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چكيده انگليسي :

Riemannian Submanifold With Prescribed Boundary Hossein Najafzadeh h najafzadeh@math iut ac ir September 2011 Master of Science Thesis in Farsi Department of Mathematical Sciences Isfahan University of Technology Isfahan 84156 83111 IranSupervisor Dr Azam Etemad 2000 MSC 53C23 53A07 Key words Non negative curvature locally convex hypersurface lling Riemannian submanifolds nite ness theorem Alexandrov space with curvature bounded below AbstractThe most basic examples of submanifolds with curvature bounded below are propably hypersurfaceswith positive sectional curvature in Euclidean space These surfaces are locally convex i e eachpoint has a neighberhood which lies locally on one side of its tangent plane In this thesis the niteness theorem states that a smooth compact submanifold of codimension 2 immersed in Rn 1 for n 3 bounds at most nitely many topologically distinct compact nonnegatively curved hyper surface To this end we de ne an Alexandrov spaces of curvature greater or equal to K then wewill prove the niteness theorem by use of this notion Speci cally in this spaces for any geodesictriangle the distance from a vertex to a point on the opposite of side is at least the distance betweencorresponding points on a geodesic triangle with the same side length in the simply connected spaceform SK of constant curvature K The niteness theorem states that without some restrictionon its boundary a positively curved hypersurfaces may assume any type of topology and be quitecomplicated geometrically as well We ask the question can one control the geometry topologyof a locally convex hypersurface by imposing conditions on its boundary Under greater regularityand by use of classical results of Cohn Vossen and Huber we prove the stronger niteness theo rem i e a nite collection of closed C 3 curves immersed in a give Riemannian manifold bounds atmost nitely many topologically distinct complete immersed C 3 surface whose total curvature areuniformly bounded below Next we use of theorems Gromov and perelman we proves the classM of Alexandrov spaces of curvature greater or equal to K contains only nitely many topologicaltypes Also We get the upper diameter bound in Gromov compactness theorem using a projectivetransformation as follows Let be a compact immersed submanifold of codimension 2 in Rn 1 Let denote the family of all compact nonnegatively curved immersed hypersurfaces having as boundary We prove that Rn 1 is projectively equivalent to a hemisphere in Sn 1 As a resultwe may regard the elements M as hypersurfaces of sectional greater or equal to 1 in Sn 1 Finally the structure of the ends for immersed Euqlidean hypersurfaces is described assuming thatcurvature positive outside a compact subset or it has curvature is non negative and the secondfundamental form at most one outside a compact subset is null We also consider complete mani folds with Ricci curvature that is nonnegative outside a compact set then it can be proved that thenumber of ends of such a manifold is nite and in particular there is upper bound for this number The niteness theorem also holds for non compact hypersurfaces This follows from a theorem ofCai which states that the number of end is nite for any complete Riemannian manifold which hasnon negative Ricci curvature o of a ball and its Ricci curvature is bounded below every where Further there is a structure theorem of Alexandrov and Currier which states that each end has aconvex representative So the ends may be clipped o Finally we suggest this question what isthe highest codimension where we have topological niteness without the diameter bound it hasto be between 2 and 15 1

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زير خمينه هاي ريماني با مرز معين

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