Geometry, Dynamics, and Rigidity

A generously illustrated textbook and reference for graduate students and researchers that integrates two major areas of mathematics Geometry, Dynamics, and Rigidity is an introduction to the modern interface between Riemannian geometry and dynamical systems. Organized in three parts, the book builds the geometric and analytic background (“Geometry”), develops the core machinery of hyperbolic dynamics (“Dynamics”), and then assembles these tools to prove three landmark rigidity theorems (“Rigidity”). Amie Wilkinson’s informal yet precise and detailed exposition balances approachability with depth: definitions and motivations appear where they are first needed, proofs are complete and carefully structured, and historical notes connect classical ideas to current research. Designed for beginning graduate students as well as researchers entering from either side of the subject, the text emphasizes concepts—including geodesic flows, curvature and entropy, Anosov and related systems, and invariant measures and regularity—while showing how they interact to force rigidity. Numerous exercises guide the reader from basic checks to open-ended problems, and cross-references allow selective reading by topic. The result is a self-contained pathway from fundamentals to state-of-the-art theorems, suitable for use in a topics course or as a long-term reference. An inviting and rigorous introduction to the meeting point of Riemannian geometry and dynamical systems designed to be used as a textbook or reference Features a self-contained, carefully cross-referenced exposition, original figures that clarify core constructions, and more than 100 color illustrations Integrates history and motivation, and presents sources otherwise scattered in the literature Includes exercises of varied difficulty to develop technique and intuition Offers flexible reading paths, encouraging readers to browse individual chapters and topics