arXiv research

New approach for principal curves on spherical data.

A new metric-based principal curve method learns 1D manifolds from spatial data.

The paper explores Bertrand and framed curves in 3D space.

When confronted with massive data streams, summarizing data with dimension reduction methods such as PCA raises theoretical and algorithmic pitfalls. Principal curves act as a nonlinear generalization of PCA and the present paper proposes a novel algorithm to automatically and sequentially learn principal curves from d…

In this paper is studied the behavior of principal curvature lines near a curve of umbilic points of a smooth surface.

We study surfaces with one constant principal curvature in Riemannian and Lorentzian three-dimensional space forms. Away from umbilic points they are characterized as one-parameter foliations by curves of constant curvature, each of these curves being centered at a point of a regular curve and contained in its normal p…

In this article, we investigate Bertrand curves corresponding to the spherical images of the tangent, binormal, principal normal and Darboux indicatrices of a space curve in Euclidean 3-space. As a result, in case of a space curve is a general helix, we show that the curves corresponding to the spherical images of its …

The study connects conic connections and torsion-free principal connections on G-structures.

A principal Higgs bundle $(P,φ)$ over a singular curve $X$ is a pair consisting of a principal bundle $P$ and a morphism $φ:X\to\text{Ad}P \otimes Ω^1_X$. We construct the moduli space of principal Higgs G-bundles over an irreducible singular curve $X$ using the theory of decorated vector bundles. More precisely, given…

The paper defines and classifies special curves in Riemannian manifolds.

Study bifurcations of curves on surfaces in Minkowski 3-space.

This paper introduces a novel monotone curve estimation framework based on convex duality.

Research on manifold learning within a density ridge estimation framework has shown great potential in recent work for both estimation and de-noising of manifolds, building on the intuitive and well-defined notion of principal curves and surfaces. However, the problem of unwrapping or unfolding manifolds has received r…

The paper studies geometric Airy curve flows on R^n and their properties.

Given a non circular spacial closed curve whose total torsion is an integer multiple of $2π$, we construct a germ of a smooth surface that contains it as a hyperbolic principal cycle.

Capacity control, the bias/variance dilemma, and learning unknown functions from data, are all concerned with identifying effective and consistent fits of unknown geometric loci to random data points. A geometric locus is a curve or surface formed by points, all of which possess some uniform property. A geometric locus…

In this paper are determined the principal curvatures and principal curvature lines on canal surfaces which are the envelopes of families of spheres with variable radius and centers moving along a closed regular curve in R^3. By means of a connection of the differential equations for these curvature lines and real Ricc…

Mannheim curves are defined for immersed curves in 3-dimensional sphere S^3 . The definition is given by considering the geodesics of S^3. First, two special geodesics, called principal normal geodesic and binormal geodesic, of S^3 are defined by using Frenet vectors of a curve immersed in S^3. Later, the curve alpha i…

We study the problem of finding the one-dimensional structure in a given data set. In other words we consider ways to approximate a given measure (data) by curves. We consider an objective functional whose minimizers are a regularization of principal curves and introduce a new functional which allows for multiple curve…

In this paper, we introduce an inclined curves according to parallel transport frame. Also, we define a vector field called Darboux vector field of an inclined curve in and we give a new characterization such as: "α: I \subset R \rightarrow E^4 is an inclined curve \Leftrightarrow k_1 \int k_1ds + k_2 \int \k_2 +k_3ds …

Study Lie algebroid connections on principal bundles over complex projective varieties.

Study of holomorphic curves and surfaces using singularity theory.

In this paper, we define a rectifying spacelike curve in the Minkowski space-time $E_1^4$ as a curve whose position vector always lies in orthogonal complement $N^{\bot}$ of its principal normal vector field $N$. In particular, we study the rectifying spacelike curves in $E_1^4$ and characterize such curves in terms of…

Principal Component Analysis (PCA) is the most common nonparametric method for estimating the volatility structure of Gaussian interest rate models. One major difficulty in the estimation of these models is the fact that forward rate curves are not directly observable from the market so that non-trivial observational e…

The seminal work of Eskin-Masur-Zorich described the principal boundary of moduli spaces of abelian differentials that parameterizes flat surfaces with a prescribed generic configuration of short parallel saddle connections. In this paper we describe the principal boundary for each configuration in terms of twisted dif…

Study the geometry of bifurcation sets for specific types of functions.

This paper develops GPCA for probability distributions using Otto-Wasserstein geometry.

Let $X$ be a compact connected Riemann surface of genus at least two, and let ${G}$ be a connected semisimple affine algebraic group defined over $\mathbb C$. For any $δ\in π_1({G})$, we prove that the moduli space of semistable principal ${G}$--bundles over $X$ of topological type $δ$ is simply connected. In contrast,…

This paper establishes the geometric structure of the lines of principal curvature of a hypersurface immersed in ${\mathbb R}^4$ in a neighborhood of the set $\mathcal{S}$ of its principal curvature singularities, consisting of the points at which atF least two principal curvatures are equal. Under generic conditions d…

The paper proves a stronger Petersen--Wilhelm conjecture for principal bundles.

We construct a connection and a curving on a bundle gerbe associated with lifting a structure group of a principal bundle to a central extension. The construction is based on certain structures on the bundle, i.e. connections and splittings. The Deligne cohomology class of the lifting bundle gerbe with the connection a…

We consider $n$-dimensional discrete motions such that any two neighbouring positions correspond in a pure rotation ("rotating motions"). In the Study quadric model of Euclidean displacements these motions correspond to quadrilateral nets with edges contained in the Study quadric ("rotation nets"). The main focus of ou…

Considering the tangent plane at a point to a surface in the four-dimensional Euclidean space, we find an invariant of a pair of two tangents in this plane. If this invariant is zero, the two tangents are said to be conjugate. When the two tangents coincide with a given tangent, then we obtain the normal curvature of t…

In this work, we study plane and spherical curves in Euclidean and Lorentz-Minkowski 3-spaces by employing rotation minimizing (RM) frames. By conveniently writing the curvature and torsion for a curve on a sphere, we show how to find the angle between the principal normal and an RM vector field for spherical curves. L…

In this study, we investigate the locus of the centers of the Meusnier spheres. Just as focal curve is the locus of the centers of the osculating spheres, we investigate the geometrical interpretation on the locus of the centers of the Meusnier spheres. We proved that if the curve is a principal line, the locus of the …

We consider the classification problem and focus on nonlinear methods for classification on manifolds. For multivariate datasets lying on an embedded nonlinear Riemannian manifold within the higher-dimensional ambient space, we aim to acquire a classification boundary for the classes with labels, using the intrinsic me…

We propose to describe the variety of galaxies from SDSS by using only one affine parameter. To this aim, we build the Principal Curve (P-curve) passing through the spine of the data point cloud, considering the eigenspace derived from Principal Component Analysis of morphological, physical and photometric galaxy prope…

A classical result attributed to Joachimsthal in 1846 states that if two surfaces intersect with constant angle along a line of curvature of one surface, then the curve of intersection is also a line of curvature of the other surface. In this note we prove a global analogue of this result, as follows. Suppose that two …

There is a natural duality between line congruences in $\mathbb{R}^3$ and surfaces in $\mathbb{R}^4$ that sends principal lines into asymptotic lines. The same correspondence takes the discriminant curve of a line congruence into the parabolic curve of the dual surface. Moreover, it takes the ridge curves to the flat r…

We first describe the numerical invariants attached to the second fundamental form of a spacelike surface in four-dimensional Minkowski space. We then study the configuration of the nu-principal curvature lines on a spacelike surface, when the normal field nu is lightlike (the lightcone configuration). Some observation…

New approach to principal groupoid bundles with connections using dg-Lie groupoids.

Unified method visualizes curvature on curves and surfaces.

A method to calculate generalized curvatures of curves in n-dimensional space.

Let X be a compact connected Riemann surface equipped with an anti-holomorphic involution σ. Let G be a connected complex reductive affine algebraic group, and let σ_G be a real form of G. We consider holomorphic principal G-bundles on X satisfying compatibility conditions with respect to σand σ_G. We prove that the po…

ccc-Autoevolutes are closed curves congruent to their evolutes, constructed via symmetry.

There are many methods developed to approximate a cloud of vectors embedded in high-dimensional space by simpler objects: starting from principal points and linear manifolds to self-organizing maps, neural gas, elastic maps, various types of principal curves and principal trees, and so on. For each type of approximator…

We compactify the classical moduli variety $A_g$ of principally polarized abelian varieties of complex dimension $g$ by attaching the moduli of flat tori of real dimensions at most $g$ in an explicit manner. Equivalently, we explicitly determine the Gromov-Hausdorff limits of principally polarized abelian varieties. Th…

Let G be a simple linear algebraic group defined over the complex numbers. Fix a proper parabolic subgroup P of G and a nontrivial antidominant character χof P. We prove that a holomorphic principal G-bundle E over a connected complex projective manifold M is semistable and the second Chern class of its adjoint bundle …