On the Hypotheses Which Lie at the Bases of Geometry

2016-04-19
On the Hypotheses Which Lie at the Bases of Geometry

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  • Author: Bernhard Riemann
  • Publisher: Birkhäuser
  • ISBN: 3319260421
  • Category : Mathematics
  • Languages : en
  • Pages : 181

This book presents William Clifford’s English translation of Bernhard Riemann’s classic text together with detailed mathematical, historical and philosophical commentary. The basic concepts and ideas, as well as their mathematical background, are provided, putting Riemann’s reasoning into the more general and systematic perspective achieved by later mathematicians and physicists (including Helmholtz, Ricci, Weyl, and Einstein) on the basis of his seminal ideas. Following a historical introduction that positions Riemann’s work in the context of his times, the history of the concept of space in philosophy, physics and mathematics is systematically presented. A subsequent chapter on the reception and influence of the text accompanies the reader from Riemann’s times to contemporary research. Not only mathematicians and historians of the mathematical sciences, but also readers from other disciplines or those with an interest in physics or philosophy will find this work both appealing and insightful.

An Introduction to Riemannian Geometry

2014-07-26
An Introduction to Riemannian Geometry

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  • Author: Leonor Godinho
  • Publisher: Springer
  • ISBN: 3319086669
  • Category : Mathematics
  • Languages : en
  • Pages : 476

Unlike many other texts on differential geometry, this textbook also offers interesting applications to geometric mechanics and general relativity. The first part is a concise and self-contained introduction to the basics of manifolds, differential forms, metrics and curvature. The second part studies applications to mechanics and relativity including the proofs of the Hawking and Penrose singularity theorems. It can be independently used for one-semester courses in either of these subjects. The main ideas are illustrated and further developed by numerous examples and over 300 exercises. Detailed solutions are provided for many of these exercises, making An Introduction to Riemannian Geometry ideal for self-study.

A Panoramic View of Riemannian Geometry

2012-12-06
A Panoramic View of Riemannian Geometry

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  • Author: Marcel Berger
  • Publisher: Springer Science & Business Media
  • ISBN: 3642182453
  • Category : Mathematics
  • Languages : en
  • Pages : 835

This book introduces readers to the living topics of Riemannian Geometry and details the main results known to date. The results are stated without detailed proofs but the main ideas involved are described, affording the reader a sweeping panoramic view of almost the entirety of the field. From the reviews "The book has intrinsic value for a student as well as for an experienced geometer. Additionally, it is really a compendium in Riemannian Geometry." --MATHEMATICAL REVIEWS

Riemannian Geometry

2009-06-22
Riemannian Geometry

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  • Author: Frank Morgan
  • Publisher: A K Peters/CRC Press
  • ISBN: 9781568814711
  • Category : Mathematics
  • Languages : en
  • Pages : 0

This classic text serves as a tool for self-study; it is also used as a basic text for undergraduate courses in differential geometry. The author's ability to extract the essential elements of the theory in a lucid and concise fashion allows the student easy access to the material and enables the instructor to add emphasis and cover special topics. The extraordinary wealth of examples within the exercises and the new material, ranging from isoperimetric problems to comments on Einstein's original paper on relativity theory, enhance this new edition.

Riemannian Geometry

2013-06-29
Riemannian Geometry

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  • Author: Peter Petersen
  • Publisher: Springer Science & Business Media
  • ISBN: 1475764340
  • Category : Mathematics
  • Languages : en
  • Pages : 443

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 specialise in Riemannian geometry. Instead of variational techniques, the author uses a unique approach, emphasising distance functions and special co-ordinate systems. He also uses standard calculus with some techniques from differential equations to provide a more elementary route. Many chapters contain material typically found in specialised texts, never before published in a single source. 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 - including sections on convergence and compactness of families of manifolds. Thus, 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.

Riemannian Geometry of Contact and Symplectic Manifolds

2013-11-11
Riemannian Geometry of Contact and Symplectic Manifolds

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  • Author: David E. Blair
  • Publisher: Springer Science & Business Media
  • ISBN: 1475736045
  • Category : Mathematics
  • Languages : en
  • Pages : 263

Book endorsed by the Sunyer Prize Committee (A. Weinstein, J. Oesterle et. al.).

Riemannian Geometry

1996-01-01
Riemannian Geometry

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  • Author: Takashi Sakai
  • Publisher: American Mathematical Soc.
  • ISBN: 9780821889565
  • Category : Mathematics
  • Languages : en
  • Pages : 378

This volume is an English translation of Sakai's textbook on Riemannian Geometry which was originally written in Japanese and published in 1992. The author's intent behind the original book was to provide to advanced undergraduate and graudate students an introduction to modern Riemannian geometry that could also serve as a reference. The book begins with an explanation of the fundamental notion of Riemannian geometry. Special emphasis is placed on understandability and readability, to guide students who are new to this area. The remaining chapters deal with various topics in Riemannian geometry, with the main focus on comparison methods and their applications.

Quantum Riemannian Geometry

2020-01-31
Quantum Riemannian Geometry

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  • Author: Edwin J. Beggs
  • Publisher: Springer Nature
  • ISBN: 3030302946
  • Category : Science
  • Languages : en
  • Pages : 826

This book provides a comprehensive account of a modern generalisation of differential geometry in which coordinates need not commute. This requires a reinvention of differential geometry that refers only to the coordinate algebra, now possibly noncommutative, rather than to actual points. Such a theory is needed for the geometry of Hopf algebras or quantum groups, which provide key examples, as well as in physics to model quantum gravity effects in the form of quantum spacetime. The mathematical formalism can be applied to any algebra and includes graph geometry and a Lie theory of finite groups. Even the algebra of 2 x 2 matrices turns out to admit a rich moduli of quantum Riemannian geometries. The approach taken is a `bottom up’ one in which the different layers of geometry are built up in succession, starting from differential forms and proceeding up to the notion of a quantum `Levi-Civita’ bimodule connection, geometric Laplacians and, in some cases, Dirac operators. The book also covers elements of Connes’ approach to the subject coming from cyclic cohomology and spectral triples. Other topics include various other cohomology theories, holomorphic structures and noncommutative D-modules. A unique feature of the book is its constructive approach and its wealth of examples drawn from a large body of literature in mathematical physics, now put on a firm algebraic footing. Including exercises with solutions, it can be used as a textbook for advanced courses as well as a reference for researchers.

Some Nonlinear Problems in Riemannian Geometry

2013-03-09
Some Nonlinear Problems in Riemannian Geometry

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  • Author: Thierry Aubin
  • Publisher: Springer Science & Business Media
  • ISBN: 3662130068
  • Category : Mathematics
  • Languages : en
  • Pages : 414

This book deals with such important subjects as variational methods, the continuity method, parabolic equations on fiber bundles, ideas concerning points of concentration, blowing-up technique, geometric and topological methods. It explores important geometric problems that are of interest to many mathematicians and scientists but have only recently been partially solved.

Riemannian Manifolds

2006-04-06
Riemannian Manifolds

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  • Author: John M. Lee
  • Publisher: Springer Science & Business Media
  • ISBN: 0387227261
  • Category : Mathematics
  • Languages : en
  • Pages : 232

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.