dc.contributor.advisor | Sheu, Albert J.-L. | |
dc.contributor.author | Chen, Wei-Da | |
dc.date.accessioned | 2018-10-24T22:35:15Z | |
dc.date.available | 2018-10-24T22:35:15Z | |
dc.date.issued | 2017-12-31 | |
dc.date.submitted | 2017 | |
dc.identifier.other | http://dissertations.umi.com/ku:15660 | |
dc.identifier.uri | http://hdl.handle.net/1808/27018 | |
dc.description.abstract | This dissertation enquires into how the theory and mechanism of Riemannian geometry can be introduced into and integrated with the existent ones in noncommutative geometry, a branch of mathematics inspired by the development of quantum physics that concentrates on C*-algebras and related research. In conformity with the Gelfand duality, a cornerstone theorem in noncommutative geometry that establishes a one-to-one correspondence between commutative C*-algebras and locally compact Hausdorff spaces, it is suggested that a noncommutative C*-algebra notionally be deemed a "virtual noncommutative space". Based on this ideology are some forms of Riemannian geometry anticipated to reincarnate on C*-algebras. J. Rosenberg demonstrated such a reincarnation on noncommutative tori. Especially, a corresponding adaptation of the Fundamental Theorem of Riemannian Geometry was attained. Moreover, based on this adaptation, he established a variant of the Gauß-Bonnet Theorem for noncommutative 2-tori. M. A. Peterka and A. J.-L. Sheu subsequently presented extensions and generalisations to the framework developed by Rosenberg. Specifically, an enhanced Gauß-Bonnet Theorem was substantiated for noncommutative 2-tori. In this dissertation, we shall first tender results that are closely related to the aforementioned work on noncommutative tori, proposing several extensions of the two Gauß-Bonnet Theorems already obtained for noncommutative 2-tori and exhibiting extensions of the theorem for two special cases on noncommutative 4-tori. Thereafter, we shall transcribe Rosenberg's framework and results for quantum discs and 2-spheres with a version of the Fundamental Theorem proved. Finally, an asymptotic behaviour of the total curvature will be demonstrated for quantum complex projective lines as an illustrative example. | |
dc.format.extent | 71 pages | |
dc.language.iso | en | |
dc.publisher | University of Kansas | |
dc.rights | Copyright held by the author. | |
dc.subject | Mathematics | |
dc.subject | Chern-Gauß-Bonnet Theorem | |
dc.subject | Levi-Civita Connections | |
dc.subject | Noncommutative Tori | |
dc.subject | Quantum Discs | |
dc.subject | Quantum Spheres | |
dc.subject | Riemann Curvatures | |
dc.title | Riemannian Geometry on Some Noncommutative Spaces | |
dc.type | Dissertation | |
dc.contributor.cmtemember | Kong, Man Cheong | |
dc.contributor.cmtemember | Shao, Shuanglin | |
dc.contributor.cmtemember | Stefanov, Atanas G. | |
dc.contributor.cmtemember | Torres, Rodolfo H. | |
dc.thesis.degreeDiscipline | Mathematics | |
dc.thesis.degreeLevel | Ph.D. | |
dc.identifier.orcid | | |
dc.rights.accessrights | openAccess | |