Riemann spaces associated with the Navier–Stokes equations: a survey of the geometric approach

We survey a geometric approach to the Navier–Stokes equations developed by the author over the past 15 years. The central object is a Riemannian space of 14 dimensions whose metric is Ricci-flat exactly on, and only on, the solutions of the Navier–Stokes system, so that the behaviour of a viscous incompressible fluid becomes a property of curvature. The space decomposes into Eulerian and dual Lagrangian coordinates together with an auxiliary flat part, which exhibits the two classical descriptions of a fluid as two halves of a single geometry. We also recall the associated six-dimensional metric, whose integrability condition is the incompressibility of the fluid; the link with the projective geometry of second-order ordinary differential equations in the sense of E. Cartan; and the use of Cartan’s invariants to construct a three-dimensional analogue of the Taylor–Green vortex. As a complete worked example in which the geometric construction yields analytical information that is hard to obtain directly from the system, the application to the equations of rotation of a rigid body (the Kovalevskaya top) is presented in detail.