This is a new and expanded edition of this textbook for first-year graduate students in mathematics and physics, discussing the basic language of Riemannian geometry and presenting some of its most fundamental theorems. It is elementary in that it only assumes a modest background from the readers, making it suitable for a wide variety of students and course structures. It features basic definitions and theorems, examples and numerous exercises. The book begins with the definition of a differential manifold and ends with one of the most important results in Riemmanian geometry - a proof of the Sphere Theorem.

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