Math 582 CA: Elementary Introduction to Metric Geometry

Dmitri Burago

Winter 2002, MWF 10:30-11:50

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Approximate list of topics:

• Introduction to length structures: basic definitions, theorems, techniques, constructions and examples;
• Introduction to curvature bounds;
• Riemannian length structures and the hyperbolic plane;
• Sub-Riemannian length structures;
• Convergence of length spaces;
• Large-scale geometry, quasi-isometries, Gromov hyperbolicity, and hyperbolic groups;
• Spaces of bounded curvature: Alexandrov spaces, Hadamard Spaces, curvatures of Riemannian manifolds.

Prerequisites: Most of the course is supposed to require only basics in elementary metric topology (continuity, completeness, compactness) and elementary calculus (derivatives, chain rule). For some topics, acquaintance with elementary topology of manifolds and basics of calculus on manifolds (the implicit function theorem, vector fields and integral curves, differential forms and Stokes Theorem) is desireable.

Textbook: D. Burago, Yu. Burgao, S. Ivanov, “A Course in Metric Geometry”, American Mathematical Society.