Riemannian Geometry on Some Noncommutative Spaces

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