Searcharxiv⌕ Search

arXiv subjects

Irina Markina

Publications and source records attributed to Irina Markina.

53 records · Page 3Linked to original sources
arXiv2010

Sub-Riemannian geodesics and heat operator on odd dimensional spheres

In this article we study the sub-Riemannian geometry of the spheres $S^{2n+1}$ and $S^{4n+3}$, arising from the principal $S^1-$bundle structure defined by the Hopf map and the principal $S^3-$bundle structure given by the quaternionic Hopf map respectively. The $S^1$ action leads to the classical contact geometry of $S^{2n+1}$, while the $S^3$ action gives another type of sub-Riemannian structure, with a distribution of corank 3. In both cases the metric is given as the restriction of the usual Riemannian metric on the respective horizontal distributions. For the contact $S^7$ case, we give an explicit form of the intrinsic sub-Laplacian and obtain a commutation relation between the sub-Riemannian heat operator and the heat operator in the vertical direction.

math.DG↗
arXiv2010

An intrinsic formulation of the rolling manifolds problem

We present an intrinsic formulation of the kinematic problem of two $n-$dimensional manifolds rolling one on another without twisting or slipping. We determine the configuration space of the system, which is an $\frac{n(n+3)}2-$dimensional manifold. The conditions of no-twisting and no-slipping are decoded by means of a distribution of rank $n$. We compare the intrinsic point of view versus the extrinsic one. We also show that the kinematic system of rolling the $n$-dimensional sphere over $\mathbb R^n$ is controllable. In contrast with this, we show that in the case of $SE(3)$ rolling over $\mathfrak{se}(3)$ the system is not controllable, since the configuration space of dimension 27 is foliated by submanifolds of dimension 12.

math.DG↗
arXiv2010

Geodesics on H-type quaternion groups with sub-Lorentzian metric and their physical interpretation

We study the existence and cardinality of normal geodesics of different causal types on H(eisenberg)-type quaternion group equipped with the sub-Lorentzian metric. We present explicit formulas for geodesics and describe reachable sets by geodesics of different causal character. We compare results with the sub-Riemannian quaternion group and with the sub-Lorentzian Heisenberg group, showing that there are similarities and distinctions. We show that the geodesics on H-type quaternion groups with the sub-Lorentzian metric satisfy the equations describing the motion of a relativistic particle in a constant homogeneous electromagnetic field.

math-ph↗
arXiv2009

Virasoro Algebra and Löwner-Kufarev Equations

Contour dynamics is a classical subject both in physics and in complex analysis. We show that the dynamics provided by the Löwner-Kufarev ODE and PDE possesses a rigid algebraic structure given by the Virasoro algebra. Namely, the `positive' Virasoro generators span the holomorphic part of the complexified vector bundle over the space of univalent functions, smooth on the boundary. In the covariant formulation they are conserved by the Löwner-Kufarev evolution. The `negative' Virasoro generators span the antiholomorphic part. They contain a conserved term and we give an iterative method to obtain them based on the Poisson structure of the Löwner-Kufarev evolution. The Löwner-Kufarev PDE provides a distribution of the tangent bundle of non-normalized univalent functions, which forms the tangent bundle of normalized ones. It also gives an explicit correspondence between the latter bundle and the holomorphic eigen space of the complexified Lie algebra of vector fields on the unit circle. Finally, we give Hamiltonian and Lagrangian formulations of the motion within the coefficient body in the field of an elliptic operator constructed by means of Virasoro generators. We also discuss relations between CFT and SLE.

math-ph↗
arXiv2009

Conformal field theory and Loewner-Kufarev evolution

One of the important aspects in recent trends in complex analysis has been the increasing degree of cross-fertilization between the latter and mathematical physics with great benefits to both subjects. Contour dynamics in the complex plane turned to be a meeting point for complex analysts, specialists in stochastic processes, and mathematical physicists. This was stimulated, first of all, by recent progress in understanding structures in the classical and stochastic Löwner evolutions, and in the Laplacian growth. The Virasoro algebra provides a basic algebraic object in conformal field theory (CFT) so it was not surprising that it turned to play an important role of a structural skeleton for contour dynamics. The present paper is a survey of recent progress in the study of the CFT viewpoint on contour dynamics, in particular, we show how the Witt and Virasoro algebras are related with the stochastic Löwner and classical Löwner-Kufarev equations.

math-ph↗
arXiv2009

Sub-Riemannian geometry of parallelizable spheres

The first aim of the present paper is to compare various sub-Riemannian structures over the three dimensional sphere $S^3$ originating from different constructions. Namely, we describe the sub-Riemannian geometry of $S^3$ arising through its right Lie group action over itself, the one inherited from the natural complex structure of the open unit ball in $\comp^2$ and the geometry that appears when considering the Hopf map as a principal bundle. The main result of this comparison is that in fact those three structures coincide. In the second place, we present two bracket generating distributions for the seven dimensional sphere $S^7$ of step 2 with ranks 6 and 4. These yield to sub-Riemannian structures for $S^7$ that are not present in the literature until now. One of the distributions can be obtained by considering the CR geometry of $S^7$ inherited from the natural complex structure of the open unit ball in $\comp^4$. The other one originates from the quaternionic analogous of the Hopf map.

math.DG↗
arXiv2009

Semi - Riemannian Geometry with Nonholonomic Constraints

In the present article the geometry of semi-Riemannian manifolds with nonholonomic constraints is studied. These manifolds can be considered as analogues to the sub-Riemannian manifolds, where the positively definite metric is substituted by a nondegenerate metric. To study properties of the exponential map the Christoffel symbols and other differential operators were introduced. We study solutions of the Hamiltonian system and their projections into the underlying manifold. The explicit formulae were found for a specific example of a semi-Riemannian manifold with nonholonomic constraints.

math.DG↗
arXiv2008

SubRiemannian geometry on the sphere $\mathbb{S}^3$

The present paper starts with an introduction to quaternions and then defines the 3-dimmensional sphere as the set of quaternions of length one. The quaternion group induces on $\mathbb{S}^3$ a structure of noncommutative Lie group. This group is compact and the results obtained in this case are very different than those obtained in the case of the Heisenberg group, which is a noncompact Lie group. Like in the Heisenberg group case, we introduce a nonintegrable distribution on the sphere and a metric on it using two of the noncommutative left invariant vector fields. This way $\mathbb{S}^3$ becomes a subRiemannian manifold. It is known that the group $SU(2) $ is isomorphic with the sphere $\mathbb{S}^3$ and represents an example of subRiemannian manifold where the elements are matrices. The main issue here is to study the connectivity by horizontal curves and its geodesics on this manifold. In this paper, we are using Lagrangian method to study the connectivity theorem on ${\mathbb S}^3$ by horizontal curves with minimal arc-length. We show that for any two points in ${\mathbb S}^3$, there exists such a geodesic joining these two points.

math.DG↗
arXiv2008

Nonholonomic Lorentzian geometry on some $\mathbb H$-type groups

We consider examples of the $\mathbb H$-type groups with the natural horizontal distribution generated by the commutation relations of the group. In the contrast with the previous studies we furnish the horizontal distribution with the Lorentzian metric, which is nondegenerate metric of index 1 instead of a positive definite quadratic form. The causal character is defined. We study the reachable set by timelike future directed curves. The parametric equations of geodesics are obtained.

math.DG↗
arXiv2008

Modified action and differential operators on the 3-D sub-Riemannian sphere

Our main aim is to present a geometrically meaningful formula for the fundamental solutions to a second order sub-elliptic differential equation and to the heat equation associated with a sub-elliptic operator in the sub-Riemannian geometry on the unit sphere $\mathbb S^3$. Our method is based on the Hamiltonian approach, where the corresponding Hamitonian system is solved with mixed boundary conditions. A closed form of the modified action is given. It is a sub-Riemannian invariant and plays the role of a distance on $\mathbb S^3$.

math.DG↗
arXiv2008

Sub-Riemannian geodesics on the 3-D sphere

The unit sphere $\mathbb S^3$ can be identified with the unitary group SU(2). Under this identification the unit sphere can be considered as a non-commutative Lie group. The commutation relations for the vector fields of the corresponding Lie algebra define a 2-step sub-Riemannian manifold. We study sub-Riemannian geodesics on this sub-Riemannian manifold making use of the Hamiltonian formalism and solving the corresponding Hamiltonian system.

math.DG↗
arXiv2008

Sub-Lorentzian Geometry on Anti-De Sitter Space

Sub-Riemannian Geometry is proved to play an important role in many applications, e.g., Mathematical Physics and Control Theory. The simplest example of sub-Riemannian structure is provided by the 3-D Heisenberg group. Sub-Riemannian Geometry enjoys major differences from the Riemannian being a generalisation of the latter at the same time, e.g., geodesics are not unique and may be singular, the Hausdorff dimension is larger than the manifold topological dimension. There exists a large amount of literature developing sub-Riemannian Geometry. However, very few is known about its natural extension to pseudo-Riemannian analogues. It is natural to begin such a study with some low-dimensional manifolds. Based on ideas from sub-Riemannian geometry we develop sub-Lorentzian geometry over the classical 3-D anti-de Sitter space. Two different distributions of the tangent bundle of anti-de Sitter space yield two different geometries: sub-Lorentzian and sub-Riemannian. It is shown that the set of timelike and spacelike `horizontal' curves is non-empty and we study the problem of horizontal connectivity in anti-de Sitter space. We also use Lagrangian and Hamiltonian formalisms for both sub-Lorentzian and sub-Riemannian geometries to find geodesics.

math.DG↗
arXiv2006

Anisotropic quaternion Carnot groups: geometric analysis and Green's function

We construct examples of 2-step Carnot groups related to quaternions and study their fine structure and geometric properties. This involves the Hamiltonian formalism, which is used to obtain explicit equations for geodesics and the computation of the number of geodesics joining two different points on these groups. We able to find the explicit lengths of geodesics. We present the fundamental solutions of the Heat and sub-Laplace equations for these anisotropic groups and obtain some estimates for them, which may be useful.

math.DG↗
arXiv2006

Quaternion $H$-type group and differential operator $Δ_λ$

We study the relations between the quaternion $H$-type group and the boundary of the unit ball on two dimensional quaternionic space. The orthogonal projection of the space of square integrable functions defined on quaternion $H$-type group into its subspace of boundary values of $q$-holomorphic functions is consider. The precise form of Cauchy-Szego kernel and the orthogonal projection operator is obtained. The fundamental solution for the operator $Δ_λ$ is found.

math.AP↗
arXiv2006

Sub-Riemannian geometry of the coefficients of univalent functions

We consider coefficient bodies $\mathcal M_n$ for univalent functions. Based on the Löwner-Kufarev parametric representation we get a partially integrable Hamiltonian system in which the first integrals are Kirillov's operators for a representation of the Virasoro algebra. Then $\mathcal M_n$ are defined as sub-Riemannian manifolds. Given a Lie-Poisson bracket they form a grading of subspaces with the first subspace as a bracket-generating distribution of complex dimension two. With this sub-Riemannian structure we construct a new Hamiltonian system and calculate regular geodesics which turn to be horizontal. Lagrangian formulation is also given in the particular case $\mathcal M_3$.

math.CV↗
arXiv2005

Generalizations of Kadanoff's solution of the Saffman-Taylor problem in a wedge

We consider a zero-surface-tension two-dimensional Hele-Shaw flow in an infinite wedge. There exists a self-similar interface evolution in this wedge, an analogue of the famous Saffman-Taylor finger in a channel, exact shape of which has been given by Kadanoff. One of the main features of this evolution is its infinite time of existence and stability for the Hadamard ill-posed problem. We derive several exact solutions existing infinitely by generalizing and perturbing the one by Kadanoff.

physics.flu-dyn↗