A Panoramic View of Riemannian Geometry - M. Berger
Riemannian geometry has become 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, with complete definitions, proofs, and so on. This contrasts sharply
with the situation in 1943, when Preissmann's dissertation 1943 [1041] 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.
We have two goals: first, to introduce the various concepts and tools of
Riemannian geometry in the most natural way; or further, to demonstrate
that one is practically forced to deal with abstract Riemannian manifolds in a
host of intuitive geometrical questions. This explains why a long first chapter
will deal with problems in the Euclidean plane. Second, 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 try to be complete, but hope that our panorama will be large enough to
show the reader a substantial part of today's Riemannian geometry.
In a panorama, you see the peaks, but you do not climb them. This is
a way of saying that we will not prove the statements we quote. But, in a
panorama, sometimes you can still see the path to a summit; analogously in
many cases we will explain the main ideas or the main ingredients for the
proof.
We hope that this form of presentation will leave many readers wanting
to climb some peak. We will give all the needed references to the literature
as the introduction and the panorama unfold. For alpinists, the equivalent
of such a book will be the refuge de haute-montagne (the base camp) where
you need to spend the night before the final climb. In the worst (we might
say, the grandest) cases, like in the Himalayas, a climber has to establish as
many as five base camps. The scientific analogue is that you need not only
books, but also original articles.
Even without proofs or definitions, some of the peaks lie very far beyond.
Distant topics will be mentioned only briefly in chapter 14. The judgement
that a peak lies far away is personal; in the present case, we mean far from
the author. His writing a book on Riemannian geometry does not indicate
that he is an expert on every topic of it, especially the recent topics.
One may ask why we study only two objects: Euclidean domains with
boundary, and Riemannian manifolds without boundary. There is a notion of
Riemannian manifold with boundary, but in the Euclidean domain the inte-
rior geometry is given, fiat and trivial, and the interesting phenomena come
from the shape of the boundary. Riemannian manifolds have no boundary,
and the geometric phenomena are those of the interior. Asking for both at
the same time risks having too much to handle (however see 14.5.1).
The present text is an introduction, so we have to refrain from saying
too much. For example, we will mainly consider compact Riemannian mani-
folds. But noncompact ones are also a very important subject; they are more
challenging and more difficult to study.
We will conform to the following principles:
- This book is not a handbook of Riemannian geometry, nor a systematic
awarding of prizes. We give only the best recent results, not all of the
intermediate ones. However, we mention when the desired type of results
started to appear, this being of historical interest and at the same time
helping the reader to realize the difficulty of the problem. We hope that
those whose results are not mentioned will pardon us.
- We present open problems as soon as they can be stated. This encourages
the reader to appreciate the difficulty and the current state of each problem.
Since this text is unusual, it is natural to expect unusual features of pre-
sentation. First, references are especially important in a book about mathe-
matical culture. But there should not be too many. Generally, we will only
give a few of the recent references. From these, the interested reader will be
able to trace back to most of the standard sources. When we are consid-
ering very basic notions (like that of manifold or billiard) we will typically
give many references. The reader might prefer to work with one more than
another. Second, since we will not give formal definitions in the text, we
thought the reader might find it useful to have most of them collected in the
final chapter.
Some words about organization: first, the immensity of the field poses a
problem of classification; in our division into chapters, necessarily arbitrary,
we did not follow any logical or historical order. We have tried to follow a
certain naturalhess and simplicity. This explains why many recent discoveries,
like those concerning the isoperimetric profile, the systolic inequalities, the
spectrum, the geodesic flow and periodic geodesics come before a host of
discoveries relating the topology of the underlying manifolds with various
assumptions on curvature, although the latter results came to light much
earlier than the former.
Second, our treatment of topics is certainly uneven, but this reflects the
tastes and knowledge of the author. Disparities appear in the choice of results
presented and in what we will offer as ideas behind the proofs. We apologize
for that. For example, everything concerning bundles over Riemannian man-
ifolds, especially spin bundles and spin geometry, will be very sketchy.
We hope that despite these weaknesses, the present book will bring plea-
sure and be of help to professional Riemannian geometers as well as those
who want to enter into the realm of Riemannian geometry, which is an amaz-
ingly beautiful, active and natural field of research today. The reader who
finds this book worthwhile will be interested in reading Dillen & Verstraelen
2000 [449].
- Verlag: Springer, Berlin
- 2003
- Ausstattung/Bilder: 2002. 875 p.
- Best.Nr. des Verlages: 10117320
- Deutsch
- Abmessung: 235mm x 155mm x 53mm
- Gewicht: 1366g
- ISBN-13: 9783540653172
- ISBN-10: 3540653171
- Best.Nr.: 11103184
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, with complete definitions, proofs, and so on. This contrasts sharply
with the situation in 1943, when Preissmann's dissertation 1943 [1041] 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.
We have two goals: first, to introduce the various concepts and tools of
Riemannian geometry in the most natural way; or further, to demonstrate
that one is practically forced to deal with abstract Riemannian manifolds in a
host of intuitive geometrical questions. This explains why a long first chapter
will deal with problems in the Euclidean plane. Second, 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 try to be complete, but hope that our panorama will be large enough to
show the reader a substantial part of today's Riemannian geometry.
In a panorama, you see the peaks, but you do not climb them. This is
a way of saying that we will not prove the statements we quote. But, in a
panorama, sometimes you can still see the path to a summit; analogously in
many cases we will explain the main ideas or the main ingredients for the
proof.
We hope that this form of presentation will leave many readers wanting
to climb some peak. We will give all the needed references to the literature
as the introduction and the panorama unfold. For alpinists, the equivalent
of such a book will be the refuge de haute-montagne (the base camp) where
you need to spend the night before the final climb. In the worst (we might
say, the grandest) cases, like in the Himalayas, a climber has to establish as
many as five base camps. The scientific analogue is that you need not only
books, but also original articles.
Even without proofs or definitions, some of the peaks lie very far beyond.
Distant topics will be mentioned only briefly in chapter 14. The judgement
that a peak lies far away is personal; in the present case, we mean far from
the author. His writing a book on Riemannian geometry does not indicate
that he is an expert on every topic of it, especially the recent topics.
One may ask why we study only two objects: Euclidean domains with
boundary, and Riemannian manifolds without boundary. There is a notion of
Riemannian manifold with boundary, but in the Euclidean domain the inte-
rior geometry is given, fiat and trivial, and the interesting phenomena come
from the shape of the boundary. Riemannian manifolds have no boundary,
and the geometric phenomena are those of the interior. Asking for both at
the same time risks having too much to handle (however see 14.5.1).
The present text is an introduction, so we have to refrain from saying
too much. For example, we will mainly consider compact Riemannian mani-
folds. But noncompact ones are also a very important subject; they are more
challenging and more difficult to study.
We will conform to the following principles:
- This book is not a handbook of Riemannian geometry, nor a systematic
awarding of prizes. We give only the best recent results, not all of the
intermediate ones. However, we mention when the desired type of results
started to appear, this being of historical interest and at the same time
helping the reader to realize the difficulty of the problem. We hope that
those whose results are not mentioned will pardon us.
- We present open problems as soon as they can be stated. This encourages
the reader to appreciate the difficulty and the current state of each problem.
Since this text is unusual, it is natural to expect unusual features of pre-
sentation. First, references are especially important in a book about mathe-
matical culture. But there should not be too many. Generally, we will only
give a few of the recent references. From these, the interested reader will be
able to trace back to most of the standard sources. When we are consid-
ering very basic notions (like that of manifold or billiard) we will typically
give many references. The reader might prefer to work with one more than
another. Second, since we will not give formal definitions in the text, we
thought the reader might find it useful to have most of them collected in the
final chapter.
Some words about organization: first, the immensity of the field poses a
problem of classification; in our division into chapters, necessarily arbitrary,
we did not follow any logical or historical order. We have tried to follow a
certain naturalhess and simplicity. This explains why many recent discoveries,
like those concerning the isoperimetric profile, the systolic inequalities, the
spectrum, the geodesic flow and periodic geodesics come before a host of
discoveries relating the topology of the underlying manifolds with various
assumptions on curvature, although the latter results came to light much
earlier than the former.
Second, our treatment of topics is certainly uneven, but this reflects the
tastes and knowledge of the author. Disparities appear in the choice of results
presented and in what we will offer as ideas behind the proofs. We apologize
for that. For example, everything concerning bundles over Riemannian man-
ifolds, especially spin bundles and spin geometry, will be very sketchy.
We hope that despite these weaknesses, the present book will bring plea-
sure and be of help to professional Riemannian geometers as well as those
who want to enter into the realm of Riemannian geometry, which is an amaz-
ingly beautiful, active and natural field of research today. The reader who
finds this book worthwhile will be interested in reading Dillen & Verstraelen
2000 [449].
