© 2018

Introduction to Riemannian Manifolds

  • Easy for instructors to adapt the topical coverage to suit their course

  • Develops an intimate acquaintance with the geometric meaning of curvature

  • Gives students strong skills via numerous exercises and problem sets

  • Adapted from Riemannian Manifolds, the new edition has been aptly expanded to cover one or two semesters


Part of the Graduate Texts in Mathematics book series (GTM, volume 176)

Table of contents

  1. Front Matter
    Pages i-xiii
  2. John M. Lee
    Pages 1-8
  3. John M. Lee
    Pages 9-54
  4. John M. Lee
    Pages 55-83
  5. John M. Lee
    Pages 85-113
  6. John M. Lee
    Pages 115-150
  7. John M. Lee
    Pages 151-191
  8. John M. Lee
    Pages 193-224
  9. John M. Lee
    Pages 225-262
  10. John M. Lee
    Pages 263-282
  11. John M. Lee
    Pages 283-317
  12. John M. Lee
    Pages 319-344
  13. John M. Lee
    Pages 345-370
  14. Back Matter
    Pages 371-437

About this book


This textbook is designed for a one or two semester graduate course on Riemannian geometry for students who are familiar with topological and differentiable manifolds. The second edition has been adapted, expanded, and aptly retitled from Lee’s earlier book, Riemannian Manifolds: An Introduction to Curvature. Numerous exercises and problem sets provide the student with opportunities to practice and develop skills; appendices contain a brief review of essential background material.

While demonstrating the uses of most of the main technical tools needed for a careful study of Riemannian manifolds, this text focuses on ensuring that the student develops an intimate acquaintance with the geometric meaning of curvature. The reasonably broad coverage begins with a treatment of indispensable tools for working with Riemannian metrics such as connections and geodesics. Several topics have been added, including an expanded treatment of pseudo-Riemannian metrics, a more detailed treatment of homogeneous spaces and invariant metrics, a completely revamped treatment of comparison theory based on Riccati equations, and a handful of new local-to-global theorems, to name just a few highlights.

Reviews of the first edition:

Arguments and proofs are written down precisely and clearly. The expertise of the author is reflected in many valuable comments and remarks on the recent developments of the subjects. Serious readers would have the challenges of solving the exercises and problems. The book is probably one of the most easily accessible introductions to Riemannian geometry. (M.C. Leung, MathReview)

The book’s aim is to develop tools and intuition for studying the central unifying theme in Riemannian geometry, which is the notion of curvature and its relation with topology. The main ideas of the subject, motivated as in the original papers, are introduced here in an intuitive and accessible way…The book is an excellent introduction designed for a one-semester graduate course, containing exercises and problems which encourage students to practice working with the new notions and develop skills for later use. By citing suitable references for detailed study, the reader is stimulated to inquire into further research. (C.-L. Bejan, zBMATH)


Riemannian geometry curvature manifold differential geometry textbook graduate mathematics textbook Riemannian geometry course textbook Riemannian metrics geodesics Levi-Cevita connection Riemannian submanifolds Gauss-Bonnet theorem Jacobi fields comparison theory curvature and topology tensor

Authors and affiliations

  1. 1.Department of MathematicsUniversity of WashingtonSeattleUSA

About the authors

​John "Jack" M. Lee is a professor of mathematics at the University of Washington. Professor Lee is the author of three highly acclaimed Springer graduate textbooks : Introduction to Smooth Manifolds, (GTM 218) Introduction to Topological Manifolds (GTM 202), and Riemannian Manifolds (GTM 176). Lee's research interests include differential geometry, the Yamabe problem, existence of Einstein metrics, the constraint equations in general relativity, geometry and analysis on CR manifolds.   

Bibliographic information

  • Book Title Introduction to Riemannian Manifolds
  • Authors John M. Lee
  • Series Title Graduate Texts in Mathematics
  • Series Abbreviated Title Graduate Texts Mathematics
  • DOI
  • Copyright Information Springer International Publishing AG 2018
  • Publisher Name Springer, Cham
  • eBook Packages Mathematics and Statistics Mathematics and Statistics (R0)
  • Hardcover ISBN 978-3-319-91754-2
  • Softcover ISBN 978-3-030-80106-9
  • eBook ISBN 978-3-319-91755-9
  • Series ISSN 0072-5285
  • Series E-ISSN 2197-5612
  • Edition Number 2
  • Number of Pages XIII, 437
  • Number of Illustrations 210 b/w illustrations, 0 illustrations in colour
  • Additional Information Originally published with title "Riemannian Manifolds: An Introduction to Curvature"
  • Topics Differential Geometry
  • Buy this book on publisher's site


“One interesting aspect of the book is the decision of which audience to target it towards. … Overall, this would make a very appropriate text for a graduate course, or a programme of individual study in Riemannian geometry, whether to give a thorough treatment of the fundamentals, or to introduce the more advanced topics in global geometry.” (Robert J. Low, Mathematical Reviews, November, 2019)

“This material is carefully developed and several useful examples and exercises are included in each chapter. The reviewer’s belief is that this excellent edition will become soon a standard text for several graduate courses as well as an frequent citation in articles.” (Mircea Crâşmăreanu, zbMATH 1409.53001, 2019)