A working knowledge of differential forms so strongly illuminates the calculus and its developments that it ought not be too long delayed in the curriculum. On the other hand, the systematic treatment of differential forms requires an apparatus of topology and algebra which is heavy for beginning undergraduates. Several texts on advanced calculus using differential forms have appeared in recent years. We may cite as representative of the variety of approaches the books of Fleming [2], (1) Nickerson-Spencer-Steenrod [3], and Spivak [6]. . Despite their accommodation to the innocence of their readers, these texts cannot lighten the burden of apparatus exactly because they offer a more or less full measure of the truth at some level of generality in a formally precise exposition. There. is consequently a gap between texts of this type and the traditional advanced calculus. Recently, on the occasion of offering a beginning course of advanced calculus, we undertook the expe- ment of attempting to present the technique of differential forms with minimal apparatus and very few prerequisites. These notes are the result of that experiment. Our exposition is intended to be heuristic and concrete. Roughly speaking, we take a differential form to be a multi-dimensional integrand, such a thing being subject to rules making change-of-variable calculations automatic. The domains of integration (manifolds) are explicitly given "surfaces" in Euclidean space. The differentiation of forms (exterior (1) Numbers in brackets refer to the Bibliography at the end.

This book aims to present to first and second year graduate students a beautiful and relatively accessible field of mathematics-the theory of singu larities of stable differentiable mappings. The study of stable singularities is based on the now classical theories of Hassler Whitney, who determined the generic singularities (or lack of them) of Rn ~ Rm (m ~ 2n - 1) and R2 ~ R2, and Marston Morse, for mappings who studied these singularities for Rn ~ R. It was Rene Thorn who noticed (in the late '50's) that all of these results could be incorporated into one theory. The 1960 Bonn notes of Thom and Harold Levine (reprinted in [42]) gave the first general exposition of this theory. However, these notes preceded the work of Bernard Malgrange [23] on what is now known as the Malgrange Preparation Theorem-which allows the relatively easy computation of normal forms of stable singularities as well as the proof of the main theorem in the subject-and the definitive work of John Mather. More recently, two survey articles have appeared, by Arnold [4] and Wall [53], which have done much to codify the new material; still there is no totally accessible description of this subject for the beginning student. We hope that these notes will partially fill this gap. In writing this manuscript, we have repeatedly cribbed from the sources mentioned above-in particular, the Thom-Levine notes and the six basic papers by Mather.

The geometry of complex hyperbolic space has not, so far, been given a comprehensive treatment in the literature. This book seeks to address this by providing an overview of this particularly rich area of research, and is largely motivated by the wide applications in other areas of mathematics and physics.

Algebraic groups and Lie groups are important in most major areas of mathematics, occuring in diverse roles such as the symmetries of differential equations and as central figures in the Langlands program for number theory. In this book, Professor Borel looks at the development of the theory of Lie groups and algebraic groups, highlighting the evolution from the almost purely local theory at the start to the global theory that we know today. As the starting point of this passage from local to global, the author takes Lie's theory of local analytic transformation groups and Lie algebras. He then follows the globalization of the process in its two most important frameworks: (transcendental) differential geometry and algebraic geometry. Chapters II to IV are devoted to the former, Chapters V to VIII, to the latter.The essays in the first part of the book survey various proofs of the full reducibility of linear representations of $SL 2M$, the contributions H. Weyl to representation and invariant theory for Lie groups, and conclude with a chapter on E. Cartan's theory of symmetric spaces and Lie groups in the large. The second part of the book starts with Chapter V describing the development of the theory of linear algebraic groups in the 19th century. Many of the main contributions here are due to E. Study, E. Cartan, and above all, to L. Maurer. After being abandoned for nearly 50 years, the theory was revived by Chevalley and Kolchin and then further developed by many others. This is the focus of Chapter VI. The book concludes with two chapters on various aspects of the works of Chevalley on Lie groups and algebraic groups and Kolchin on algebraic groups and the Galois theory of differential fields. The author brings a unique perspective to this study. As an important developer of some of the modern elements of both the differential geometric and the algebraic geometric sides of the theory, he has a particularly deep appreciation of the underlying mathematics. His lifelong involvement and his historical research in the subject give him a special appreciation of the story of its development.

This paper is a contribution to the topological study of vector fields on manifolds. In particular we shall be concerned with the problems of exist ence of r linearly independent vector fields. For r = 1 the classical result of H. Hopf asserts that the vanishing of the Euler characteristic is the necessary and sufficient condition, and our results will give partial extens ions of Hopf's theorem to the case r > 1. Arecent article by E. Thomas [10] gives a good survey of work in this general area. Our approach to these problems is based on the index theory of elliptic differential operators and is therefore rather different from the standard topological approach. Briefly speaking, what we do is to observe that certain invariants of a manifold (Euler characteristic, signature, etc. ) are indices of elliptic operators (see [5]) and the existence of a certain number of vector fields implies certain symmetry conditions for these operators and hence corresponding results for their indices. In this way we obtain certain necessary conditions for the existence of vector fields and, more generally , for the existence of fields of tangent planes. For example, one of our results is the following THEOREM (1. 1). Let X be a compact oriented smooth manifold 0/ dimension 4 q, and assume that X possesses a tangent fteld of oriented 2-planes (that is, an oriented 2-dimensional sub-bundle 0/ the tangent vector bundle).

Providing a succinct yet comprehensive treatment of the essentials of modern differential geometry and topology, this book's clear prose and informal style make it accessible to advanced undergraduate and graduate students in mathematics and the physical sciences. The text covers the basics of multilinear algebra, differentiation and integration on manifolds, Lie groups and Lie algebras, homotopy and de Rham cohomology, homology, vector bundles, Riemannian and pseudo-Riemannian geometry, and degree theory. It also features over 250 detailed exercises, and a variety of applications revealing fundamental connections to classical mechanics, electromagnetism (including circuit theory), general relativity and gauge theory. Solutions to the problems are available for instructors at www.cambridge.org/9781107042193.

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.

From Backlund to Darboux, this monograph presents a comprehensive journey through the transformation theory of constrained Willmore surfaces, a topic of great importance in modern differential geometry and, in particular, in the field of integrable systems in Riemannian geometry. The first book on this topic, it discusses in detail a spectral deformation, Backlund transformations and Darboux transformations, and proves that all these transformations preserve the existence of a conserved quantity, defining, in particular, transformations within the class of constant mean curvature surfaces in 3-dimensional space-forms, with, furthermore, preservation of both the space-form and the mean curvature, and bridging the gap between different approaches to the subject, classical and modern. Clearly written with extensive references, chapter introductions and self-contained accounts of the core topics, it is suitable for newcomers to the theory of constrained Wilmore surfaces. Many detailed computations and new results unavailable elsewhere in the literature make it also an appealing reference for experts.

The most immediate one-dimensional variation problem is certainly the problem of determining an arc of curve, bounded by two given and having a smallest possible length. The problem of deter points mining and investigating a surface with given boundary and with a smallest possible area might then be considered as the most immediate two-dimensional variation problem. The classical work, concerned with the latter problem, is summed up in a beautiful and enthusiastic manner in DARBOUX'S Theorie generale des surfaces, vol. I, and in the first volume of the collected papers of H. A. SCHWARZ. The purpose of the present report is to give a picture of the progress achieved in this problem during the period beginning with the Thesis of LEBESGUE (1902). Our problem has always been considered as the outstanding example for the application of Analysis and Geometry to each other, and the recent work in the problem will certainly strengthen this opinion. It seems, in particular, that this recent work will be a source of inspiration to the Analyst interested in Calculus of Variations and to the Geometer interested in the theory of the area and in the theory of the conformal maps of general surfaces. These aspects of the subject will be especially emphasized in this report. The report consists of six Chapters. The first three Chapters are important tools or concerned with investigations which yielded either important ideas for the proofs of the existence theorems reviewed in the last three Chapters."

This book presents, for the first time in English, the papers of Beltrami, Klein, and Poincare that brought hyperbolic geometry into the mainstream of mathematics. A recognition of Beltrami comparable to that given the pioneering works of Bolyai and Lobachevsky seems long overdue - not only because Beltrami rescued hyperbolic geometry from oblivion by proving it to be logically consistent, but because he gave it a concrete meaning (a model) that made hyperbolic geometry part of ordinary mathematics. The models subsequently discovered by Klein and Poincare brought hyperbolic geometry even further down to earth and paved the way for the current explosion of activity in low-dimensional geometry and topology.By placing the works of these three mathematicians side by side and providing commentaries, this book gives the student, historian, or professional geometer a bird's-eye view of one of the great episodes in mathematics. The unified setting and historical context reveal the insights of Beltrami, Klein, and Poincare in their full brilliance.

The present volume gives a systematic treatment of potential functions. It takes its origin in two courses, one elementary and one advanced, which the author has given at intervals during the last ten years, and has a two-fold purpose: first, to serve as an introduction for students whose attainments in the Calculus include some knowledge of partial derivatives and multiple and line integrals; and secondly, to provide the reader with the fundamentals of the subject, so that he may proceed immediately to the applications, or to the periodical literature of the day. It is inherent in the nature of the subject that physical intuition and illustration be appealed to freely, and this has been done. However, in order that the book may present sound ideals to the student, and also serve the mathematician, both for purposes of reference and as a basis for further developments, the proofs have been given by rigorous methods. This has led, at a number of points, to results either not found elsewhere, or not readily accessible. Thus, Chapter IV contains a proof for the general regular region of the divergence theorem (Gauss', or Green's theorem) on the reduction of volume to surface integrals. The treatment of the fundamental existence theorems in Chapter XI by means of integral equations meets squarely the difficulties incident to the discontinuity of the kernel, and the same chapter gives an account of the most recent developments with respect to the Dirichlet problem.

This book contains an up-to-date survey and self-contained chapters on complex slant submanifolds and geometry, authored by internationally renowned researchers. The book discusses a wide range of topics, including slant surfaces, slant submersions, nearly Kaehler, locally conformal Kaehler, and quaternion Kaehler manifolds. It provides several classification results of minimal slant surfaces, quasi-minimal slant surfaces, slant surfaces with parallel mean curvature vector, pseudo-umbilical slant surfaces, and biharmonic and quasi biharmonic slant surfaces in Lorentzian complex space forms. Furthermore, this book includes new results on slant submanifolds of para-Hermitian manifolds. This book also includes recent results on slant lightlike submanifolds of indefinite Hermitian manifolds, which are of extensive use in general theory of relativity and potential applications in radiation and electromagnetic fields. Various open problems and conjectures on slant surfaces in complex space forms are also included in the book. It presents detailed information on the most recent advances in the area, making it valuable for scientists, educators and graduate students.

The 2019 'Australian-German Workshop on Differential Geometry in the Large' represented an extraordinary cross section of topics across differential geometry, geometric analysis and differential topology. The two-week programme featured talks from prominent keynote speakers from across the globe, treating geometric evolution equations, structures on manifolds, non-negative curvature and Alexandrov geometry, and topics in differential topology. A joy to the expert and novice alike, this proceedings volume touches on topics as diverse as Ricci and mean curvature flow, geometric invariant theory, Alexandrov spaces, almost formality, prescribed Ricci curvature, and Kahler and Sasaki geometry.

This book consists of contributions from the participants of the Abel Symposium 2019 held in Alesund, Norway. It was centered about applications of the ideas of symmetry and invariance, including equivalence and deformation theory of geometric structures, classification of differential invariants and invariant differential operators, integrability analysis of equations of mathematical physics, progress in parabolic geometry and mathematical aspects of general relativity. The chapters are written by leading international researchers, and consist of both survey and research articles. The book gives the reader an insight into the current research in differential geometry and Lie theory, as well as applications of these topics, in particular to general relativity and string theory.

While Eugenio Calabi is best known for his contributions to the theory of Calabi-Yau manifolds, this Steele-Prize-winning geometer's fundamental contributions to mathematics have been far broader and more diverse than might be guessed from this one aspect of his work. His works have deep influence and lasting impact in global differential geometry, mathematical physics and beyond. By bringing together 47 of Calabi's important articles in a single volume, this book provides a comprehensive overview of his mathematical oeuvre, and includes papers on complex manifolds, algebraic geometry, Kahler metrics, affine geometry, partial differential equations, several complex variables, group actions and topology. The volume also includes essays on Calabi's mathematics by several of his mathematical admirers, including S.K. Donaldson, B. Lawson and S.-T. Yau, Marcel Berger; and Jean Pierre Bourguignon. This book is intended for mathematicians and graduate students around the world. Calabi's visionary contributions will certainly continue to shape the course of this subject far into the future.

This book contains all research papers published by the distinguished Brazilian mathematician Elon Lima. It includes the papers from his PhD thesis on homotopy theory, which are hard to find elsewhere. Elon Lima wrote more than 40 books in the field of topology and dynamical systems. He was a profound mathematician with a genuine vocation to teach and write mathematics.

Cet ouvrage traite de la transformation fondamentale survenue dans la pensee mathematique a la suite de la decouverte de la geometrie non euclidienne. Cette transformation a eu comme consequence celle d'admettre que, non seulement pouvaient exister plusieurs geometries, mais encore plusieurs espaces mathematiques et plusieurs espaces physiques differents. La recherche s'attache en grande partie a analyser les etapes qui ont conduit a cette nouvelle conception et aux idees mathematiques qui en sont le fondement. Le livre cherche egalement a en elucider la signification epistemologique et a mettre en evidence la nature et le role de l'espace dans la constitution de certaines theories mathematiques et dans la recherche des principes essentiels de la physique.

This volume collects lecture notes from courses offered at several conferences and workshops, and provides the first exposition in book form of the basic theory of the Kahler-Ricci flow and its current state-of-the-art. While several excellent books on Kahler-Einstein geometry are available, there have been no such works on the Kahler-Ricci flow. The book will serve as a valuable resource for graduate students and researchers in complex differential geometry, complex algebraic geometry and Riemannian geometry, and will hopefully foster further developments in this fascinating area of research.
The Ricci flow was first introduced by R. Hamilton in the early 1980s, and is central in G. Perelman's celebrated proof of the Poincare conjecture. When specialized for Kahler manifolds, it becomes the Kahler-Ricci flow, and reduces to a scalar PDE (parabolic complex Monge-Ampere equation).
As a spin-off of his breakthrough, G. Perelman proved the convergence of the Kahler-Ricci flow on Kahler-Einstein manifolds of positive scalar curvature (Fano manifolds). Shortly after, G. Tian and J. Song discovered a complex analogue of Perelman's ideas: the Kahler-Ricci flow is a metric embodiment of the Minimal Model Program of the underlying manifold, and flips and divisorial contractions assume the role of Perelman's surgeries."

This book revisits the mathematical foundations of thermodynamics and gauge theory by using new differential geometric methods coming from the formal theory of systems of partial differential equations and Lie pseudogroups. The gauge theory of gravity is also established, in which spinorial and ventorial matter fields serve as gravitating sources. The potential applications of the present gauge theory of gravity, including quantum-vacuum-energy gravity, cosmological constant problem and gravity-gauge unification is also addressed. The third chapter focuses on a gravitational gauge theory with spin connection and vierbein as fundamental variables of gravity. Next, the place and physical significance of Poincare gauge theory of gravity (PGTG) in the framework of gauge approach to gravitation is discussed. A cutoff regularization method in gauge theory is discussed in Chapter Five. The remaining chapters in the book focus on differential geometry, in particular, the authors show how fractional differential derived from fractional difference provides a basis to expand a theory of fractional differential geometry which would apply to non-differentiable manifolds; a review of the infinitesimal Baker-Campbell-Hausdorff formula is provided and the book concludes with a short communication where the authors focus on local stability, and describe how this leads naturally into the question of finite-time singularities and generalized soliton solutions.

This text, the first of two volumes, provides a comprehensive and self-contained introduction to a wide range of fundamental results from ergodic theory and geometric measure theory. Topics covered include: finite and infinite abstract ergodic theory, Young's towers, measure-theoretic Kolmogorov-Sinai entropy, thermodynamics formalism, geometric function theory, various kinds of conformal measures, conformal graph directed Markov systems and iterated functions systems, semi-local dynamics of analytic functions, and nice sets. Many examples are included, along with detailed explanations of essential concepts and full proofs, in what is sure to be an indispensable reference for both researchers and graduate students.

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.

This book is devoted to investigating the spinor structures in particle physics and in polarisation optics. In fact, it consists of two parts joined by the question: Which are the manifestations of spinor structures in different branches of physics. It is based on original research. The main idea is the statement that the physical understanding of geometry should be based on physical field theories. The book contains numerous topics with the accent on field theory, quantum mechanics and polarisation optics of the light, and on the spinor approach.

In this book detailed analytical treatment and exact solutions are given to a number of problems of classical electrodynamics and boson field theory in simplest non-Euclidean space-time models, open Bolyai and Lobachevsky space H3 and closed Riemann space S3, and (anti) de Sitter space-times. The main attention is focused on new themes created by non-vanishing curvature in the following topics: electrodynamics in curved spacetime and modelling of the media, MajoranaOppenheimer approach in curved space time, spin 1 field theory, tetrad based DuffinKemmer-Petiau formalism, SchroedingerPauli limit, DiracKahler particle, spin 2 field, anomalous magnetic moment, plane wave, cylindrical, and spherical solutions, spin 1 particle in a magnetic field, spin 1 field and cosmological radiation in de Sitter space-time, electromagnetic field and Schwarzschild black hole.

Geometry in ancient Greece is said to have originated in the curiosity of mathematicians about the shapes of crystals, with that curiosity culminating in the classification of regular convex polyhedra addressed in the final volume of Euclid's Elements. Since then, geometry has taken its own path and the study of crystals has not been a central theme in mathematics, with the exception of Kepler's work on snowflakes. Only in the nineteenth century did mathematics begin to play a role in crystallography as group theory came to be applied to the morphology of crystals. This monograph follows the Greek tradition in seeking beautiful shapes such as regular convex polyhedra. The primary aim is to convey to the reader how algebraic topology is effectively used to explore the rich world of crystal structures. Graph theory, homology theory, and the theory of covering maps are employed to introduce the notion of the topological crystal which retains, in the abstract, all the information on the connectivity of atoms in the crystal. For that reason the title Topological Crystallography has been chosen. Topological crystals can be described as "living in the logical world, not in space," leading to the question of how to place or realize them "canonically" in space. Proposed here is the notion of standard realizations of topological crystals in space, including as typical examples the crystal structures of diamond and lonsdaleite. A mathematical view of the standard realizations is also provided by relating them to asymptotic behaviors of random walks and harmonic maps. Furthermore, it can be seen that a discrete analogue of algebraic geometry is linked to the standard realizations. Applications of the discussions in this volume include not only a systematic enumeration of crystal structures, an area of considerable scientific interest for many years, but also the architectural design of lightweight rigid structures. The reader therefore can see the agreement of theory and practice.