arXiv research

Clarifies Einstein-Cartan gravitation with Dirac spinor on generalized frame bundle.

In this paper, the parallel transport frames over non-lightlike curves in Minkowski 3-space are introduced. Evolution equations of these frames with respect to arc length and time are calculated over the space of these curves. Then the equivalence of the non-linear Schrödinger equation and non-linear heat system to the…

In this paper, we study spinor Bishop equations of curves in E^3. We research the spinor formulations of curves according to Bishop frames in E^3. Also, the relation between spinor formulations of Bishop frames and Frenet frame are expressed.

Local generalization of frame bundles using a weakened Maurer-Cartan equation.

Presented spherical symmetric teleparallel geometry frames and field equations.

We study variational systems for space curves, for which the Lagrangian or action principle has a Euclidean symmetry, using the Rotation Minimising frame, also known as the Normal, Parallel or Bishop frame. Such systems have previously been studied using the Frenet-Serret frame. The Rotation Minimising frame has many a…

It is shown that the integrability conditions of the equations satisfied by the local Frenet frame associated with a holomorphic curve in a complex Grassmann manifold coincide with a special class of nonabelian Toda equations. A local moving frame of a holomorphic immersion of a Riemann surface into a complex Grassmann…

This paper is devoted to apply the equivariant moving frame method to study the local equivalence problem of third order ordinarily differential equation under the pseudo-group of fiber preserving transformations.

The paper studies curve evolution using the PLR equation and its solutions.

We define spin frames, with the aim of extending spin structures from the category of (pseudo-)Riemannian manifolds to the category of spin manifolds with a fixed signature on them, though with no selected metric structure. Because of this softer requirements, transformations allowed by spin frames are more general tha…

New flow generates surfaces with constant curvature.

The paper studies geometric properties of soliton surfaces using an extended Darboux frame field.

We prove that the Einstein equations can be solved in a very general form for arbitrary spacetime dimensions and various types of vacuum and non-vacuum cases following a geometric method of anholonomic frame deformations for constructing exact solutions in gravity. The main idea of this method is to introduce on (pseud…

Introduces formal frames for manifolds and their properties.

Universal bi-Hamiltonian hierarchies of group-invariant (multicomponent) soliton equations are derived from non-stretching geometric curve flows $\map(t,x)$ in Riemannian symmetric spaces $M=G/H$, including compact semisimple Lie groups $M=K$ for $G=K\times K$, $H={\rm diag} G$. The derivation of these soliton hierarch…

New quantum invariant for framed 3-manifolds using ideal triangulations.

Study curvature of piecewise metrics using moving frames.

In this paper, we introduce two new classes of representations of the framed braid groups. One is the homological representation constructed as the action of a mapping class group on a certain homology group. The other is the monodromy representation of the confluent KZ equation, which is a generalization of the KZ equ…

In this paper we study the general affine geometry of curves in affine space $A^2$. For a regular plane curves we define two kinds of moving frames. The first is of minimal order in all moving frames.The second is the Frenet moving frame. We get the moving equations of these moving frames. And we prove that curvature a…

An explicit model of fiber bundle with local fibers being disinct copies of vector 3-space is introduced. They are endowed with frames which are used as local isotopic ones. The field local of isotopic frames is considered as gauge field itself while the form of gauge connections is derived from it. A covariant equatio…

Generalizes Hasimoto transformation to arbitrary flows on space curves.

Paper derives invariantised Euler-Lagrange equations for Herglotz problems.

We analyze two reduction methods for nonholonomic systems that are invariant under the action of a Lie group on the configuration space. Our approach for obtaining the reduced equations is entirely based on the observation that the dynamics can be represented by a second-order differential equations vector field and th…

The paper examines soliton surfaces using a parallel transport frame field in 4D space.

Study contact 3-manifolds with special frames, deriving curvature bounds.

Nous montrons que les équations du repère mobile des surfaces de Bonnet conduisent à une paire de Lax matricielle isomonodromique d'ordre deux pour la sixième équation de Painlevé. We show that the moving frame equations of Bonnet surfaces can be extrapolated to a second order, isomonodromic matrix Lax pair of the sixt…

We construct a canonical frame for an arbitrary Gl(2)-structure thus solving the equivalence problem for Gl(2)-structures. Our treatment includes also a problem of contact equivalence of ordinary differential equations and applies to certain classes of vector distributions. Additionally we characterise Gl(2)-structures…

This preliminary report studies immersed surfaces of constant mean curvature in $H^3$ through their {\it adjusted Gauss maps} (as harmonic maps in $S^2$) and their {\it adjusted frames} in SU(2). Lawson's correspondence between Euclidean CMC surfaces and their hyperbolic cousins is interpreted here under a different pe…

In this paper we show that if one writes down the structure equations for the evolution of a curve embedded in an (n)-dimensional Riemannian manifold with constant curvature this leads to a symplectic, a Hamiltonian and an hereditary operator. This gives us a natural connection between finite dimensional geometry, infi…

Discover equations of motion from distorted video frames.

We propose a twistor construction of surfaces in Lie sphere geometry based on the linear system which copies equations of Wilczynski's projective frame. In the particular case of Lie-applicable surfaces this linear system describes joint eigenfunctions of a pair of commuting Schrödinger operators with magnetic fields.

We consider a problem of equivalence of generic pairs $(X,V)$ on a manifold $M$, where $V$ is a distribution of rank $m$ and $X$ is a distribution of rank one. We construct a canonical bundle with a canonical frame. We prove that two pairs are equivalent if and only if the corresponding frames are diffeomorphic. As a p…

We present in this paper the formalism for the splitting of a four-dimensional Lorentzian manifold by a set of time-like integral curves. Introducing the geometrical tensors characterizing the local spatial frames induced by the congruence (namely, the spatial metric tensor, the extrinsic curvature tensor and the Riema…

We show how pairs of isothermic surfaces are given by curved flats in a pseudo Riemannian symmetric space and vice versa. Calapso's fourth order partial differential equation is derived and, using a solution of this equation, a Möbius invariant frame for an isothermic surface is built.

This paper develops efficient algorithms for multibody dynamics using screw and Lie group theory.

Proves well-posedness for hard phase model in general relativity.

Study tangle equations linking enzyme actions to knot theory.

The study shows how to regularize weakly harmonic maps using Sobolev norms and Coulomb frames.

We define a conformal reference frame, i.e., a special projection of the six-dimensional sky bundle of a Lorentzian manifold (or the five-dimensional twistor space) to a three-dimensional manifold. We construct an example, a conformal compactification, for Minkowski space. Based on the complex structure on the skies, w…

Study inextensible flows of curves in 4D pseudo-Galilean space and defines energy functions.

The article classifies liftings of connections on differential manifolds for geodesic modeling.

The paper defines Vn-slant helices in a lightlike cone and their curvature functions.

We address the geometric Cauchy problem for surfaces associated to the membrane shape equation describing equilibrium configurations of vesicles formed by lipid bilayers. This is the Euler-Lagrange equation of the Canham-Helfrich-Evans elastic curvature energy subject to constraints on the enclosed volume and the surfa…

Motivated by the geometric theory of differential equations and the variational approach to the equivalence problem for geometric structures on manifolds, we consider the problem of equivalence for distributions with fixed submanifolds of flags on each fiber. We call them flag structures. The construction of the canoni…

Study of curves in Lie sphere geometry using moving frames and variational principles.

New framework for gravitational perturbations of Kerr spacetimes, focusing on stability.

Simplified construction of contact triad connection for analysis.

In this work, we are interested in the differential geometry of curves in the simply isotropic and pseudo-isotropic 3-spaces, which are examples of Cayley-Klein geometries whose absolute figure is given by a plane at infinity and a degenerate quadric. Motivated by the success of rotation minimizing (RM) frames in Eucli…