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- Type of Document: M.Sc. Thesis
- Language: Farsi
- Document No: 45088 (02)
- University: Sharif University of Technology
- Department: Mathematical Sciences
- Advisor(s): Bahraini, Alireza; Ranjbar Motlagh, Alireza
- Abstract:
- One of the important questions in mathematics is generalized. In case of metric spaces, one of the questions is: Can we extend notions of Riemannian geometry to arbitrary metric spaces smoothly? Ricci curvature is one of the most important concepts in differential geometry. Ricci curvature is defined for the Riemannian manifold and has many applications in mathematics and physics like Einstein’s equation in relativity theory. There is a good notion for a metric space having “sectional curvature bounded below by K” or “sectional curvature bounded above by K”, due to Alexandrov. Can we define the concept of Ricci curvature in metric space? A motivation for this question comes from Gromov’s precomactness theorem. Let M denote the set of compact metric spaces (module isometry) with Gromov-Hausdorff topology. The precompactness theorem says that given N 2 Z+,D < and K 2 R, the subset of M consisting of closed Riemannian manifolds (M; g) with dim(M) = N,RicKg and diamD, is precomact. This thesis is based on work of Cedric Villani and John Lott and two objectives are pursued in this thesis. First, we talk about the Optimal Transport on length space and then we use these notions to generalize the Ricci curvature
- Keywords:
- Ricci Curvature ; Metric-Measure Space ; Optimal Transport ; Entropy ; Wasserstein Space
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