© 2016

Riemannian Geometry


Part of the Graduate Texts in Mathematics book series (GTM, volume 171)

Table of contents

  1. Front Matter
    Pages i-xviii
  2. Peter Petersen
    Pages 1-39
  3. Peter Petersen
    Pages 41-76
  4. Peter Petersen
    Pages 77-114
  5. Peter Petersen
    Pages 115-163
  6. Peter Petersen
    Pages 165-229
  7. Peter Petersen
    Pages 231-273
  8. Peter Petersen
    Pages 275-311
  9. Peter Petersen
    Pages 313-332
  10. Peter Petersen
    Pages 333-363
  11. Peter Petersen
    Pages 365-394
  12. Peter Petersen
    Pages 395-442
  13. Peter Petersen
    Pages 443-490
  14. Back Matter
    Pages 491-499

About this book


Intended for a one year course, this text serves as a single source, introducing readers to the important techniques and theorems, while also containing enough background on advanced topics to appeal to those students wishing to specialize in Riemannian geometry. This is one of the few Works to combine both the geometric parts of  Riemannian geometry and the analytic aspects of the theory. The book will appeal to a readership that have a basic knowledge of standard manifold theory, including tensors, forms, and Lie groups.

Important revisions to the third edition include:

  • a substantial addition of unique and enriching exercises scattered throughout the text;
  • inclusion of an increased number of coordinate calculations of connection and curvature;
  • addition of general formulas for curvature on Lie Groups and submersions;
  • integration of variational calculus into the text allowing for an early treatment of the Sphere theorem using a proof by Berger;
  • incorporation of several recent results about manifolds with positive curvature;
  • presentation of a new simplifying approach to the Bochner technique for tensors with application to bound topological quantities with general lower curvature bounds.

From reviews of the first edition:

"The book can be highly recommended to all mathematicians who want to get a more profound idea about the most interesting achievements in Riemannian geometry. It is one of the few comprehensive sources of this type."

―Bernd Wegner, ZbMATH


Riemannian geometry textbook adoption tensor geometry geodesics distance geometry Hopf fibration Jacobi fields Hadamard-Cartan theorem Sobolev constants Lie groups Killing fields Hölder spaces Riemannian metrics Riemannian geometry text adoption Derivatives Curvature Lichenerowicz Laplacians Bochner technique Hodge theory holonomy Betti numbers

Authors and affiliations

  1. 1.Los AngelesUSA

About the authors

Peter Petersen is a Professor of Mathematics at UCLA. His current research is on various aspects of Riemannian geometry. Professor Petersen has authored two important textbooks for Springer: Riemannian Geometry in the GTM series and Linear Algebra in the UTM series.

Bibliographic information

  • Book Title Riemannian Geometry
  • Authors Peter Petersen
  • Series Title Graduate Texts in Mathematics
  • Series Abbreviated Title Graduate Texts Mathematics
  • DOI
  • Copyright Information Springer International Publishing AG 2016
  • Publisher Name Springer, Cham
  • eBook Packages Mathematics and Statistics Mathematics and Statistics (R0)
  • Hardcover ISBN 978-3-319-26652-7
  • Softcover ISBN 978-3-319-79989-6
  • eBook ISBN 978-3-319-26654-1
  • Series ISSN 0072-5285
  • Series E-ISSN 2197-5612
  • Edition Number 3
  • Number of Pages XVIII, 499
  • Number of Illustrations 49 b/w illustrations, 1 illustrations in colour
  • Topics Differential Geometry
  • Buy this book on publisher's site


“This is a very advanced textbook on metric and algebraic proofs of critical theorems in the field of metric spaces involving manifolds and other 3D structures. … First, definitions, theorems, proofs, and exercises abound throughout every section of this 500 page mathematics book. The history of development in the area is comprehensive. … The experts will find this a useful research tool. … I recommend this book for researchers having a strong background to begin with.” (Joseph J. Grenier,, June, 2016)