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<p>A link between first-order ordinary differential equations (ODEs) and 2-dimensional Riemannian manifolds is explored. Given a first-order ODE, an associated Riemannian metric on the variable space is defined, and some properties of the resulting surface are studied, in relation to the integrability of the equation. Next, deformations of the associated surfaces are considered. A relation between relative Jacobi fields on the deformed surface and integrability of the ODE is established, showing that this class of vector fields are useful for solving first-order ODEs. As a consequence, it is proven that if the associated surface is of constant curvature, or alternatively it can be deformed into one of constant curvature, then the ODE can be integrated by quadratures. In particular, the search for an integrating factor for the ODE is interpreted as a deformation of the associated surface into a flat one.</p>

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Surfaces Associated with First-Order ODEs

  • 期刊论文

  • 发表日期：

    2025年6月11日

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  • A. J. Pan-Collantes,
  • J. A. Álvarez-García

摘要

A link between first-order ordinary differential equations (ODEs) and 2-dimensional Riemannian manifolds is explored. Given a first-order ODE, an associated Riemannian metric on the variable space is defined, and some properties of the resulting surface are studied, in relation to the integrability of the equation. Next, deformations of the associated surfaces are considered. A relation between relative Jacobi fields on the deformed surface and integrability of the ODE is established, showing that this class of vector fields are useful for solving first-order ODEs. As a consequence, it is proven that if the associated surface is of constant curvature, or alternatively it can be deformed into one of constant curvature, then the ODE can be integrated by quadratures. In particular, the search for an integrating factor for the ODE is interpreted as a deformation of the associated surface into a flat one.