Universal geometries underpinning linear second order ordinary differential equations
Łukasz Rudnicki
TL;DR
The paper generalizes the known link between linear second-order ODEs $u''(z)+h(z)u(z)=0$ and 2D hyperbolic geometry to complex settings by developing three parallel geometric realizations: a Lorentzian 2D (anti-)de Sitter framework, a 2D complex Riemannian framework with a holomorphic metric, and a 4D pseudo-Riemannian Kähler-Norden framework. In each setting, ODE solutions are encoded as geodesics, with Riccati equations $\Theta'+\Theta^2+h=0$ (and their complex analogues) playing a central role in connecting differential and geometric data. The results show that, locally, the complex ODEs correspond to a complex sphere geometry with constant holomorphic curvature $-1$, and in 4D KN geometry the equation is realized as a maximal Einstein metric with two 2D submanifolds reproducing the real hyperbolic and AdS pictures; a singular Riccati–geodesic regime also emerges in the AdS/complex contexts. The work thereby provides a unified local geometric underpinning for complex extensions of Rudnicki's real case, highlighting the deep interplay between ODE theory and multiple geometric formalisms.
Abstract
A deep relationship [arXiv:2503.17816v1] between real linear second order ordinary differential equations $u''\left(x\right)+h\left(x\right)u\left(x\right)=0$, with differentiable $h(x)$, and two dimensional hyperbolic geometry is generalized in a multitude of ways. First, I present an equivalent relationship in which the hyperbolic geometry is replaced by a two dimensional (anti-)de Sitter geometry. I show that this equation everywhere admits a pair of linearly independent solutions locally expressed in terms of an arbitrary non-vertical geodesic curve in this geometry. I also show that every solution of a corresponding Ricatti equation $ Θ'\left(x\right)+Θ^2\left(x\right)+h(x)=0$ obtained through $u'\left(x\right)=Θ\left(x\right)u\left(x\right)$ itself is a geodesic curve in the two dimensional (anti-)de Sitter geometry. Next, after promoting $h(x)$ to a holomorphic function $h(z)$, I express two linearly independent solutions of $u''\left(z\right)+h\left(z\right)u\left(z\right)=0$ in virtually the same way as for the real scenario and hyperbolic geometry. In this case, the curves used to build the solutions are geodesic in a two dimensional complex Riemannian geometry of a sphere. Analogous results for the complex Ricatti equation follow. This geometric interpretation is independent of the function $h(z)$, while the holomorphic metric assumes the same functional form as the hyperbolic metric discovered in [arXiv:2503.17816v1]. Finally, I show that the equation in question is in an equivalent relationship with four dimensional pseudo Riemannian Kähler-Norden geometry. The added value of working with real geometry turns out to be that certain two dimensional submanifold of the Kähler-Norden manifold render the hyperbolic and the (anti-)de Sitter scenario, both relevant for the real equation.