Riemannian Geometry is an expanded edition of a highly acclaimed and successful textbook originally published in Portuguese for first year graduate students in mathematics and physics The author s treatment goes very directly to the basic language of Riemannian geometry and immediately presents some of its most fundamental theorems It is elementary, assuming only a modest background from readers, making it suitable for a wide variety of students and course structures Its selection of topics has been deemed superb by teachers who have used the text A significant feature of the book is its powerful and revealing structure, beginning simply with the definition of a differentiable manifold and ending with one of the most important results in Riemannian geometry, a proof of the Sphere Theorem The text abounds with basic definitions and theorems, examples, applications, and numerous exercises to test the student s understanding and extend knowledge and insight into the subject Instructors and students alike will find the work to be a significant contribution to this highly applicable and stimulating subject