Quasigeodesic mapping and Riemannian gauge structure. (English. Russian original) Zbl 0703.53025

Igoshin, V. A.; Shapiro, Ya. L.

Sov. Math., Dokl. 39, No. 2, 351-354 (1989); translation from Dokl. Akad. Nauk SSSR 305, No. 5, 1035-1038 (1989).

A special case of a mapping preserving the trajectories of a second dynamical system is considered. From the viewpoint of principal bundles it leads to the inclusion of an “electromagnetic field” determined by the equation \[ \frac{\nabla}{dt}\frac{dx}{dt}=2F^{\#}(x,\frac{dx}{dt})-1/2 \text{grad} f(x) \] in a Riemannian gauge structure \(k=((\bar M,\bar g),\Phi,M,G)\). The basic result of the paper contains necessary and sufficient conditions for the “electromagnetic field” to be included in a Riemannian gauge structure. It is also proved that the cohomology class [F] \((\in H^ 2(M,R))\) of the inclusion of the “magnetic” field F is an invariant.

Reviewer: J.Mikeš

MSC:

53C10	\(G\)-structures
53B20	Local Riemannian geometry
83C50	Electromagnetic fields in general relativity and gravitational theory

Keywords:

second dynamical system; electromagnetic field; Riemannian gauge structure