Correction to: A Geometric View on the Generalized Proudman-Johnson and r-Hunter-Saxton Equations. A Geometric View on the Generalized Proudman–Johnson and r-Hunter–Saxton Equations (Journal of Nonlinear Science, (2022), 32, 1, (17), 10.1007/s00332-021-09775-5)

The article “A geometric view on the generalized Proudman-Johnson and r-Hunter-Saxton equations” Bauer et al. (2022) concerns the study of two families of equations: the generalized inviscid Proudman-Johnson equation, and the r-Hunter-Saxton equation. There, we investigate these equations both on the real line and the circle and claim that they are equivalent for both cases. This statement was wrong, as the equations are only equivalent when considered on the real line. As a consequence the results on the circle (Section 3) are only valid for the r-Hunter-Saxton equations and do not hold for the generalized inviscid Proudman-Johnson equations. The results on the real line (Section 2) are valid as written.