How many linearly independent smooth nowhere-zero vector fields can be on an n-sphere
This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. Find sources: "Vector fields on spheres" – news · newspapers · books · scholar · JSTOR(May 2012) (Learn how and when to remove this message)
In mathematics, the discussion of vector fields on spheres was a classical problem of differential topology, beginning with the hairy ball theorem, and early work on the classification of division algebras.
Specifically, the question is how many linearly independent smooth nowhere-zero vector fields can be constructed on a sphere in -dimensional Euclidean space. A definitive answer was provided in 1962 by Frank Adams. It was already known,[1] by direct construction using Clifford algebras, that there were at least such fields (see definition below). Adams applied homotopy theory and topological K-theory[2] to prove that no more independent vector fields could be found. Hence is the exact number of pointwise linearly independent vector fields that exist on an ()-dimensional sphere.
^James, I. M. (1957). "Whitehead products and vector-fields on spheres". Proceedings of the Cambridge Philosophical Society. 53 (4): 817–820. doi:10.1017/S0305004100032928. S2CID 119646042.
^Adams, J. F. (1962). "Vector Fields on Spheres". Annals of Mathematics. 75 (3): 603–632. doi:10.2307/1970213. JSTOR 1970213. Zbl 0112.38102.
and 25 Related for: Vector fields on spheres information
discussion of vectorfieldsonspheres was a classical problem of differential topology, beginning with the hairy ball theorem, and early work on the classification...
integral calculus extend naturally to vectorfields. When a vectorfield represents force, the line integral of a vectorfield represents the work done by a force...
tangent vectorfieldon even-dimensional n-spheres. For the ordinary sphere, or 2‑sphere, if f is a continuous function that assigns a vector in R3 to...
In mathematics, a Killing vectorfield (often called a Killing field), named after Wilhelm Killing, is a vectorfieldon a Riemannian manifold (or pseudo-Riemannian...
means that, for two vector spaces over a given field and with the same dimension, the properties that depend only on the vector-space structure are exactly...
section. A vectorfieldon M {\displaystyle M} is therefore a section of the tangent bundle of M {\displaystyle M} . The set of all vectorfieldson M {\displaystyle...
of classical electromagnetism. Assume that f is a scalar, vector, or tensor field defined on a surface S. To find an explicit formula for the surface integral...
time-varying electric currents. Electric fields and magnetic fields are both manifestations of the electromagnetic field, Electromagnetism is one of the four...
contexts. Adams introduced them in a 1962 paper to solve the famous vectorfieldsonspheres problem. Subsequently he used them to investigate the Adams conjecture...
family of nested round spheres. There are several different types of coordinate chart which are adapted to this family of nested spheres; the best known is...
orthogonal state vectors. The north and south poles of the Bloch sphere are typically chosen to correspond to the standard basis vectors | 0 ⟩ {\displaystyle...
standard Riemannian metric on the unit radius 2-sphere. That is, these nested coordinate spheres do in fact represent geometric spheres with surface area A =...
including pressure vessels and most curved mirrors and lenses are based onspheres. Spheres roll smoothly in any direction, so most balls used in sports and...
which the fiber may have other structures; for example sphere bundles are fibered by spheres. A vector bundle (E, p, M) is smooth, if E and M are smooth manifolds...
parallelizable if there exist smooth vectorfields { V 1 , … , V n } {\displaystyle \{V_{1},\ldots ,V_{n}\}} on the manifold, such that at every point...
known objects that allow such a solution: spheres, concentric spheres, infinite cylinders, clusters of spheres and clusters of cylinders. There are also...
after Ernst Witt, is the Lie algebra of meromorphic vectorfields defined on the Riemann sphere that are holomorphic except at two fixed points. It is...
and pseudo-Riemannian manifold. The vector Laplacian operator, a generalization of the Laplacian to vectorfields. The Laplacian in differential geometry...
infinitesimal version of symplectomorphisms gives the symplectic vectorfields. A vectorfield X ∈ Γ ∞ ( T M ) {\displaystyle X\in \Gamma ^{\infty }(TM)} is...
field, maximises when it is antiparallel, and is zero when it is perpendicular. The symbol "×" refers to the vector cross product. The E-fieldvector...
In mathematics, given a vector space X with an associated quadratic form q, written (X, q), a null vector or isotropic vector is a non-zero element x...
element of the algebra, not just vectors and other quaternions, but also lines, planes, circles, spheres, rays, and so on. In the conformal model of Euclidean...
ISBN 978-3-540-08158-6. (§0.26 on page 6) sci.math.research 1993 thread "Spheres fibred by spheres" Friedman, John L. (June 2015). "Historical note on fiber bundles"...
Global Information
