Geodesic Webs and PDE Systems of Euler Equations
  We find necessary and sufficient conditions for the foliation defined by level sets of a function f(x_{1},...,x_{n}) to be totally geodesic in a torsion-free connection and apply them to find the conditions for d-webs of hypersurfaces to be geodesic, and in the case of flat connections, for d-webs (d > n) of hypersurfaces to be hyperplanar webs. These conditions are systems of generalized Euler equations, and for flat connections we give an explicit construction of their solutions. 
Vladislav V. Goldberg (New Jersey Institute of Technology, Newark, NJ,   USA) and Valentin V. Lychagin (University of Tromso...