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Differential Geometry


An Introduction to Manifolds (Universitext)

An Introduction to Manifolds (Universitext) Lowest new price: $33.63
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Author: Loring W. Tu
Brand: Tu Loring W

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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  • An Introduction to Manifolds

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Riemannian Geometry

Riemannian Geometry Lowest new price: $38.07
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Author: Manfredo Perdigao do Carmo
Brand: Manfredo P Do Carmo

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.

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  • Riemannian Geometry

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Riemannian Manifolds: An Introduction to Curvature (Graduate Texts in Mathematics)

Riemannian Manifolds: An Introduction to Curvature (Graduate Texts in Mathematics) Lowest new price: $37.71
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Author: John M. Lee

This text focuses on developing an intimate acquaintance with the geometric meaning of curvature and thereby introduces and demonstrates all the main technical tools needed for a more advanced course on Riemannian manifolds. It covers proving the four most fundamental theorems relating curvature and topology: the Gauss-Bonnet Theorem, the Cartan-Hadamard Theorem, Bonnet’s Theorem, and a special case of the Cartan-Ambrose-Hicks Theorem.

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Lectures on the Topology of 3-Manifolds: An Introduction to the Casson Invariant (De Gruyter Textbook)

Lectures on the Topology of 3-Manifolds: An Introduction to the Casson Invariant (De Gruyter Textbook) Lowest new price: $118.64
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Author: Nikolai Saveliev
Brand: Brand: De Gruyter

Progress in low-dimensional topology has been very fast over the last two decades, leading to the solutions of many difficult problems. One of the consequences of this "acceleration of history" is that many results have only appeared in professional journals and monographs. These are hardly accessible to students who have completed only a basic course in algebraic topology, or even to some researchers whose immediate area of expertise is not topology.

Among the highlights of this period are Casson’s results on the Rohlin invariant of homotopy 3-spheres, as well as his l-invariant. The purpose of this book is to provide a much-needed bridge to these modern topics. The book covers some classical topics, such as Heegaard splittings, Dehn surgery, and invariants of knots and links. It proceeds through the Kirby calculus and Rohlin’s theorem to Casson’s invariant and its applications, and gives a brief sketch of links with the latest developments in low-dimensional topology and gauge theory.

The book will be accessible to graduate students in mathematics and theoretical physics familiar with some elementary algebraic topology, including the fundamental group, basic homology theory, and Poncaré duality on manifolds.

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  • Used Book in Good Condition

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Analysis On Manifolds (Advanced Books Classics)

Analysis On Manifolds (Advanced Books Classics) Lowest new price: $61.30
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Author: James R. Munkres
Brand: James R Munkres

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.

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  • Analysis on Manifolds

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Differential Geometry: Connections, Curvature, and Characteristic Classes (Graduate Texts in Mathematics)

Differential Geometry: Connections, Curvature, and Characteristic Classes (Graduate Texts in Mathematics) Lowest new price: $54.50
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Author: Loring W. Tu

This text presents a graduate-level introduction to differential geometry for mathematics and physics students. The exposition follows the historical development of the concepts of connection and curvature with the goal of explaining the Chern–Weil theory of characteristic classes on a principal bundle. Along the way we encounter some of the high points in the history of differential geometry, for example, Gauss' Theorema Egregium and the Gauss–Bonnet theorem. Exercises throughout the book test the reader’s understanding of the material and sometimes illustrate extensions of the theory. Initially, the prerequisites for the reader include a passing familiarity with manifolds. After the first chapter, it becomes necessary to understand and manipulate differential forms. A knowledge of  de Rham cohomology is required for the last third of the text.

Prerequisite material is contained in author's text An Introduction to Manifolds, and can be learned in one semester. For the benefit of the reader and to establish common notations, Appendix A recalls the basics of manifold theory. Additionally, in an attempt to make the exposition more self-contained, sections on algebraic constructions such as the tensor product and the exterior power are included.

Differential geometry, as its name implies, is the study of geometry using differential calculus. It dates back to Newton and Leibniz in the seventeenth century, but it was not until the nineteenth century, with the work of Gauss on surfaces and Riemann on the curvature tensor, that differential geometry flourished and its modern foundation was laid. Over the past one hundred years, differential geometry has proven indispensable to an understanding of the physical world, in Einstein's general theory of relativity, in the theory of gravitation, in gauge theory, and now in string theory. Differential geometry is also useful in topology, several complex variables, algebraic geometry, complex manifolds, and dynamical systems, among other fields.  The field has even found applications to group theory as in Gromov's work and to probability theory as in Diaconis's work. It is not too far-fetched to argue that differential geometry should be in every mathematician's arsenal.

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Geometric Analysis of the Bergman Kernel and Metric (Graduate Texts in Mathematics)

Geometric Analysis of the Bergman Kernel and Metric (Graduate Texts in Mathematics) Lowest new price: $58.16
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Author: Steven G. Krantz
Brand: Steven G Krantz

This text provides a masterful and systematic treatment of all the basic analytic and geometric aspects of Bergman's classic theory of the kernel and its invariance properties. These include calculation, invariance properties, boundary asymptotics, and asymptotic expansion of the Bergman kernel and metric. Moreover, it presents a unique compendium of results with applications to function theory, geometry, partial differential equations, and interpretations in the language of functional analysis, with emphasis on the several complex variables context. Several of these topics appear here for the first time in book form. Each chapter includes illustrative examples and a collection of exercises which will be of interest to both graduate students and experienced mathematicians.

Graduate students who have taken courses in complex variables
and have a basic background in real and functional analysis will find this textbook appealing. Applicable courses for either main or supplementary usage include those in complex variables, several complex variables, complex differential geometry, and partial differential equations. Researchers in complex analysis, harmonic analysis, PDEs, and complex differential geometry will also benefit from the thorough treatment of the many exciting aspects of Bergman's theory.

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  • Geometric Analysis of the Bergman Kernel and Metric

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Michael Atiyah: Collected Works: Volume 4: Index Theory: 2 Volume 4: Index Theory: 2

Michael Atiyah: Collected Works: Volume 4: Index Theory: 2 Volume 4: Index Theory: 2 Lowest new price: $91.89
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Author: Michael Atiyah
Brand: Brand: Oxford University Press, USA

Professor Atiyah is one of the greatest living mathematicians and is well known throughout the mathematical world. He is a recipient of the Fields Medal, the mathematical equivalent of the Nobel Prize, and is still at the peak of his career. His huge number of published papers, focusing on the areas of algebraic geometry and topology, have here been collected into six volumes, divided thematically for easy reference by individuals interested in a particular subject. Volumes III and IV cover papers written in 1963-84 and are the result of a long collaboration with I. M. Singer on the Index Theory of elliptic operators.

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  • Used Book in Good Condition


Benjamin Harrison: The American Presidents Series: The 23rd President, 1889-1893

Benjamin Harrison: The American Presidents Series: The 23rd President, 1889-1893 Lowest new price: $22.14
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Author: Charles W. Calhoun
Brand: Calhoun, Charles W./ Schesliner, Arthur M., Jr. (EDT)

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.

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Differential Geometry of Curves and Surfaces (Undergraduate Texts in Mathematics)

Differential Geometry of Curves and Surfaces (Undergraduate Texts in Mathematics) Lowest new price: $49.60
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Author: Kristopher Tapp

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. Throughout this book, applications, metaphors and visualizations are tools that motivate and clarify the rigorous mathematical content, but never replace it. 

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