Geodesic Webs and PDE Systems of Euler Equations

TL;DR

This paper establishes conditions under which webs of hypersurfaces are geodesic or hyperplanar, using systems of generalized Euler equations, and provides explicit solutions in the flat connection case.

Contribution

It derives necessary and sufficient conditions for geodesic webs in torsion-free connections and offers explicit solutions for flat connections.

Findings

Conditions for totally geodesic foliations are characterized by generalized Euler equations.

Explicit solutions for geodesic webs in flat connections are constructed.

Criteria for hyperplanar webs in flat connections are established.

Abstract

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.