Collapsed Riemannian and non-Riemannian spaces
We will present a topic course that covers on the collapsing theory in Metric Riemannian geometry. We will start with the Gromov’s almost ﬂat manifolds, the nilpotent structure theory of Cheeger-Fukaya-Gromov. We will discuss similar nilpotent structures found on collapsed Riemannian manifolds with local Ricci bounded covering geometry, and on Alexandrov spaces (non-Riemannian spaces) with local bounded covering geometry. CONTENTS: I. Collapsed Manifolds with Bounded Sectional Curvature - 1.1. Examples 1.2. The almost ﬂat manifolds 1.3. The nilpotent ﬁber bundle theorem 1.4. The singular nilpotent ﬁbration theorem 1.5. Applications - II. Collapsed Manifolds with Local Bounded Ricci Covering Geometry - 2.1. Examples 2.2. The Cheeger-Colding-Naber theory on Ricci limit spaces 2.3. The Margulis lemma 2.4. Maximally collapsed manifolds with local bounded Ricci covering geometry 2.5. The nilpotent ﬁbration theorem 2.6. Applications - III. Collapsed Alexandrov Spaces - 3.1. Basic Alexandrov spaces 3.2. The ﬁbration theorem 3.3. Maximally collapsed Alexandrov space with local bounded covering geometry 3.4. The nilpotent ﬁbration theorem 3.5. Singular ﬁbration structures 3.6. Applications
Reference: Notes on convergence and collapsing theorems in Riemannian geometry, Handbook of Geometric Analysis (Vol II) ALM 13, pp. 193-298.
Schedule of Sections
- Fall 2020 Prof. F. Luo
- Fall 2017 Prof. J. Song