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Both classical geometry and modern differential geometry have
been active subjects of research throughout the 20th century and
lie at the heart of many recent advances in mathematics and
physics. The underlying motivating concept for the present book is
that it offers readers the elements of a modern geometric culture
by means of a whole series of visually appealing unsolved (or
recently solved) problems that require the creation of concepts and
tools of varying abstraction. Starting with such natural, classical
objects as lines, planes, circles, spheres, polygons, polyhedra,
curves, surfaces, convex sets, etc., crucial ideas and above all
abstract concepts needed for attaining the results are elucidated.
These are conceptual notions, each built "above" the preceding and
permitting an increase in abstraction, represented metaphorically
by Jacob's ladder with its rungs: the 'ladder' in the Old
Testament, that angels ascended and descended...
In all this, the aim of the book is to demonstrate to readers
the unceasingly renewed spirit of geometry and that even so-called
"elementary" geometry is very much alive and at the very heart of
the work of numerous contemporary mathematicians. It is also shown
that there are innumerable paths yet to be explored and concepts to
be created. The book is visually rich and inviting, so that readers
may open it at random places and find much pleasure throughout
according their own intuitions and inclinations.
Marcel Berger is the author of numerous successful books on
geometry, this book once again is addressed to all students and
teachers of mathematics with an affinity for geometry.
The textbook Geometry, published in French by CEDICjFernand Nathan
and in English by Springer-Verlag (scheduled for 1985) was very
favorably re ceived. Nevertheless, many readers found the text too
concise and the exercises at the end of each chapter too difficult,
and regretted the absence of any hints for the solution of the
exercises. This book is intended to respond, at least in part, to
these needs. The length of the textbook (which will be referred to
as B] throughout this book) and the volume of the material covered
in it preclude any thought of publishing an expanded version, but
we considered that it might prove both profitable and amusing to
some of our readers to have detailed solutions to some of the
exercises in the textbook. At the same time, we planned this book
to be independent, at least to a certain extent, from the textbook;
thus, we have provided summaries of each of its twenty chapters,
condensing in a few pages and under the same titles the most
important notions and results, used in the solution of the
problems. The statement of the selected problems follows each
summary, and they are numbered in order, with a reference to the
corresponding place in B]. These references are not meant as
indications for the solutions of the problems. In the body of each
summary there are frequent references to B], and these can be
helpful in elaborating a point which is discussed too cursorily in
this book."
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.
This book consists of two parts, different in form but similar in
spirit. The first, which comprises chapters 0 through 9, is a
revised and somewhat enlarged version of the 1972 book Geometrie
Differentielle. The second part, chapters 10 and 11, is an attempt
to remedy the notorious absence in the original book of any
treatment of surfaces in three-space, an omission all the more
unforgivable in that surfaces are some of the most common
geometrical objects, not only in mathematics but in many branches
of physics. Geometrie Differentielle was based on a course I taught
in Paris in 1969- 70 and again in 1970-71. In designing this course
I was decisively influ enced by a conversation with Serge Lang, and
I let myself be guided by three general ideas. First, to avoid
making the statement and proof of Stokes' formula the climax of the
course and running out of time before any of its applications could
be discussed. Second, to illustrate each new notion with
non-trivial examples, as soon as possible after its introduc tion.
And finally, to familiarize geometry-oriented students with
analysis and analysis-oriented students with geometry, at least in
what concerns manifolds."
Both classical geometry and modern differential geometry have been
active subjects of research throughout the 20th century and lie at
the heart of many recent advances in mathematics and physics. The
underlying motivating concept for the present book is that it
offers readers the elements of a modern geometric culture by means
of a whole series of visually appealing unsolved (or recently
solved) problems that require the creation of concepts and tools of
varying abstraction. Starting with such natural, classical objects
as lines, planes, circles, spheres, polygons, polyhedra, curves,
surfaces, convex sets, etc., crucial ideas and above all abstract
concepts needed for attaining the results are elucidated. These are
conceptual notions, each built "above" the preceding and permitting
an increase in abstraction, represented metaphorically by Jacob's
ladder with its rungs: the 'ladder' in the Old Testament, that
angels ascended and descended... In all this, the aim of the book
is to demonstrate to readers the unceasingly renewed spirit of
geometry and that even so-called "elementary" geometry is very much
alive and at the very heart of the work of numerous contemporary
mathematicians. It is also shown that there are innumerable paths
yet to be explored and concepts to be created. The book is visually
rich and inviting, so that readers may open it at random places and
find much pleasure throughout according their own intuitions and
inclinations. Marcel Berger is t he author of numerous successful
books on geometry, this book once again is addressed to all
students and teachers of mathematics with an affinity for geometry.
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.
This book consists of two parts, different in form but similar in
spirit. The first, which comprises chapters 0 through 9, is a
revised and somewhat enlarged version of the 1972 book Geometrie
Differentielle. The second part, chapters 10 and 11, is an attempt
to remedy the notorious absence in the original book of any
treatment of surfaces in three-space, an omission all the more
unforgivable in that surfaces are some of the most common
geometrical objects, not only in mathematics but in many branches
of physics. Geometrie Differentielle was based on a course I taught
in Paris in 1969- 70 and again in 1970-71. In designing this course
I was decisively influ enced by a conversation with Serge Lang, and
I let myself be guided by three general ideas. First, to avoid
making the statement and proof of Stokes' formula the climax of the
course and running out of time before any of its applications could
be discussed. Second, to illustrate each new notion with
non-trivial examples, as soon as possible after its introduc tion.
And finally, to familiarize geometry-oriented students with
analysis and analysis-oriented students with geometry, at least in
what concerns manifolds."
The textbook Geometry, published in French by CEDICjFernand Nathan
and in English by Springer-Verlag (scheduled for 1985) was very
favorably re ceived. Nevertheless, many readers found the text too
concise and the exercises at the end of each chapter too difficult,
and regretted the absence of any hints for the solution of the
exercises. This book is intended to respond, at least in part, to
these needs. The length of the textbook (which will be referred to
as B] throughout this book) and the volume of the material covered
in it preclude any thought of publishing an expanded version, but
we considered that it might prove both profitable and amusing to
some of our readers to have detailed solutions to some of the
exercises in the textbook. At the same time, we planned this book
to be independent, at least to a certain extent, from the textbook;
thus, we have provided summaries of each of its twenty chapters,
condensing in a few pages and under the same titles the most
important notions and results, used in the solution of the
problems. The statement of the selected problems follows each
summary, and they are numbered in order, with a reference to the
corresponding place in B]. These references are not meant as
indications for the solutions of the problems. In the body of each
summary there are frequent references to B], and these can be
helpful in elaborating a point which is discussed too cursorily in
this book."
The DD6 Symposium was, like its predecessors DD1 to DD5 both a
research symposium and a summer seminar and concentrated on
differential geometry. This volume contains a selection of the
invited papers and some additional contributions. They cover recent
advances and principal trends in current research in differential
geometry.
This is a reproduction of a book published before 1923. This book
may have occasional imperfections such as missing or blurred pages,
poor pictures, errant marks, etc. that were either part of the
original artifact, or were introduced by the scanning process. We
believe this work is culturally important, and despite the
imperfections, have elected to bring it back into print as part of
our continuing commitment to the preservation of printed works
worldwide. We appreciate your understanding of the imperfections in
the preservation process, and hope you enjoy this valuable book.
This scarce antiquarian book is a selection from Kessinger
Publishing's Legacy Reprint Series. Due to its age, it may contain
imperfections such as marks, notations, marginalia and flawed
pages. Because we believe this work is culturally important, we
have made it available as part of our commitment to protecting,
preserving, and promoting the world's literature. Kessinger
Publishing is the place to find hundreds of thousands of rare and
hard-to-find books with something of interest for everyone!
This scarce antiquarian book is a selection from Kessinger
Publishing's Legacy Reprint Series. Due to its age, it may contain
imperfections such as marks, notations, marginalia and flawed
pages. Because we believe this work is culturally important, we
have made it available as part of our commitment to protecting,
preserving, and promoting the world's literature. Kessinger
Publishing is the place to find hundreds of thousands of rare and
hard-to-find books with something of interest for everyone!
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