On the transmutation and annihilation of pencil-generated spacetime dimensions

A spacetime manifold generated by the pencil of conics defined by two distinct pairs of complex-conjugated lines and a pair of real lines is considered. The manifold, originally endowed with two spatial and two temporal dimensions, is shown to substantially change its properties as we change the affine properties of the pencil. Two kinds of transformation are of particular interest. A dimensionality-preserving process, characterized by the transmutation of a temporal coordinate into a spatial one and leading to familiar (3+1)D spacetime, and a dimensionality-reducing scenario, featuring simultaneous annihilation of one temporal and one spatia dimension and ending up with a (1+1)D spacetime. A striking difference between the nature of temporal and spatial is revealed; whereas we find purely spatial manifolds, those comprising exclusively temporal dimensions donot exist.