Parallelogram space (Q1893822)

The geometrical interpretation of gravitation can be formulated with two main operations: measuring lengths of intervals and parallel transfer of a tensor along a curve. In the standard theory, the parallel transfer occurs with a symmetric connection, hence with zero torsion. Recently, however, many changes to the standard theory have included nonzero torsion. While torsion may be useful, it is well known that torsion acts to pull apart parallelograms. In an affine space with torsion, starting at the endpoint of one short geodesic interval and following along it and the three remaining sides of two pairs of parallel intervals does not bring one back to the original point. The gap found in such a construction is linear in the torsion components. Thus to make small parallelograms closed and therefore make them like the parallelograms of Euclidean spaces, the torsion in an affine space has to vanish. In an attempt to construct a parallelogram, the author concentrates on labeling points in an \(n\)-dimensional space by taking one set of coordinates for each point. Then different paths from an origin to a point give that point different ``coordinates''. Since this changes the meaning of the term ``coordinates'', the author introduces the new quantities ``\(p\)- coordinates'' for their path dependence to define parallelogram spaces. In the parallelogram space the author derives an expression for the round trip changes along a small closed curve and considers vector- and tensor- like quantities, \(p\)-vectors, \(p\)-tensors, and \(p\)-curvatures.