5063

4760

كارشناسي ارشد

هندسه

اصفهان: دانشگاه صنعتي اصفهان،دانشكده علوم رياضي

1388

[هشت]،98ص.

ص.ع.به: فارسي و انگليسي

اعظم اعتماد

فريد بهرامي

88/12/9

رضا ميرزايي،منصور آقاسي

رياضي

ID4760

به فارسي و انگليسي: قابل رويت در نسخه ديجيتالي

The Charactrization of Spherical Submanifolds in Euclidean Spaces Mona Effati mona e ati@gmail com Janury 11 2010 Master of Science Thesis in Farsi Department of Mathematical Sciences Isfahan University of Technology Isfahan 84156 83111 IranSupervisors Dr Azam Etemad ae110mat@cc iut ac ir2000 MSC Primary 53C20 Secondary 53C40 Key words Spherical Submanifolds Mean Curvature Vector Field Ricci Curvature Umbilical Section First Eigenvalue of Laplacian Operator Abstract In this thesis we present an expanded account of a characterization of sphere in Euclideanspace and spherical submanifolds in a Euclidean space based on two article by H Alodanand S Deshmukh 2007 Given an n dimensional submanifold M of a Euclidean space Rn pwith immersion M Rn p the position vector eld plays an important role instudying the geometry of the submanifolds One of the interesting questions in the theory ofcurves in R3 is to characterize spherical curves that is to obtain conditions under which aunit speed curve lies on a sphere S 2 c R3 There are several important characterizationfor spherical curves In 4 the question of extending the study of spherical curves in R3 tosubmanifolds in Euclidean space is considered namely to obtain conditions under which acompact submanifold M Rn p satis es M S n p 1 c where S n p 1 c is thehypersphere of constant sectional curvature c in the Euclidean space Rn p In chapter 3 rst we obtain two conditions characterizing spherical submanifolds and then obtain a con dition under which the normal component of the position vector eld is umbilical sectionand use it to prove that for umbilic section on a compact Einstein submanifol M either M S n p 1 c or else M is isometric to a sphere furthermore prove that a compactsubmanifold M of Rn p with constant scalar curvature S and umbilical normal section either lies on a hypersphere S n p 1 c or else the rst nonzero eigenvalue 1 of the Laplacianoperator on M satis es S n 1 1 The class of compact hypersurfaces with positive scalar curvature in a Euclidean space Rn isquite large and therefore it is an interesting question in Geometry to obtain conditions whichcharacterize spheres in this class It is Known that compact positively curved hypersurfaceswith constant mean curvature in Rn are spheres as well as that embedded compact hyper surfaces with constant scalar curvature in Rn are spheres For an orientable compact andconnected hypersurface in Euclidean space R4 with positive scalar curvature S the shapeoperator A and the mean curvature it is known that the inequality 6 detA 2 S 4 2implies that the hypersurface is a sphere where is the gradient of A similar charac terization is also obtain for spheres the Euclidean space R3 1

اعظم اعتماد

فريد بهرامي

رضا ميرزايي،منصور آقاسي