Peter Petersen 
Riemannian Geometry [PDF ebook] 

Intended for a one year course, this volume serves as a single source, introducing students 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, while also presenting the most up-to-date research. Important additions to this new edition include: — A completely new coordinate free formula that is easily remembered, and is, in fact, the Koszul formula in disguise — An increased number of coordinate calculations of connection and curvature — General fomulas for curvature on Lie Groups and submersions — Variational calculus has been integrated into the text, which allows for an early treatment of the Sphere theorem using a forgottten proof by Berger — Several recent results about manifolds with positive curvature