Riemann spaces associated with the Navier-Stokes equations: a survey of the geometric approach
Abstract
We survey a geometric approach to the Navier-Stokes equations developed by the author over the past fifteen years. The central object is a Riemannian space of fourteen dimensions whose metric is Ricci-flat precisely 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 a flat six-dimensional part and two dual quadruples of coordinates, Eulerian and Lagrangian, 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. An application of the same construction to the equations of rotation of a rigid body is indicated.
Related
- Asymptotic behavior of generic Riemann solutions to one-dimensional full compressible Navier-Stokes-Poisson equations
- On the effects of rotation on stationary Navier--Stokes equations in the half spaces in scale-critical spaces
- Stability Analysis of Bounded Solutions to 2-D Incompressible Navier-Stokes Equations in L^2_uloc Spaces
- Blow-Up of Modified Navier--Stokes Equations in Nonhomogeneous Fourier Spaces
- Global regularity and sharp decay rates for the one-dimensional hypo-viscous compressible Navier-Stokes equations in critical Besov spaces
BibTeX
@misc{dryuma2026riemann,
title = {{Riemann spaces associated with the Navier-Stokes equations: a survey of the geometric approach}},
author = {Valerii S. Dryuma},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.13314},
eprint = {2609.13314},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.13314},
}
RIS
TY - UNPB
TI - Riemann spaces associated with the Navier-Stokes equations: a survey of the geometric approach
AU - Valerii S. Dryuma
PY - 2026
DA - 2026/09/10
PB - arXiv
DO - 10.48550/arxiv.2609.13314
UR - https://arxiv.org/abs/2609.13314
AB - We survey a geometric approach to the Navier-Stokes equations developed by the author over the past fifteen years. The central object is a Riemannian space of fourteen dimensions whose metric is Ricci-flat precisely 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 a flat six-dimensional part and two dual quadruples of coordinates, Eulerian and Lagrangian, 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. An application of the same construction to the equations of rotation of a rigid body is indicated.
ER -