Arnold-type invariants of curves and wave fronts on surfaces

University dissertation from Uppsala : Acta Universitatis Upsaliensis

Abstract: This thesis is devoted to the study of invariants of generic curves and wave fronts on surfaces. The invariants J± and St were axiomatically defined by Arnold as numerical characteristics of generic curves (immersions of the circle)on ℝ2 he introduced J± in the case of generic planar wave fronts. The generalization of St to this case was independently obtained by F. Aicardi and M. Polyak.In the first two chapters of this thesis I construct generalizations of the three Arnold's invariants to the case of generic curves and wave fronts on an arbitrary surface (not necessarily ℝ2). I explicitly describe all the invariants satisfying axioms, which naturally generalize the axioms used by Arnold.To prove existence of these invariants I use certain properties of the fundamental group of the space of curves on a surface. All the homotopy groups of this space are calculated in the third chapter of the thesis.

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