1A Comprehensive Introduction to Differential Geometry, Vol. 1
Eric Weinstein
Managing Director/Thiel Capital
Eric Weinstein recommended this book on Twitter. [source]
100 Best Differential Geometry Books of All Time
We've ranked the best differential geometry books using expert recommendations, sales data, and millions of reader ratings. At Shortform, we know books. Our book guides are the best in the world. Learn why.
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2Calculus On Manifolds: A Modern Approach To Classical Theorems Of Advanced Calculus
This little book is especially concerned with those portions of ”advanced calculus” in which the subtlety of the concepts and methods makes rigor difficult to attain at an elementary level. The approach taken here uses elementary versions of modern methods found in sophisticated mathematics. The formal prerequisites include only a term of linear algebra, a nodding acquaintance with the notation of set theory, and a respectable first-year calculus course (one which at least mentions the least upper bound (sup) and greatest lower bound (inf) of a set of real numbers). Beyond this a certain (perhaps latent) rapport with abstract mathematics will be found almost essential.
Recommended by Eric Weinstein.Eric Weinstein
Managing Director/Thiel Capital
Eric Weinstein recommended this book on Twitter. [source]
3Differential Geometry of Curves and Surfaces: Revised and Updated Second Edition
One of the most widely used texts in its field, this volume introduces the differential geometry of curves and surfaces in both local and global aspects. The presentation departs from the traditional approach with its more extensive use of elementary linear algebra and its emphasis on basic geometrical facts rather than machinery or random details. Many examples and exercises enhance the clear, well-written exposition, along with hints and answers to some of the problems.
The treatment begins with a chapter on curves, followed by explorations of regular surfaces, the geometry of the Gauss map, the intrinsic geometry of surfaces, and global differential geometry. Suitable for advanced undergraduates and graduate students of mathematics, this text's prerequisites include an undergraduate course in linear algebra and some familiarity with the calculus of several variables. For this second edition, the author has corrected, revised, and updated the entire volume.
The treatment begins with a chapter on curves, followed by explorations of regular surfaces, the geometry of the Gauss map, the intrinsic geometry of surfaces, and global differential geometry. Suitable for advanced undergraduates and graduate students of mathematics, this text's prerequisites include an undergraduate course in linear algebra and some familiarity with the calculus of several variables. For this second edition, the author has corrected, revised, and updated the entire volume.
4Differential Geometry
An introductory textbook on the differential geometry of curves and surfaces in 3-dimensional Euclidean space, presented in its simplest, most essential form, but with many explanatory details, figures, and examples, and in a manner that conveys the theoretical and practical importance of the different concepts, methods, and results involved. With problems and solutions. Includes 99 illustrations.
5A Comprehensive Introduction to Differential Geometry, Volume 5
Recommended by Eric Weinstein.Eric Weinstein
Managing Director/Thiel Capital
Eric Weinstein recommended this book on Twitter. [source]
6Differential Forms in Algebraic Topology
Developed from a first-year graduate course in algebraic topology, this text is an informal introduction to some of the main ideas of contemporary homotopy and cohomology theory. The materials are structured around four core areas: de Rham theory, the Cech-de Rham complex, spectral sequences, and characteristic classes. By using the de Rham theory of differential forms as a prototype of cohomology, the machineries of algebraic topology are made easier to assimilate. With its stress on concreteness, motivation, and readability, this book is equally suitable for self-study and as a one-semester course in topology.
Recommended by Eric Weinstein.Eric Weinstein
Managing Director/Thiel Capital
Eric Weinstein recommended this book on Twitter. [source]
7Introduction to Smooth Manifolds
This book is an introductory graduate-level textbook on the theory of smooth manifolds. Its goal is to familiarize students with the tools they will need in order to use manifolds in mathematical or scientific research--- smooth structures, tangent vectors and covectors, vector bundles, immersed and embedded submanifolds, tensors, differential forms, de Rham cohomology, vector fields, flows, foliations, Lie derivatives, Lie groups, Lie algebras, and more. The approach is as concrete as possible, with pictures and intuitive discussions of how one should think geometrically about the abstract concepts, while making full use of the powerful tools that modern mathematics has to offer.
This second edition has been extensively revised and clarified, and the topics have been substantially rearranged. The book now introduces the two most important analytic tools, the rank theorem and the fundamental theorem on flows, much earlier so that they can be used throughout the book. A few new topics have been added, notably Sard's theorem and transversality, a proof that infinitesimal Lie group actions generate global group actions, a more thorough study of first-order partial differential equations, a brief treatment of degree theory for smooth maps between compact manifolds, and an introduction to contact structures.
Prerequisites include a solid acquaintance with general topology, the fundamental group, and covering spaces, as well as basic undergraduate linear algebra and real analysis.
This second edition has been extensively revised and clarified, and the topics have been substantially rearranged. The book now introduces the two most important analytic tools, the rank theorem and the fundamental theorem on flows, much earlier so that they can be used throughout the book. A few new topics have been added, notably Sard's theorem and transversality, a proof that infinitesimal Lie group actions generate global group actions, a more thorough study of first-order partial differential equations, a brief treatment of degree theory for smooth maps between compact manifolds, and an introduction to contact structures.
Prerequisites include a solid acquaintance with general topology, the fundamental group, and covering spaces, as well as basic undergraduate linear algebra and real analysis.
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8A Comprehensive Introduction to Differential Geometry, Volume 4
Recommended by Eric Weinstein.Eric Weinstein
Managing Director/Thiel Capital
Eric Weinstein recommended this book on Twitter. [source]
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9Lectures on Differential Geometry
In 1984, the authors gave a series of lectures on differential geometry in the Institute for Advanced Studies in Princeton, USA. These lectures are published in this volume, which describes the major achievements in the field.
10An Introduction to Manifolds
Manifolds, the higher-dimensional analogs of smooth curves and surfaces, are fundamental objects in modern mathematics. Combining aspects of algebra, topology, and analysis, manifolds have also been applied to classical mechanics, general relativity, and quantum field theory. In this streamlined introduction to the subject, the theory of manifolds is presented with the aim of helping the reader achieve a rapid mastery of the essential topics. By the end of the book the reader should be able to compute, at least for simple spaces, one of the most basic topological invariants of a manifold, its de Rham cohomology. Along the way, the reader acquires the knowledge and skills necessary for further study of geometry and topology. The requisite point-set topology is included in an appendix of twenty pages; other appendices review facts from real analysis and linear algebra. Hints and solutions are provided to many of the exercises and problems. This work may be used as the text for a one-semester graduate or advanced undergraduate course, as well as by students engaged in self-study. Requiring only minimal undergraduate prerequisites, 'Introduction to Manifolds' is also an excellent foundation for Springer's GTM 82, 'Differential Forms in Algebraic Topology'.
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11A Comprehensive Introduction to Differential Geometry, Volume 2
Recommended by Eric Weinstein.Eric Weinstein
Managing Director/Thiel Capital
Eric Weinstein recommended this book on Twitter. [source]
12Lost in Math: How Beauty Leads Physics Astray
A contrarian argues that modern physicists' obsession with beauty has given us wonderful math but bad science.
Whether pondering black holes or predicting discoveries at CERN, physicists believe the best theories are beautiful, natural, and elegant, and this standard separates popular theories from disposable ones. This is why, Sabine Hossenfelder argues, we have not seen a major breakthrough in the foundations of physics for more than four decades. The belief in beauty has become so dogmatic that it now conflicts with scientific objectivity: observation has been unable to confirm mindboggling theories, like supersymmetry or grand unification, invented by physicists based on aesthetic criteria. Worse, these "too good to not be true" theories are actually untestable and they have left the field in a cul-de-sac. To escape, physicists must rethink their methods. Only by embracing reality as it is can science discover the truth.
Whether pondering black holes or predicting discoveries at CERN, physicists believe the best theories are beautiful, natural, and elegant, and this standard separates popular theories from disposable ones. This is why, Sabine Hossenfelder argues, we have not seen a major breakthrough in the foundations of physics for more than four decades. The belief in beauty has become so dogmatic that it now conflicts with scientific objectivity: observation has been unable to confirm mindboggling theories, like supersymmetry or grand unification, invented by physicists based on aesthetic criteria. Worse, these "too good to not be true" theories are actually untestable and they have left the field in a cul-de-sac. To escape, physicists must rethink their methods. Only by embracing reality as it is can science discover the truth.
Recommended by Barbara Kiser.Barbara Kiser
This is a firecracker of a book—a shot across the bows of theoretical physics. Sabine Hossenfelder, a theoretical physicist working on quantum gravity and blogger, confronts failures in her field head-on. [source]
13Geometry of Differential Forms (Translations of Mathematical Monographs, Vol. 201)
14The Geometry of Physics: An Introduction
Theodore Frankel explains those parts of exterior differential forms, differential geometry, algebraic and differential topology, Lie groups, vector bundles and Chern forms essential to a better understanding of classical and modern physics and engineering. Key highlights of his new edition are the inclusion of three new appendices that cover symmetries, quarks, and meson masses; representations and hyperelastic bodies; and orbits and Morse-Bott Theory in compact Lie groups. Geometric intuition is developed through a rather extensive introduction to the study of surfaces in ordinary space. First Edition Hb (1997): 0-521-38334-X First Edition Pb (1999): 0-521-38753-1
15Schaum's Outline of Differential Geometry
Confusing Textbooks? Missed Lectures? Not Enough Time?
Fortunately for you, there's Schaum's Outlines. More than 40 million students have trusted Schaum's to help them succeed in the classroom and on exams. Schaum's is the key to faster learning and higher grades in every subject. Each Outline presents all the essential course information in an easy-to-follow, topic-by-topic format. You also get hundreds of examples, solved problems, and practice exercises to test your skills.
This Schaum's Outline gives you Practice problems with full explanations that reinforce knowledge Coverage of the most up-to-date developments in your course field In-depth review of practices and applications
Fully compatible with your classroom text, Schaum's highlights all the important facts you need to know. Use Schaum's to shorten your study time-and get your best test scores!
Schaum's Outlines-Problem Solved.
Fortunately for you, there's Schaum's Outlines. More than 40 million students have trusted Schaum's to help them succeed in the classroom and on exams. Schaum's is the key to faster learning and higher grades in every subject. Each Outline presents all the essential course information in an easy-to-follow, topic-by-topic format. You also get hundreds of examples, solved problems, and practice exercises to test your skills.
This Schaum's Outline gives you Practice problems with full explanations that reinforce knowledge Coverage of the most up-to-date developments in your course field In-depth review of practices and applications
Fully compatible with your classroom text, Schaum's highlights all the important facts you need to know. Use Schaum's to shorten your study time-and get your best test scores!
Schaum's Outlines-Problem Solved.
16Introduction to Differential Geometry of Space Curves and Surfaces
This is the balck and white version of the book. The book, which consists of 260 pages, is about differential geometry of space curves and surfaces. The formulation and presentation are largely based on a tensor calculus approach. It can be used as part of a course on tensor calculus as well as a textbook or a reference for an intermediate-level course on differential geometry of curves and surfaces. The book is furnished with an index, extensive sets of exercises and many cross references, which are hyperlinked for the ebook users, to facilitate linking related concepts and sections. The book also contains a considerable number of 2D and 3D graphic illustrations to help the readers and users to visualize the ideas and understand the abstract concepts. We also provided an introductory chapter where the main concepts and techniques needed to understand the offered materials of differential geometry are outlined to make the book fairly self-contained and reduce the need for external references.
17Differential Geometry of Curves and Surfaces
This is a textbook on differential geometry well-suited to a variety of courses on this topic. For readers seeking an elementary text, the prerequisites are minimal and include plenty of examples and intermediate steps within proofs, while providing an invitation to more excursive applications and advanced topics. For readers bound for graduate school in math or physics, this is a clear, concise, rigorous development of the topic including the deep global theorems. For the benefit of all readers, the author employs various techniques to render the difficult abstract ideas herein more understandable and engaging.
Over 300 color illustrations bring the mathematics to life, instantly clarifying concepts in ways that grayscale could not. Green-boxed definitions and purple-boxed theorems help to visually organize the mathematical content. Color is even used within the text to highlight logical relationships.
Applications abound! The study of conformal and equiareal functions is grounded in its application to cartography. Evolutes, involutes and cycloids are introduced through Christiaan Huygens' fascinating story: in attempting to solve the famous longitude problem with a mathematically-improved pendulum clock, he invented mathematics that would later be applied to optics and gears. Clairaut s Theorem is presented as a conservation law for angular momentum. Green s Theorem makes possible a drafting tool called a planimeter. Foucault s Pendulum helps one visualize a parallel vector field along a latitude of the earth. Even better, a south-pointing chariot helps one visualize a parallel vector field along any curve in any surface.
In truth, the most profound application of differential geometry is to modern physics, which is beyond the scope of this book. The GPS in any car wouldn t work without general relativity, formalized through the language of differential geometry. The above-mentioned applications don t purport to match the significance of modern physics, but instead they serve a crucial pedagogical role within this book: to ground each abstract idea in something concrete. Search YouTube for southpointing chariot and you will learn about a fascinating toy, but in this book it s more than that it s a concrete devise that buttresses the rigorous definition of a parallel vector field and that motivates the variational formulas for arclength. Throughout this book, applications, metaphors and visualizations are tools that motivate and clarify the rigorous mathematical content, but never replace it. "
Over 300 color illustrations bring the mathematics to life, instantly clarifying concepts in ways that grayscale could not. Green-boxed definitions and purple-boxed theorems help to visually organize the mathematical content. Color is even used within the text to highlight logical relationships.
Applications abound! The study of conformal and equiareal functions is grounded in its application to cartography. Evolutes, involutes and cycloids are introduced through Christiaan Huygens' fascinating story: in attempting to solve the famous longitude problem with a mathematically-improved pendulum clock, he invented mathematics that would later be applied to optics and gears. Clairaut s Theorem is presented as a conservation law for angular momentum. Green s Theorem makes possible a drafting tool called a planimeter. Foucault s Pendulum helps one visualize a parallel vector field along a latitude of the earth. Even better, a south-pointing chariot helps one visualize a parallel vector field along any curve in any surface.
In truth, the most profound application of differential geometry is to modern physics, which is beyond the scope of this book. The GPS in any car wouldn t work without general relativity, formalized through the language of differential geometry. The above-mentioned applications don t purport to match the significance of modern physics, but instead they serve a crucial pedagogical role within this book: to ground each abstract idea in something concrete. Search YouTube for southpointing chariot and you will learn about a fascinating toy, but in this book it s more than that it s a concrete devise that buttresses the rigorous definition of a parallel vector field and that motivates the variational formulas for arclength. Throughout this book, applications, metaphors and visualizations are tools that motivate and clarify the rigorous mathematical content, but never replace it. "
18General Relativity: The Theoretical Minimum
19Introduction to Tensor Analysis and the Calculus of Moving Surfaces
This textbook is distinguished from other texts on the subject by the depth of the presentation and the discussion of the calculus of moving surfaces, which is an extension of tensor calculus to deforming manifolds.
Designed for advanced undergraduate and graduate students, this text invites its audience to take a fresh look at previously learned material through the prism of tensor calculus. Once the framework is mastered, the student is introduced to new material which includes differential geometry on manifolds, shape optimization, boundary perturbation and dynamic fluid film equations.
The language of tensors, originally championed by Einstein, is as fundamental as the languages of calculus and linear algebra and is one that every technical scientist ought to speak. The tensor technique, invented at the turn of the 20th century, is now considered classical. Yet, as the author shows, it remains remarkably vital and relevant. The author's skilled lecturing capabilities are evident by the inclusion of insightful examples and a plethora of exercises. A great deal of material is devoted to the geometric fundamentals, the mechanics of change of variables, the proper use of the tensor notation and the discussion of the interplay between algebra and geometry. The early chapters have many words and few equations. The definition of a tensor comes only in Chapter 6 - when the reader is ready for it. While this text maintains a consistent level of rigor, it takes great care to avoid formalizing the subject.
The last part of the textbook is devoted to the Calculus of Moving Surfaces. It is the first textbook exposition of this important technique and is one of the gems of this text. A number of exciting applications of the calculus are presented including shape optimization, boundary perturbation of boundary value problems and dynamic fluid film equations developed by the author in recent years. Furthermore, the moving surfaces framework is used to offer new derivations of classical results such as the geodesic equation and the celebrated Gauss-Bonnet theorem.
Designed for advanced undergraduate and graduate students, this text invites its audience to take a fresh look at previously learned material through the prism of tensor calculus. Once the framework is mastered, the student is introduced to new material which includes differential geometry on manifolds, shape optimization, boundary perturbation and dynamic fluid film equations.
The language of tensors, originally championed by Einstein, is as fundamental as the languages of calculus and linear algebra and is one that every technical scientist ought to speak. The tensor technique, invented at the turn of the 20th century, is now considered classical. Yet, as the author shows, it remains remarkably vital and relevant. The author's skilled lecturing capabilities are evident by the inclusion of insightful examples and a plethora of exercises. A great deal of material is devoted to the geometric fundamentals, the mechanics of change of variables, the proper use of the tensor notation and the discussion of the interplay between algebra and geometry. The early chapters have many words and few equations. The definition of a tensor comes only in Chapter 6 - when the reader is ready for it. While this text maintains a consistent level of rigor, it takes great care to avoid formalizing the subject.
The last part of the textbook is devoted to the Calculus of Moving Surfaces. It is the first textbook exposition of this important technique and is one of the gems of this text. A number of exciting applications of the calculus are presented including shape optimization, boundary perturbation of boundary value problems and dynamic fluid film equations developed by the author in recent years. Furthermore, the moving surfaces framework is used to offer new derivations of classical results such as the geodesic equation and the celebrated Gauss-Bonnet theorem.
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20A Comprehensive Introduction to Differential Geometry, Volume 3
Recommended by Eric Weinstein.Eric Weinstein
Managing Director/Thiel Capital
Eric Weinstein recommended this book on Twitter. [source]
21An Introduction to Differential Geometry
Suitable for advanced undergraduate and graduate students of mathematics, physics, and engineering, this text employs vector methods to explore the classical theory of curves and surfaces. Subsequent topics include the basic theory of tensor algebra, tensor calculus, the calculus of differential forms, and elements of Riemannian geometry. 1959 edition.
22Differential Geometry: Connections, Curvature, and Characteristic Classes (Graduate Texts in Mathematics)
23Geometry, Topology and Physics (Graduate Student Series in Physics)
This series of books in physics and related subjects is designed to meet the needs of graduate students. Although not primarily research texts, they point out the direction which research is currently taking and where it is expected to lead.
24Visual Differential Geometry and Forms: A Mathematical Drama in Five Acts
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25The Wheel of Time: The Shamans of Ancient Mexico, Their Thoughts about Life, Death and the Universe
World-renowned bestselling author Carlos Castaneda's Selection of his wrtings on the shamans of ancient Mexico.
Originally drawn to Yaqui Indian spiritual leader don Juan Matus for his knowledge of mind-altering plants, bestselling author Carlos Castaneda soon immersed himself in the sorcerer’s magical world entirely. Ten years after his first encounter with the shaman, Castaneda examines his field notes and comes to understand what don Juan knew all along—that these plants are merely a means to understanding the alternative realities that one cannot fully embrace on one’s own. In Journey to Ixtlan, Carlos Castaneda introduces readers to this new approach for the first time and explores, as he comes to experience it himself, his own final voyage into the teachings of don Juan, sharing with us what it is like to truly “stop the world” and perceive reality on his own terms.
Originally drawn to Yaqui Indian spiritual leader don Juan Matus for his knowledge of mind-altering plants, bestselling author Carlos Castaneda soon immersed himself in the sorcerer’s magical world entirely. Ten years after his first encounter with the shaman, Castaneda examines his field notes and comes to understand what don Juan knew all along—that these plants are merely a means to understanding the alternative realities that one cannot fully embrace on one’s own. In Journey to Ixtlan, Carlos Castaneda introduces readers to this new approach for the first time and explores, as he comes to experience it himself, his own final voyage into the teachings of don Juan, sharing with us what it is like to truly “stop the world” and perceive reality on his own terms.
Recommended by Aubrey Marcus.
26Special Relativity and Classical Field Theory: The Theoretical Minimum
A funny, insightful, and self-contained guide to Einstein's relativity theory and classical field theories--including electromagnetism
Physicist Leonard Susskind and data engineer Art Friedman are back. This time, they introduce readers to Einstein's special relativity and Maxwell's classical field theory. Using their typical brand of real math, enlightening drawings, and humor, Susskind and Friedman walk us through the complexities of waves, forces, and particles by exploring special relativity and electromagnetism. It's a must-read for both devotees of the series and any armchair physicist who wants to improve their knowledge of physics' deepest truths.
Physicist Leonard Susskind and data engineer Art Friedman are back. This time, they introduce readers to Einstein's special relativity and Maxwell's classical field theory. Using their typical brand of real math, enlightening drawings, and humor, Susskind and Friedman walk us through the complexities of waves, forces, and particles by exploring special relativity and electromagnetism. It's a must-read for both devotees of the series and any armchair physicist who wants to improve their knowledge of physics' deepest truths.
27A Student's Guide to Maxwell's Equations
Gauss's law for electric fields, Gauss's law for magnetic fields, Faraday's law, and the Ampere-Maxwell law are four of the most influential equations in science. In this guide for students, each equation is the subject of an entire chapter, with detailed, plain-language explanations of the physical meaning of each symbol in the equation, for both the integral and differential forms. The final chapter shows how Maxwell's equations may be combined to produce the wave equation, the basis for the electromagnetic theory of light. This book is a wonderful resource for undergraduate and graduate courses in electromagnetism and electromagnetics. A website hosted by the author at www.cambridge.org/9780521701471 contains interactive solutions to every problem in the text as well as audio podcasts to walk students through each chapter.
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28An Introduction to Geometric Topology
This book provides a self-contained introduction to the topology and geometry of surfaces and three-manifolds. The main goal is to describe Thurston's geometrisation of three-manifolds, proved by Perelman in 2002. The book is divided into three parts: the first is devoted to hyperbolic geometry, the second to surfaces, and the third to three-manifolds. It contains complete proofs of Mostow's rigidity, the thick-thin decomposition, Thurston's classification of the diffeomorphisms of surfaces (via Bonahon's geodesic currents), the prime and JSJ decomposition, the topological and geometric classification of Seifert manifolds, and Thurston's hyperbolic Dehn filling Theorem.
29Differential Geometry of Curves and Surfaces
This volume covers local as well as global differential geometry of curves and surfaces.
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30Topology from the Differentiable Viewpoint
This elegant book by distinguished mathematician John Milnor, provides a clear and succinct introduction to one of the most important subjects in modern mathematics. Beginning with basic concepts such as diffeomorphisms and smooth manifolds, he goes on to examine tangent spaces, oriented manifolds, and vector fields. Key concepts such as homotopy, the index number of a map, and the Pontryagin construction are discussed. The author presents proofs of Sard's theorem and the Hopf theorem.
31Riemannian Geometry
Intended for a one year course, this volume serves as a single source, introducing students to the important techniques and theorems, while also containing enough background on advanced topics to appeal to those students wishing to specialize in Riemannian geometry. This is one of the few works to combine both the geometric parts of Riemannian geometry and the analytic aspects of the theory, while also presenting the most up-to-date research. This book will appeal to readers with a knowledge of standard manifold theory, including such topics as tensors and Stokes theorem. Various exercises are scattered throughout the text, helping motivate readers to deepen their understanding of the subject.
Important additions to this new edition include:
* A completely new coordinate free formula that is easily remembered, and is, in fact, the Koszul formula in disguise;
* An increased number of coordinate calculations of connection and curvature;
* General fomulas for curvature on Lie Groups and submersions;
* Variational calculus has been integrated into the text, which allows for an early treatment of the Sphere theorem using a forgottten proof by Berger;
* Several recent results about manifolds with positive curvature.
From reviews of the first edition:
"The book can be highly recommended to all mathematicians who want to get a more profound idea about the most interesting
achievements in Riemannian geometry. It is one of the few comprehensive sources of this type."
- Bernd Wegner, Zentralblatt
Important additions to this new edition include:
* A completely new coordinate free formula that is easily remembered, and is, in fact, the Koszul formula in disguise;
* An increased number of coordinate calculations of connection and curvature;
* General fomulas for curvature on Lie Groups and submersions;
* Variational calculus has been integrated into the text, which allows for an early treatment of the Sphere theorem using a forgottten proof by Berger;
* Several recent results about manifolds with positive curvature.
From reviews of the first edition:
"The book can be highly recommended to all mathematicians who want to get a more profound idea about the most interesting
achievements in Riemannian geometry. It is one of the few comprehensive sources of this type."
- Bernd Wegner, Zentralblatt
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32Methods of Information Geometry
There has been much progress in the field of information geometry in recent years and yet there are few textbooks which reflect these developments, except for those which deal with statistics. Information geometry provides a new method applicable to various areas including information sciences and physical sciences. It has emerged from investigating the geometrical structures of the manifold of probability distributions, and has been applied successfully to statistical inference problems. However, it has been proved that information geometry opens a new paradigm useful for elucidation of information systems, intelligent systems, physical systems and mathematical systems.
33Riemannian Geometry: Theory & Applications
Riemannian Geometry is an expanded edition of a highly acclaimed and successful textbook (originally published in Portuguese) for first-year graduate students in mathematics and physics. The author's treatment goes very directly to the basic language of Riemannian geometry and immediately presents some of its most fundamental theorems. It is elementary, assuming only a modest background from readers, making it suitable for a wide variety of students and course structures. Its selection of topics has been deemed "superb" by teachers who have used the text.
A significant feature of the book is its powerful and revealing structure, beginning simply with the definition of a differentiable manifold and ending with one of the most important results in Riemannian geometry, a proof of the Sphere Theorem. The text abounds with basic definitions and theorems, examples, applications, and numerous exercises to test the student's understanding and extend knowledge and insight into the subject. Instructors and students alike will find the work to be a significant contribution to this highly applicable and stimulating subject.
A significant feature of the book is its powerful and revealing structure, beginning simply with the definition of a differentiable manifold and ending with one of the most important results in Riemannian geometry, a proof of the Sphere Theorem. The text abounds with basic definitions and theorems, examples, applications, and numerous exercises to test the student's understanding and extend knowledge and insight into the subject. Instructors and students alike will find the work to be a significant contribution to this highly applicable and stimulating subject.
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34Gauge Fields, Knots and Gravity
This is an introduction to the basic tools of mathematics needed to understand the relation between knot theory and quantum gravity. The book begins with a rapid course on manifolds and differential forms, emphasizing how these provide a proper language for formulating Maxwell's equations on arbitrary spacetimes. The authors then introduce vector bundles, connections and curvature in order to generalize Maxwell theory to the Yang-Mills equations. The relation of gauge theory to the newly discovered knot invariants such as the Jones polynomial is sketched. Riemannian geometry is then introduced in order to describe Einstein's equations of general relativity and show how an attempt to quantize gravity leads to interesting applications of knot theory.
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35Geometry with an Introduction to Cosmic Topology
Geometry With An Introduction To Cosmic Topology Is Motivated By Questions That Have Ignited The Imagination Of Stargazers Since Antiquity. What Is The Shape Of The Universe? Does The Universe Have An Edge? Is It Infinitely Big? Dr. Hitchman Aims To Clarify This Fascinating Area Of Mathematics And Focuses On The Mathematical Tools Used To Investigate The Shape Of The Universe. The Text Follows The Erlangen Program, Which Develops Geometry In Terms Of A Space And A Group Of Transformations Of That Space. This Approach To Non-Euclidean Geometry Provides Excellent Material By Which Students Can Learn The More Sophisticated Modes Of Thinking Necessary In Upper-Division Mathematics Courses. This Unique Text Is Organized Into Three Natural Parts: Chapter 1 Introduces The Geometric Perspective Taken In The Text And The Motivation For The Material That Comes From Cosmology. Chapters 2-7 Contain The Core Mathematical Content Of The Text, Developing Hyperbolic Elliptic, And Euclidean Geometry From The Complex Plane And Subgroups Of Mobius Transformations. Other Topics Include The Topology And Geometry Of Surfaces And Dirichlet Domains. Finally, Chapter 8 Explores The Topic Of Cosmic Topology Through The Geometry Learned In The Preceding Chapters.
36Geometrical Methods of Mathematical Physics
In recent years the methods of modern differential geometry have become of considerable importance in theoretical physics and have found application in relativity and cosmology, high-energy physics and field theory, thermodynamics, fluid dynamics and mechanics. This textbook provides an introduction to these methods - in particular Lie derivatives, Lie groups and differential forms - and covers their extensive applications to theoretical physics. The reader is assumed to have some familiarity with advanced calculus, linear algebra and a little elementary operator theory. The advanced physics undergraduate should therefore find the presentation quite accessible. This account will prove valuable for those with backgrounds in physics and applied mathematics who desire an introduction to the subject. Having studied the book, the reader will be able to comprehend research papers that use this mathematics and follow more advanced pure-mathematical expositions.
37A Geometric Approach to Differential Forms
The modern subject of differential forms subsumes classical vector calculus. This text presents differential forms from a geometric perspective accessible at the undergraduate level. The book begins with basic concepts such as partial differentiation and multiple integration and gently develops the entire machinery of differential forms. The author approaches the subject with the idea that complex concepts can be built up by analogy from simpler cases, which, being inherently geometric, often can be best understood visually. Each new concept is presented with a natural picture that students can easily grasp. Algebraic properties then follow. This facilitates the development of differential forms without assuming a background in linear algebra. Throughout the text, emphasis is placed on applications in 3 dimensions, but all definitions are given so as to be easily generalized to higher dimensions. Containing excellent motivation, numerous illustrations and solutions to selected problems in an appendix, the material has been tested in the classroom along all three potential course tracks.
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38Differential Geometry
Designed for advanced undergraduate or beginning graduate study, this text contains an elementary introduction to continuous groups and differential invariants; an extensive treatment of groups of motions in euclidean, affine, and riemannian geometry; and development of the method of integral formulas for global differential geometry.
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39The Variational Theory of Geodesics
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40Spaces of Constant Currature
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41Differential Geometry: Manifolds, Curves, And Surfaces
This book is an introduction to modern differential geometry. The authors begin with the necessary tools from analysis and topology, including Sard's theorem, de Rham cohomology, calculus on manifolds, and a degree theory. The general theory is illustrated and expanded using the examples of curves and surfaces. In particular, the book contains the classical local and global theory of surfaces, including the fundamental forms, curvature, the Gauss-Bonnet formula, geodesics, and minimal surfaces.
42Calculus: An Intuitive and Physical Approach
Application-oriented introduction relates the subject as closely as possible to science. In-depth explorations of the derivative, the differentiation and integration of the powers of x, and theorems on differentiation and antidifferentiation lead to a definition of the chain rule and examinations of trigonometric functions, logarithmic and exponential functions, techniques of integration, polar coordinates, much more. Clear-cut explanations, numerous drills, illustrative examples. 1967 edition. Solution guide available upon request.
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43An Introduction to Tensors and Group Theory for Physicists
"An Introduction to Tensors and Group Theory for Physicists" provides both an intuitive and rigorous approach to tensors and groups and their role in theoretical physics and applied mathematics. A particular aim is to demystify tensors and provide a unified framework for understanding them in the context of classical and quantum physics. Connecting the component formalism prevalent in physics calculations with the abstract but more conceptual formulation found in many mathematical texts, the work will be a welcome addition to the literature on tensors and group theory.Advanced undergraduate and graduate students in physics and applied mathematics will find clarity and insight into the subject in this textbook."
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44Elementary Differential Geometry (Springer Undergraduate Mathematics Series)
Elementary Differential Geometry presents the main results in the differential geometry of curves and surfaces suitable for a first course on the subject. Prerequisites are kept to an absolute minimum - nothing beyond first courses in linear algebra and multivariable calculus - and the most direct and straightforward approach is used throughout. New features of this revised and expanded second edition include: a chapter on non-Euclidean geometry, a subject that is of great importance in the history of mathematics and crucial in many modern developments. The main results can be reached easily and quickly by making use of the results and techniques developed earlier in the book. Coverage of topics such as: parallel transport and its applications; map colouring; holonomy and Gaussian curvature. Around 200 additional exercises, and a full solutions manual for instructors, available via www.springer.com
45Information Geometry and Its Applications
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46Differential Geometric Structures
Useful for independent study and as a reference work, this introduction to differential geometry features many examples and exercises. It defines geometric structure by specifying the parallel transport in an appropriate fiber bundle, focusing on the simplest cases of linear parallel transport in a vector bundle.
The treatment opens with an introductory chapter on fiber bundles that proceeds to examinations of connection theory for vector bundles and Riemannian vector bundles. Additional topics include the role of harmonic theory, geometric vector fields on Riemannian manifolds, Lie groups, symmetric spaces, and symplectic and Hermitian vector bundles. A consideration of other differential geometric structures concludes the text, including surveys of characteristic classes of principal bundles, Cartan connections, and spin structures.
The treatment opens with an introductory chapter on fiber bundles that proceeds to examinations of connection theory for vector bundles and Riemannian vector bundles. Additional topics include the role of harmonic theory, geometric vector fields on Riemannian manifolds, Lie groups, symmetric spaces, and symplectic and Hermitian vector bundles. A consideration of other differential geometric structures concludes the text, including surveys of characteristic classes of principal bundles, Cartan connections, and spin structures.
47Analysis on Manifolds
A readable introduction to the subject of calculus on arbitrary surfaces or manifolds. Accessible to readers with knowledge of basic calculus and linear algebra. Sections include series of problems to reinforce concepts.
48Differential Geometry for Machine Learning Step-by-Step Workbook: Guided Problems for Manifolds, Tangent Spaces, Metrics, Geodesics, Curvature, ... Embeddings, and Geometric Deep Learning
A Step-by-Step Workbook for Learning the Geometry Behind Modern Machine LearningDifferential Geometry for Machine Learning is a guided workbook for practitioners and students who keep running into manifolds, embeddings, and constrained optimization in their work but have never had a clear, computational path to actually understanding the geometry behind them. Instead of presenting abstract theorems, this workbook walks through each concept using solved examples, guided problems, and complete solutions, so you build real intuition alongside the math.Many machine learning tasks quietly assume flat, Euclidean space when the underlying data is not flat at all — normalized embeddings, probability distributions, rotations, and covariance matrices all live on curved spaces. This workbook explains why that mismatch causes subtle errors, and how differential geometry gives you the tools to reason about it correctly.What's InsideManifolds, Charts, and Atlases — how to describe curved spaces with local coordinates and understand why a single coordinate system often isn't enough.Tangent Spaces and Local Linearization — finding valid local directions and understanding which updates preserve a constraint.Differential Maps, Jacobians, Pushforwards, and Pullbacks — how tangent information moves through encoders, decoders, and other maps.Metrics and Measuring Geometry — how to define length, angle, and distance in a way that actually matches the structure of your data.Gradients on Manifolds — computing constraint-aware gradients instead of naive Euclidean ones.Geodesics, Exponential Maps, and Retractions — the correct way to interpolate and take optimization steps on curved spaces.Connections, Parallel Transport, and Curvature — comparing vectors at different points and understanding how curvature shapes learning spaces.Applications to Riemannian Optimization, Information Geometry, Lie Groups, Hyperbolic Embeddings, and Geometric Deep Learning — connecting the math directly to real modeling choices.How the Workbook Is StructuredEach chapter follows the same pattern: a clear explanation of the concept, worked examples that show the reasoning step by step, a list of common mistakes, and a set of practice problems with full solutions. Chapters build on each other, moving from the motivation for geometric thinking in Chapter 1 through coordinates, tangent spaces, metrics, gradients, and geodesics, before combining everything into a complete geometry-aware machine learning workflow in the final chapter.Appendices provide a symbol glossary, formula reference cards, a catalog of common machine learning manifolds, and cumulative review problems, so the workbook can also serve as an ongoing reference once you've worked through the main chapters.Who This Workbook Is ForMachine learning engineers working with embeddings, normalized representations, or constrained parametersGraduate students and researchers studying geometric deep learning or Riemannian optimizationPractitioners who have encountered terms like tangent space, geodesic, or pullback metric in papers and want to understand them properlyAnyone comfortable with linear algebra and multivariable calculus who wants a structured, problem-based path into differential geometryThis is a workbook meant to be worked through, not just read. Each solved problem is designed to be reworked by hand before checking the solution, and each chapter checklist helps confirm the ideas have actually settled in before moving forward.Scroll up and get your copy to start building a geometric foundation for your machine learning work.
49Benjamin Harrison (The American Presidents, #23)
The scion of a political dynasty ushers in the era of big government
Politics was in Benjamin Harrison's blood. His great-grandfather signed the Declaration and his grandfather, William Henry Harrison, was the ninth president of the United States. Harrison, a leading Indiana lawyer, became a Republican Party champion, even taking a leave from the Civil War to campaign for Lincoln. After a scandal-free term in the Senate-no small feat in the Gilded Age-the Republicans chose Harrison as their presidential candidate in 1888. Despite losing the popular vote, he trounced the incumbent, Grover Cleveland, in the electoral college.
In contrast to standard histories, which dismiss Harrison's presidency as corrupt and inactive, Charles W. Calhoun sweeps away the stereotypes of the age to reveal the accomplishments of our twenty-third president. With Congress under Republican control, he exemplified the activist president, working feverishly to put the Party's planks into law and approving the first billion-dollar peacetime budget. But the Democrats won Congress in 1890, stalling his legislative agenda, and with the First Lady ill, his race for reelection proceeded quietly. (She died just before the election.) In the end, Harrison could not beat Cleveland in their unprecedented rematch.
With dazzling attention to this president's life and the social tapestry of his times, Calhoun compellingly reconsiders Harrison's legacy.
Politics was in Benjamin Harrison's blood. His great-grandfather signed the Declaration and his grandfather, William Henry Harrison, was the ninth president of the United States. Harrison, a leading Indiana lawyer, became a Republican Party champion, even taking a leave from the Civil War to campaign for Lincoln. After a scandal-free term in the Senate-no small feat in the Gilded Age-the Republicans chose Harrison as their presidential candidate in 1888. Despite losing the popular vote, he trounced the incumbent, Grover Cleveland, in the electoral college.
In contrast to standard histories, which dismiss Harrison's presidency as corrupt and inactive, Charles W. Calhoun sweeps away the stereotypes of the age to reveal the accomplishments of our twenty-third president. With Congress under Republican control, he exemplified the activist president, working feverishly to put the Party's planks into law and approving the first billion-dollar peacetime budget. But the Democrats won Congress in 1890, stalling his legislative agenda, and with the First Lady ill, his race for reelection proceeded quietly. (She died just before the election.) In the end, Harrison could not beat Cleveland in their unprecedented rematch.
With dazzling attention to this president's life and the social tapestry of his times, Calhoun compellingly reconsiders Harrison's legacy.
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50First Steps in Differential Geometry: Riemannian, Contact, Symplectic
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51Foundations of Differential Geometry, Volume 1
This two-volume introduction to differential geometry, part of Wiley's popular Classics Library, lays the foundation for understanding an area of study that has become vital to contemporary mathematics. It is completely self-contained and will serve as a reference as well as a teaching guide. Volume 1 presents a systematic introduction to the field from a brief survey of differentiable manifolds, Lie groups and fibre bundles to the extension of local transformations and Riemannian connections. The second volume continues with the study of variational problems on geodesics through differential geometric aspects of characteristic classes. Both volumes familiarize readers with basic computational techniques.
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52Elements of Differential Geometry
This text is intended for an advanced undergraduate (having taken linear algebra and multivariable calculus). It provides the necessary background for a more abstract course in differential geometry. The inclusion of diagrams is done without sacrificing the rigor of the material. For all readers interested in differential geometry.
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53Riemannian Geometry
In his classic work of geometry, Euclid focused on the properties of flat surfaces. In the age of exploration, mapmakers such as Mercator had to concern themselves with the properties of spherical surfaces. The study of curved surfaces, or non-Euclidean geometry, flowered in the late nineteenth century, as mathematicians such as Riemann increasingly questioned Euclid's parallel postulate, and by relaxing this constraint derived a wealth of new results. These seemingly abstract properties found immediate application in physics upon Einstein's introduction of the general theory of relativity.
In this book, Eisenhart succinctly surveys the key concepts of Riemannian geometry, addressing mathematicians and theoretical physicists alike.
In this book, Eisenhart succinctly surveys the key concepts of Riemannian geometry, addressing mathematicians and theoretical physicists alike.
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54Synthetic Differential Geometry
Synthetic Differential Geometry is a method of reasoning in differential geometry and differential calculus, based on the assumption of sufficiently many nilpotent elements on the number line, in particular numbers d such that d2=0. The use of nilpotent elements allows one to replace the limit processes of calculus by purely algebraic calculations and notions. For the first half of the book, first published in 2006, familiarity with differential calculus and abstract algebra is presupposed during the development of results in calculus and differential geometry on a purely axiomatic/synthetic basis. In the second half basic notions of category theory are presumed in the construction of suitable Cartesian closed categories and the interpretation of logical formulae within them. This is a second edition of Kock's classical text from 1981. Many notes have been included, with comments on developments in the field from the intermediate years, and almost 100 new bibliographic entries have been added.
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55Applied Differential Geometry
This is a self-contained introductory textbook on the calculus of differential forms and modern differential geometry. The intended audience is physicists, so the author emphasises applications and geometrical reasoning in order to give results and concepts a precise but intuitive meaning without getting bogged down in analysis. The large number of diagrams helps elucidate the fundamental ideas. Mathematical topics covered include differentiable manifolds, differential forms and twisted forms, the Hodge star operator, exterior differential systems and symplectic geometry. All of the mathematics is motivated and illustrated by useful physical examples.
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56Applications of Differential Geometry to Econometrics
Differential geometry has become a standard tool in the analysis of statistical models, offering a deeper appreciation of existing methodologies and highlighting the issues that can be hidden in an algebraic development of a problem. This volume is the first to apply these techniques to econometrics. An introductory chapter provides a brief tutorial for those unfamiliar with the tools of differential geometry. The following chapters offer applications of geometric methods to practical solutions and offer insight into problems of econometric inference.
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57Information Geometry: Near Randomness and Near Independence
The main motivation for this book lies in the breadth of applications in which a statistical model is used to represent small departures from, for example, a Poisson process. Our approach uses information geometry to provide a c- mon context but we need only rather elementary material from di?erential geometry, information theory and mathematical statistics. Introductory s- tions serve together to help those interested from the applications side in making use of our methods and results. We have available Mathematica no- books to perform many of the computations for those who wish to pursue their own calculations or developments. Some 44 years ago, the second author ?rst encountered, at about the same time, di?erential geometry via relativity from Weyl's book [209] during - dergraduate studies and information theory from Tribus [200, 201] via spatial statistical processes while working on research projects at Wiggins Teape - searchandDevelopmentLtd-cf. theForewordin[196]and[170,47,58]. H- ing started work there as a student laboratory assistant in 1959, this research environment engendered a recognition of the importance of international c- laboration, and a lifelong research interest in randomness and near-Poisson statistical geometric processes, persisting at various rates through a career mainly involved with global di?erential geometry. From correspondence in the 1960s with Gabriel Kron [4, 124, 125] on his Diakoptics, and with Kazuo Kondo who in?uenced the post-war Japanese schools of di?erential geometry and supervised Shun-ichi Amari's doctorate [6], it was clear that both had a much wider remit than traditionally pursued elsewhere.
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58Introduction to Arithmetic Groups
This book provides a gentle introduction to the study of arithmetic subgroups of semisimple Lie groups. This means that the goal is to understand the group SL(n, Z) and certain of its subgroups. Among the major results discussed in the later chapters are the Mostow Rigidity Theorem, the Margulis Superrigidity Theorem, Ratner's Theorems, and the classification of arithmetic subgroups of classical groups. As background for the proofs of these theorems, the book provides primers on lattice subgroups, arithmetic groups, real rank and Q-rank, ergodic theory, unitary representations, amenability, Kazhdan's property (T), and quasi-isometries. Numerous exercises enhance the book's usefulness both as a textbook for a second-year graduate course and for self-study. In addition, notes at the end of each chapter have suggestions for further reading. (Proofs in this book often consider only an illuminating special case.) Readers are expected to have some acquaintance with Lie groups, but appendices briefly review the prerequisite background. A PDF file of the book is available on the internet. This inexpensive printed edition is for readers who prefer a hardcopy.
59Fundamentals of Differential Geometry
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60Manifolds, Tensors, and Forms
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.
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61Schaum's Outline of Tensor Calculus
Confusing d104books? Missed Lectures?
Not Enough Time? Fortunately for you, there's Schaum's.
More than 40 million students have trusted Schaum's to help them succeed in the classroom and on exams. Schaum's is the key to faster learning and higher grades in every subject. Each Outline presents all the essential course information in an easy-to-follow, topic-by-topic format. You also get hundreds of examples, solved problems, and practice exercises to test your skills.
This Schaum's Outline gives you
Practice problems with full explanations that reinforce knowledge
Coverage of the most up-to-date developments in your course field
In-depth review of practices and applications
Fully compatible with your classroom text, Schaum's highlights all the important facts you need to know. Use Schaum's to shorten your study time-and get your best test scores!
Schaum's Outlines-Problem Solved.
Not Enough Time? Fortunately for you, there's Schaum's.
More than 40 million students have trusted Schaum's to help them succeed in the classroom and on exams. Schaum's is the key to faster learning and higher grades in every subject. Each Outline presents all the essential course information in an easy-to-follow, topic-by-topic format. You also get hundreds of examples, solved problems, and practice exercises to test your skills.
This Schaum's Outline gives you
Practice problems with full explanations that reinforce knowledge
Coverage of the most up-to-date developments in your course field
In-depth review of practices and applications
Fully compatible with your classroom text, Schaum's highlights all the important facts you need to know. Use Schaum's to shorten your study time-and get your best test scores!
Schaum's Outlines-Problem Solved.
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62Modern Geometry -- Methods and Applications: Part I: The Geometry of Surfaces, Transformation Groups, and Fields
This is the first volume of a three-volume introduction to modern geometry which emphasizes applications to other areas of mathematics and theoretical physics. Topics covered include tensors and their differential calculus, the calculus of variations in one and several dimensions, and geometric field theory. This new edition offers substantial revisions, and the material is written in concrete language with terminology acceptable to physicists.
Recommended by Eric Weinstein.Eric Weinstein
Managing Director/Thiel Capital
Eric Weinstein recommended this book on Twitter. [source]
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63The Geometry of Kerr Black Holes
This unique monograph by a noted UCLA professor examines in detail the mathematics of Kerr black holes, which possess the properties of mass and angular momentum but carry no electrical charge. Suitable for advanced undergraduates and graduate students of mathematics, physics, and astronomy as well as professional physicists, the self-contained treatment constitutes an introduction to modern techniques in differential geometry.
The text begins with a substantial chapter offering background on the mathematics needed for the rest of the book. Subsequent chapters emphasize physical interpretations of geometric properties such as curvature, geodesics, isometries, totally geodesic submanifolds, and topological structure. Further investigations cover relativistic concepts such as causality, Petrov types, optical scalars, and the Goldberg-Sachs theorem. Four helpful appendixes supplement the text.
The text begins with a substantial chapter offering background on the mathematics needed for the rest of the book. Subsequent chapters emphasize physical interpretations of geometric properties such as curvature, geodesics, isometries, totally geodesic submanifolds, and topological structure. Further investigations cover relativistic concepts such as causality, Petrov types, optical scalars, and the Goldberg-Sachs theorem. Four helpful appendixes supplement the text.
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64Lectures on Differential Geometry
This book is a translation of an authoritative introductory text based on a lecture series delivered by the renowned differential geometer, Professor S S Chern in Beijing University in 1980. The original Chinese text, authored by Professor Chern and Professor Wei-Huan Chen, was a unique contribution to the mathematics literature, combining simplicity and economy of approach with depth of contents. The present translation is aimed at a wide audience, including (but not limited to) advanced undergraduate and graduate students in mathematics, as well as physicists interested in the diverse applications of differential geometry to physics. In addition to a thorough treatment of the fundamentals of manifold theory, exterior algebra, the exterior calculus, connections on fiber bundles, Riemannian geometry, Lie groups and moving frames, and complex manifolds (with a succinct introduction to the theory of Chern classes), and an appendix on the relationship between differential geometry and theoretical physics, this book includes a new chapter on Finsler geometry and a new appendix on the history and recent developments of differential geometry, the latter prepared specially for this edition by Professor Chern to bring the text into perspectives.
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65Modern Calculus and Analytic Geometry
A self-contained text for an introductory course, this volume places strong emphasis on physical applications. Key elements of differential equations and linear algebra are introduced early and are consistently referenced, all theorems are proved using elementary methods, and numerous worked-out examples appear throughout. The highly readable text approaches calculus from the student's viewpoint and points out potential stumbling blocks before they develop. A collection of more than 1,600 problems ranges from exercise material to exploration of new points of theory — many of the answers are found at the end of the book; some of them worked out fully so that the entire process can be followed. This well-organized, unified text is copiously illustrated, amply cross-referenced, and fully indexed.
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66Riemannian Manifolds: An Introduction to Curvature
This text is designed for a one-quarter or one-semester graduate course on Riemannian geometry. It 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 study of Riemannian manifolds. The book begins with a careful treatment of the machinery of metrics, connections, and geodesics, and then introduces the curvature tensor as a way of measuring whether a Riemannian manifold is locally equivalent to Euclidean space. Submanifold theory is developed next in order to give the curvature tensor a concrete quantitative interpretation. The remainder of the text is devoted to proving the four most fundamental theorems relating curvature and topology: the Gauss-Bonnet Theorem, the Cartan-Hadamard Theorem, Bonnet's Theorem, and the characterization of manifolds of constant curvature. This unique volume will appeal especially to students by presenting a selective introduction to the main ideas of the subject in an easily accessible way. The material is ideal for a single course, but broad enough to provide students with a firm foundation from which to pursue research or develop applications in Riemannian geometry and other fields that use its tools.
67Differential Manifolds
The concepts of differential topology form the center of many mathematical disciplines such as differential geometry and Lie group theory. Differential Manifolds presents to advanced undergraduates and graduate students the systematic study of the topological structure of smooth manifolds. Author Antoni A. Kosinski, Professor Emeritus of Mathematics at Rutgers University, offers an accessible approach to both the h-cobordism theorem and the classification of differential structures on spheres.
"How useful it is," noted the Bulletin of the American Mathematical Society, "to have a single, short, well-written book on differential topology." This volume begins with a detailed, self-contained review of the foundations of differential topology that requires only a minimal knowledge of elementary algebraic topology. Subsequent chapters explain the technique of joining manifolds along submanifolds, the handle presentation theorem, and the proof of the h-cobordism theorem based on these constructions. There follows a chapter on the Pontriagin Construction—the principal link between differential topology and homotopy theory. The final chapter introduces the method of surgery and applies it to the classification of smooth structures of spheres. The text is supplemented by numerous interesting historical notes and contains a new appendix, "The Work of Grigory Perelman," by John W. Morgan, which discusses the most recent developments in differential topology.
"How useful it is," noted the Bulletin of the American Mathematical Society, "to have a single, short, well-written book on differential topology." This volume begins with a detailed, self-contained review of the foundations of differential topology that requires only a minimal knowledge of elementary algebraic topology. Subsequent chapters explain the technique of joining manifolds along submanifolds, the handle presentation theorem, and the proof of the h-cobordism theorem based on these constructions. There follows a chapter on the Pontriagin Construction—the principal link between differential topology and homotopy theory. The final chapter introduces the method of surgery and applies it to the classification of smooth structures of spheres. The text is supplemented by numerous interesting historical notes and contains a new appendix, "The Work of Grigory Perelman," by John W. Morgan, which discusses the most recent developments in differential topology.
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68Foundations of Differential Geometry, Volume 2
This two-volume introduction to differential geometry, part of Wiley's popular Classics Library, lays the foundation for understanding an area of study that has become vital to contemporary mathematics. It is completely self-contained and will serve as a reference as well as a teaching guide. Volume 1 presents a systematic introduction to the field from a brief survey of differentiable manifolds, Lie groups and fibre bundles to the extension of local transformations and Riemannian connections. The second volume continues with the study of variational problems on geodesics through differential geometric aspects of characteristic classes. Both volumes familiarize readers with basic computational techniques.
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69Morse Theory. (Am-51), Volume 51
One of the most cited books in mathematics, John Milnor's exposition of Morse theory has been the most important book on the subject for more than forty years. Morse theory was developed in the 1920s by mathematician Marston Morse. (Morse was on the faculty of the Institute for Advanced Study, and Princeton published his Topological Methods in the Theory of Functions of a Complex Variable in the Annals of Mathematics Studies series in 1947.) One classical application of Morse theory includes the attempt to understand, with only limited information, the large-scale structure of an object. This kind of problem occurs in mathematical physics, dynamic systems, and mechanical engineering. Morse theory has received much attention in the last two decades as a result of a famous paper in which theoretical physicist Edward Witten relates Morse theory to quantum field theory.
Milnor was awarded the Fields Medal (the mathematical equivalent of a Nobel Prize) in 1962 for his work in differential topology. He has since received the National Medal of Science (1967) and the Steele Prize from the American Mathematical Society twice (1982 and 2004) in recognition of his explanations of mathematical concepts across a wide range of scienti.c disciplines. The citation reads, The phrase sublime elegance is rarely associated with mathematical exposition, but it applies to all of Milnor's writings. Reading his books, one is struck with the ease with which the subject is unfolding and it only becomes apparent after re.ection that this ease is the mark of a master.?
Milnor has published five books with Princeton University Press.
Milnor was awarded the Fields Medal (the mathematical equivalent of a Nobel Prize) in 1962 for his work in differential topology. He has since received the National Medal of Science (1967) and the Steele Prize from the American Mathematical Society twice (1982 and 2004) in recognition of his explanations of mathematical concepts across a wide range of scienti.c disciplines. The citation reads, The phrase sublime elegance is rarely associated with mathematical exposition, but it applies to all of Milnor's writings. Reading his books, one is struck with the ease with which the subject is unfolding and it only becomes apparent after re.ection that this ease is the mark of a master.?
Milnor has published five books with Princeton University Press.
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70Modern Differential Geometry for Physicists (World Scientific Lecture Notes in Physics)
71Differential Equations for Dummies
The fun and easy way to understand and solve complex equations Many of the fundamental laws of physics, chemistry, biology, and economics can be formulated as differential equations. This plain-English guide explores the many applications of this mathematical tool and shows how differential equations can help us understand the world around us. Differential Equations For Dummies is the perfect companion for a college differential equations course and is an ideal supplemental resource for other calculus classes as well as science and engineering courses. It offers step-by-step techniques, practical tips, numerous exercises, and clear, concise examples to help readers improve their differential equation-solving skills and boost their test scores.
72Regular Polytopes
Foremost book available on polytopes, incorporating ancient Greek and most modern work done on them. Beginning with polygons and polyhedrons, the book moves on to multi-dimensional polytopes in a way that anyone with a basic knowledge of geometry and trigonometry can easily understand. Definitions of symbols. Eight tables plus many diagrams and examples. 1963 edition.
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73The Absolute Differential Calculus (Calculus of Tensors)
Written by a towering figure of twentieth-century mathematics, this classic examines the mathematical background necessary for a grasp of relativity theory. Tullio Levi-Civita provides a thorough treatment of the introductory theories that form the basis for discussions of fundamental quadratic forms and absolute differential calculus, and he further explores physical applications.
Part one opens with considerations of functional determinants and matrices, advancing to systems of total differential equations, linear partial differential equations, algebraic foundations, and a geometrical introduction to theory. The second part addresses covariant differentiation, curvature-related Riemann's symbols and properties, differential quadratic forms of classes zero and one, and intrinsic geometry. The final section focuses on physical applications, covering gravitational equations and general relativity.
Part one opens with considerations of functional determinants and matrices, advancing to systems of total differential equations, linear partial differential equations, algebraic foundations, and a geometrical introduction to theory. The second part addresses covariant differentiation, curvature-related Riemann's symbols and properties, differential quadratic forms of classes zero and one, and intrinsic geometry. The final section focuses on physical applications, covering gravitational equations and general relativity.
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74The Geometry of Spacetime: An Introduction to Special and General Relativity
In 1905, Albert Einstein offered a revolutionary theory--special relativity--to explain some of the most troubling problems in current physics concerning electromagnetism and motion. Soon afterwards, Hermann Minkowski recast special relativity essentially as a new geometric structure for spacetime. These ideas are the subject of the first part of the book. The second part develops the main implications of Einstein's general relativity as a theory of gravity rooted in the differential geometry of surfaces. The author explores the way an individual observer views the world and how a pair of observers collaborate to gain objective knowledge of the world. To encompass both the general and special theory, he uses the geometry of spacetime as the unifying theme of the book. To read it, one needs only a first course in linear algebra and multivariable calculus and familiarity with the physical applications of calculus.
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758th grade Math Workbook: CommonCore Math Workbook
8th Grade Common Core Math: Practice Workbook - Practice Questions, Answers & Explanations - Recommended by Teachers - Ace Academic Publishing-Based on Common Core State Standards: Similar to a standardized exam, you can find questions of all types, including multiple-choice, fill-in-the-blank, true or false, match the correct answer and free-response questions.-High Standards of Questions: Each of these questions are divided into Chapters and Sub-Topics. The contents of this Math workbook include multiple chapters and units covering all the required Common Core Standards for this grade level.-Recommended by Teachers: These carefully written questions aim to help students reason abstractly and quantitatively using various models, strategies, and problem-solving techniques.-Identifying trouble spots and fixing it: This workbook will help students overcome any deficiencies in their understanding of critical concepts and also will help you identify the specific topic that students may require additional practice.-Practice all types of questions including free response: The book is comprised of multiple tests for each topic so that your child can have an abundant amount of test on the same topic.-Includes 2 Comprehensive Full-Length Tests: to practice the entire syllabus to prepare for the common core standardized exams.-Includes detailed step-by-step answers: The detailed answer explanations in the back of the book help the students understand the topics and gain confidence in solving similar problems and take the Common Core Standardized Exam with Confidence.
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76Lectures on Symplectic Geometry
These notes approximately transcribe a 15-week course on symplectic geometry I taught at UC Berkeley in the Fall of 1997. The course at Berkeley was greatly inspired in content and style by Victor Guillemin, whose masterly teaching of beautiful courses on topics related to s- plectic geometry at MIT, I was lucky enough to experience as a graduate student. I am very thankful to him! That course also borrowed from the 1997 Park City summer courses on symplectic geometry and topology, and from many talks and discussions of the symplectic geometry group at MIT. Among the regular participants in the MIT - formal symplectic seminar 93-96, I would like to acknowledge the contributions of Allen Knutson, Chris Woodward, David Metzler, Eckhard Meinrenken, Elisa Prato, Eugene Lerman, Jonathan Weitsman, Lisa Jeffrey, Reyer Sjamaar, Shaun Martin, Stephanie Singer, Sue Tolman and, last but not least, Yael Karshon. Thanks to everyone sitting in Math 242 in the Fall of 1997 for all the c- ments they made, and especially to those who wrote notes on the basis of which I was better able to reconstruct what went on: Alexandru Scorpan, Ben Davis, David Martinez, DonBarkauskas, EzraMiller, HenriqueBursztyn, John-PeterLund, Laura De Marco, Olga Radko, Peter P? rib ?k, Pieter Collins, Sarah Packman, Stephen Bigelow, Susan Harrington, Tolga Etgu ] and Yi Ma.
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77Mathematical Discovery on Understanding, Learning, and Teaching Problem Solving, Volume I
George Polya was a Hungarian mathematician. Born in Budapest on 13 December 1887, his original name was Polya Gyorg. He wrote perhaps the most famous book of mathematics ever written, namely "How to Solve It." However, "How to Solve It" is not strictly speaking a math book. It is a book about how to solve problems of any kind, of which math is just one type of problem. The same techniques could in principle be used to solve any problem one encounters in life (such as how to choose the best wife ). Therefore, Polya wrote the current volume to explain how the techniques set forth in "How to Solve It" can be applied to specific areas such as geometry."
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78Geometry and Physics
"Geometry and Physics" addresses mathematicians wanting to understand modern physics, and physicists wanting to learn geometry. It gives an introduction to modern quantum field theory and related areas of theoretical high-energy physics from the perspective of Riemannian geometry, and an introduction to modern geometry as needed and utilized in modern physics. Jurgen Jost, a well-known research mathematician and advanced textbook author, also develops important geometric concepts and methods that can be used for the structures of physics. In particular, he discusses the Lagrangians of the standard model and its supersymmetric extensions from a geometric perspective.
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79Elementary Differential Geometry
Written primarily for students who have completed the standard first courses in calculus and linear algebra, ELEMENTARY DIFFERENTIAL GEOMETRY, REVISED SECOND EDITION, provides an introduction to the geometry of curves and surfaces.
The Second Edition maintained the accessibility of the first, while providing an introduction to the use of computers and expanding discussion on certain topics. Further emphasis was placed on topological properties, properties of geodesics, singularities of vector fields, and the theorems of Bonnet and Hadamard.
This revision of the Second Edition provides a thorough update of commands for the symbolic computation programs Mathematica or Maple, as well as additional computer exercises. As with the Second Edition, this material supplements the content but no computer skill is necessary to take full advantage of this comprehensive text.
• Fortieth anniversary of publication! Over 36,000 copies sold worldwide
• Accessible, practical yet rigorous approach to a complex topic—also suitable for self-study
• Extensive update of appendices on Mathematica and Maple software packages
• Thorough streamlining of second edition's numbering system
• Fuller information on solutions to odd-numbered problems
• Additional exercises and hints guide students in using the latest computer modeling tools
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80Lectures on Classical Differential Geometry
Excellent brief introduction presents fundamental theory of curves and surfaces and applies them to a number of examples. Topics include curves, theory of surfaces, fundamental equations, geometry on a surface, envelopes, conformal mapping, minimal surfaces, more. Well-illustrated, with abundant problems and solutions. Bibliography.
81Differential Forms
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82An Introduction to Differential Geometry - With the Use of Tensor Calculus
Many of the earliest books, particularly those dating back to the 1900s and before, are now extremely scarce and increasingly expensive. We are republishing these classic works in affordable, high quality, modern editions, using the original text and artwork.
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83Topics in Physical Mathematics
This title adopts the view that physics is the primary driving force behind a number of developments in mathematics. Previously, science and mathematics were part of natural philosophy and many mathematical theories arose as a result of trying to understand natural phenomena. This situation changed at the beginning of last century as science and mathematics diverged. These two fields are collaborating once again; 'Topics in Mathematical Physics' takes the reader through this journey.
The author discusses topics where the interaction of physical and mathematical theories has led to new points of view and new results in mathematics. The area where this is most evident is that of geometric topology of low dimensional manifolds. These include the theories of Donaldson, Chern-Simons, Floer-Fukaya, Seiberg-Witten, and Topological (Quantum) Field Theory.
The author also discusses the interaction of CFT, Supersymmetry, String Theory and Gravity with diverse areas of mathematics. Several of these ideas have led to new insights into old mathematical structures and some have led to surprising new results The term "Physical Mathematics'' has been coined to describe collectively these new and fast growing areas of research, and regards the work of Donaldson and Witten as belonging to this new area of physical mathematics. Study of this work forms an important part of this book.
The author discusses topics where the interaction of physical and mathematical theories has led to new points of view and new results in mathematics. The area where this is most evident is that of geometric topology of low dimensional manifolds. These include the theories of Donaldson, Chern-Simons, Floer-Fukaya, Seiberg-Witten, and Topological (Quantum) Field Theory.
The author also discusses the interaction of CFT, Supersymmetry, String Theory and Gravity with diverse areas of mathematics. Several of these ideas have led to new insights into old mathematical structures and some have led to surprising new results The term "Physical Mathematics'' has been coined to describe collectively these new and fast growing areas of research, and regards the work of Donaldson and Witten as belonging to this new area of physical mathematics. Study of this work forms an important part of this book.
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84Introduction to Differentiable Manifolds
Author is well-known and established book author (all Serge Lang books are now published by Springer); Presents a brief introduction to the subject; All manifolds are assumed finite dimensional in order not to frighten some readers; Complete proofs are given; Use of manifolds cuts across disciplines and includes physics, engineering and economics
85A Visual Introduction to Differential Forms and Calculus on Manifolds
This book explains and helps readers to develop geometric intuition as it relates to differential forms. It includes over 250 figures to aid understand and enable readers to visualize the concepts being discussed. The author gradually builds up to the basic ideas and concepts so that definitions, when made, do not appear out of nowhere, and both the importance and role that theorems play is evident as or before they are presented. With a clear writing style and easy-to- understand motivations for each topic, this book is primarily aimed at second- or third-year undergraduate math and physics students with a basic knowledge of vector calculus and linear algebra.
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86The Mystery of Space; A Study of the Hyperspace Movement in the Light of the Evolution of New Psychic Faculties and an Inquiry Into the Genesis and Essential Nature of Space
This work has been selected by scholars as being culturally important, and is part of the knowledge base of civilization as we know it. This work was reproduced from the original artifact, and remains as true to the original work as possible. Therefore, you will see the original copyright references, library stamps (as most of these works have been housed in our most important libraries around the world), and other notations in the work. This work is in the public domain in the United States of America, and possibly other nations. Within the United States, you may freely copy and distribute this work, as no entity (individual or corporate) has a copyright on the body of the work.As a reproduction of a historical artifact, this work may contain missing or blurred pages, poor pictures, errant marks, etc. Scholars believe, and we concur, that this work is important enough to be preserved, reproduced, and made generally available to the public. We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant.
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87Differential Topology
Originally published: Englewood Cliffs, N.J.: Prentice-Hall, 1974.
Recommended by Eric Weinstein.Eric Weinstein
Managing Director/Thiel Capital
Folks frequently ask “What are the books that changed your life?” If I tell them, they are usually radically disappointed. I find that curious. I just cleared out of an office, and these are 4 shelves of spines of books that mattered enough to me to bring home. So here they are. [source]
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88The Concept of a Riemann Surface
This classic on the general history of functions was written by one of the twentieth century's best-known mathematicians. Hermann Weyl, who worked with Einstein at Princeton, combined function theory and geometry in this high-level landmark work, forming a new branch of mathematics and the basis of the modern approach to analysis, geometry, and topology.
The author intended this book not only to develop the basic ideas of Riemann's theory of algebraic functions and their integrals but also to examine the related ideas and theorems with an unprecedented degree of rigor. Weyl's two-part treatment begins by defining the concept and topology of Riemann surfaces and concludes with an exploration of functions of Riemann surfaces. His teachings illustrate the role of Riemann surfaces as not only devices for visualizing the values of analytic functions but also as indispensable components of the theory.
The author intended this book not only to develop the basic ideas of Riemann's theory of algebraic functions and their integrals but also to examine the related ideas and theorems with an unprecedented degree of rigor. Weyl's two-part treatment begins by defining the concept and topology of Riemann surfaces and concludes with an exploration of functions of Riemann surfaces. His teachings illustrate the role of Riemann surfaces as not only devices for visualizing the values of analytic functions but also as indispensable components of the theory.
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89An Introduction To Differential Manifolds
This invaluable book, based on the many years of teaching experience of both authors, introduces the reader to the basic ideas in differential topology. Among the topics covered are smooth manifolds and maps, the structure of the tangent bundle and its associates, the calculation of real cohomology groups using differential forms (de Rham theory), and applications such as the Poincaré-Hopf theorem relating the Euler number of a manifold and the index of a vector field. Each chapter contains exercises of varying difficulty for which solutions are provided. Special features include examples drawn from geometric manifolds in dimension 3 and Brieskorn varieties in dimensions 5 and 7, as well as detailed calculations for the cohomology groups of spheres and tori.
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90Topology and Geometry for Physicists
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91Projective Geometry
In Euclidean geometry, constructions are made with ruler and compass. Projective geometry is simpler: its constructions require only a ruler. In projective geometry one never measures anything, instead, one relates one set of points to another by a projectivity. The first two chapters of this book introduce the important concepts of the subject and provide the logical foundations. The third and fourth chapters introduce the famous theorems of Desargues and Pappus. Chapters 5 and 6 make use of projectivities on a line and plane, respectively. The next three chapters develop a self-contained account of von Staudt's approach to the theory of conics. The modern approach used in that development is exploited in Chapter 10, which deals with the simplest finite geometry that is rich enough to illustrate all the theorems nontrivially. The concluding chapters show the connections among projective, Euclidean, and analytic geometry.
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92Curvature in Mathematics and Physics
This original Dover textbook is based on an advanced undergraduate course taught by the author for more than 50 years. It introduces semi-Riemannian geometry and its principal physical application, Einstein's theory of general relativity, using the Cartan exterior calculus as a principal tool. Prerequisites include linear algebra and advanced calculus. 2012 edition.
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93Java: Programming Basics for Absolute Beginners (Step-By-Step Java Book 1)
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94Differential Geometry
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95Differential Geometry: Curves - Surfaces - Manifolds (Student Mathematical Library, Volume 16)
Our first knowledge of differential geometry usually comes from the study of the curves and surfaces in $I\!\!R^3$ that arise in calculus. Here we learn about line and surface integrals, divergence and curl, and the various forms of Stokes' Theorem. If we are fortunate, we may encounter curvature and such things as the Serret-Frenet formulas. With just the basic tools from multivariable calculus, plus a little knowledge of linear algebra, it is possible to begin a much richer and rewarding study of differential geometry, which is what is presented in this book. It starts with an introduction to the classical differential geometry of curves and surfaces in Euclidean space, then leads to an introduction to the Riemannian geometry of more general manifolds, including a look at Einstein spaces. An important bridge from the low-dimensional theory to the general case is provided by a chapter on the intrinsic geometry of surfaces. The first half of the book, covering the geometry of curves and surfaces, would be suitable for a one-semester undergraduate course. The local and global theories of curves and surfaces are presented, including detailed discussions of surfaces of rotation, ruled surfaces, and minimal surfaces. The second half of the book, which could be used for a more advanced course, begins with an introduction to differentiable manifolds, Riemannian structures, and the curvature tensor. Two special topics are treated in detail: spaces of constant curvature and Einstein spaces. The main goal of the book is to get started in a fairly elementary way, then to guide the reader toward more sophisticated concepts and more advanced topics. There are many examples and exercises to helpalong the way. Numerous figures help the reader visualize key concepts and examples, especially in lower dimensions. For the second edition, a number of errors were corrected and some text and a number of figures have been added.
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96Introduction to Tensor Calculus, Relativity and Cosmology
Elementary introduction pays special attention to aspects of tensor calculus and relativity that students find most difficult. Contents include tensors in curved spaces and application to general relativity theory; black holes; gravitational waves; application of general relativity principles to cosmology. Numerous exercises. Solution guide available upon request. 1982 edition.
97Differential Geometry: Bundles, Connections, Metrics and Curvature
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98Geometric Fundamentals of Robotics
• Provides an elegant introduction to the geometric concepts that are important to applications in robotics
* Includes significant state-of-the art material that reflects important advances, connecting robotics back to mathematical fundamentals in group theory and geometry
* An invaluable reference that serves a wide audience of grad students and researchers in mechanical engineering, computer science, and applied mathematics
* Includes significant state-of-the art material that reflects important advances, connecting robotics back to mathematical fundamentals in group theory and geometry
* An invaluable reference that serves a wide audience of grad students and researchers in mechanical engineering, computer science, and applied mathematics
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99Differential Geometry and Mathematical Physics: Part I. Manifolds, Lie Groups and Hamiltonian Systems
Starting from undergraduate level, this book systematically develops the basics of - Analysis on manifolds, Lie groups and G-manifolds (including equivariant dynamics) - Symplectic algebra and geometry, Hamiltonian systems, symmetries and reduction, - Integrable systems, Hamilton-Jacobi theory (including Morse families, the Maslov class and caustics). The first item is relevant for virtually all areas of mathematical physics, while the second item provides the basis of Hamiltonian mechanics. The last item introduces to important special areas. Necessary background knowledge on topology is provided in an appendix. The aim of this book is to enable the reader to access research monographs on more advanced topics. The style of this book is that of a mathematics textbook, with full proofs given in the text or as exercises. All material is illustrated by detailed examples, a number of which is taken up repeatedly for demonstrating how the methods evolve and interact.
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100Riemannian Geometry (Universitext)
This book covers the topics of differential manifolds, Riemannian metrics, connections, geodesics and curvature, with special emphasis on the intrinsic features of the subject. It treats in detail classical results on the relations between curvature and topology. The book features numerous exercises with full solutions and a series of detailed examples are picked up repeatedly to illustrate each new definition or property introduced.