# Airy stress, Beltrami stress and gauge fields

The following problem comes from the theory of elasticity, but reduces to a pure geometric problem. Consider a $$d$$-dimensional Riemannian manifold $$(M,g)$$ with boundary representing the intrinsic geometry of an elastic body. A configuration of that body is an immersion $$f:M\to R^d$$. For simplicity assume that $$M$$ is compact, orientable and simply-connected.

Equilibrium configurations are critical points of an energy functional; the Euler-Lagrange equation is of the form $$\delta^\nabla \tau = 0,$$
where $$\tau\in\Omega^1(M;f^*TR^d)$$ is the tension field and $$\delta^\nabla$$ is the covariant codifferential associated with the induced connection on $$f^*TR^d$$. The tension field $$\tau$$ is determined by the precise form of the energy functional. A typical choice of the energy functional yields $$\tau = df((df)^Tdf - Id_{TM}).$$

Regardless of the form of the energy functional a tension field $$\tau$$ satisfying the Euler-Lagrange equation can be represented as $$\tau = df\circ \delta^\nabla(\delta^\nabla\Psi)^T\circ(df^Tdf)^{-1} \, \det df,$$ where $$\Psi\in\Omega^2(M;\wedge^2 TM)$$ and the connection here is the pullback of the Euclidean connection on $$TM$$. The stress function" $$\Psi$$ is of the type of a curvature operator, i.e., has $$d^2(d^2-1)/12$$ degrees of freedom. In turn, $$\tau$$ determines $$df^Tdf$$ which determines the pullback metric on $$TM$$; since the latter must be flat, one obtains a system of $$d^2(d^2-1)/12$$ compatibility equations for the vanishing of the Riemann curvature tensor of the pullback metric. These equations are highly nonlinear, but this doesn't matter for the sake of the ensuing discussion.

For $$d=2$$, $$\Psi$$ can be represented by a scalar field (known as the Airy stress function) and one obtains a single scalar compatibility condition for the vanishing of the Gaussian curvature of the pullback metric. For $$d\ge 3$$, one has a gauge freedom, as the operator $$\Psi \mapsto \delta^\nabla(\delta^\nabla \Psi)^T$$ has a non-trivial kernel. For $$d=3$$, this has been known for over a century, yielding the so-called Maxwell and Morera representations of the stress function.

I am trying to figure out the reduced representation in general dimension. The first two questions that come in mind are:

1. What is the symmetry at the heart of this gauge freedom?
2. Is there a natural coordinate-free way to obtain a reduced set of compatibility equations for a reduced representation of $$\Psi$$.