6325

5913

كارشناسي ارشد

رياضي محض

اصفهان: دانشگاه صنعتي اصفهان، دانشكده علوم رياضي

1389

شش،109ص.: مصور

ص.ع.به فارسي و انگليسي

اعظم اعتماد

امير هاشمي

8/8/90

محمدتقي جهانديده، محمدرضا كوشش

رياضي

ID5913

به فارسي و انگليسي: قابل رويت در نسخه ديجيتالي

Regular Homotopy and Total Curvature Circle Immersions into Surfaces Elahe Shahsavri Pour elahe shahsavari@gmail com 2011 Master of Science Thesis Department of Mathematical Sciences Isfahan University of Technology Isfahan 84156 83111 IranSupervisor Dr Azam Etemad ae110mat@cc iut ac irAdvisor Dr Amir Hashemi amir hashemi@cc iut ac ir2000 MSC 53C42 53A04 57R42Key words total absolute geodesic curvature PGC curves PGC homotopy Locally convex curve Abstract In this thesis we consider propertis of total absolute geodesic curvature functional on circle immersionsinto a Riemann surface based on the article T Ekholm Regular Homotopy and Total curvature I circleimmersions into surfaces 2006 In particular we study the behavior of under regular homotopies it isin ma in regular homotopy classes and the homotopy types of spaces of it is local minima An immersion ofmanifold is map with everywhere injective di erential Two immersions are regular homotopic if there exists acontinous 1 parameter family of immersions conecting one to the other In this thesis we study some aspectsof the di erential geometry of immersions and regular homotopies in the most basic cases of codimensionone immersions Let a riemann surface ie an orientable 2 manifold with a riemannian metric and letc S be a immersion of the circle parametrized by arc length The total absolute geodesic curvature of a circle immersion c into Riemann surface is given by the integral c g ds where g is the geodesic ccurvature of c and where ds denotes the arc length element We consider the simplest Riemann surface ofconstant curvature First we compute the rst variation of total absolute geodesic curvature We use the resultto classify local minima We de ne piecewise geodesic curves with curvature concentrations and show thatcircle immersions can be approximated by such curves without increasing the total absolute geodesic curvature A piecewise geodesic curve in is a continous curve c S1 which is a nite union of geodesic segments We construct special PGC homotopies of PGC curves on at Riemann surfaces and on the hyperbolic planewhich decrease the total absolute geodesic curvature of a given intial curve and which ends at a curve ofcertain standard shape For curves in the hyperbolic plane di ers from the at case in a essential way thelimit curve which arises as the end of a decreasing PGC homotopy is not a PGC curve in the hyperbolicplane in fact it often has in nte length To deal with this phenomenon we de ne a generalized PGC curve inthe hyperbolic plane as a PGC curve which is allowed to have vertices at in nity More concretely consider 4 dx2 dx2 the disk model of the hyperbolic plane D x x1 x2 R2 x 1 ds2 1 2 and add to it the 1 x 2 2

اعظم اعتماد

امير هاشمي

محمدتقي جهانديده، محمدرضا كوشش