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Advances in Mathematical Physics
Volume 2018 (2018), Article ID 7590847, 8 pages
https://doi.org/10.1155/2018/7590847
Research Article

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

1Department of Physics, Ege University, 35040 İzmir, Turkey
2Department of Mathematics, Akdeniz University, 07058 Antalya, Turkey

Correspondence should be addressed to Z. Ok Bayrakdar

Received 12 October 2017; Revised 13 December 2017; Accepted 21 December 2017; Published 8 February 2018

Academic Editor: Boris G. Konopelchenko

Copyright © 2018 Z. Ok Bayrakdar and T. Bayrakdar. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.