Related papers: Vector fields on RP^m x RP^n

Let X be a smooth projective variety of dimension n in P^r. We study the fibers of a general linear projection pi: X --> P^{n+c}, with c > 0. When n is small it is classical that the degree of any fiber is bounded by n/c+1, but this fails…

Let $P$ be a polygon with $r>0$ reflex vertices and possibly with holes and islands. A subsuming polygon of $P$ is a polygon $P'$ such that $P \subseteq P'$, each connected component $R$ of $P$ is a subset of a distinct connected component…

We construct an algorithm for solving the following problem: given a number field $K$, a positive integer $N$, and a positive real number $B$, determine all points in $\mathbb P^N(K)$ having relative height at most $B$. A theoretical…

Sommese has conjectured a classification of smooth projective varieties X containing, as an ample divisor, a P^d-bundle Y over a smooth variety Z. This conjecture is known if d>1, if dim(X)<5, or if Z admits a finite morphism to an Abelian…

A vector field s on a Riemannian manifold M is said to be harmonic if there exists a member of a 2-parameter family of generalised Cheeger-Gromoll metrics on TM with respect to which s is a harmonic section. If M is a simply-connected…

A unit-vector field n on a convex three-dimensional polyhedron P is tangent if, on the faces of P, n is tangent to the faces. A homotopy classification of tangent unit-vector fields continuous away from the vertices of P is given. The…

We study the space of link maps, which are smooth maps from the disjoint union of manifolds P and Q to a manifold N such that the images of P and Q are disjoint. We give a range of dimensions, interpreted as the connectivity of a certain…

Let $M$ be a complex projective manifold of dimension $n+1$ and $f$ a meromorphic function on $M$ obtained by a generic pencil of hyperplane sections of $M$. The $n$-th cohomology vector bundle of $f_0=f|_{M-\RR}$, where $\RR$ is the set of…

This is a survey on the Darboux theory of integrability for polynomial vector fields, first in $\R^n$ and second in the $n$-dimensional sphere $\sss^n$. We also provide new results about the maximum number of parallels and meridians that a…

Let $M$ be any $n$ dimensional smooth manifold and $PM$ be the space of all smooth paths, then we showed that $PM$ is a smooth manifold modelled over a complete normable space. We discussed many geometric structure on Path spaces and its…

Polynomial sequences $p_n(x)$ of binomial type are a principal tool in the umbral calculus of enumerative combinatorics. We express $p_n(x)$ as a \emph{path integral} in the ``phase space'' $\Space{N}{} \times {[-\pi,\pi]}$. The Hamiltonian…

In the vector-field guided path-following problem, a sufficiently smooth vector field is designed such that its integral curves converge to and move along a one-dimensional geometric desired path. The existence of singular points where the…

We consider a plane polynomial vector field $P(x,y)dx+Q(x,y)dy$ of degree $m>1$. To each algebraic invariant curve of such a field we associate a compact Riemann surface with the meromorphic differential $\omega=dx/P=dy/Q$. The asymptotic…

Indices of vector fields on (complex analytic) singular varieties have been considered by various authors from several different viewpoints. All these indices coincide with the classical local index of Poincar\'e-Hopf when the ambient…

The construction (by Kapranov) of the space of infinitesimal paths on a manifold is extended to include higher dimensional infinitesimal objects, encoding contractions of infinitesimal loops. This full infinitesimal groupoid is shown to…

A convex polygon is defined as a sequence (V_0,...,V_{n-1}) of points on a plane such that the union of the edges [V_0,V_1],..., [V_{n-2},V_{n-1}], [V_{n-1},V_0] coincides with the boundary of the convex hull of the set of vertices…

We study linearly independent complex line fields on almost-complex manifolds, which is a topic of long-standing interest in differential topology and complex geometry. A necessary condition for the existence of such fields is the vanishing…

We prove that every automorphism of an infinite-dimensional vector space over a field is the product of four involutions, a result that is optimal in the general case. We also characterize the automorphisms that are the product of three…

All spaces below are Tychonov. We define the projective pi-character p(X) of a space X as the supremum of the values $\pi\chi(Y)$ where Y ranges over all continuous images of X. Our main result says that every space X has a pi-base whose…

A map Y -> P^n is determined by a line bundle quotient of (O_Y)^{n+1}. In this paper, we generalize this description to the case of maps from Y to an arbitrary smooth toric variety. The data needed to determine such a map consists of a…