An Introduction to Geometric Inverse Problems

This textbook serves as an introduction to inverse problems for geodesics of Riemannian manifolds. Welcoming readers new to the area, the authors build from foundational material through to the spectacular results of recent decades. Appendices supplement specialist knowledge, creating a self-contained introduction to modern developments in the area for students and researchers alike. Organized into four parts, the book begins with an introduction to Riemannian geometry that focuses on the dynamical and variational theory of geodesics. From there, the authors introduce the X-ray transform from integral geometry, a crucial tool in the study of rigidity problems for geodesics. The following part focuses on the boundary rigidity problem for simple Riemannian disks. Moving to the closed setting, the final part of the book explores the length spectrum rigidity problem for closed Riemannian manifolds, including concise introductions to general Anosov flows, Riemannian manifolds without conjugate points, and Riemannian surfaces of negative curvature. An Introduction to Geometric Inverse Problems is intended for a broad audience. Readers are assumed to know the basic notions of differential geometry, complex and functional analysis, distributions, and algebraic topology. The appendices cover material needed from symplectic geometry and pseudo-differential operators on compact manifolds.