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We prove that both local and non-local formulations of the Degasperis-Procesiequation possess a pseudospherical nature. As a result, solutions determined byCauchy problems with non-trivial initial data and a minimal specific regularitydefine an orthonormal coframe for a pseudospherical metric within a designatedstrip. While the region is entirely described by the initial condition, theone-forms are determined by the corresponding solutions. Moreover, we establishthat any non-trivial initial condition yields a second fundamental form, whichis locally determined within the strip. The surface may collapse within afinite-height strip when the initial condition leads to wave breaking. Wedetermine conditions for the coframe to be defined on the upper plane. Weinvestigate possible integrability structures related to the triad offundamental one-forms.

Local isometric immersions and breakdown of manifolds determined by Cauchy problems of the Degasperis-Procesi equation