In this paper we consider the harmonicity of the 1-parameter group of local infinitesimal transformations associated to a vector field on a (pseudo-) Riemannian manifold to study this class of vector fields, which we call harmonic-Killing vector fields.

where n = dim M and 6» = ith Betti number of M ( = dim of Hi(M; Q)). Thus the geometric property of M having a nonzero vector field is expressed in terms of the algebraic invariant xM. We will discuss extensions of this idea to vector ^-fields, fields of ^-planes, and foliations of manifolds. All manifolds considered will be connected, smooth and without boundary; all maps will be continuous. F...

Vector fields can arise in the cosmological context in different ways, and we discuss both abelian and nonabelian sector. In the abelian sector vector fields of the geometrical origin (from dimensional reduction and Einstein–Eddington modification of gravity) can provide a very non-trivial dynamics, which can be expressed in terms of the effective dilaton-scalar gravity with the specific potent...

In this paper we will address the question of how many nonvanishing, linearly independent tangent vector fields can exist on a sphere Sn−1 ⊆ R. By this we mean the following, a tangent vector field on Sn−1 = {x ∈ R : ‖x‖ = 1} is a map v : Sn−1 → R such that v(x) ⊥ x for all x ∈ Sn−1. However, by assumption v is nonvanishing, so we can normalize such that ‖v(x)‖ = 1 and we obtain a map v : Sn−1 ...

We propose statistically self-similar and rotation-invariant models for vector fields, study some of the more significant properties of these models, and suggest algorithms and methods for reconstructing vector fields from numerical observations, using the same notions of self-similarity and invariance that give rise to our stochastic models. We illustrate the efficacy of the proposed schemes b...

We present alternative ways of looking at vector elds that complement existing ow visualization methods. These techniques are based on bump mapping and are simple, easy to compute and fast to render. For example, in one of the techniques a surface from the vector eld is extracted and the directions of the vector values on the surface are normalized and used as surface normals. The shading of th...

This paper presents a solution to the problem of finding the maximum number of linearly independent vector fields that can be placed on a sphere. To produce the correct upper bound, we make use of K-theory. After briefly recapitulating the basics of K-theory, we introduce Adams operations and compute the K-theory of the complex and real projective spaces. We then define the characteristic class...

We introduce a scheme of control polygons to design topological skeletons for vector fields of arbitrary topology. Based on this we construct piecewise linear vector fields of exactly the topology specified by the control polygons. This way a controlled construction of vector fields of any topology is possible. Finally we apply this method for topology-preserving compression of vector fields co...

These notes cover some of the recent developments in the renormalization of quasiperiodic flows. This includes skew flows over tori, Hamiltonian flows, and other flows on T×R. After stating some of the problems and describing alternative approaches, we focus on the definition and basic properties of a single renormalization step. A second part deals with the construction of conjugacies and inva...