Topological Torsion A^dA # 0

For 1-dimensional subsets, the concept of Frenet torsion is intuitively related to the twisting of the thread that approximates any given space curve. For 2-dimension subsets, note that a twisted tube is different from an untwisted tube, and leads to the concept of a distribution parameter in the differential geometry of surfaces. For 3-dimensional subsets in space-time, things becomes more complex, for now the concept of Topological Torsion leads to visual topological defects, the production of lines of intersection, and separated parts. Topological Torsion does not exist in less than three dimensions. Topological torsion is a "point defect" in three dimensional spaces, and a line "defect" in four dimensional spaces.

Topological Torsion cannot be zero in a chaotic dynamical system. In hydrodynamic systems a non-zero value for the helicity density is a signature of topological torsion. In space time, the helicity density, V .curl V/(V.V), is the fourth component of the Torsion Current.
The arguments above apply to a 4 dimensional variety, like space time, but the concepts can be extended to N forms dA^dA... and N-1 forms A^dA...


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