Riemannian geometry has today become a vast and important subject. This
new book of Marcel Berger sets out to introduce readers to most of the
living topics of the field and convey them quickly to the main results
known to date. These results are stated without detailed proofs but the
main ideas involved are described and motivated. This enables the reader
to obtain a sweeping panoramic view of almost the entirety of the field.
However, since a Riemannian manifold is, even initially, a subtle
object, appealing to highly non-natural concepts, the first three
chapters devote themselves to introducing the various concepts and tools
of Riemannian geometry in the most natural and motivating way, following
in particular Gauss and Riemann.