruent distinct curved foldings were established. In this paper, we determine the possible values for the number N of congruence classes of curved foldings with the same crease and crease pattern. As a consequence, if C is a non-closed simple arc, then $$N=4$$ N = 4 if and only if both $$\Gamma $$ Γ and C do not admit any symmetries. On the other hand, when C is a closed curve, there are infinitely many distinct possibilities for curved foldings with the same crease and crease pattern, in general.

Shortest closed curve to inspect a sphere

We show that in Euclidean 3-space any closed curve γ which lies outside the unit sphere and contains the sphere within its convex hull has length ≥ 4 ⁢ π {\geq 4\pi} . Equality holds only when γ is composed of four semicircles of length π, arranged in the shape of a baseball seam, as conjectured by V. A. Zalgaller in 1996.

Construction of Initial Data Sets for Lorentzian Manifolds with Lightlike Parallel Spinors

Lorentzian manifolds with parallel spinors are important objects of study in several branches of geometry, analysis and mathematical physics. Their Cauchy problem has recently been discussed by Baum, Leistner and Lischewski, who proved that the problem locally has a unique solution up to diffeomorphisms, provided that the intial data given on a space-like hypersurface satisfy some constraint equations. In this article we provide a method to solve these constraint equations. In particular, any curve (resp. closed curve) in the moduli space of Riemannian metrics on M with a parallel spinor gives rise to a solution of the constraint equations on $$M\times (a,b)$$ M × ( a , b ) (resp. $$M\times S^1$$ M × S 1 ).

A topological extension of movement primitives for curvature modulation and sampling of robot motion

This paper proposes to enrich robot motion data with trajectory curvature information. To do so, we use an approximate implementation of a topological feature named writhe, which measures the curling of a closed curve around itself, and its analog feature for two closed curves, namely the linking number. Despite these features have been established for closed curves, their definition allows for a discrete calculation that is well-defined for non-closed curves and can thus provide information about how much a robot trajectory is curling around a line in space. Such lines can be predefined by a user, observed by vision or, in our case, inferred as virtual lines in space around which the robot motion is curling. We use these topological features to augment the data of a trajectory encapsulated as a Movement Primitive (MP). We propose a method to determine how many virtual segments best characterize a trajectory and then find such segments. This results in a generative model that permits modulating curvature to generate new samples, while still staying within the dataset distribution and being able to adapt to contextual variables.

Compact connected components in relative character varieties of punctured spheres

We prove that some relative character varieties of the fundamental group of a punctured sphere into the Hermitian Lie groups $\mathrm{SU}(p,q)$ admit compact connected components. The representations in these components have several counter-intuitive properties. For instance, the image of any simple closed curve is an elliptic element. These results extend a recent work of Deroin and the first author, which treated the case of $\textrm{PU}(1,1) = \mathrm{PSL}(2,\mathbb{R})$. Our proof relies on the non-Abelian Hodge correspondance between relative character varieties and parabolic Higgs bundles. The examples we construct admit a rather explicit description as projective varieties obtained via Geometric Invariant Theory.

Current Algebra, a U(1) Gauge Theory and the Wess-Zumino-Witten Model

In this note we realise current algebra with anomalous terms in terms of a U(1) gauge theory, in the space of maps M, from S1 into a compact Lie group corresponding to the current algebra. The Wilson loop around a closed curve in M is shown to be the Wess-Zumino-Witten term. This discussion enables a simple understanding of the non-Abelian anomaly in the Schrödinger picture.

Influence of an L^{p}-perturbation on Hardy-Sobolev inequality with singularity a curve

We consider a bounded domain \(\Omega\) of \(\mathbb{R}^N\), \(N \geq 3\), \(h\) and \(b\) continuous functions on \(\Omega\). Let \(\Gamma\) be a closed curve contained in \(\Omega\). We study existence of positive solutions \(u \in H^1_0(\Omega)\) to the perturbed Hardy-Sobolev equation: \[-\Delta u+hu+bu^{1+\delta}=\rho^{-\sigma}_{\Gamma} u^{2^*_{\sigma}-1} \quad \textrm{ in } \Omega,\] where \(2^*_{\sigma}:=\frac{2(N-\sigma)}{N-2}\) is the critical Hardy-Sobolev exponent, \(\sigma\in [0,2)\), \(0\lt\delta\lt\frac{4}{N-2}\) and \(\rho_{\Gamma}\) is the distance function to \(\Gamma\). We show that the existence of minimizers does not depend on the local geometry of \(\Gamma\) nor on the potential \(h\). For \(N=3\), the existence of ground-state solution may depends on the trace of the regular part of the Green function of \(-\Delta+h\) and or on \(b\). This is due to the perturbative term of order \(1+\delta\).