Abstract
A geometric approach to the study of natural two-component generalizations of the periodic Hunter–Saxton is presented. We give rigorous evidence of the fact that these systems can be realized as geodesic equations with respect to symmetric linear connections on the semidirect product of a suitable subgroup of the diffeomorphism group of the circle \({{\text{\sc Diff}}(\mathbb{S})}\) with the space of smooth functions on the circle. An immediate consequence of this approach is a well-posedness result of the corresponding Cauchy problems in the smooth category.
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