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(x1,..., xn) to be totally geodesic in a torsion-free con-nection 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+ 1) 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