Riemannian metric for a first-order ODE

Given a first-order ODE

u^{'} = ϕ (x, u)

with associated vector field $A = \partial_{x} + ϕ \partial_{u}$ , we define a Riemannian metric on the $(x, u)$ -plane by

g = (1 + ϕ^{2}) d x \otimes d x - ϕ d x \otimes d u - ϕ d u \otimes d x + d u \otimes d u,

or in matrix form

g = (\begin{matrix} 1 + ϕ^{2} & - ϕ \\ - ϕ & 1 \end{matrix}) .

Properties:

$∥ A ∥ = 1$
$∥ \partial_{u} ∥ = 1$
$A ⊥ \partial_{u}$

Thus ${A, \partial_{u}}$ is an orthonormal frame, with dual coframe

ω^{1} = d x, ω^{2} = - ϕ d x + d u .

The geodesics of this surface are closely related to the solutions of the ODE: when $ϕ_{u} \neq 0$ , a curve $γ_{f} (t) = (t, f (t))$ is a geodesic iff $f$ solves the ODE. Consequently, $A$ is a geodesic vector field.

There are two kinds of geodesics on this surface:

ODE-solution geodesics — integral curves of $A$ , graphs of solutions of $u^{'} = ϕ$ . These are the geodesics in the $A$ -direction.
Non-ODE geodesics — geodesics whose tangent is not parallel to $A$ . They are characterized as pregeodesics (Pan-Álvarez & Álvarez-García 2023, Prop. 4.3): a curve $γ_{f} (t) = (t, f (t))$ is a pregeodesic iff

u^{″} = A (ϕ) - ϕ_{u} (u^{'} - ϕ)^{3} .

When $ϕ_{u} \neq 0$ , any geodesic can be reparametrized to a curve of this form, so this equation describes all non-ODE geodesics up to reparametrization.

The same two-type classification extends to the complex case on $C^{2}$ : there are ODE-solution geodesics ( $Δ = z_{2}^{'} - z_{1}^{'} f = 0$ , $z_{1}$ affine) and non-ODE geodesics ( $Δ \neq 0$ ), governed by the coupled system $z_{1}^{″} = - f_{z_{2}} Δ^{2}, Δ^{'} = z_{1}^{'} f_{z_{2}} Δ$ .

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