A Panoramic View Of Riemannian Geometry (Springer, 2002)(T)(872S).pdf

Marcel Berger
A Panoramic View of Riemannian
Geometry
27th August 2002
Springer
Berlin Heidelberg NewYork
Barcelona Hong Kong
London Milan Paris
Tokyo
Preface
VIII Preface
Riemannian geometry has e an important and vast subject. It deserves
an encyclopedia, rather than a modest length book. It is therefore impossible
to present Riemannian geometry in a book in the standard fashion of math-
ematics, plete definitions, proofs, and so on. This contrasts sharply
with the situation in 1943, when Preissmann’s dissertation 1943 [1015] pre-
sented all the global results of Riemannian geometry (but for the theory of
symmetric spaces) including new ones, with proofs, in only forty pages.
Moreover, even at the root of the subject, the idea of a Riemannian mani-
fold is subtle, appealing to unnatural concepts. Consequently, all recent books
on Riemannian geometry, however good they may be, can only present two
or three topics, having to spend quite a few pages on the foundations. Since
our aim is to introduce the reader to most of the living topics of the field,
we have had to follow the only possible path: to present the results without
proofs.
Our aim is twofold, as announced by our subtitle. The first is to introduce
the various concepts and tools of Riemannian geometry in the most natural
way; or more, to demonstrate that one is practically forced to deal with
abstract Riemannian manifolds in a host of intuitive geometrical questions.
This explains the word “introduction” and the fact that a long first chapter
will deal with problems in the Euclidean plane. Secondly, once equipped with
the concept of Riemannian manifold, we will present a panorama of current
day Riemannian geometry. A panorama is never a full 360 degrees, so we will
not pretend to plete, but hope that our panorama will be large enough
to show the reader a substantial part of today’s Riemannian geometry.
And in a panorama, you see the peaks, but you do not climb them. This