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Curvature tensors of higher-spin gauge theories have been known for sometime. In the past, they were postulated using a generalization of the symmetryproperties of the Riemann tensor (curl on each index of a totally symmetricrank-$n$ field for each spin-$n$). For this reason they are sometimes referredto as the generalized 'Riemann' tensors. In this article, a method for derivingthese curvature tensors from first principles is presented; the derivation iscompleted without any a priori knowledge of the existence of the Riemanntensors or the curvature tensors of higher-spin gauge theories. To perform thisderivation, a recently developed procedure for deriving exactly gauge invariantLagrangian densities from quadratic combinations of $N$ order of derivativesand $M$ rank of tensor potential is applied to the $N = M = n$ case under thespin-$n$ gauge transformations. This procedure uniquely yields the Lagrangianfor classical electrodynamics in the $N = M = 1$ case and the Lagrangian forhigher derivative gravity (`Riemann' and `Ricci' squared terms) in the $N = M =2$ case. It is proven here by direct calculation for the $N = M = 3$ case thatthe unique solution to this procedure is the spin-3 curvature tensor and itscontractions. The spin-4 curvature tensor is also uniquely derived for the $N =M = 4$ case. In other words, it is proven here that, for the most generallinear combination of scalars built from $N$ derivatives and $M$ rank of tensorpotential, up to $N=M=4$, there exists a unique solution to the resultingsystem of linear equations as the contracted spin-$n$ curvature tensors.Conjectures regarding the solutions to the higher spin-$n$ $N = M = n$ arediscussed.

Curvature tensors of higher-spin gauge theories derived from general Lagrangian densities