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

Abstract

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 / = ( , ) 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.

A geometric description for simple and damped harmonic oscillators

“…By this means, we consider the equations of harmonic motions in the framework of Riemannian geometry associated with second-order ODEs. This work can be seen as the continuation of the recent work of the authors dealing with a first order differential equation as a curved space in jet manifold in the context of Riemannian geometry [26]. Our consideration in this paper is based on constructing Riemannian metric and connection 1-form from the exterior differential system encoding a second-order ODE in the secondorder jet space [2].…”

“…As it is considered in [2,26], an ODE can be treated as a submanifold of an appropriate jet space and Riemannian structure may be given by the sum of squares of the elements of the exterior differential system encoding given differential equation. Our aim in this paper is to uncover the geometric role of the physical quantities oscillation frequency and damping coefficient in equations of harmonic motions.…”

Geometry of submanifolds of all classes of third-order ODEs as a Riemannian manifold

“…It has been shown in Bayrakdar et al [6], the Gaussian curvature of a surface corresponding to a first-order ODE is given by certain Burgers' equations and it is possible to obtain two-dimensional spaces of constant curvature from some integrable PDEs.…”