Related papers: On the flat geometry of the cuspidal edge
Here we investigate some geometric properties of the contactomorphism group $\mathcal{D}_\theta(M)$ of a compact contact manifold with the $L^2$ metric on the stream functions. Viewing this group as a generalization to the…
One of the main questions in the theory of normal surface singularities is to understand the relations between their geometry and topology. The lattice cohomology is an important tool in the study of topological properties of a plumbed…
We study scaling function geometry. We show the existence of the scaling function of a geometrically finite one-dimensional mapping. This scaling function is discontinuous. We prove that the scaling function and the asymmetries at the…
This paper examines the relationship between the knotting of an embedded surface in $\R^3$ and the knotting of its fold curves, formed by the singular set of projection to a plane. The first result shows that every surface, no matter how…
By recasting metrical geometry in a purely algebraic setting, both Euclidean and non-Euclidean geometries can be studied over a general field with an arbitrary quadratic form. Both an affine and a projective version of this new theory are…
We study focal surfaces of (wave) fronts associated to unbounded principal curvatures near non-degenerate singular points of initial fronts. We give characterizations of singularities of those focal surfaces in terms of types of…
Given a surface $S$ in a 3D contact sub-Riemannian manifold $M$, we investigate the metric structure induced on $S$ by $M$, in the sense of length spaces. First, we define a coefficient $\widehat K$ at characteristic points that determines…
One considers geometry with the intransitive equaivalence relation. Such a geometry is a physical geometry, i.e. it is described completely by the world function, which is a half of the squared distance function. The physical geometry…
Given a null-cobordant oriented framed link $L$ in a closed oriented $3$--manifold $M$, we determine those links in $M \setminus L$ which can be realized as the singular point set of a generic map $M \to \mathbb{R}^2$ that has $L$ as an…
In this paper, we classify the algebraic isomonodromic deformations that can be obtained through restriction to generic lines of logarithmic flat connections on the complex projective plane $\mathbb{P}^2_\mathbb{C}$ whose singular locus is…
This article establishes several remarkably simple identities relating certain metric invariants of level curves of real and complex functions. In particular, we relate lengths of level curves to their curvature and to the gradient field of…
In the paper, some geometric properties of the plane interception curve defined by a nonlinear ordinary differential equation are discussed. Its parametric representation is used to find the limits of some triangle elements associated with…
We assume that the sample height measured using AFM is reduced due to contact deformation. The Herz contact theory is applied for the quantitative description. General numerical solution is found. Analytical approximations for specific…
The geometry of closed surfaces equipped with a Euclidean metric with finitely many conical points of arbitrary angle is studied. The main result is that the set of closed geodesics is dense in the space of geodesics.
We consider flat surfaces and the points of their metric completions, particularly the singularities to which the flat structure of the surface does not extend. The local behavior near a singular point x can be partially described by a…
We use normal sections to relate the curvature locus of regular (resp. singular corank 1) 3-manifolds in $\mathbb{R}^6$ (resp. $\mathbb R^5$) with regular (resp. singular corank 1) surfaces in $\mathbb R^5$ (resp. $\mathbb R^4$). For…
Estimating correspondences between deformed shape instances is a long-standing problem in computer graphics; numerous applications, from texture transfer to statistical modelling, rely on recovering an accurate correspondence map. Many…
We begin the study of completeness of affine connections, especially those on statistical manifolds as well as on affine hypersurfaces. We collect basic facts, prove new theorems and provide examples with remarkable properties.
We survey determinantal singularities, their deformations, and their topology. This class of singularities generalizes the well studied case of complete intersections in several different aspects, but exhibits a plethora of new phenomena…
Consider the Euclidean space $\mathbb{R}^3$ endowed with a canonical semi-symmetric non-metric connection determined by a vector field $\mathsf{C}\in\mathfrak{X}(\mathbb{R}^3)$. We study surfaces when the sectional curvature with respect to…