Generic bifurcations of framed curves in a space form and their envelopes

The generic singularities and bifurcations are classified for one-parameter families of curves with frames in the space forms E n + 1 , S n + 1 , H n + 1 . Two kinds of frames are considered; adapted frames and osculating frames. In particular, we give the classification results on the singularities of envelopes associated to framed curves. The associated envelopes and their singularities are classified. characterised in term of geometric invariants of framed curves. We apply to the global problem of framed curves and to the extension problem of surfaces with boundaries in three space. generalising the results obtained in Ishikawa (2010) [12] .