Universal geometries underpinning linear second order ordinary differential equations

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 $ \Theta'\left(x\right)+\Theta^2\left(x\right)+h(x)=0$ obtained through $u'\left(x\right)=\Theta\left(x\right)u\left(x\right)$ 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 K\"ahler-Norden geometry. The added value of working with real geometry turns out to be that certain two dimensional submanifold of the K\"ahler-Norden manifold render the hyperbolic and the (anti-)de Sitter scenario, both relevant for the real equation.