On functions with a conjugate (Q748359)

A smooth function \(f\) defined on a 2-dimensional Riemannian manifold \(M\) admits a conjugate -- that is, a function \(g\) on \(M\) whose gradient has same length and is orthogonal to the gradient of \(f\) -- if and only if \(f\) is harmonic. The authors investigate how such a statement may be extended to the case where the dimension of \(M\) is higher, in particular when it is an open subset of \(\mathbb R^3\). They provide necessary and sufficient conditions for a function \(f\) to have a conjugate, in terms of inequalities satisfied by particular conformal differential invariants. Those are expressed as polynomials in the derivatives of \(f\). The authors then discuss special cases involving the multiplicity of conjugates and provide examples.