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RUcore Resource Object
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TitleControlled geometry via volumes on Alexandrov spaces
NameLi, Nan (author), Rong, Xiaochun (chair), Luo, Feng (internal member), Han, Zhengchao (internal member), Cao, Jianguo (outside member), Rutgers University, Graduate School - New Brunswick,
Degree Date2010-10
Date Created2010
SubjectMathematics,
Geodesics (Mathematics),
Geometry, Differential,
Rigidity (Geometry)
DescriptionIn this thesis we recall the basic definitions and properties for Alexandrov space and describe two geometry phenomenons controlled via volume (Hausdorff measure or rough volume) conditions. (1) For a path in X [Greek letter epsilon] 2 Alex [superscript n] (Greek letter kappa) (the compact n-dimensional Alexandrov spaces with curvature [greater than or equal to kappa].), the sum of the length and the turning angle is bounded from below in terms of [kappa], n, diameter and volume of X. This generalizes a basic estimate by Cheeger on the length of a closed geodesic in closed Riemannian manifold ([Ch]). (2) Let [epsilon]p be the space of directions at p 2 [epsilon] X and the pointed radius R = inf{r : X C B[subscript r](p)}. If X [epsilon] Alex[superscript n](kappa), then vol(X) [less than or equal to symbol] vol(CR/k (Epsilon subscript p). where (CR/k (Epsilon subscript p) is the metric R-ball at the vertex in the [kappa]-suspension (CR/k (Epsilon subscript p). We give an isometric classification of X X [epsilon] Alex[superscript n](kappa) whose volume achieves the maximal possible value (CR/k (Epsilon subscript p). We also determine homeomorphic types of such X when X is a topological manifold. These results are natural extension of K. Grove and P. Petersen's work in 1992 ([GP 92]).
NotePh.D.
NoteIncludes bibliographical references
NoteIncludes vita
Noteby Nan Li
Genretheses
Persistent URLhttp://hdl.rutgers.edu/1782.1/rucore10001600001.ETD.000056482
Languageeng
CollectionGraduate School - New Brunswick Electronic Theses and Dissertations
Organization NameRutgers, The State University of New Jersey
RightsThe author owns the copyright to this work.