  In this paper we construct a functor from the category of two-dimensional
Riemannian manifolds to the category of three-dimensional manifolds with
generalized metric tensors. For each two-dimensional oriented Riemannian
manifold $(M,g)$ we construct a metric tensor $\hat g$ (in general, with
singularities) on the total space $SO(M,g)$ of the principal bundle of the
positively oriented orthonormal frames on $M$. We call the metric $\hat g$ the
Wagner lift of $g$. We study the relation between the isometry groups of
$(M,g)$ and $(SO(M,g),\hat g)$. We prove that the projections of the geodesics
of $(SO(M,g),\hat g)$ onto $M$ are the curves which satisfy the equation
\begin{equation*} \nabla_{\frac{d\gamma}{dt}}\frac{d\gamma}{dt} = C K J
(\dot\gamma) - C^2 K grad K, \end{equation*} where $K$ is the curvature of
$(M,g)$, $J$ is the operator of the complex structure associated with $g$, and
$C$ is a constant. We find the properties of the solutions of this equation, in
particular, for the case when $(M,g)$ is a surface of revolution.
