scholarly journals About homotopy classes of non-singular vector fields on the three-sphere

2002 ◽  
Vol 3 (2) ◽  
pp. 361-376 ◽  
Author(s):  
Emmanuel Dufraine
Keyword(s):  
Vector Fields ◽  
Singular Vector ◽  
Homotopy Classes ◽  
Three Sphere

STRUCTURED GROUPS AND LINK-HOMOTOPY

1993 ◽  
Vol 02 (01) ◽  
pp. 37-63 ◽  
Author(s):  
JAMES R. HUGHES
Keyword(s):  
Automorphism Groups ◽  
Original Version ◽  
Theoretic Approach ◽  
Homotopy Classes ◽  
Structure Preserving ◽  
Three Sphere ◽  
Complex Program ◽  
Link Homotopy ◽  
Structured Groups

We study link-homotopy classes of links in the three sphere using reduced groups endowed with peripheral structures derived from meridian-longitude pairs. Two types of peripheral structures are considered — Milnor’s original version (called “pre-peripheral structures” in Levine’s terminology) and Levine’s refinement (called simply “peripheral structures”). We show here that pre-peripheral structures are not strong enough to classify links up to link-homotopy, and that Levine’s peripheral structures, although strong enough to distinguish those classes not distinguished by pre-peripheral structures, are also in all likelihood not strong enough to distinguish all link-homotopy classes. Following Levine’s classification program, we compare structure-preserving and realizable automorphisms, using an obstruction-theoretic approach suggested by work of Habegger and Lin. We find that these automorphism groups are in general different, so that a more complex program for classification by structured groups is required.

Immersions with non-zero normal vector fields

1992 ◽  
Vol 112 (2) ◽  
pp. 281-285 ◽  
Author(s):  
Bang-He Li ◽  
Gui-Song Li
Keyword(s):  
Vector Fields ◽  
Normal Vector ◽  
Homotopy Classes ◽  
Natural Action ◽  
Singularity Method ◽  
One To One ◽  
Regular Homotopy

Let M be a smooth n-manifold, X be a smooth (2n − 1)-manifold, and g:M → X be a map. It was proved in [6] that g is always homotopic to an immersion. The set of homotopy classes of monomorphisms from TM into g*TX, which is denoted by Sg, may be enumerated either by the method of I. M. James and E. Thomas or by the singularity method of U. Koschorke (see [1] and references therein). When the natural action of π1(XM, g) on Sg is trivial, for example, if X is euclidean, the set Sg is in one-to-one correspondence with the set of regular homotopy classes of immersions homotopic to g (see e.g. [4]).

A simple global characterization for normal forms of singular vector fields

1987 ◽  
Vol 29 (1-2) ◽  
pp. 95-127 ◽  
Author(s):  
C. Elphick ◽  
E. Tirapegui ◽  
M.E. Brachet ◽  
P. Coullet ◽  
G. Iooss
Keyword(s):  
Vector Fields ◽  
Normal Forms ◽  
Singular Vector

scholarly journals Extended Fuller index, sky catastrophes and the Seifert conjecture

2018 ◽  
Vol 29 (13) ◽  
pp. 1850096 ◽  
Author(s):  
Yasha Savelyev
Keyword(s):  
Vector Field ◽  
Periodic Orbits ◽  
General Class ◽  
Smooth Manifold ◽  
Vector Fields ◽  
Singular Vector ◽  
Natural Question ◽  
Reeb Vector ◽  
Fuller Index

We extend the classical Fuller index, and use this to prove that for a certain general class of vector fields [Formula: see text] on a compact smooth manifold, if a homotopy of smooth non-singular vector fields starting at [Formula: see text] has no sky catastrophes as defined by the paper, then the time 1 limit of the homotopy has periodic orbits. This class of vector fields includes the Hopf vector field on [Formula: see text]. A sky catastrophe is a kind of bifurcation originally discovered by Fuller. This answers a natural question that existed since the time of Fuller’s foundational papers. We also put strong constraints on the kind of sky-catastrophes that may appear for homotopies of Reeb vector fields.

scholarly journals ENERGY OF GLOBAL FRAMES

2008 ◽  
Vol 84 (2) ◽  
pp. 155-162
Author(s):  
FABIANO G. B. BRITO ◽  
PABLO M. CHACÓN
Keyword(s):  
Riemannian Manifold ◽  
Vector Field ◽  
Lower Bound ◽  
Vector Fields ◽  
Singular Vector ◽  
Unit Vector ◽  
Energy Functional ◽  
Closed Riemannian Manifold ◽  
Total Scalar Curvature ◽  
Dimension 2

AbstractThe energy of a unit vector field X on a closed Riemannian manifold M is defined as the energy of the section into T1M determined by X. For odd-dimensional spheres, the energy functional has an infimum for each dimension 2k+1 which is not attained by any non-singular vector field for k>1. For k=1, Hopf vector fields are the unique minima. In this paper we show that for any closed Riemannian manifold, the energy of a frame defined on the manifold, possibly except on a finite subset, admits a lower bound in terms of the total scalar curvature of the manifold. In particular, for odd-dimensional spheres this lower bound is attained by a family of frames defined on the sphere minus one point and consisting of vector fields parallel along geodesics.

On the Minimal Sets of Non-Singular Vector Fields

1966 ◽  
Vol 84 (3) ◽  
pp. 529 ◽  
Author(s):  
F. Wesley Wilson
Keyword(s):  
Vector Fields ◽  
Singular Vector ◽  
Minimal Sets
Sign in / Sign up

Export Citation Format

Share Document