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We construct metric connection associated with a first-order differential equation by means of the generator set of a Pfaffian system on a submanifold of an appropriate first-order jet bundle. We firstly show that the inviscid and viscous Burgers’ equations describe surfaces attached to an ODE of the form

A simple dynamical transport phenomenon such as transport of a pollutant in a fluid, transportation of information, wave propagation, traffic flow, population density, chemical reactions, and gas dynamics is described by a certain first-order partial differential equation (PDE) of space and time variables, the so-called transport equation [

Geometric study of differential equations is a notable subject in differential geometry and dates back to Tresse, Lie, and Cartan [

With this regard, in this paper, we construct

In this subsection, we shall give the basic notions of the formulation of the Riemannian geometry in an orthonormal frame. We assume that all considerations are local and all objects are smooth. For details in the subject we refer to [

Let

Consider a first-order ODE of the form

Similar to ODEs, any first-order PDE with one dependent and two independent variables, in the form

Consider three-dimensional submanifold

The vector fields

We shall define a Riemannian metric on a coordinate neighbourhood of

In the formulation that we use, the one-forms

The structure equations for the coframe field

Accordingly, from definition (

Curvature of the connection

Gaussian curvature of the metric connection (

This representation for the curvature is worthwhile to word in some aspects: By means of expression (

From (

A linear first-order equation

The proposition follows by a direct calculation by taking

On the other hand, to see what happens with our representation if the viscosity is added, identity (

Another interesting integrable model is Korteweg-de Vries (KdV) equation

Analogues to the previous subsection, we shall define a Riemannian metric on a coordinate neighbourhood of

In order to construct

The

By definition (

As it is seen directly from (

Any integral surface of (

Due to

Now, let us state and prove the theorem manifesting that the scalar curvature of the Riemannian manifold

The scalar curvature of the Riemannian manifold

The proof is based on the straightforward calculation. From the second structure equations

As a consequence of this theorem, the following are immediate:

Scalar curvature of the Riemannian manifold

For all points on an integral manifold of (

It is also suitable to note here that a Riemannian manifold

Riemannian manifold

It follows from this corollary that in the case of a flat manifold there exists a unique integral surface passing through each point of

The authors declare that there are no conflicts of interest regarding the publication of this paper.

^{2}y/dx

^{2}= F(x, y, dy/dx)