Torsion of Translation vs. Torsion of Expansion-Rotation

Projective Matrices of functions over a variety (Base variables) are restricted to those base points where the determinant of functions does not vanish. Over the reals, the general situation implies two disconnected matrix components. Over complex variables, the domain can be connected.

The equivalence class of projective matrices can be used as Frame Fields on a domain, defining a global basis for vector spaces over the domain. From this "Repere Mobile" of projective matrices of functions, Cartan methods can be applied to compute the Cartan torsion and Cartan curvature 2-forms (structural equations) induced on subspaces. The domain of support for the projective matrices of non-zero determinant form a space of absolute parallelism. The vector of Cartan Torsion 2-forms and the matrix of Cartan curvature 2-forms are zero on a space of absolute parallelism. This does not mean that the space of absolute parallelism is free of "affine" torsion (not the same as the Cartan torsion 2-forms) or metric curvature. For example, the sphere S3 in 4 dimensions is parallelizable. The subspaces of a space of absolute parallelism may be simple (such as a euclidean space) or complicated with different and distinct topological structures due to non-zero Cartan curvature and Cartan torsion (of the subspaces). The methods in matrix (rather than in tensor) form are to be found at the download projfram.pdf.

The Projective matrices can be constrained to equivalence classes that are of interest to physics. In particular (using R4 over{x,y,z,t} as an example), Projective transformations have 16 functions ( with a family constraint such that det <>0), or 16 "degrees of freedom". Elimination of 3 degrees of freedom (set to zero the first 3 elements of the right hand column) produces what is known as the Affine transformations on R4. The Affine transformation is the epitomy of the sophomore rule, "the absolute velocity is the velocity of the center of mass, plus the velocity about the center of mass". Herein this specialization of projective transformations will be called the "particle Affine map".

However, a different matrix group is given when the first three elements of the bottom row are set to zero. The result is another type of affine map, called herein the "Wave Affine map".

Neither Particle Affine or Wave Affine maps alone (when used as Frame Fields on the domain) produce curvature 2-forms on the subspace of 3D (x,y,z). So points of R4 confined or constrained to PA or to WA equivalence classes are 3D flat. These constrained spaces can admit affine torsion coefficients.

When the map has both Affine and Wave (projective) properties, then curvature can be induced on the 3D subspace (induced by Frame Fields of the projective equivalence class).

The differences between such equivalence classes of Frame Field constraints, and computations of the induced torsion and curvature 2-forms by use of Maple and the matrix methods, will be found in the download affine.pdf.

Also see
Space-Time as a Minkowski Fluid for R4 constrained by the Lorentz equivalence class of Frame Fields, and Shipov and Spaces of Absolute Parallelism