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Riemannian Geometry on Some Noncommutative Spaces
Chen, Wei-Da
Chen, Wei-Da
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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.
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Date
2017-12-31
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University of Kansas
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This item contains archived web content.
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Keywords
Mathematics, Chern-Gauß-Bonnet Theorem, Levi-Civita Connections, Noncommutative Tori, Quantum Discs, Quantum Spheres, Riemann Curvatures