Burgers' Equations in the Riemannian Geometry Associated with First-Order Differential Equations

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 dx/dt = u(t,x) with certain Gaussian curvatures. In the case of PDEs, we show that the scalar curvature of a three-dimensional manifold encoding a system of first-order PDEs is determined in terms of the integrability condition and the Gaussian curvatures of the surfaces corresponding to the integral curves of the vector fields which are annihilated by the contact form. We see that an integral manifold of any PDE defines intrinsically flat and totally geodesic submanifold.