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Books > Science & Mathematics > Mathematics > Geometry > Differential & Riemannian geometry
This collection of papers constitutes a wide-ranging survey of
recent developments in differential geometry and its interactions
with other fields, especially partial differential equations and
mathematical physics. This area of mathematics was the subject of a
special program at the Institute for Advanced Study in Princeton
during the academic year 1979-1980; the papers in this volume were
contributed by the speakers in the sequence of seminars organized
by Shing-Tung Yau for this program. Both survey articles and
articles presenting new results are included. The articles on
differential geometry and partial differential equations include a
general survey article by the editor on the relationship of the two
fields and more specialized articles on topics including harmonic
mappings, isoperimetric and Poincare inequalities, metrics with
specified curvature properties, the Monge-Arnpere equation, L2
harmonic forms and cohomology, manifolds of positive curvature,
isometric embedding, and Kraumlhler manifolds and metrics. The
articles on differential geometry and mathematical physics cover
such topics as renormalization, instantons, gauge fields and the
Yang-Mills equation, nonlinear evolution equations, incompleteness
of space-times, black holes, and quantum gravity. A feature of
special interest is the inclusion of a list of more than one
hundred unsolved research problems compiled by the editor with
comments and bibliographical information.
The book aims to present a comprehensive survey on biharmonic
submanifolds and maps from the viewpoint of Riemannian geometry. It
provides some basic knowledge and tools used in the study of the
subject as well as an overall picture of the development of the
subject with most up-to-date important results.Biharmonic
submanifolds are submanifolds whose isometric immersions are
biharmonic maps, thus biharmonic submanifolds include minimal
submanifolds as a subclass. Biharmonic submanifolds also appeared
in the study of finite type submanifolds in Euclidean
spaces.Biharmonic maps are maps between Riemannian manifolds that
are critical points of the bienergy. They are generalizations of
harmonic maps and biharmonic functions which have many important
applications and interesting links to many areas of mathematics and
theoretical physics.Since 2000, biharmonic submanifolds and maps
have become a vibrant research field with a growing number of
researchers around the world, with many interesting results have
been obtained.This book containing basic knowledge, tools for some
fundamental problems and a comprehensive survey on the study of
biharmonic submanifolds and maps will be greatly beneficial for
graduate students and beginning researchers who want to study the
subject, as well as researchers who have already been working in
the field.
In der Reihe "TEUBNER-ARCHIV zur Mathematik" werden bedeutende
klassische Arbeiten kommentiert, mit aktuellen Anmerkungen versehen
und durch Literaturhin- weise ergiinzt. Dieser erste Band enthillt
fotomechanische Nachdrucke von vier Beitragen der Mathe- matiker C.
F. GAUSS, B. RIEMANN und H. MINKOWSKI. Diese Arbeiten waren grund-
legend filr die Entwicklung und Weiterentwicklung der
Differentialgeometrie als innere Geometrie bis zur allgemeinen Rel,
ativitatstheorie. Es ist gewiB nicht nur ein Zufall, daB sich filr
diese drei Manner die produktive Zeit des Wirkens auf dem genannten
Gebiet der Geometrie in der Universitiitsstadt Gottingen vollzog.
Durch die folgenden Satze ALBERT EINSTEINS aus seiner Abhandlung
tiber die Grund- ztige der Relativitatstheorie aus dem Jahre 1922
lassen sich in einfacher und klarer Weise die diesbeztiglichen
Verdienste dieser drei Mathematiker charakterisieren: "GAUSS hat in
seiner Fliichentheorie die metrischen Eigenschaften einer in einem
dreidimensionalen euklidischen Raum eingebetteten Fliiche
untersucht und gezeigt, daB diese durch Begriffe beschrieben werden
konnen, die sich nur auf die Flache selbst, nicht aber auf die Ein-
bettung beziehen . . . RIEMANN dehnte den GauBschen Gedankengang
auf Kontinua beliebiger Dimensionszahl aus; er hat die
physikalische Bedeutung dieser Verallgemei- nerung der Geometrie
EUKLIDS mit prophetischem Blick vorausgesehen . . . Durch die
Einfilhrung der imaginiiren Zeitvariable X4 = it hat MINKOWSKI die
Invariantentheorie des vierdimensionalen Kontinuums des
physikalischen Geschehens der des dreidimen- sionalen Kontinuums
des euklidischen Raumes vollig analog gemacht.
The present volume contains all but two of the papers read at the
conference, as well as a few papers and short notes submitted
afterwards. We hope that it reflects faithfully the present state
of research in the fields covered, and that it may provide an
access to these fields for future investigations.
Differential Geometry in Physics is a treatment of the mathematical
foundations of the theory of general relativity and gauge theory of
quantum fields. The material is intended to help bridge the gap
that often exists between theoretical physics and applied
mathematics. The approach is to carve an optimal path to learning
this challenging field by appealing to the much more accessible
theory of curves and surfaces. The transition from classical
differential geometry as developed by Gauss, Riemann and other
giants, to the modern approach, is facilitated by a very intuitive
approach that sacrifices some mathematical rigor for the sake of
understanding the physics. The book features numerous examples of
beautiful curves and surfaces often reflected in nature, plus more
advanced computations of trajectory of particles in black holes.
Also embedded in the later chapters is a detailed description of
the famous Dirac monopole and instantons. Features of this book: *
Chapters 1-4 and chapter 5 comprise the content of a one-semester
course taught by the author for many years. * The material in the
other chapters has served as the foundation for many master's
thesis at University of North Carolina Wilmington for students
seeking doctoral degrees. * An open access ebook edition is
available at Open UNC (https://openunc.org) * The book contains
over 80 illustrations, including a large array of surfaces related
to the theory of soliton waves that does not commonly appear in
standard mathematical texts on differential geometry.
This book collects papers presented in the Invited Workshop,
"Liutex and Third Generation of Vortex Definition and
Identification for Turbulence," from CHAOS2020, June 9-12, 2020,
which was held online as a virtual conference. Liutex is a new
physical quantity introduced by Prof. Chaoqun Liu of the University
of Texas at Arlington. It is a vector and could give a unique and
accurate mathematical definition for fluid rotation or vortex. The
papers in this volume include some Liutex theories and many
applications in hydrodynamics, aerodynamics and thermal dynamics
including turbine machinery. As vortex exists everywhere in the
universe, a mathematical definition of vortex or Liutex will play a
critical role in scientific research. There is almost no place
without vortex in fluid dynamics. As a projection, the Liutex
theory will play an important role on the investigations of the
vortex dynamics in hydrodynamics, aerodynamics, thermodynamics,
oceanography, meteorology, metallurgy, civil engineering,
astronomy, biology, etc. and to the researches of the generation,
sustenance, modelling and controlling of turbulence.
Aus dem Vorwort: "Globale Probleme der Differentialgeometrie
erfreuen sich eines immer noch wachsenden Interesses. Gerade in der
Riemannschen Geometrie hat die Frage nach Beziehungen zwischen
Riemannscher und topologischer Struktur in neuerer Zeit zu vielen
schonen und uberraschenden Einsichten gefuhrt. Dabei denken wir
hier vor allem an den Problemkreis: Welche topologischen
Invarianten werden charakterisiert durch eine der wichtigsten
isometrischen Invarianten, die Krummung? Ziel der folgenden Noten
ist, einige zentrale Resultate in dieser Richtung darzustellen....
Wir haben uns bemuht, die Darstellung moglichst elementar und in
sich abgeschlossen zu halten und einen einfachen leistungsfahigen
Kalkul zu entwickeln.""
1. Innere Produkte Wir fUhren im Ramne ein kartesisches
Koordinatensystem ein, dessen Achsen so orientiert sind, wie das in
der Fig. 1 angedeutet ist. Die drei Koordinaten eines Punktes
bezeichnen wir mit XI, X, x - Alle betrach- 2 3 teten Punkte setzen
wir, falls nicht ausdrucklich etwas anderes gesagt wird, als reell
voraus. Xz Xl Fig.1. Zwei in bestimmter Reihenfolge angeordnete
Punkte und t) des Raumes mit den Koordinaten XI' X, x3 und YI' Y2,
Y3 bestimmen eine 2 von nach t) fuhrende gerichtete Strecke. Zwei
zu den Punktepaaren, t) und i, gehOrende gerichtete Strecken sind
dann und nur dann gleichsinnig parallel und gleich lang, wenn die
entsprechenden Koordi- natendifferenzen alle ubereinstimmen: (1) Yi
- Xi = Yi - Xi (i = 1, 2, 3). Wir bezeichnen das System aller von
den samtlichen Punkten des Rau- mes auslaufenden gerichteten
Strecken von einer und derselben Rich- tung, demselben Sinn und der
gleichen Lange als einen Vektor. Da fUr diese Strecken die
Koordinatendifferenzen der beiden Endpunkte immer die gleichen
sind, k6nnen wir diese drei Differenzen dem Vektor als seine 2
Einleitung Komponenten zuordnen, und zwar entsprechen die
verschiedenen Systeme der als Vektorkomponenten genommenen
Zahlentripel eineindeutig den verschiedenen Vektoren. An den
Vektoren ist bemerkenswert, daB ihre Komponenten sich bei einer
Parallelverschiebung des Koordinaten- systems nicht andern im
Gegensatz zu den Koordinaten der Punkte.
This text focuses on developing an intimate acquaintance with the
geometric meaning of curvature and thereby introduces and
demonstrates all the main technical tools needed for a more
advanced course on Riemannian manifolds. It covers proving the four
most fundamental theorems relating curvature and topology: the
Gauss-Bonnet Theorem, the Cartan-Hadamard Theorem, Bonnet's
Theorem, and a special case of the Cartan-Ambrose-Hicks Theorem.
Discrete Differential Geometry (DDG) is an emerging discipline at
the boundary between mathematics and computer science. It aims to
translate concepts from classical differential geometry into a
language that is purely finite and discrete, and can hence be used
by algorithms to reason about geometric data. In contrast to
standard numerical approximation, the central philosophy of DDG is
to faithfully and exactly preserve key invariants of geometric
objects at the discrete level. This process of translation from
smooth to discrete helps to both illuminate the fundamental meaning
behind geometric ideas and provide useful algorithmic guarantees.
This volume is based on lectures delivered at the 2018 AMS Short
Course ``Discrete Differential Geometry,'' held January 8-9, 2018,
in San Diego, California. The papers in this volume illustrate the
principles of DDG via several recent topics: discrete nets,
discrete differential operators, discrete mappings, discrete
conformal geometry, and discrete optimal transport.
A sequel to Lectures on Riemann Surfaces (Mathematical Notes,
1966), this volume continues the discussion of the dimensions of
spaces of holomorphic cross-sections of complex line bundles over
compact Riemann surfaces. Whereas the earlier treatment was limited
to results obtainable chiefly by one-dimensional methods, the more
detailed analysis presented here requires the use of various
properties of Jacobi varieties and of symmetric products of Riemann
surfaces, and so serves as a further introduction to these topics
as well. The first chapter consists of a rather explicit
description of a canonical basis for the Abelian differentials on a
marked Riemann surface, and of the description of the canonical
meromorphic differentials and the prime function of a marked
Riemann surface. Chapter 2 treats Jacobi varieties of compact
Riemann surfaces and various subvarieties that arise in determining
the dimensions of spaces of holomorphic cross-sections of complex
line bundles. In Chapter 3, the author discusses the relations
between Jacobi varieties and symmetric products of Riemann surfaces
relevant to the determination of dimensions of spaces of
holomorphic cross-sections of complex line bundles. The final
chapter derives Torelli's theorem following A. Weil, but in an
analytical context. Originally published in 1973. The Princeton
Legacy Library uses the latest print-on-demand technology to again
make available previously out-of-print books from the distinguished
backlist of Princeton University Press. These editions preserve the
original texts of these important books while presenting them in
durable paperback and hardcover editions. The goal of the Princeton
Legacy Library is to vastly increase access to the rich scholarly
heritage found in the thousands of books published by Princeton
University Press since its founding in 1905.
Noncommutative geometry combines themes from algebra, analysis and
geometry and has significant applications to physics. This book
focuses on cyclic theory, and is based upon the lecture courses by
Daniel G. Quillen at the University of Oxford from 1988-92, which
developed his own approach to the subject. The basic definitions,
examples and exercises provided here allow non-specialists and
students with a background in elementary functional analysis,
commutative algebra and differential geometry to get to grips with
the subject. Quillen's development of cyclic theory emphasizes
analogies between commutative and noncommutative theories, in which
he reinterpreted classical results of Hamiltonian mechanics,
operator algebras and differential graded algebras into a new
formalism. In this book, cyclic theory is developed from motivating
examples and background towards general results. Themes covered are
relevant to current research, including homomorphisms modulo powers
of ideals, traces on noncommutative differential forms, quasi-free
algebras and Chern characters on connections.
Dieser Buchtitel ist Teil des Digitalisierungsprojekts Springer
Book Archives mit Publikationen, die seit den Anfangen des Verlags
von 1842 erschienen sind. Der Verlag stellt mit diesem Archiv
Quellen fur die historische wie auch die disziplingeschichtliche
Forschung zur Verfugung, die jeweils im historischen Kontext
betrachtet werden mussen. Dieser Titel erschien in der Zeit vor
1945 und wird daher in seiner zeittypischen politisch-ideologischen
Ausrichtung vom Verlag nicht beworben.
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