  We consider the consistency condition for ``strong'' sustained rapid turn
inflation with third order slow roll (SRRT) in two-field cosmological models
with oriented scalar manifold as a geometric PDE which constrains the metric
and potential of such models. When supplemented by appropriate boundary
conditions, the equation determines one of these objects in terms of the other
and hence selects ``fiducial'' models for strong SRRT inflation. When the
scalar potential is given, the equation can be simplified by fixing the
conformal class of the scalar field metric (equivalently, fixing a complex
structure which makes the scalar manifold into a complex Riemann surface). Then
the consistency equation becomes a contact Hamilton-Jacobi PDE which determines
the scalar field metric within the given conformal class. We analyze this
equation with standard methods of PDE theory, discuss its approximation near a
nondegenerate critical point of the scalar potential and extract natural
asymptotic conditions for its solutions at such points. We also give numerical
examples of solutions to a simple Dirichlet problem. For the case of elliptic
curves relevant to two-field axion cosmology, we determine the general
symmetry-adapted solution of the equation for potentials with a single charge
vector.
