This carefully written book is an introduction to the beautiful ideas and results of differential geometry. The first half covers the geometry of curves and surfaces, which provide much of the motivation and intuition for the general theory. The second part studies the geometry of general manifolds, with particular emphasis on connections and curvature. The text is illustrated with many figures and examples. The prerequisites are undergraduate analysis and linear algebra. This new edition provides many advancements, including more figures and exercises, and—as a new feature—a good number of solutions to selected exercises. This new edition is an improved version of what was already an excellent and carefully written introduction to both differential geometry and Riemannian geometry. In addition to a variety of improvements, the author has included solutions to many of the problems, making the book even more appropriate for use in the classroom. —Colin Adams, Williams College This book on differential geometry by Kühnel is an excellent and useful introduction to the subject. … There are many points of view in differential geometry and many paths to its concepts. This book provides a good, often exciting and beautiful basis from which to make explorations into this deep and fundamental mathematical subject. — Louis Kauffman, University of Illinois at Chicago