Beginner's book for Riemannian geometry

Could you recommend some beginner books for Riemannian geometry to me?

I am completely new to Riemannian geometry, but have some basic knowledge of differential geometry.

I am looking for a book in Riemannian which is similarly as light as Tu's "Introduction to Manifolds" for differential geometry. I know that the classical reference is the book of do Carmo, but I have heard some students complaining that its not a book for the absolute beginner.

For the moment, I am just looking for a source to introduce me Riemannian metrics, Riemannian manifolds, curvature, geodesics in a way as clear as possible.

Thank you

I'm a fan of Lee's Riemannian Manifolds: An Introduction to Curvature. It is definitely an introductory book; there are many deeper topics that it doesn't mention (compare to Peterson's Riemannian Geometry). Here is an excerpt from the preface:

"I have selected a set of topics that can reasonably be covered in ten to fifteen weeks, instead of making any attempt to provide an encyclopedic treatment of the subject."

One of the features that I really like about this book is the careful treatment of tensors and tensor fields (chapter 2). Understanding exactly what these objects are is one of the potential obstructions to learning Riemannian geometry.

• Btw, there is a new edition for this book. See here. Definitely better than the first edition. – Sou Apr 2 at 5:04

By far Gallot et al is a very good choice.

Have you tried Riemannian Geometry: A Beginners Guide, by Frank Morgan?

Actually there is a book by Tu himself about Riemannian geometry. It was just released last year. It is called Differential Geometry - Connections, Curvature and Characteristic Classes and is published by Springer.

Milnor "Morse Theory" contains an extremely well written introduction to the subject.

Riemannian Geometry by S.Gallot, D.Hulin, J.Lafontaine is a great book which contains a lot more than one could learn in an introductory course in Riemannian geometry. I would suggest this book if you already have some idea in the subject. Reading this book for someone new to the subject might be cumbersome and intimidating.

Have fun :)