Relations between Bochner identities and higher-dimensional Grushin Bochner formulas

Investigate the deeper relations between the Bochner identities for harmonic mappings in Grushin planes and the previously known Bochner identities, and establish Bochner identities for harmonic mappings in higher-dimensional Grushin spaces.

Background

The paper derives a Bochner identity for harmonic mappings from an alpha-Grushin plane to a beta-Grushin plane. The resulting formula contains terms involving the horizontal Hessian, commutators of the Grushin vector fields, and interactions between the two component functions of the mapping.

The authors explicitly identify two unresolved directions: clarifying how their identity is related to analogous Bochner identities in other geometric settings, and extending the theory from Grushin planes to higher-dimensional Grushin spaces. These questions are included because the paper does not resolve either the deeper comparative relationship or the higher-dimensional extension.