Surface deformation of ODEs

Given a first-order ODE $u^{\prime }=\varphi (x,u)$, the associated surface can be deformed while preserving $\parallel A\parallel =1$ and $A\perp {\partial }_u$. The family of metrics is parametrized by $ϵ(x,u)\in C^{\mathrm{\infty }}(U)$:

with orthonormal frame $\{A,e^{-ϵ}{\partial }_u\}$. The original surface corresponds to $ϵ=0$.

Curvature under deformation

The Gaussian curvature of $S_{ϵ}$ is

The key observation: the second-order operator acting on normal variations along the geodesic flow $A$ factorises as

into two first-order deformation operators (they encode the effect of $ϵ$ through ${\mathrm{\Delta }}_{ϵ}$):

This is the same algebraic structure as ladder operators in SUSY QM. The function ${\mathrm{\Delta }}_{ϵ}$ plays the role of a superpotential: $A^2+{\mathcal{K}}_{ϵ}$ is a Schrödinger-type operator factorised as $(A+{\mathrm{\Delta }}_{ϵ})(A-{\mathrm{\Delta }}_{ϵ})$.

Why the factorisation matters

The Jacobi equation for a normal component $\delta$ along $A$ is $A^2(\delta )+{\mathcal{K}}_{ϵ}\delta =0$, i.e.

Hence either ${\mathfrak{S}}_{ϵ}(\delta )=0$ or ${\mathfrak{T}}_{ϵ}({\mathfrak{S}}_{ϵ}(\delta ))=0$, and each condition yields an integrating factor (see Lemma in paper):

• ${\mathfrak{T}}_{ϵ}(h)=0⟺e^{ϵ}h$ is an integrating factor for the ODE
• ${\mathfrak{S}}_{ϵ}(h)=0⟺e^{ϵ}{h}^{-1}$ is an integrating factor

Thus a non-trivial relative Jacobi field on any $S_{ϵ}$ unlocks integrability by quadratures. In particular, constant curvature ${\mathcal{K}}_{ϵ}=k$ makes $\delta$ explicit (solve ${\delta }^{″}+k\delta =0$), so the deformed surface always produces a Jacobi field — see constant curvature integrability.

Conceptual summary

Deformation expands the search space: instead of requiring the original surface to have special geometry (flatness, constant curvature), we can deform it via $ϵ$ until $S_{ϵ}$ acquires the needed property. The deformation operators ${\mathfrak{T}}_{ϵ},{\mathfrak{S}}_{ϵ}$ track how the curvature changes and directly factor the Jacobi operator, linking geometry to integrability.

Related notes: relative Jacobi field, constant curvature integrability, Riemannian metric for a first-order ODE, Gauss curvature of an ODE-surface

Reference: @panalvarez2023surfaces