Universal geometries underpinning linear second order ordinary differential equations
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915213042778112 |
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| author | Rudnicki, Łukasz |
| author_facet | Rudnicki, Łukasz |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_19415 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Universal geometries underpinning linear second order ordinary differential equations Rudnicki, Łukasz Classical Analysis and ODEs Complex Variables Differential Geometry 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. |
| title | Universal geometries underpinning linear second order ordinary differential equations |
| topic | Classical Analysis and ODEs Complex Variables Differential Geometry |
| url | https://arxiv.org/abs/2503.19415 |