Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userFibrations of $mathbb{R}^3$ by oriented lines
219
0
0.0
(
0
)
Authors
Michael Harrison
Ask ChatGPT about the research
No Arabic abstract
A fibration of $mathbb{R}^3$ by oriented lines is given by a unit vector field $V : mathbb{R}^3 to S^2$, for which all of the integral curves are oriented lines. A line fibration is called skew if no two fibers are parallel. Skew fibrations have been the focus of recent study, in part due to their close relationships with great circle fibrations of $S^3$ and with tight contact structures on $mathbb{R}^3$. Both geometric and topological classifications of the space of skew fibrations have appeared; these classifications rely on certain rigid geometric properties exhibited by skew fibrations. Here we study these properties for line fibrations which are not necessarily skew, and we offer some partial answers to the question: in what sense do nonskew fibrations look and behave like skew fibrations? We develop and utilize a technique, called the parallel plane pushoff, for studying nonskew fibrations. In addition, we summarize the known relationship between line fibrations and contact structures, and we extend these results to give a complete correspondence. Finally, we develop a technique for generating nonskew fibrations and offer a number of examples.
rate research
Read More
A smooth fibration of $mathbb{R}^3$ by oriented lines is given by a smooth unit vector field $V$ on $mathbb{R}^3$, for which all of the integral curves are oriented lines. Such a fibration is called skew if no two fibers are parallel, and it is called nondegenerate if $ abla V$ vanishes only in the direction of $V$. Nondegeneracy is a form of local skewness, though in fact any nondegenerate fibration is globally skew. Nondegenerate and skew fibrations have each been recently studied, from both geometric and topological perspectives, in part due to their close relationship with great circle fibrations of $S^3$. Any fibration of $mathbb{R}^3$ by oriented lines induces a plane field on $mathbb{R}^3$, obtained by taking the orthogonal plane to the unique line through each point. We show that the plane field induced by any nondegenerate fibration is a tight contact structure. For contactness we require a new characterization of nondegenerate fibrations, whereas the proof of tightness employs a recent result of Etnyre, Komendarczyk, and Massot on tightness in contact metric 3-manifolds. We conclude with some examples which highlight relationships among great circle fibrations, nondegenerate fibrations, skew fibrations, and the contact structures associated to fibrations.
In the following article we discuss Delaunay triangulations for a point cloud on an embedded surface in $mathbb{R}^3$. We give sufficient conditions on the point cloud to show that the diagonal switch algorithm finds an embedded Delaunay triangulation.
It has been known for a long time that the fundamental group of the quotient of $RR ^3$ by the Case-Chamberlin continuum is nontrivial. In the present paper we prove that this group is in fact, uncountable.
Let $M$ be a closed orientable irreducible $3$-manifold with a left orderable fundamental group, and $M_0 = M - Int(B^{3})$. We show that there exists a Reebless co-orientable foliation $mathcal{F}$ in $M_0$, whose leaves may be transverse to $partial M_0$ or tangent to $partial M_0$ at their intersections with $partial M_0$, such that $mathcal{F}$ has a transverse $(pi_1(M_0),mathbb{R})$ structure, and $mathcal{F}$ is analogue to taut foliations (in closed $3$-manifolds) in the following sense: there exists a compact $1$-manifold (i.e. a finite union of properly embedded arcs and/or simple closed curves) transverse to $mathcal{F}$ that intersects every leaf of $mathcal{F}$. We conjecture that $mathcal{F}$ is obtained from removing a $3$-ball foliated with horizontal disks from a taut foliation in $M$.
In this paper we describe the local limits under conjugation of all closed connected subgroups of $SL_3(mathbb{R})$ in the Chabauty topology.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
