Related papers: The Structure of Stable Vector Fields on Surfaces

We construct new substantive examples of non-autonomous vector fields on 3-dimensional sphere having a simple dynamics but non-trivial topology. The construction is based on two ideas: the theory of diffeomorpisms with wild separatrix…

We investigate the open Closing Lemma problem for vector fields on the 2-dimensional torus. Under the assumption of bounded type rotation number, the $C^r$ Closing Lemma is verified for smooth vector fields that are area-preserving at all…

Let $g:\mathbb{R}^2\to\mathbb{R}$ be a homogeneous polynomial of degree $p>1$, $G=(-g'_{y}, g'_{x})$ be its Hamiltonian vector field, and $G_t$ be the local flow generated by $G$. Denote by $E(G,O)$ the space of germs of $C^{\infty}$…

We develop a geometric approach to stable homotopy groups of spheres in the spirit of the work of Pontrjagin and Rokhlin. A new proof of the Hopf Invariant One Theorem by J.F.Adams is obtained in all dimensions except 15 and 31. To prove…

We discuss criteria for the nonexistence, existence and computation of invariant algebraic surfaces for three-dimensional complex polynomial vector fields, thus transferring a classical problem of Poincar\'e from dimension two to dimension…

We give the classification of globally generated vector bundles of rank $2$ on a smooth quadric surface with $c_1\le (2,2)$ in terms of the indices of the bundles, and extend the result to arbitrary higher rank case. We also investigate…

In this paper a geometric approach toward stable homotopy groups of spheres, based on the Pontrjagin-Thom construction is proposed. From this approach a new proof of Hopf Invariant One Theorem by J.F.Adams for all dimensions except…

We show in this work that homology in degree d of a congruence group, in a very general framework, defines a weakly polynomial functor of degree at most 2d and we describe this functor modulo polynomial functors of smaller degree. Our main…

In this paper, we use Conley index theory to examine the Poincare index of an isolated invariant set. We obtain some limiting conditions on a critical point of a planar vector field to be an isolated invariant set. As a result we show the…

We study the structure of $C^1$-interiors of sets of smooth vector fields with various properties of shadowing of pseudotrajectories. It is shown for which classes of reparametrizations of shadowing trajectories the corresponding interiors…

Let p be a saddle fixed point for an orientation-preserving surface diffeomorphism f admitting a homoclinic point q. Let V be an open 2-cell bounded by a simple loop formed by two arcs joining p to q lying respectively in the stable and…

In this paper we show that every sufficiently large family of convex bodies in the plane has a large subfamily in convex position provided that the number of common tangents of each pair of bodies is bounded and every subfamily of size five…

Gradient vector fields are fundamental objects from both theoretical and practical perspectives, since various phenomena can be modeled within this framework. The ``moduli space'' of such vector fields provides the foundation for describing…

This note shows that the module of smooth vector fields on ${\mathbb{R}}^n$, which are invariant under the linear action of a compact Lie group $G$ is finitely generated by polynomial vector fields on ${\mathbb{R}}^n$ which are invariant…

The space of degree d single-variable monic and centered complex polynomial vector fields can be decomposed into loci in which the vector fields have the same topological structure. We analyze the geometric structure of these loci and…

Oftentimes, Stokes' theorem is derived by using, more or less explicitly, the invariance of the curl of the vector field with respect to translations and rotations. However, this invariance -- which is oftentimes described as the curl being…

We investigate the standard stable manifold theorem in the context of a partially hyperbolic singu-larity of a vector field depending on a parameter. We prove some estimates on the size of the neighbourhood where the local stable manifold…

Let $S_{p,q}$ be the hypersurface in $\mathbb{R}^{p+q+1}$ defined by the following: $$ S_{p,q} := \left\lbrace (x_1,\ldots,x_{p+1},x_{p+2},\ldots,x_{p+q+1}) \in \mathbb{R}^{p+q+1} \big| \left( \sum_{i=1}^{p+1} x_i^2 - a^2 \right)^2 +…

We prove that stably isomorphic vector bundles of rank d-1 on a smooth affine d-fold X over an algebraically closed field k are indeed isomorphic, provided d! is invertible in k. This answers an old conjecture of Suslin.

Let $C$ be an algebraic smooth complex curve of genus $g>1$. The object of this paper is the study of the birational structure of certain moduli spaces of vector bundles and of coherent systems on $C$ and the comparison of different type of…