Topological Torsion

Consider an evolutionary process that visually mimics a fluid flow. Streak line measurements indicate that in certain situations (streamline flow) the flow lines can be transversely connected in a global manner. The connecting surface (often of equal time points) is orthogonal to the flow lines. The velocity field in such a case is said to be integrable, and the flow lines are proportional to a gradient of that single scalar function, which represents the normal to the transversal surface. For such systems the Action 1-form is at most of Pfaff dimension 2. A^dA=0.

The 1-form is said to satisfy the Frobenius theorem of unique integrability.

The Topological Torsion is ZERO.

Now consider those Action 1-forms for which the Frobenius condition is not satisfied. Then it is possible that the flow lines can be determined in terms of the intersection of TWO surfaces (not a unique single surface). In such a case the 3-form A^dA is not zero. The flow has Topological Torsion. If the two surfaces have a cuspoidal contact, the line(s) of intersection form an edge of regression. It is remarkable that linear displacements along the line generated as an edge of regression if not perfect will extend onto the neighborhood of (say) the top surface, where linear displacements in the reverse direction will extend onto the neighborhood of the other (bottom) surface. Motion in foreward direction is not the same as motion in the backwards direction. Truely an indication of irreversibility, which only occurs when A^dA<>0 and dA^dA<>0.

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