Differential Geometry is a wide field. We have chosen to concentrate upon certain aspects that are appropriate for an introduction to the subject; we have not attempted an encyclopedic treatment. In Book I, we focus on preliminaries. Chapter 1 provides an introduction to multivariable calculus and treats the Inverse Function Theorem, Implicit Function Theorem, the theory of the Riemann Integral, and the Change of Variable Theorem. Chapter 2 treats smooth manifolds, the tangent and cotangent bundles, and Stokes' Theorem. Chapter 3 is an introduction to Riemannian geometry. The Levi'Civita connection is presented, geodesics introduced, the Jacobi operator is discussed, and the Gauss-Bonnet Theorem is proved. The material is appropriate for an undergraduate course in the subject. We have given some different proofs than those that are classically given and there is some new material in these volumes. For example, the treatment of the Chern-Gauss-Bonnet Theorem for pseudo-Riemannian manifolds with boundary is new.

**Click here for Book II**.

### Table of Contents

Preface

Acknowledgments

Basic Notions and Concepts

Manifolds

Riemannian and Pseudo-Riemannian Geometry

Bibliography

Authors' Biographies

Index

### About the Author(s)

**Peter B. Gilkey**, University of Oregon

Peter B. Gilkey is a Professor of Mathematics and a member of the Institute of Theoretical Science at the University of Oregon. He is a fellow of the American Mathematical Society and is a member of the editorial board of Results in Mathematics, J. Differential Geometry and Applications, and J. Geometric Analysis. He received his Ph.D. in 1972 from Harvard University under the direction of of L. Nirenberg. His research specialties are Differential Geometry, Elliptic Partial Differential Equations, and Algebraic topology. He has published more than 250 research articles and books.

**Jeong Hyoeng Park**, Sungkyunkan University (Korea)

Jeong Hyeong Park is a Professor of Mathematics at Sungkyunkan University and is an associate member of the KIAS Korea). She received her Ph.D. in 1990 from Kanazawa University in Japan under the direction of H. Kitahara. Her research specialties are spectral geometry of Riemannian submersion and geometric structures on manifolds like eta-Einstein manifolds and H-contact manifolds. She organized the geometry section of AMC 2013 (The Asian Mathematical Conference 2013) and the ICM 2014 satellite conference on Geometric analysis. She has published more than 71 research articles and books.

**Ramon Vazquez-Lorenzo**, University of Santiago de Compostela (Spain)

Ramon Vazquez-Lorenzo is a member of the research group in Riemannian Geometry at the Department of Geometry and Topology of the University of Santiago de Compostela (Spain). He is a member of the Spanish Research Network on Relativity and Gravitation. He received his Ph.D. in 1997 from the University of Santiago de Compostela under the direction of E. Garcia-Rio. His research focuses mainly on Differential Geometry with special emphasis on the study of the curvature and the algebraic properties of curvature operators in the Lorentzian and in the higher signature settings. He has published more than 50 research articles and books.

### Related Series

Mathematics and Statistics

### Reviews

The ordering in both books is very good and both books are well written. That is why it is a good idea to use them as textbooks for some courses. While the first book can be thought of as an undergraduate textbook, the second one can be used in the first years of graduate studies and may also be used for some advanced courses with some small adjustments. The notations used in both books are standard. Since the written form is pleasant, both books are quite attractive.
- Bedia Akyar (in Mathematical Reviews, October 2016)