Geometry Books

Book SynopsisBoth 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.Trade Review“It is a must own book for anyone serious about developing a conceptual understanding of the interconnected web of modern geometry and the ever-growing intertwining of geometry with practically all other branches of mathematics. … It is remarkable for a book to provide such a detailed glimpse of contemporary geometry via well developed discussions of so many questions of current interest. It provides the most extensive exposition of geometric thinking I’ve ever seen in a book at this level.” (William H. Barker, MAA Reviews, August, 2017)“Geometry Revealed is to give the reader a feel for the conceptual frameworks of modern geometry, attempting to reach as far as possible with a minimum of assumed knowledge and formal scaffolding. … Geometry Revealed being useful for research mathematicians as a still reasonably up-to-date survey. … Geometry Revealed offered an ascent into the wonders of a new world.” (Danny Yee, Danny Yee’s Book Reviews, dannyreviews.com, July, 2015)“By considering a hierarchy of ‘natural’ geometrical objects … it sets out to investigate significant geometrical problems which are either unsolved or were solved only recently. … it is undoubtedly a major tour de force, and if you really want to gain an idea of where geometry is going in the 21st century, you will find plenty of exquisite material here.” (Gerry Leversha, The Mathematical Gazette, Vol. 96 (356), July, 2012)“The book contains twelve chapters, each of them is a collection of such problems about geometric objects with more and more complexity … . The chapters are independent from each other, any of them can serve as a course. Researchers in geometry can use it as a source for further research. … the book is accessible to a wide audience of people who are interested in geometry.” (János Kincses, Acta Scientiarum Mathematicarum (Szeged), Vol. 78 (1-2), 2012)“‘Geometry Revealed’ is a massive text of 831 pages which is organized in twelve chapters and which additionally provides indices for names, subjects and symbols … throughout the author quite carefully lays out the historical perspective. … a typical chapter starts with an observation or a problem in elementary geometry. Large parts of the text are very accessible, and a reader who likes (mathematical) physics will often get something extra.” (Michael Joswig, Zentralblatt MATH, Vol. 1232, 2012)“The author provides the reader with an enormous amount of detailed information and thus yields deep insight into the various topics. … All in all an overwhelming book which is a must … for everyone having sufficient mathematical knowledge.” (G. Kowol, Monatshefte für Mathematik, Vol. 164 (2), October, 2011)“The book is a very readable account of several branches of geometry, classical and modern, elementary and advanced. … Every chapter is extremely interesting and alive. … The book is rich in ideas, written in an informal style, with no formulae and no unnecessary technical details. … Every part of this book is interesting and should be accessible to a wide audience of mathematicians. … Every mathematician will experience great pleasure in reading this book.” (Athanase Papadopoulos, Mathematical Reviews, Issue 2011 m)Table of ContentsPoints and lines in the plane.- Circles and spheres.- The sphere by itself: can we distribute points on it evenly?.- Conics and quadrics.- Plane curves.- Smooth surfaces.- Convexity and convex sets.- Polygons, polyhedra, polytopes.- Lattices, packings and tilings in the plane.- Lattices and packings in higher dimensions.- Geometry and dynamics I: billiards.- Geometry and dynamics II: geodesic flow on a surface.

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Book SynopsisEinstein's equations stem from General Relativity. In the context of Riemannian manifolds, an independent mathematical theory has developed around them. This is the first book which presents an overview of several striking results ensuing from the examination of Einstein’s equations in the context of Riemannian manifolds. Parts of the text can be used as an introduction to modern Riemannian geometry through topics like homogeneous spaces, submersions, or Riemannian functionals.Trade ReviewFrom the reviews: "[...] an efficient reference book for many fundamental techniques of Riemannian geometry. [...] despite its length, the reader will have no difficulty in getting the feel of its contents and discovering excellent examples of all interaction of geometry with partial differential equeations, topology, and Lie groups. Above all, the book provides a clear insight into the scope and diversity of problems posed by its title."S.M. Salamon in MathSciNet 1988 "It seemed likely to anyone who read the previous book by the same author, namely Manifolds all of whose geodesic are closed, that the present book would be one of the most important ever published on Riemannian geometry. This prophecy is indeed fulfilled."T.J. Wilmore in Bulletin of the London Mathematical Society 1987 "Einstein Manifolds is accordingly described as Besse’s second book … . there is no doubt that Einstein Manifolds is a magnificient work of mathematical scholarship. … It is truly a seminal work on an incomparably fascinating and important subject." (Michael Berg, MathDL, March, 2008) "The present book is intended to be a complete reference book. … The book under review serves several purposes. It is an efficient reference for many fundamental techniques of Riemannian geometry as well as excellent examples of the interaction of geometry with partial differential equations, topology and Lie groups. Certainly the monograph provides a clear insight into the scope and diversity of problems posed by its title." (Adela-Gabriela Mihai, Zentralblatt MATH, Vol. 1147, 2008)Table of ContentsBasic Material.- Basic Material (Continued): Kähler Manifolds.- Relativity.- Riemannian Functionals.- Ricci Curvature as a Partial Differential Equation.- Einstein Manifolds and Topology.- Homogeneous Riemannian Manifolds.- Compact Homogeneous Kähler Manifolds.- Riemannian Submersions.- Holonomy Groups.- Kähler-Einstein Metrics and the Calabi Conjecture.- The Moduli Space of Einstein Structures.- Self-Duality.- Quaternion-Kähler Manifolds.- A Report on the Non-Compact Case.- Generalizations of the Einstein Condition.

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Book SynopsisThis volume deals with the theory of finite topological spaces and its relationship with the homotopy and simple homotopy theory of polyhedra. The interaction between their intrinsic combinatorial and topological structures makes finite spaces a useful tool for studying problems in Topology, Algebra and Geometry from a new perspective. In particular, the methods developed in this manuscript are used to study Quillen's conjecture on the poset of p-subgroups of a finite group and the Andrews-Curtis conjecture on the 3-deformability of contractible two-dimensional complexes. This self-contained work constitutes the first detailed exposition on the algebraic topology of finite spaces. It is intended for topologists and combinatorialists, but it is also recommended for advanced undergraduate students and graduate students with a modest knowledge of Algebraic Topology.Trade ReviewFrom the reviews:“This book deals with the algebraic topology of finite topological spaces and its applications, and includes well-known results on finite spaces and original results developed by the author. The book is self-contained and well written. It is understandable and enjoyable to read. It contains a lot of examples and figures which help the readers to understand the theory.” (Fumihiro Ushitaki, Mathematical Reviews, March, 2014)“This book illustrates convincingly the idea that the study of finite non-Hausdorff spaces from a homotopical point of view is useful in many areas and can even be used to study well-known problems in classical algebraic topology. … This book is a revised version of the PhD Thesis of the author. … All the concepts introduced with the chapters are usefully illustrated by examples and the recollection of all these results gives a very nice introduction to a domain of growing interest.” (Etienne Fieux, Zentralblatt MATH, Vol. 1235, 2012)Table of Contents1 Preliminaries.- 2 Basic topological properties of finite spaces.- 3 Minimal finite models.- 4 Simple homotopy types and finite spaces.- 5 Strong homotopy types.- 6 Methods of reduction.- 7 h-regular complexes and quotients.- 8 Group actions and a conjecture of Quillen.- 9 Reduced lattices.- 10 Fixed points and the Lefschetz number.- 11 The Andrews-Curtis conjecture.

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Book SynopsisIn this textbook the authors present first-year geometry roughly in the order in which it was discovered. The first five chapters show how the ancient Greeks established geometry, together with its numerous practical applications, while more recent findings on Euclidian geometry are discussed as well. The following three chapters explain the revolution in geometry due to the progress made in the field of algebra by Descartes, Euler and Gauss. Spatial geometry, vector algebra and matrices are treated in chapters 9 and 10. The last chapter offers an introduction to projective geometry, which emerged in the 19thcentury.Complemented by numerous examples, exercises, figures and pictures, the book offers both motivation and insightful explanations, and provides stimulating and enjoyable reading for students and teachers alike.Trade ReviewFrom the book reviews:Choice - Outstanding Academic Title in 2012“This is an excellent, challenging textbook as well as a valuable resource for historical information, problems, and student projects. The historical content is broad based, comprehensive, and reliable. Each chapter has extensive exercises, many taken directly from or based on historical sources … . Hints and solutions for all problems are given in an appendix. Thorough bibliography. Summing Up: Highly recommended. Lower-division undergraduates and above.” (C. A. Gorini, Choice, Vol. 50 (3), November, 2012)“The book under review is a treasure chest of interesting theorems and problems in geometry together with their illuminating histories. … This is the kind of book that one would enjoy browsing through and reading while sitting relaxedly in an armchair without any paper or pencil and starting at almost any page or paragraph. It should be on the shelf of every lover of geometry.” (Mowaffaq Hajja, zbMATH, Vol. 1288, 2014)“This book belongs on the bookshelf of every geometer. … The authors have penned their book with students of geometry as well as science in mind. In fact, the book would serve well as a second year mathematics course in a classical liberal arts setting. … the book treats many interesting and beautiful problems, introducing powerful concepts along the way, and yet is written at a level suitable for an introductory course of geometry or even advanced mathematics.” (Alan S. McRae, Mathematical Reviews, February, 2013)“There is a lot of interesting material in this book, supplemented by a lot of very nice artwork and many interesting exercises … . I would think that any other college instructor … with an interest in geometry would also want a copy on his or her shelf.” (Mark Hunacek, The Mathematical Association of America, June, 2012)Table of ContentsPreface.- Part I: Classical Geometry.- Thales and Pythagoras.- The Elements of Euclid.- Conic Sections.- Further Results on Euclidean Geometry.- Trigonometry.- Part II: Analytic Geometry.- Descartes' Geometry.- Cartesian Coordinates.- To be Constructible, or not to be.- Spatial Geometry and Vector Algebra.- Matrices and Linear Mappings.- Projective Geometry.- Solutions to Exercises.- References.- Figure Source and Copyright.- Index.

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Book SynopsisDieser Buchtitel ist Teil des Digitalisierungsprojekts Springer Book Archives mit Publikationen, die seit den Anfängen des Verlags von 1842 erschienen sind. Der Verlag stellt mit diesem Archiv Quellen für die historische wie auch die disziplingeschichtliche Forschung zur Verfügung, die jeweils im historischen Kontext betrachtet werden müssen. 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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Book SynopsisIn July 2004, a conference on graph theory was held in Paris in memory of Claude Berge, one of the pioneers of the field. The event brought together many prominent specialists on topics such as perfect graphs and matching theory, upon which Claude Berge's work has had a major impact. This volume includes contributions to these and other topics from many of the participants.Table of ContentsClaude Berge — Sculptor of Graph Theory.- ?-path-connectivity and mk-generation: an Upper Bound on m.- Automated Results and Conjectures on Average Distance in Graphs.- Brambles, Prisms and Grids.- Dead Cell Analysis in Hex and the Shannon Game.- Ratios of Some Domination Parameters in Graphs and Claw-free Graphs.- Excessive Factorizations of Regular Graphs.- Odd Pairs of Cliques.- Recognition of Perfect Circular-arc Graphs.- On Edge-maps whose Inverse Preserves Flows or Tensions.- On the Extremal Number of Edges in 2-Factor Hamiltonian Graphs.- Generalized Colourings (Matrix Partitions) of Cographs.- A Note on [k, l]-sparse Graphs.- Even Pairs in Bull-reducible Graphs.- Kernels in Orientations of Pretransitive Orientable Graphs.- Nonrepetitive Graph Coloring.- A Characterization of the 1-well-covered Graphs with no 4-cycles.- A Graph-theoretical Generalization of Berge’s Analogue of the Erd?s-Ko-Rado Theorem.- Independence Polynomials and the Unimodality Conjecture for Very Well-covered, Quasi-regularizable, and Perfect Graphs.- Precoloring Extension on Chordal Graphs.- On the Enumeration of Bipartite Minimum Edge Colorings.- Kempe Equivalence of Colorings.- Acyclic 4-choosability of Planar Graphs with Girth at Least 5.- Automorphism Groups of Circulant Graphs — a Survey.- Hypo-matchings in Directed Graphs.- On Reed’s Conjecture about ?,? and ?.- On the Generalization of the Matroid Parity Problem.- Reconstruction of a Rank 3 Oriented Matroids from its Rank 2 Signed Circuits.- The Normal Graph Conjecture is True for Circulants.- Two-arc Transitive Near-polygonal Graphs.- Open Problems.

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Book SynopsisThis book develops the geometric intuition of the reader by examining the symmetries (or rigid motions) of the space in question. This approach introduces in turn all the classical geometries: Euclidean, affine, elliptic, projective and hyperbolic. The main focus is on the mathematically rich two-dimensional case, although some aspects of 3- or $n$-dimensional geometries are included. Basic notions of algebra and analysis are used to convey better understanding of various concepts and results. Concepts of geometry are presented in a very simple way, so that they become easily accessible: the only pre-requisites are calculus, linear algebra and basic analytic geometry.Trade ReviewFrom the reviews:“This book offers an introduction to classical geometry. … The book contains many figures, which help the reader to develop geometric intuition, and lots of exercises. To follow the presentation, only a basic background in analysis and linear algebra is required, some necessary facts are collected in an appendix.” (A. Cap, Monatshefte für Mathematik, Vol. 158 (2), October, 2009)Table of ContentsEuclidean geometry.- Affine geometry.- Projective geometry.- Hyperbolic geometry.- Appendices.

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Book SynopsisThis book arose out of original research on the extension of well-established applications of complex numbers related to Euclidean geometry and to the space-time symmetry of two-dimensional Special Relativity. The system of hyperbolic numbers is extensively studied, and a plain exposition of space-time geometry and trigonometry is given. Commutative hypercomplex systems with four unities are studied and attention is drawn to their interesting properties.Trade ReviewFrom the reviews: “It is worth pointing out that the book is mainly a text about commutative hypercomplex numbers and some of their applications to a 2-dimensional Minkowski spacetime. … This book should be interesting to anybody who is interested in applications of hypercomplex numbers … . In conclusion, I recommend this book to anyone who wants to learn about hypercomplex numbers.” (Emanuel Gallo, Mathematical Reviews, Issue 2010 d)Table of ContentsThe Mathematics of Minkowski Space-Time: 1 N-Dimensional Hypercomplex Numbers and the associated Geometries.- Commutative Hypercomplex Number Systems.- The General Two-Dimensional System.- Linear Transformations and Geometries.- The Geometries Associated with Hypercomplex Numbers.- Conclusions.- 2 Trigonometry in the Minkowski Plane.- Geometrical Representation of Hyperbolic Numbers.- Basics of Hyperbolic Trigonometry.- Geometry in Pseudo-Euclidean Cartesian Plane.- Trigonometry in the Pseudo-Euclidean Plane.- Theorems on Equilateral Hyperbolas in the Pseudo-Euclidean Plane.- Some Examples of Triangle Solutions in the Minkowski Plane.- Conclusions.- 3 Uniform and Accelerated Motions in the Minkowski Space-Time (Twin Paradox).- Inertial Motions.- Inertial and Uniformly Accelerated Motions.- Non-uniformly Accelerated Motions.- Conclusions.- 4 General Two-Dimensional Hypercomplex Numbers.-Geometrical Representation.- Geometry and Trigonometry in Two-Dimensional Algebras.- Some Properties of Fundamental Conic Section.- Numerical Examples.- 5 Functions of a Hyperbolic Variable.- Some Remarks on Functions of a Complex Variable.- Functions of Hypercomplex Variables.- The Functions of a Hyperbolic Variable.- The Elementary Functions of a Canonical Hyperbolic Variable.- H-Conformal Mappings.- Commutative Hypercomplex Systems with Three Unities.- 6 Hyperbolic Variables on Lorentz Surfaces.- Introduction.- Gauss: Conformal Mapping of Surfaces.- Extension of Gauss Theorem: Conformal Mapping of Lorentz Surfaces.- Beltrami: Complex Variables on a Surface.- Beltrami’s Integration of Geodesic Equations.- Extension of Beltrami’s Equation to Non-Definite Differential Forms.- 7 Constant Curvature Lorentz Surfaces.- Introduction.- Constant Curvature RiemannSurfaces.- Constant Curvature Lorentz Surfaces.- Geodesics and Geodesic Distances on Riemann and Lorentz Surfaces.- Conclusions.- 8 Generalization of Two-Dimensional Special Relativity (Hyperbolic Transformations and the Equivalence Principle).- Physical Meaning of Transformations by Hyperbolic Functions.- Physical Interpretation of Geodesics on Riemann and Lorentz Surfaces with Positive Constant Curvature.- Einstein’s Way to General Relativity.- Conclusions.- II An Introduction to Commutative Hypercomplex Numbers.- 9 Commutative Segre’s Quaternions.- Introduction.- Hypercomplex Systems with Four Units.- Historical Introduction of Segre’s Quaternion.- Algebraic Properties of Commutative Quaternions.- Functions of a Quaternion Variable.- Mapping by Means of Quaternion Functions.- Elementary Functions of the Quaternions.- Elliptic-Hyperbolic Quaternions.- Elliptic-Parabolic Generalized Segre’s Quaternions.- 10 Constant Curvature Segre’s Quaternion Spaces.- Introduction.- Quaternion differential geometry and geodesic equations.- Orthogonality in Segre’s Quaternion Space.- Constant Curvature Quaternion Spaces.- Geodesic Equations in Quaternion Space.- Beltrami’s Integration Method for Quaternion Spaces.- Beltrami’s Integration Method for Quaternion Spaces.- Conclusions.- 11 A Matrix Formalization for Commutative Hypercomplex Systems.- Mathematical Operations.- Properties of the Characteristic Matrix M.- Functions of Hypercomplex Variable.- Functions of a Two-Dimensional Hypercomplex Variable.- Derivatives of a Hypercomplex Function.- Characteristic Differential Equation.- A Equivalence Between the Formalizations of Hypercomplex Numbers.

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