Almost all real linear second order ordinary differential equations are solved by geodesic curves in two dimensional Riemannian hyperbolic geometry

Abstract

I show that a real linear second order ordinary differential equation $u^{\prime \prime }\left(x\right)+h\left(x\right)u\left(x\right)=0$, with differentiable $h(x)$, locally admits two linearly independent solutions which exist on an open interval around any $x_0\in \mathbb{R}$: \[ u_\mathtt{top}(x)=\exp\left[\int_{x_0}^{x}\!\!dξ\,Φ\left(ξ\right)\frac{Φ'\left(ξ\right)-\sqrt{\left[h\left(ξ\right)-Φ^{2}\left(ξ\right)\right]^{2}+\left[Φ'\left(ξ\right)\right]^{2}}}{h\left(ξ\right)-Φ^{2}\left(ξ\right)}\right], \] \[ u_\mathtt{bot}(x)=\exp\left[\int_{x_0}^{x}\!\!dξ\,Φ\left(ξ\right)\frac{Φ'\left(ξ\right)+\sqrt{\left[h\left(ξ\right)-Φ^{2}\left(ξ\right)\right]^{2}+\left[Φ'\left(ξ\right)\right]^{2}}}{h\left(ξ\right)-Φ^{2}\left(ξ\right)}\right], \] where $\Phi (x)$ is any geodesic curve in a two dimensional hyperbolic geometry of a Riemannian manifold ${\mathbb{M}}_h$, which is non-vertical at $x_0$. I define ${\mathbb{M}}_h$ to be an upper half plane $\{\left(x,\Phi \right)\in {\mathbb{R}}^2|\Phi >0\}$, with points in which ${\Phi }^2=h(x)$ being removed, equipped with metric $g_h=\left[{\left(h(x)-{\Phi }^2\right)}^{2}d{x}^2+d{\Phi }^2\right]/{\Phi }^2$. A non-trivial character of the presented result stems from the fact that $g_h$ is solely defined in terms of the function $h(x)$. I also show that a local diffeomorphism between ${\mathbb{M}}_h$ and Poincaré upper half plane $\mathbb{H}$ is induced by any pair of linearly independent solutions of $u^{\prime \prime }\left(x\right)+h\left(x\right)u\left(x\right)=0$. If this pair is selected to be $u_{\mathtt{top}}(x)$ and $u_{\mathtt{bot}}(x)$, the associate geodesic curve $\Phi (x)$ is mapped to a vertical geodesic curve on $\mathbb{H}$. Thus, I establish a link between linear second order ordinary differential equations and two dimensional hyperbolic geometry.