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This introductory graduate level text provides a relatively quick path to a special topic in classical differential geometry: principal bundles. While the topic of principal bundles in differential geometry has become classic, even standard, material in the modern graduate mathematics curriculum, the unique approach taken in this text presents the material in a way that is intuitive for both students of mathematics and of physics. The goal of this book is to present important, modern geometric ideas in a form readily accessible to students and researchers in both the physics and mathematics communities, providing each with an understanding and appreciation of the language and ideas of the other.

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Introduction.- Basics of Manifolds.- Vector Bundles.- Vectors and Covectors.- Differential Forms.- Lie Derivatives.- Lie Groups.- Frobenius Theorem.- Principle Bundles.- Connections on Principle Bundles.- Curvature of a Connection.- Classical Electromagnetism.- Yang-Mills Theory.- Gauge Theory.- The Dirac Monopole.- Instantons.- What Next?.- Discussion of the Exercises.

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"He has written a book about principal bundles in the classical sense which is of great interest in and of itself ... . a textbook which can be used in an advanced one-year course or for self-learning. ... the book is also interesting for a physicist, because one can find the geometric basis of many mathematical tools used in physics. ... reviewer has greatly enjoyed reading the book and acknowledges the author´s bravery in writing another text on differential geometry!" (Fernando Etayo Gordejuela, Mathematical Reviews, November, 2015)

"The present book deals with principle bundles and their relevance in physics with a ground work on differential geometry. ... The book will be helpful to the graduate and under graduate students of mathematics and physics. It can also be an informative hand book of the researchers in differential geometry and physics." (Uday Chand De, zbMATH 1321.53004, 2015)

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Stephen Bruce Sontz is a Professor at Centro de Investigacion en Matematicas (CIMAT). He received his Ph.D. from the University of Virginia, USA, in 1994.

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