My goals in thiook on Riemannian geometry are essentially the same as those which guided me in my Eigenvalues in Riemannian Geometry [69], to introduce the subject, to coherently present a number of itasic techniques and results with a mind to future work, and to present some of the results that are attractive in their own right. Thiook differs from Eigenvalues in that it starts at a more basic level,and therefore, it must present a broader view of the ideas from which all the various directions emerge. At the same time, other treatments of Riemannian geometry are available at varying levels and interests,so I need not introduce everything. I have, therefore, attempted a viable introduction to Riemannian geometry for a very broad group of students, with emphases and developments in areas not covered by other books.

The launch of this Advanced Lectures in Mathematics series is aimed at kegmathematicians informed of the latest developments in mathematics, as well asto aid in the learning of new mathematical topics by students all over the world.Each volume consists of either an expository monograph or a collection of signifi-cant introductions to important topics. This series emphasizes the history andsources of motivation for the topics under discussion, and also gives an overviewof the current status of research in each particular field. These volumes are thefirst source to which people will turn in order to learn new subjects and to dis-cover the latest results of many cutting-edge fields in mathematics.