Representations of the Lie algebra of vector fields on a sphere

For an affine algebraic variety X we study a category of modules that admit compatible actions of both the algebra A of functions on X and the Lie algebra of vector fields on X. In particular, for the case when X is the sphere ${\mathbb{S}}^2$, we construct a set of simple modules that are finitely generated over A. In addition, we prove that the monoidal category that these modules generate is equivalent to the category of finite-dimensional rational ${\mathrm{GL}}_2$-modules.     

Invariant circles for homogeneous polynomial vector fields on the 2-dimensional sphere

LexX be a homogeneous polynomial vector field of degreen≥3 on S² having finitely many invariant circles. Then, for such a vector fieldX we find upper bounds for the number of invariant circles, invariant great circles, invariant circles intersecting at a same point and parallel circles with the same director vector. We give examples of homogeneous polynomial vector fields of degree 3 on S² having finitely many invariant circles which are not great circles, which are limit cycles, but are not great circles and invariant great circles that are limit cycles. Moreover, for the casen=3 we determine the maximum number of parallel invariant circles with the same director vector. The authors are partially supported by a MCYT grant BFM2002-04236-C02-02 and by a CIRIT grant number 2001SGR 00173.     

Proportional concentration phenomena on the sphere

In this paper we establish concentration phenomena for subspaces with arbitrary dimension. Namely, we display conditions under which the Haar measure on the sphere concentrates on a neighborhood of the intersection of the sphere with a subspace ofR ⁿ of a given dimension. We display applications to a problem of projections of points on the sphere, and to the duality of entropy numbers conjecture. Research was partially supported by the Israel Science Foundation.     

Conformal vector fields on spacetimes

We study conformal vector fields and their zeros on spacetimes which are non-conformally-flat. Depending on the Petrov type, we classify all conformal vector fields with zeros. The problems of reducing a conformal vector field to a homothetic vector field are considered. We show that a spacetime admitting a proper homothetic vector field is (locally) a plane wave. This precises a well-known theorem of {Alekseevski}, where all these spacetimes are determined in a more general form.     

Conformal vector fields on tangent bundle of a Riemannian manifold

Let M be an n-dimensional Riemannian manifold and TM its tangent bundle. The conformal and fiber preserving vector fields on TM have well-known physical interpretations and have been studied by physicists and geometers using some Riemannian and pseudo-Riemannian lift metrics on TM. Here we consider the Riemannian or pseudo-Riemannian lift metric G on TM which is in some senses more general than other lift metrics previously defined on TM, and seems to complete these works. Next we study the lift conformal vector fields on (TM,G).     

Smoothly versal deformations of coarse vector fields on a circumference

Translated from Matematicheskie Zametki, Vol. 49, No. 5, pp. 26–31, May, 1991.     

The smooth classification of coarse vector fields on a circumference

Translated from Teoriya Funktsii, Funktsional'nyi Analiz i Ikh Prilozheniya, No. 52, pp. 110–113, 1989.     

Like-linearizations of vector fields

We characterize the n-dimensional vector fields (with or without null linear parts) which can be transformed, under conjugation or orbital equivalence, into their quasi-homogeneous parts of minimum degree and, therefore, have the same dynamics. We give several examples of nilpotent and degenerate systems.     

Fredholm vector fields and a transversality theorem

We define the notion of a Fredholm vector field and prove a transversality result giving conditions under which a vertical family of such vector fields generically have nondegenerate zeros. Many geometric objects like minimal surfaces, geodesics, and harmonic maps arise as the zeros of a Fredholm vector field.     

Harmonic vector fields on Landsberg manifolds

Let (M,F)$(M,F)$ be a compact boundaryless Landsberg manifold. In this work, a necessary and sufficient condition for a vector field on (M,F)$(M,F)$ to be harmonic is obtained. Next, on a compact boundaryless Finsler manifold of zero flag curvature, a necessary and sufficient condition for a vector field to be harmonic is found. Furthermore, the nonexistence of harmonic vector fields on a compact Landsberg manifold is studied and an upper bound for the first de Rham cohomology group is obtained.