This volume is devoted to various aspects of Alexandrov Geometry for those wishing to get a detailed picture of the advances in the field. It contains enhanced versions of the lecture notes of the two mini-courses plus those of one research talk given at CIMAT.
Peter Petersen’s part aims at presenting various rigidity results about Alexandrov spaces in a way that facilitates the understanding by a larger audience of geometers of some of the current research in the subject. They contain a brief overview of the fundamental aspects of the theory of Alexandrov spaces with lower curvature bounds, as well as the aforementioned rigidity results with complete proofs.
The text from Fernando Galaz-García’s minicourse was completed in collaboration with Jesús Nuñez-Zimbrón. It presents an up-to-date and panoramic view of the topology and geometry of 3-dimensional Alexandrov spaces, including the classification of positively and non-negatively curved spaces and the geometrization theorem. They also present Lie group actions and their topological and equivariant classifications as well as a brief account of results on collapsing Alexandrov spaces.
Jesús Nuñez-Zimbrón’s contribution surveys two recent developments in the understanding of the topological and geometric rigidity of singular spaces with curvature bounded below.
Author(s): Gerardo Arizmendi Echegaray, Luis Hernández-Lamoneda, Rafael Herrera Guzmán
Series: CIMAT Lectures in Mathematical Sciences
Publisher: Birkhäuser
Year: 2022
Language: English
Pages: 118
City: Cham
Preface
Contents
Rigidity of Alexandrov Spaces
1 Introduction
2 The Basic Theory
2.1 Spaces and Geodesics
2.2 Lower Curvature Bounds
2.3 Angles
2.4 Examples
2.5 Dimension and Volume
2.6 The Space of Directions and the Tangent Cone
3 Calculus on Alexandrov Spaces
3.1 Calculus of Continuous Functions
3.2 Calculus on Alexandrov Spaces
3.3 Special Curves
3.4 The Fibration and Stability Theorems
4 Rigidity Results
4.1 Bonnet-Myers' and Toponogov's Diameter/Radius Theorem
4.2 Controlling the Boundary
4.3 The Positive Mass Conjectures
4.4 Lytchak's Problem
4.5 Examples
4.6 Generalizations of Lytchak's Problem
4.7 The Boundary Conjecture
5 The Weak Inner Regularity Theorem
5.1 Basic Structure
5.2 Proofs of Rigidity Results
References
Three-Dimensional Alexandrov Spaces: A Survey
1 Introduction
2 Alexandrov Spaces
2.1 Basic Definitions
2.2 Curvature Bounded Below
2.3 Examples and Constructions
Complete Riemannian Manifolds with ps: [/EMC pdfmark [/Subtype /Span /ActualText (secant greater than or equals k) /StPNE pdfmark [/StBMC pdfmarksec≥kps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark
Convex Sets
Convex Surfaces
Gromov–Hausdorff Limits
Cartesian Products
Cones
Suspensions
Joins
Quotients
Doubles and Glued Spaces
2.4 Local Structure
2.5 A Riemannian Digression
3 Three-Dimensional Alexandrov Spaces
3.1 Basic Structure
4 Spaces with Positive or Non-negative Curvature
4.1 Spaces with Positive or Non-negative Ricci Curvature
5 Topological Results
5.1 Geometrization
5.2 Simply-Connected Spaces
5.3 Aspherical Spaces and the Borel Conjecture
6 Alexandrov 3-Spaces with Compact Lie Group Actions
6.1 Setup
6.2 Homogeneous Spaces
6.3 Cohomogeneity One Spaces
6.4 Cohomogeneity Two Spaces
6.5 Spaces with Local Circle Actions
7 Collapse
References
Topological and Geometric Rigidity
1 Introduction
2 Spaces with Curvature Bounded Below
3 Topological Rigidity of Alexandrov 3-Spaces
4 Volume Entropy Rigidity of RCD*-Spaces
References
