Projectively and affinely invariant PDEs on hypersurfaces

In Communications in Contemporary Mathematics 24 3, (2022),the authors have developed a method for constructing G -invariant partial differential equations (PDEs) imposed on hypersurfaces of an $(n+1)$ -dimensional homogeneous space $G/H$ , under mild assumptions on the Lie group G . In the present paper, the method is applied to the case when G=PGL(n+1) (respectively, G=Aff(n+1) ) and the homogeneous space $G/H$ is the $(n+1)$ -dimensional projective Pⁿ⁺¹ (respectively, affine Aⁿ⁺¹ ) space, respectively. The main result of the paper is that projectively or affinely invariant PDEs with n independent and one unknown variables are in one-to-one correspondence with invariant hypersurfaces of the space of trace-free cubic forms in n variables with respect to the group CO(d,n-d) of conformal transformations of Rd,n-d .