# Equivariant vector fields on spheres

Written in English

## Edition Notes

Classifications The Physical Object Statement by Unni Namboodiri. LC Classifications Microfilm 82/471 Format Microform Pagination iii, 70 leaves. Number of Pages 70 Open Library OL3070831M LC Control Number 82169088

The case is considered of a critical fixed point of a diffeomorphism of codimension 2 whose linear part has the ing to ideas developed by Takens and Arnol'd, to deformations of such diffeomorphisms there correspond families of vector fields invariant with respect to an involution of the plane, namely, a reflection relative to a line passing through the fixed point. Motivation. In Heinz Hopf used Clifford parallels to construct the Hopf map: →, and proved that is essential, i.e., not homotopic to the constant map, by using the fact that the linking number of the circles − (), − ⊂ is equal to 1, for any ≠ ∈.. It was later shown that the homotopy group is the infinite cyclic group generated , Jean-Pierre Serre proved that the. Section Vector Fields A vector field, $$\FF\text{,}$$ is a function whose output is a vector at each point in the domain of the function. Each such vector should be thought of as "living" (having its tail at) the point at which it is defined, rather than at the origin. The dual space of a vector space is the set of real valued linear functions on the vector space. The cotangent space at a point is the dual of the tangent space at that point, and the cotangent bundle is the collection of all cotangent spaces.. Like the tangent bundle, the cotangent bundle is again a differentiable manifold. The Hamiltonian is a scalar on the cotangent bundle.

Cohomology in any of its various forms is one of the most important inventions of the twentieth century. A functor from topological spaces to rings, cohomology turns a geometric problem into an easier algebraic problem. Equivariant cohomology is a cohomology theory that takes into account the.   The purpose of Theorem is (1) to give readers a feel for what it means dynamically for two vector fields to be isomorphic and (2) to show that the notion of equivalence of vector fields has been around implicitly in equivariant dynamics literature for quite some time. AU - Field, M. AU - Swift, J. W. PY - /12/1. Y1 - /12/1. N2 - We present techniques for studying the local dynamics generated by an equivariant Hopf bifurcation. We show that under general hypotheses we can expect the formation of a branch of attracting invariant spheres . Vector fields represent fluid flow (among many other things). They also offer a way to visualize functions whose input space and output space have the same dimension. If you're seeing this message, it means we're having trouble loading external resources on our website.

The vector multiplication operation is $${\otimes}$$, and thus the infinite-dimensional tensor algebra is associative. In fact, the tensor algebra can alternatively be defined as the free associative algebra on $${V}$$, with juxtaposition indicated by the tensor product. Bifurcations of Planar Vector Fields and Hilberts Sixteenth. 11 sep this is a bifurcation of a fixed point of an autonomous vector field we consider the following nonlinear. In [AGRS] a multiplicity one theorem is proven for general linear groups, orthogonal groups and unitary groups (, and) over -adic local fields. That is to say that when we have a.

## Recent

EQUIVARIANT VECTOR FIELDS ON SPHERES In order to state our main results, we Equivariant vector fields on spheres book some terminology. If «is a positive integer written as (2a + l)2c+4i/, where a, d > 0 and 0, we define three functions a, ß and y by the following formulas: a(n) = 8d+2c- 1 =p(n) - 1, ß(n) = %d+2c+ 1, %d + c + 3 if c ¥=3.

1. Introduction. The relationship between Euler characteristics, zeros of vector fields, and self-maps of spheres has been understood since the presence of an action of a compact Lie group, however, complicates matters: the Euler characteristic is no longer an integer, transversality fails in general, and the possible self-maps of a sphere depend heavily on the action of the by: 1.

Equivariant vector fields and self-maps of spheres Article in Journal of Pure and Applied Algebra (1) March with 10 Reads How we measure 'reads'. CiteSeerX - Document Details (Isaac Councill, Lee Giles, Pradeep Teregowda): Abstract. We identify, for actions of a compact Lie group, the generalization of the Euler characteristic that is the sole obstruction to the existence of a nowhere-zero smooth equivariant tangent vector field.

We use this characteristic to calculate the monoid of self-maps of the unit sphere of a representation. This chapter describes the concept of vector fields on spheres. The non-existence of vector fields on even dimensional spheres is a very simple question from current standpoint.

The natural question is to determine, which spheres admit several vector fields that are linearly independent at each point. The vector field problem (for the spheres) then find the maximal number of linearly independent F-vector fields on the spheres SF n. We shall apply KG-theory to the equivariant map (), and so we shall recall from [3] and [10] a few facts about equivariant K-theory.

Fix a finite group G, and for a compact G-space X let KG(X) denote the Grothendieck group of complex G-vector bundles over X. The tensor product of G-vector bundles makes KG(X) a ring. As an application of this reduction, we give a generalization of the results of Namboodiri [U.

Namboodiri, Equivariant vector fields on spheres, Trans. Amer. Math. Soc. (2) () – INTO SPHERES GABOR TOTH Abstract. This note studies nonrigidity of equivariant harmonic maps /: M -» S" G-invariant vector field v and complex structure J on Rn + 1.

Choose a Killing vector field A on Equivariant vector fields on spheres book such that /*(A") = Y°f for a nonzero y g so(«+ 1). Setting. _____, Parametrized functions, bifurcation and vector fields on spheres, Anniversary Volume in Honor of Mitropolsky, vol.

Naukova Dumka,pp. 15– MathSciNet Google Scholar [AY3]. Lecture 1. Introduction to vector elds on spheres We shall discuss some classical topics in homotopy theory. A great deal of what we will cover owes much to the work of Frank Adams.

Today we look at a few of the problems (which turn out to have deep general signi cance), and begin with vector elds on spheres. We de ne Sn1 = fx2Rnjkxk= 1g. The idea is to. Vector fields, torus actions and equivariant cohomology Article (PDF Available) in Pacific Journal of Mathematics (1) March with 43 Reads How we measure 'reads'.

LetG be a finite group and letM be a unitary representation space ofG. We consider the existence problem of equivariant frame fields on the unit sphereS(M) whose orthogonal complements in the tangent bundleT(S(M)) admitG-equivariant complex structures.

Under mild fixed point conditions we give a complete solution for this problem whenG is either ℤ/2ℤ or a finite group of odd order. 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 N -dimensional Euclidean space. The series is devoted to the publication of monographs and high-level textbooks in mathematics, mathematical methods and their applications.

APart from covering important areas of current interest, a major aim is to make topics of an interdisciplinary nature accessible to the non-specialist. THe works in this series are addressed to advanced students and researchers in mathematics and. This gauge equivariant convolution takes as input a number of feature fields on M of various types (analogous to matter fields in physics), and produces as output new feature fields.

Each field is represented by a number of feature maps, whose activations are interpreted as the coefficients of a geometrical object (e.g. scalar, vector, tensor. stand for the tension fields of with respect to the metrics h and hˆ, respectively.

The geodesic curvature at s is defined by: h where denotes the unit normal vector field to. S1-Equivariant CMC-Immersion. For a curve: J X S. EQUIVARIANT VECTOR FIELDS ON SPHERES BY UNNI NAMBOODIRI1 ABSTRACT.

We address the following question: If G is a compact Lie group and S(M) is the unit sphere of an R[G]-module M, then how many orthonormal G-invariant vector fields can be found on S(M). We call this number the G-field. Topology and its Applications 28 () 29 North-Holland EQUIVARIANT SKK AND VECTOR FIELD BORDISM Stefan WANER and Yihren WU Department of Mathematics, Hofstra University, Hempstead, NYU.S.A.

Received 2 May Revised 1 October We use a notion of equivariant Euler characteristic in order to extend classical results on controllable cutting and pasting, and vector field.

Rotation equivariant vector ﬁeld networks Diego Marcos ∗1, Michele Volpi1, Nikos Komodakis2, and Devis Tuia1 1University of Zurich, 2Ecole des Ponts, Paris Tech Abstract In many computer vision tasks, we expect a particular behavior of the output with respect to rotations of the input.

Here Ind G (v) denotes the equivariant index of the field v, the v-induced Morse stratification (see [10]) of the boundary ∂X, and the class of the (n - k)-manifold in A(G). We examine some applications of this formula to the equivariant real algebraic fields v in compact domains X ⊂ ℝ n defined via a generic polynomial inequality.

EQUIVARIANT VECTOR FIELDS 3 the discriminant of q. Since det(B) = det(P)2 det(A), the discriminant is only determined up to a quadratic factor.

Two quadratic forms q: V → K and q0: V0 → K are equivalent, if there exists a vector space isomorphism f: V → V0 such that q(x) = q0(f(x)) for all x ∈ V. Vector fields on spheres. Annals of Math. 75(), – MathSciNet CrossRef zbMATH Google Scholar. Homology of classical groups over finite fields and their associated infinite loop spaces.

Lecture Notes in Mathematics Vol () Equivariant algebraic K-theory. In: Dennis R.K. (eds) Algebraic K-Theory. Lecture Notes in. The form $${\varphi_{P}}$$ is called an equivariant form if this mapping is equivariant with respect to these actions, i.e.

if $$\displaystyle g^{*}\varphi_{P}=g^{-1}\left(\varphi_{P}\right).$$ If $${\varphi_{P}}$$ is also horizontal, then it is called a horizontal equivariant form (AKA basic form, tensorial form). If we pull back a horizontal. This paper discusses the dynamics and bifurcation theory of equivariant dynamical systemsnear relative equilibria, that is, group orbits invariant under the flow of an equivariant vector field.

The theory developed here applies, in particular, to secondary steady-state bifurcations from invariant equilibria. Let G be a finite group and let M be a unitary representation space of G.A solution to the existence problem of G-equivariant cross sections of the complex Stiefel manifold W k (M) of unitary k-frames over the unit sphere S(M) is given under mild restrictions on G and on fixed point sets.

In the case G is an even ordered group, some sufficient conditions for the existence of G-equivariant real. They managed to attain the simplicity of matrices, and require readers to travel that far to meet them.

They cover preliminaries, equivariant degree, equivariant homotopy groups of spheres, and applications. Appendices cover equivariant matrices and periodic solutions of linear systems. Real, complex and quaternionic equivariant vector fields on spheres.

(English summary) Topology Appl. (), no. 12, – Let F = R, C, or H, G a compact Lie group, M an F-representation space of G with G-equivariant F-inner product, and S(M) the unit sphere in M. G-invariant vector field v and complex structure J on Rn + 1. Choose a Killing vector field A on M such that /*(A") = Y°f for a nonzero y g so(«+ 1).

An appendix gives a proof of the equivariant de Rham theorem, demonstrating that equivariant cohomology can be computed using equivariant differential forms. Examples and calculations illustrate new concepts. Exercises include hints or solutions, making this book suitable for self-study.

References: The article Kac-Moody and Virasoro algebras in relation to quantum physics by Goddard and Olive claims that the diffeomorphism group of the circle has the infinitesimal symmetries of the Witt algebra. Schottenloher's book A Mathematical Introduction to Conformal Field Theory claims, in sectionthat there is no complex Virasoro group and also no complex Witt group.

U. Namboodiri, Equivariant vector fields on spheres, Thesis, Chicago Google Scholar [29] U. Namboodiri, Equivariant vector fields on spheres, Trans. Amer.

Math. Soc., to appear. Google Scholar eBook Packages Springer Book Archive; Buy this book on publisher's site; Reprints and Permissions; Personalised recommendations.First-order, time-reversible n-body problems in three-space whose velocity fields consist of sums of identical two-body interactions are studied under a set of natural symmetry assumptions.

Up to linearization about maximally symmetric equilibria, the entire class is shown to be represented by a two-parameter normal form. The symmetries of the class are used to find formulas for the.harmonic maps into spheres.

Given a harmonic ma S",p n /: M ^ 2 - [5> ] of a compact Riemannian manifold M into the Euclidean n-sphere S" the (finite dimensional) vector space K(f) of all divergence free Jacobi fields along / [7] contains the vector space of infinitesimal isometric deformation so{n +1)s o .