Higher order obstructions to Riccati-type equations

We develop new techniques in order to deal with Riccati-type equations, subject to a further algebraic constraint, on Riemannian manifolds $(M^3,g)$. We find that the obstruction to solve the aforementioned equation has order $4$ in the metric coefficients and is fully described by an homogeneous polynomial in $\mathrm{Sym}^{16}TM$. Techniques from real algebraic geometry, reminiscent of those used for the "PositiveStellen-Satz " problem, allow determining the geometry in terms of the connection coefficients and a class of Hessian-type equations. Analysis of the latter shows flatness for the metric $g$; in particular we complete the classification of asymptotically harmonic manifolds of dimension $3$, establishing those are either flat or real hyperbolic spaces.