Gauss curvature of an ODE-surface

For the surface associated to a first-order ODE $u^{\prime }=\varphi (x,u)$, the Gaussian curvature is

where $A={\partial }_x+\varphi {\partial }_u$.

This formula implies that first-order ODEs with curvature $\mathcal{K}(x,u)$ satisfy the inhomogeneous inviscid Burgers equation

Constant curvature examples:

• $\mathcal{K}=1$: $u^{\prime }=-1+\sqrt{1-(x+u{)}^2}$ (spherical geometry)
• $\mathcal{K}=-1$: $u^{\prime }=-1+\sqrt{1+(x+u{)}^2}$ (hyperbolic geometry)

Related notes: constant curvature integrability, flatness-integrability theorem, surface deformation of ODEs, curvature-dependent Riccati connection, u-independent curvature ODEs

Reference: @panalvarez2023surfaces
@panalvarezcurvature