On equivariant binary differential equations

TL;DR

This paper studies symmetries in binary differential equations (BDEs), revealing how group actions affect solutions and providing algebraic methods to classify symmetric quadratic forms, with applications to equivariant BDEs under orthogonal groups.

Contribution

It introduces a new algebraic framework for understanding symmetries in BDEs and develops an algorithm for classifying equivariant quadratic forms under compact group actions.

Findings

Derived a formula relating symmetry group homomorphisms in BDEs.

Developed an algorithm to compute invariant quadratic forms.

Classified equivariant quadratic 1-forms for subgroups of O(2).

Abstract

This paper introduces the study of occurrence of symmetries in binary differential equations (BDEs). These are implicit differential equations given by the zeros of a quadratic 1-form, $a(x,y)dy^2 + b(x,y)dxdy + c(x,y)dx^2 = 0,$ for $a, b, c$ smooth real functions defined on an open set of $\mathbb{R}^2$. Generically, solutions of a BDE are given as leaves of a pair of foliations, and the appropriate way to define the action of a symmetry must depend not only whether it preserves or inverts the plane orientation, but also whether it preserves or interchanges the foliations. The first main result reveals this dependence, which is given algebraically by a formula relating three group homomorphisms defined on the symmetry group of the BDE. The second main result adapts algebraic methods from invariant theory for representations of compact Lie groups on the space of quadratic forms on $\mathbb{R}^n$, $n \geq 2$. With that we obtain an algorithm to compute general expressions of quadratic forms. Now, symmetric quadratic 1-forms are in one-to-one corrspondence with equivariant quadratic forms on the plane, so these are treated here as a particular case. We then apply the result to obtain the general forms of equivariant quadratic 1-forms under each compact subgroup of the orthogonal group $\mathbf{O}(2)$.