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Does a Riemann Manifold Possess Energy?

Taking our cue from elasticity theory, the strain can be expressed as the symmetric gradient of a generalized displacement, and the curl of the curl of the strain field. The compatibility condition is a highflautin way of expressing Maxwell's relations that guarantees the existence of a scalar potential.

In linearized elasticity theory, the incompatibility condition is equivalent to Riemann's tensor, and it says that the space is Euclidean. The elastic energy is determined by the difference between two configurations: one perturbed by an external force and the reference configuration. These are the tidal forces, and the compatibility conditions imply that the space in which they operate must be flat.

An analogous situation occurs in electromagnetism. As long as the system is static, the electric field, an example of a displacement field, is the gradient of the Coulomb potential. In order to create a magnetic field, the "tidal" forces must be destroyed by adding the time rate of change of a vector potential which determines the magnetic field.

So the existence of a potential energy associated with the electric field is tantamount to requiring the time rate of change of the vector potential be zero. If the incompatibility condition is not satisfied, there is no potential and no tidal forces. A time dependent vector potential would be incompatible with the existence of tidal forces.

Viewed in these terms, the paper by Eells and Sampson, "Harmonic mappings of Riemann manifolds" appears troublesome. Surely, on can form a bilinear sum of the gradient of the mapping from one Riemann space to another, and its transpose, to be a sort of kinetic energy whose mass would be the metric itself. But the manifolds themselves are not shown to be the operation of a body force that transforms a reference metric into a new one. In other words, curvature is not determined by some external force that does work on the system resulting in the creation of a potential energy. In other words, it does not satisfy Clapeyron's theorem that the total strain energy of a body is equal to the work done by the external forces. The harmonic mapping between two Riemann manifolds occurs not by the work to create one manifold from another, but by a a simple mapping.

This should have a profound effect on the theory of Ricci flows where energetical considerations are introduced into the evolution of the Ricci tensor of positive curvature to prove Poincare's conjecture. Something like a vector potential, that occurs in electromagnetism, is required to treat such a situation. More will be said in a future blog.

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