arXiv research

The study classifies spaces with specific conformal vector fields.

The $π$-exterior derivative $ød$, which is the Finslerian generalization of the (usual) exterior derivative $d$ of Riemannian geometry, is defined. The notion of a $ød$-closed vector field is introduced and investigated. Various characterizations of $ød$-closed vector fields are established. Some results concerning $ød…

The abstract proves that certain Reeb vector fields on 3-manifolds have Birkhoff sections.

In this paper we examine different aspects of the geometry of closed conformal vector fields on Riemannian manifolds. We begin by getting obstructions to the existence of closed conformal and nonparallel vector fields on complete manifolds with nonpositive Ricci curvature, thus generalizing a theorem of T. K. Pan. Then…

Killing vector fields of a closed homogeneous and isotropic universe are studied. It is shown that in general case there is no time-like Killing vector fields in such a universe. Two exceptional cases are revealed.

We introduce G_2-vector fields, Rochesterian 1-forms and Rochesterian vector fields on manifolds with a closed G_2-structure as analogues of symplectic vector fields, Hamiltonian functions and Hamiltonian vector fields respectively, and we show that the spaces of G_2-vector fields and of Rochesterian vector fields are …

The study classifies gradient Ricci solitons with specific vector fields.

The paper proves conditions for compact vacuum static spaces to be isometric to spheres.

Geodesic vector fields on flat 3-manifolds are related to contact structures.

Paper studies non-gradient almost Yamabe solitons and their properties.

Study proves 3-manifolds with parallel vector fields have odd Betti numbers.

The paper is an informal report on joint work with Stefan Haller on Dynamics in relation with Topology and Spectral Geometry. By dynamics one means a smooth vector field on a closed smooth manifold; the elements of dynamics of concern are the rest points, instantons and closed trajectories. One discusses their counting…

In 1984, Anatole Katok conjectured that the only closed orientable manifolds that support cohomology-free vector fields are tori and these vector fields are smoothly conjugated to Diophantine (constant) ones. In this work we present a proof of Katok conjecture for 3-manifolds.

The study counts periodic orbits on smooth manifolds, adding ghost orbits for completeness.

The study extends and generalizes a result about quasi Einstein manifolds, proving conditions for Killing vector fields.

Study foliations on Riemannian manifolds with specific vector fields, focusing on geometric properties.

The paper examines timelike conformal fields on 3-manifolds and finds they are rigidly tied to specific geometric structures.

Classifies vector fields in the kernel of a 1-form, up to equivalence.

The abstract discusses vector fields on curved spaces and conservation laws.

In this paper we introduce and study a geometric heat flow to find Killing vector fields on closed Riemannian manifolds with positive sectional curvature. We study its various properties, prove the global existence of the solution of this flow, discuss its convergence and possible applications, and its relation to the …

New proof confirms periodic orbit conjecture for Eulerisable flows.

The paper explores how vector fields relate to volume in geometric contexts.

Generalizes Poincaré-Hopf Theorem for piecewise smooth boundaries.

The normal map of curves is analyzed as a vector field on a cylinder.

Study projective symmetries in Finsler spaces, showing reductions and constant flag curvature.

The stability of the 3-dimensional Hopf vector field, as a harmonic section of the unit tangent bundle, is viewed from a number of different angles. The spectrum of the vertical Jacobi operator is computed, and compared with that of the Jacobi operator of the identity map on the 3-sphere. The variational behaviour of t…

Rescaling expansiveness proven for k*-expansive vector fields.

Theorems on the existence of vector fields with given sets of Indexes of isolated Singular points are proved for the cases of closed manifolds, pairs of manifolds, manifolds with boundary, and gradient fields. It is proved that, on a two-dimensional manifold, an index of an isolated Singular point of the gradient field…

Conditions for Riemannian manifolds to be Euclidean spheres or spaces.

Floer constructs homology from flow lines in generalized dynamical systems and combinatorial vector fields.

In this paper, it is proved that a connected 3-dimensional Riemannian manifold or a closed connected semi-Riemannian manifold $M^n$($n>1$) admitting a projective vector field with a non-linearizable singularity is projectively flat.

The complete invariant for gradient like Morse-Smale dynamical systems (vector fields and diffeomorphisms) on closed 4-manifolds are constructed. It is same as Kirby diagram in a case of polar vector field without fixed points of index 3.

Given a unit vector field on a closed Euclidean hypersurface, we define a map from the hypersurface to a sphere in the Euclidean space. This application allows us to exhibit a list of topological invariants which combines the second fundamental form of the hypersurface and the vector field itself. We show how these inv…

In the paper, we show that for a generic $C^1$ vector field $X$ on a closed three dimensional manifold $M$, any isolated transitive set of $X$ is singular hyperbolic. It is a partial answer of the conjecture in \cite{MP}.

Example shows no global coordinates on 2-torus's cover.

Two Kähler metrics on a complex manifold are called c-projectively equivalent if their $J$-planar curves coincide. These curves are defined by the property that the acceleration is complex proportional to the velocity. We give an explicit local description of all pairs of c-projectively equivalent Kähler metrics of arb…

We show that every Lie algebra is equipped with a natural $(1,1)$-variant tensor field, the "canonical endomorphism field", naturally determined by the Lie structure, and satisfying a certain Nijenhuis bracket condition. This observation may be considered as complementary to the Kirillov-Kostant-Souriau theorem on symp…

Flow of curves with curvature and forcing vector field exists.

Let M denote a compact, orientable, 3-dimensional manifold and let a denote a contact 1-form on M; thus the wedge product of a with da is nowhere zero. This article explains how the Seiberg-Witten Floer homology groups as defined for any given Spin-C structure on M give closed, integral curves of the vector field that …

Study investigates Hamiltonian systems in fibered almost-symplectic manifolds.

New proof finds three divergence-free vector fields for any 3D manifold.

Study on null hypersurfaces with constant angle in Lorentzian manifolds.

In this article we investigate the relations between three kinds of vector fields with close connection to each other. A compact orientable manifold enables us to integrate over it, which is very different from noncompact manifolds, and this gives difference of those relationships between on compact and noncompact mani…

Study stabilizes second-order systems to first-order dynamics.

Study shows connection-preserving vector fields are equivalent to certain algebroid structures.

We determine several necessary and sufficient conditions for a closed almost-complex orbifold $Q$ with cyclic local groups to admit a nonvanishing vector field. These conditions are stated separately in terms of the orbifold Euler-Satake characteristics of $Q$ and its sectors, the Euler characteristics of the underlyin…

This paper describes the construction of a canonical compactification of the space of trajectories and of the unstable/stable sets of a generic gradient like vector field on a closed manifold as well as a canonical structure of a smooth manifold with corners of these spaces. As an application we discuss the geometric c…

Study on symplectic semi-characteristic using cohomology and vector fields.