This book introduces and explores some of the deep connections between Einstein's theory of gravitation and differential geometry. As an outgrowth of graduate summer schools, the presentation is aimed at graduate students in mathematics and mathematical physics, starting from the foundations of special and general relativity, and moving to more advanced results in geometric analysis and the Einstein constraint equations. Topics include the formulation of the Einstein field equation and the Einstein constraint equations; gluing construction of initial data sets which are Schwarzschild near infinity; and an introduction to the Riemannian Penrose inequality. While the book assumes a background in differential geometry and real analysis, a number of basic results in geometry are provided. There are well over 100 exercises, many woven into the fabric of the chapters as well as others collected at the end of chapters, to give readers a chance to engage and extend the text.

‘This book will be a valuable source for any student planning to specialize in differential geometry, and especially for those preparing to work in mathematical relativity. It begins with a thorough introduction to relativistic spacetimes and the Einstein equations, and continues with some of the basic geometric results in the subject including the Penrose singularity theorem, the positive mass theorem, and geometric properties of solutions of the Einstein constraint equations.’

Richard M. Schoen - Stanford University

‘Corvino and Miao have written an excellent introduction to Special and General Relativity, from a geometric point of view not usually presented by most books on these topics. I recommend beginning students read this book to gain a beautiful overview of the subject, followed by Dan Lee's Geometric Relativity to master additional topics and details.’

Hubert Lewis Bray - Duke University

‘An impressive entry for graduate students into some of the deeper, and very important, topics in mathematical GR, written with care by leading experts in the field.’

Greg Galloway - University of Miami (Emeritus), Fellow of the American Mathematical Society

‘This book provides a rigorous and robust foundation for those seriously pursuing the mathematical theory of general relativity, with a particular focus on the Riemannian geometry of initial data sets. It meticulously develops the required mathematical framework and explains key concepts with clarity and precision. Readers will be well-prepared for more advanced studies in this fascinating field.’

Mu-Tao Wang - Columbia University