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 2forms (structural
equations) induced on subspaces. The domain of support for the projective
matrices of nonzero determinant form a space of absolute parallelism. The
vector of Cartan Torsion 2forms and the matrix of Cartan curvature 2forms 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 2forms) 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 nonzero 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 2forms 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 2forms by use of Maple and the matrix methods, will be found in the download affine.pdf.
Also see
