The curvature tensor and the einstein equations for a four-dimensional nonholonomic distribution

The space of possible particle velocities is a four-dimensional nonholonomic distribution on a manifold of higher dimension, say, M ⁴ × ?¹. This distribution is determined by the 4-potential of the electromagnetic field. The equations of admissible (horizontal) geodesics for this distribution are the same as those of the motion of a charged particle in general relativity theory. On the distribution, a metric tensor with Lorentzian signature (+, ?, ?, ?) is defined, which gives rise to the causal structure, as in general relativity theory. Covariant differentiation (a linear connection) and the curvature tensor for this distribution are introduced. The Einstein equations are obtained from the variational principle for the scalar curvature of the distribution. It is proved that the Dirac operator for the four-dimensional distribution can be extended to functions defined on the manifold M ⁴ × S ¹, where S ¹ is the circle. For such functions, electric charges are topologically quantized.