Related papers: Vector fields with big and small volume on the 2-s…

We introduce the notion of $V$-minimality, for $V$ a smooth vector field on a Riemannian manifold, a natural extension of the classical notion of minimality, and we prove several basic properties. One featured example is given for locally…

Let $\Omega\subseteq \mathbb{R}^n$ be an $m$-dimensional closed submanifold of class $C^2$, $d$ be a positive integer between 1 and $m$. We will study the geometric and topological proprieties of quasiminimal sets in $\Omega$, and show that…

Let \Delta be a (d-1)-dimensional homology sphere on n vertices with m minimal non-faces. We consider the invariant \alpha := m - (n-d) and prove that for a given value of \alpha, there are only finitely many homology spheres that cannot be…

The aim of this article is to investigate the presence of a conformal vector $\xi$ with conformal factor $\rho$ on a compact Riemannian manifold $M$ with or without boundary $\partial M$. We firstly prove that a compact Riemannian manifold…

The objective of the present paper (the second in a series of four) is to give a theory of multivector and extensor fields on a smooth manifold M of arbitrary topology based on the powerful geometric algebra of multivectors and extensors.…

Consider a codimension $1$ submanifold $N^n\subset M^{n+1}$, where $M^{n+1}\subset\mathbb{R}^{n+2}$ is a hypersurface. The envelope of tangent spaces of $M$ along $N$ generalizes the concept of tangent developable surface of a surface along…

We study orbifolds of two-dimensional topological field theories using defects. If the TFT arises as the twist of a superconformal field theory, we recover results on the Neveu-Schwarz and Ramond sectors of the orbifold theory as well as…

Let $M$ be a non-compact connected manifold with a cocompact and properly discontinuous action of a discrete group $G$. We establish a Poincar\'{e}-Hopf theorem for a bounded vector field on $M$ satisfying a mild condition on zeros. As an…

Vector fields can arise in the cosmological context in different ways, and we discuss both abelian and nonabelian sector. In the abelian sector vector fields of the geometrical origin (from dimensional reduction and Einstein-Eddington…

In this note, we find a sharp bound for the minimal number (or in general, indexing set) of subspaces of a fixed (finite) codimension needed to cover any vector space V over any field. If V is a finite set, this is related to the problem of…

We study ordinary differential equations in the complex domain given by meromorphic vector fields on K\"ahler compact complex surfaces. We prove that if such an equation has a maximal single valued solution with Zariski-dense image (in…

Natural metric structures on tangent bundles and tangent sphere bundles enclose many important problems, from the topology of the base to the determination of their holonomy. We make here a brief study of the topic. We find the…

This work investigates biharmonic and interpolating sesqui-harmonic vector fields on the tangent bundle of a para-K\"ahler--Norden manifold (M, varphi, g) endowed with the varphi-Sasaki metric. We derive the first variation of the bienergy…

Given a Riemannian $\mathbb{RP}^3$ with a bumpy metric or a metric of positive Ricci curvature, we show that there either exist four distinct minimal real projective planes, or exist one minimal real projective plane together with two…

In this work we extend some of the results of Ignat and Jerrard for Ginzburg-Landau vortices of tangent vector fields on two-dimensional Riemannian manifolds to the setting of complex hermitian line bundles. In particular, we elucidate the…

Let $M$ be a compact, orientable, $n$-dimensional Riemannian manifold, $n\geq2$, and let $F$ be the energy functional acting on the space $\Xi (M)$ of $C^{\infty }$ vector fields of $M$, \[ F(X):=\frac{\int_{M}\left\Vert \nabla X\right\Vert…

We study existence and structure of $P-$area minimizing surfaces in the Heisenberg group under Dirichlet and Neumann boundary conditions. We show that there exists an underlying vector field $N$ that characterized existence and structure of…

This paper investigates a real-valued topological invariant of 3-manifolds called topological volume. For a given 3-manifold M it is defined as the smallest volume of the complement of a (possibly empty) hyperbolic link in M. Various…

We study moduli spaces of vector bundles on a two-dimensional neighbourhood $Z_k$ of an irreducible curve $\ell = CP^1$ with $\ell^2 = -k$ and give an explicit construction of these moduli as stratified spaces. We give sharp bounds for the…

Let $(M^n,g)$ be an $n$-dimensional compact connected Riemannian manifold with smooth boundary. We show that the presence of a nontrivial conformal gradient vector field on $M$, with an appropriate control on the Ricci curvature makes $M$…