On One-Parameter Families of Dido Riemannian Problems

Locally, isoperimetric problems on Riemannian surfaces are sub-Riemannian problems in dimension 3. The particular case of Dido problems corresponds to a class of singular contact sub-Riemannian metrics : metrics which have the charateristic vertor field as symmetry. We give a classification of the generic conjugate loci (i.e. classification of generic singularities of the exponential mapping) of a 1-parameter family of 3-d contact sub-Riemannian metrics associated to a 1-parameter family of Dido Riemannian problems.