Arnold-type invariants of curves and wave fronts on surfaces
1998 (English)Doctoral thesis, comprehensive summary (Other academic)
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.
Place, publisher, year, edition, pages
Uppsala: Acta Universitatis Upsaliensis , 1998. , 58 p.
Uppsala Dissertations in Mathematics, ISSN 1401-2049 ; 10
Mathematics, Immersion of the circle, generic immersion, curves on surfaces, Legendrian immersion, Legendrian knot, wave fronts on surfaces, Whitney index, Maslov index, regular homotopy, perestroikas of plane curves and fronts, Arnold's basic invariants of plane curves and fronts, finite order invariants, Vassiliev invariants, homotopy groups of the space of curves on a surface
Research subject Mathematics
IdentifiersURN: urn:nbn:se:uu:diva-1274ISBN: 91-506-1265-4OAI: oai:DiVA.org:uu-1274DiVA: diva2:160837
1998-05-06, Room 347, Polacksbacken, Uppsala University, Uppsala, 10:15