JGSP 38 (2015) 1–23
APPLICATIONS OF THE LOCAL ALGEBRAS OF VECTOR FIELDS
TO THE MODELLING OF PHYSICAL PHENOMENA
IGOR V. BAYAK
Communicated by Izu Vaisman
Abstract. In this paper we discuss the local algebras of linear vector fields that can
be used in the mathematical modelling of physical space by building the dynamical
flows of vector fields on eight-dimensional cylindrical or toroidal manifolds. It is
shown that the topological features of the vector fields obey the Dirac equation
when moving freely within the surface of a pseudo-sphere in the eight-dimensional
pseudo-Euclidean space.
MSC : 57R25
Keywords : vector fields, algebra of linear vector fields, dynamic flow, topological
features
Contents
1
Introduction
1
2
Local Algebra of Vector Fields
4
3
The Geometry of the Algebra of Vector Fields
9
4
The Dynamics of the Algebra of Vector Fields
14
References
23
1. Introduction
The paper contains a collection of algebraic, geometric and dynamical facts concerning linear vector fields on simple locally affine manifolds (R n , cylinders, tori)
where the main ingredient is the algebra defined by these vector fields with the
product X ? Y = ∇X Y , where ∇ is the flat, torsionless connection of the locally
affine structure of the manifold.
doi: 10.7546/jgsp-38-2015-1-23
1