Constant curvature integrability

Theorem. If the deformed surface $S_{ϵ}$ associated to a first-order ODE $u^{\prime }=\varphi (x,u)$ has constant Gaussian curvature ${\mathcal{K}}_{ϵ}=k\in \mathbb{R}$, then the ODE is integrable by quadratures.

Proof. Solving ${\delta }^{″}+k\delta =0$ gives

so $J=\delta e^{-ϵ}{\partial }_u$ is a non-trivial relative Jacobi field on $S_{ϵ}$. The main result on relative Jacobi fields then yields an integrating factor.

The flat case $k=0$ is a particular instance of this result, though it also admits a direct proof (see flatness-integrability theorem).

Related notes: relative Jacobi field, surface deformation of ODEs, flatness-integrability theorem, Gauss curvature of an ODE-surface

Reference: @panalvarez2023surfaces
@panalvarezcurvature