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I. A. Taimanov

Publications and source records attributed to I. A. Taimanov.

At least 55 records · Page 3Linked to original sources
arXiv2006

Orthogonal curvilinear coordinate systems corresponding to singular spectral curves

We study the limiting case of the Krichever construction of orthogonal curvilinear coordinate systems when the spectral curve becomes singular. We show that the case when the curve is reducible and all its irreducible components are rational curves the construction procedure reduces to solving systems of linear equations and simple computations with elementary functions. Therewith we demonstrate how such well-known coordinate systems as the polar coordinates, the cylindrical coordinates in the three-space, and the spherical coordinates in the Euclidean spaces fit in this scheme.

math.DG↗
arXiv2005

An example of a jump from chaos to integrability for magnetic geodesic flows

It is proved that the motion of a charge particle on a hyperbolic oriented two-dimensional surface in a magnetic field given by the volume form of the hyperbolic metric is completely integrable on the energy levels E < 1/2 in terms of real-analytic integrals. However it was known that on the level E=1/2 every trajectory is transitive and the restrictions of this flow onto the levels E>1/2 are Anosov flows

math.DS↗
arXiv2004

Integrability of the n-centre problem at high energies

It is shown that for generic configuration of the centres at high energy levels the n-centre problem is completely integrable by using $C^\infty$ integrals of the motion however it is not integrable in terms of real analytic functions

math.DS↗
arXiv2004

Finite gap theory of the Clifford torus

We construct the spectral curve and the Baker--Akhiezer function for the Dirac operator which corresponds to the Clifford torus via the Weierstrass representation. By constructing this Baker--Akhiezer function we demonstrate a general procedure for constructing Dirac operators and their Baker--Akhiezer functions corresponding to singular spectral curves

math-ph↗
arXiv2002

Tame integrals of motion and o-minimal structures

We expose some ideas from mathematical logics, i.e. the background of the theory of o-minimal structures, and demonstrate how they lead to the notion of a tame integral of motion and some extensions and clarifications of previous results on obstructions to integrability of geodesic flows.

math.DS↗
arXiv2000

Dirac operators and conformal invariants of tori in 3-space

We show how to assign to any immersed torus in $\R^3$ or $S^3$ a Riemann surface such that the immersion is described by functions defined on this surface. We call this surface the spectrum or the spectral curve of the torus. The spectrum contains important information about conformally invariant properties of the torus and, in particular, relates to the Willmore functional. We propose a simple proof that for isothermic tori in $\R^3$ (this class includes constant mean curvature tori and tori of revolution) the spectrum is invariant with respect to conformal transformations of $\R^3$. We show that the spectral curves of minimal tori in $S^3$ introduced by Hitchin and of constant mean curvature tori in $\R^3$ introduced by Pinkall and Sterling are particular cases of this general spectrum. The construction is based on the Weierstrass representation of closed surfaces in $\R^3$ and the construction of the Floquet--Bloch varieties of periodic differential operators.

math.DG↗
arXiv1999

Integrable geodesic flows on the suspensions of toric automorphisms

For any toric automorphism with only real eigenvalues a Riemannian metric with an integrable geodesic flow on the suspension of this automorphism is constructed. A qualitative analysis of such a flow on a three-solvmanifold constructed by the authors in math.DG/9905078 is done. This flow is an example of the geodesic flow, which has vanishing Liouville entropy and, moreover, is integrable but has positive topological entropy. The authors also discuss some open problems on integrability of geodesic flows and related subjects.

math.DG↗
arXiv1999

Massey products in symplectic manifolds

The paper is devoted to study of Massey products in symplectic manifolds. Theory of generalized and classical Massey products and a general construction of symplectic manifolds with nontrivial Massey products of arbitrary large order are exposed. The construction uses the symplectic blow-up and is based on the author results, which describe conditions under which the blow-up of a symplectic manifold X along its submanifold Y inherits nontrivial Massey products from X ot Y. This gives a general construction of nonformal symplectic manifolds.

math.SG↗
arXiv1996

Integrable geodesic flows of non-holonomic metrics

Normal geodesic flows flows of Carnot-Caratheodory are discussed from the point of view of the theory of Hamiltonian systems. The geodesic flows corresponding to left-invariant metrics and left- and -right-invariant rank 2 distributions on the three-dimensional Heisenberg group are analysed as integrable systems. The flows corresponding to left-invariant metrics and left-invariant distributions on Lie groups are reduced to Euler equations on Lie groups. Relation of these constructions to problems of analytical mechanics is discussed.

dg-ga↗
arXiv1996

Surfaces of revolution in terms of solitons

The detailed analysis of the generalised Weierstrass representation of surfaces of revolution and their deformations induced by the modified Korteweg--de Vries (mKdV) equations is done. In particular, it is shown that these deformations preserve tori. The geometric meaning of the potential of surface is discussed and the deformations of potentials induced by geometric deformations (the Moebius inversion and a passing to a dual surface) are described. The conjecture of invariance of first integrals of the mKdV hierarchy, considering as invariants of surfaces, with respect to confomal changes of the metric of the ambient space is introduced. Some numeric computations confirming it are also presented. This paper is the sequel to the previous author's paper "Modified Novikov-Veselov equation and differential geometry of surfaces" (dg-ga/9511005) where generic surfaces are considered. Here the surfaces of revolution are regarded as a toy model for the general case.

dg-ga↗
arXiv1996

Secants of Abelian Varieties, Theta functions, and Soliton Equations

This paper is a survey on relations between secant identities and soliton equations and applications of soliton equations to problems of algebraic geometry, i.e., the Riemann-Schottky problem and its analogues. A short introduction into the analytic theory of theta functions is also given.

alg-geom↗
arXiv1995

Periodic orbits in magnetic fields and Ricci curvature of Lagrangian systems

We consider a periodic problem for the motion of a charged particle in a magnetic field. Introducing a notion of Ricci curvature for such Lagrangian systems and using the methods of the calculus of variations in the large, we prove the existence of periodic motions for such particles under a condition of positivity of the Ricci curvature.

dg-ga↗
arXiv1995

A remark on positively curved manifolds of dimensions 7 and 13

Totally geodesically embeddings of infinitely many closed 7-manifolds into 13-dimensional positively curved closed Riemannian manifolds are constructed. The problems of computing pinching constants and existence of other totally geodesical embeddings are discussed.

dg-ga↗
arXiv1995

Generalized Weierstrass formulae, soliton equations and Willmore surfaces. I. Tori of revolution and the mKdV equation

A new approach is proposed for study structure and properties of the total squared mean curvature $W$ of surfaces in ${\bf R}^3$. It is based on the generalized Weierstrass formulae for inducing surfaces. The quantity $W$ (Willmore functional) is shown to be invariant under the modified Novikov--Veselov hierarchy of integrable flows. The $1+1$--dimensional case and, in particular, Willmore tori of revolution, are studied in details. The Willmore conjecture is proved for the mKDV--invariant Willmore tori.

dg-ga↗