A generalization of Frenet's frame for non-degenerate quadratic forms with any index

Contents 1 Introduction 1.1 Statement of the problem 1.2 Some définitions and notations 1.3 Statement of the main resuit 2 A preliminary study of kernels 3 The geome trical fondations of the construction 3.1 First step: when the kernel grows 3.2 Second step: when the kernel decreases 4 The construction of our basis 4.1 The case where k < fcmax 4.2 The case where fc = fcmax 5 The matrix of dérivatives 5.1 The case where A; < fcmax 5.2 The case where k = Jfcmax 6 The change of basis 7 Parametrizationofcurves 8 Remarks on the invariants x*, Yjt.i and Çk* 9 Generalizing the construction to arbitrary pseudo-Riemannian manifolds Classification math.: 53B30, 53A04.102 L BÉRARD BERGERY & X.CHARUEL DÉFINITION 1.2 A curve will be said "r-pseudo-regular" if 1. is r-regular, Le.F r -\ £ F r = Fr+\ 2. for all k < r, the function gk is either positive, identically zero, or négative.From now on, the curve c will always be assumed pseudo-regular.