  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, including a connection between
Jacobi fields and Lie point symmetries. In particular, it is proven that if the
associated surface is flat, then the ODE can be integrated by quadratures.
  Next, deformations of the associated surfaces are considered. A relation
between some Jacobi fields on the deformed surface and integrability of the ODE
is established, showing that there is a class of vector fields, beyond Lie
point symmetries, which are useful for solving first-order ODEs. As a result,
it is concluded that the deformation into a constant curvature surface leads to
the integrability of the given ODE.
