Related papers: Integrability and adapted complex structures to sm…
We obtain general solutions for some flat Friedmann universes filled with a scalar field in induced gravity models and models including the Hilbert-Einstein curvature term plus a scalar field conformally coupled to gravity. As is well…
We study the analytic and topological invariants associated with complex normal surface singularities. Our goal is to provide topological formulae for several discrete analytic invariants whenever the analytic structure is generic (with…
We introduce a framework to prove integral rigidity results for the Seiberg-Witten invariants of a closed $4$-manifold $X$ containing a non-separating hypersurface $Y$ satisfying suitable (chain-level) Floer theoretic conditions. As a…
Given a finite collection of $C^1$ complex vector fields on a $C^2$ manifold $M$ such that they and their complex conjugates span the complexified tangent space at every point, the classical Newlander-Nirenberg theorem gives conditions on…
We enlarge the class of open Riemann surfaces known to be holomorphically embeddable into the plane by allowing them to have additional isolated punctures compared to the known embedding results.
We address some global solvability issues for classes of smooth nonsingular vector fields $L$ in the plane related to cohomological equations $Lu=f$ in geometry and dynamical systems. The first main result is that $L$ is not surjective in…
We consider complex rational vector fields that admit a first integral whose logarithmic derivative lies in a finite extension of the rational function field $K$. In view of the Prelle-Singer theorem, these are the rational vector fields…
We give a self-contained introduction to the relations between Integrable Systems and the Geometry of Riemann Surfaces. We start from a historical introduction to the topic of integrable systems. Afterwards, we study the polynomial…
We study phase portraits and singular points of vector fields of a special type, that is, vector fields whose components are fractions with a common denominator vanishing on a smooth regular hypersurface in the phase space. We assume also…
In this paper we focus on various aspects of singular complex plane curves, mostly in the context of their homological properties and the associated combinatorial structures. We formulate some challenging open problems that can point to new…
Motivated by the notion of integrability introduced by Bogoyavlenskij for vector fields, we propose a definition of smooth integrability for general diffeomorphisms. In brief, we say that a diffeomorphism is integrable if it commutes with…
Finite-dimensional subalgebras of a Lie algebra of smooth vector fields on a circle, as well as piecewise-smooth global transformations of a circle on itself, are considered. A canonical forms of realizations of two- and three-dimensional…
This short report establishes some basic properties of smooth vector fields on product manifolds. The main results are: (i) On a product manifold there always exists a direct sum decomposition into horizontal and vertical vector fields.…
A complex-analytic structure within the unit disk of the complex plane is presented. It can be used to represent and analyze a large class of real functions. It is shown that any integrable real function can be obtained by means of the…
A regularization procedure developed in [1] for the integral curvature invariants on manifolds with conical singularities is generalized to the case of squashed cones. In general, the squashed conical singularities do not have rotational…
We study flows of smooth vector fields $X$ over invariant surfaces $M$ which are levels of rational first integrals. It leads us to study constrained systems, that is, systems with impasses. We identify a subset $\mathcal{I} \subset M$…
This work investigates the existence of complex structures on 2-step nilpotent Lie algebras arising from finite graphs. We introduce the notion of adapted complex structure, namely a complex structure that maps vertices and edges of the…
Unless another thing is stated one works in the $C^\infty$ category and manifolds have empty boundary. Let $X$ and $Y$ be vector fields on a manifold $M$. We say that $Y$ tracks $X$ if $[Y,X]=fX$ for some continuous function $f\colon…
New splitting theorems in a semi-Riemannian manifold which admits an irrotational vector field (not necessarily a gradient) with some suitable properties are obtained. According to the extras hypothesis assumed on the vector field, we can…
We initiate the study of inflection curves of rational vector fields on the Riemann sphere. For a rational vector field $v_R=-R(z)\frac{\partial}{\partial z}, \qquad R(z)=\frac{Q(z)}{P(z)} $ we define its affine regular inflection locus by…