Hyperspin manifolds

Riemannian manifolds are but one of three ways to extrapolate from four-dimensional Minkowskian manifolds to spaces of higher dimension, and not the most plausible. If we take seriously a certain construction of time space from spinors, and replace the underlying binary spinors byN-ary hyperspinors with new internal components besides the usual two external ones, this leads to a second line, the hyperspin manifolds and their tangent spaces , different in structure and symmetry group from the Riemannian line, except that the binary spaces (Minkowski time space) and (Minkowskian manifold) lie on both. and have dimensionn=N ². In hyperspin manifolds the energies of modes of motion multiply instead of adding their squares, and theN-ary chronometric form is not quadratic, butN-ic, with determinantal normal form. For the nine-dimensional ternary hyperspin manifold, we construct the trino, trine-Gordon, and trirac equations and their mass spectra in flat time space. It is possible that our four-dimensional time space sits in a hyperspin manifold rather than in a Kaluza-Klein Riemannian manifold. If so, then gauge quanta with spin-3 exist.