u-independent curvature ODEs

A first-order ODE $u^{'} = ϕ (x, u)$ whose associated surface has Gaussian curvature depending only on $x$ :

K (x, u) = κ (x) .

Why this matters

The Jacobi equation for the normal component $δ$ along the geodesic flow $A$ is

A^{2} (δ) + K δ = 0.

When $K$ is $u$ -independent, $A$ acts as $d / d x$ along integral curves $(x, u (x))$ , so the equation reduces to an ordinary second-order linear ODE in $x$ alone:

δ^{″} (x) + κ (x) δ (x) = 0.

This is always solvable in principle — every solution $δ$ yields a relative Jacobi field $δ \partial_{u}$ (on the original surface $S_{0}$ ).

Results

ODE embedding into linear equations — every solution of the nonlinear ODE also satisfies a fixed second-order linear equation $u^{″} + κ (x) u = c (x)$ . This embeds the solution set into an affine space $\ker (d^{2} / d x^{2} + κ (x))$ , linking the ODE to linear theory and differential Galois theory.
curvature-dependent Riccati connection — the divergence $p = ϕ_{u}$ along solutions satisfies $p^{'} + p^{2} + κ (x) = 0$ , a Riccati equation that linearises to $y^{″} + κ (x) y = 0$ . This bridges the u-independent condition to the same linear operator $d^{2} / d x^{2} + κ (x)$ .

Both results flow from the curvature condition $K = κ (x)$ : the nonlinear ODE inherits the linear structure of $L = d^{2} / d x^{2} + κ (x)$ .

Relation to other curvature classes

constant curvature integrability: $κ (x) = k \in R$ — the ODE is integrable by quadratures. The $u$ -independent case is a strict generalisation: $κ (x)$ need not be constant, yet a Jacobi field still exists.
flatness-integrability theorem: $κ (x) = 0$ — the associated surface is flat, the simplest instance of $u$ -independent curvature.
surface deformation of ODEs: if $K_{0}$ depends on $u$ , we may deform via $ϵ$ until $K_{ϵ}$ becomes $u$ -independent (or constant), unlocking a Jacobi field.

Reference: @panalvarez2023surfaces
@panalvarez2024integration