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This volume is an English translation of Sakai's textbook on Riemannian Geometry which was originally written in Japanese and published in 1992. The author's intent behind the original book was to provide to advanced undergraduate and graudate students an introduction to modern Riemannian geometry that could also serve as a reference.

Intended for a one year course, this text serves as a single source, introducing readers to the important techniques and theorems, while also containing enough background on advanced topics to appeal to those students wishing to specialize in Riemannian geometry. This is one of the few Works to combine both the geometric parts of Riemannian geometry and the analytic aspects of the theory. The book will appeal to a readership that have a basic knowledge of standard manifold theory, including tensors, forms, and Lie groups. “This is a very advanced textbook on metric and algebraic proofs of critical theorems in the field of metric spaces involving manifolds and other 3D structures. First, definitions, theorems, proofs, and exercises abound throughout every section of this 500 page mathematics book. The history of development in the area is comprehensive. The experts will find this a useful research tool.

I recommend this book for researchers having a strong background to begin with.” (Joseph J. Grenier, Amazon.com, June, 2016).