Relative Jacobi field

Let $X$ be a geodesic vector field on a surface $S$. A vector field $J$ on $S$ is a Jacobi field relative to $X$ if its restriction to every integral curve $\gamma$ of $X$ is a Jacobi field along $\gamma$. Equivalently,

For the surface associated to $u^{\prime }=\varphi (x,u)$, with $A$ as geodesic vector field:

• Lie point symmetries orthogonal to $A$ are relative Jacobi fields.
• A symmetrizing factor $\delta$ for ${\partial }_u$ satisfies $A^2(\delta )+\mathcal{K}\delta =0$.

The operator $A^2+\mathcal{K}$ acting on the ${\partial }_u$-component $\delta$ is a Schrödinger-type equation that decomposes into two first-order operators (see Jacobi operator factorization).

Main result

Theorem. The knowledge of a non-trivial relative Jacobi field on the original surface $S_0$ or on any deformed surface $S_{ϵ}$ yields an integrating factor for the ODE, hence integrability by quadratures.

Corollary (constant curvature integrability). If ${\mathcal{K}}_{ϵ}$ is constant ($k\in \mathbb{R}$), solving ${\delta }^{″}+k\delta =0$ gives $\delta (x)$, and $J=\delta e^{-ϵ}{\partial }_u$ is a non-trivial relative Jacobi field. The flat case $k=0$ (flatness-integrability theorem) is the simplest instance.

In the curvature class $\mathcal{K}(x,u)=\kappa (x)$, every solution $\delta$ of ${\delta }^{″}+\kappa (x)\delta =0$ produces a relative Jacobi field $\delta e^{-ϵ}{\partial }_u$.

Related notes: Jacobi operator factorization, surface deformation of ODEs, constant curvature integrability, flatness-integrability theorem, Jacobi equation

Reference: @panalvarez2024integration