Geometric Mechanics on Riemannian Manifolds
Geometric Mechanics on Riemannian Manifolds

Author: Ovidiu Calin

Publisher: Springer Science & Business Media

Published: 2006-03-15

Total Pages: 285

ISBN-13: 0817644210

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* A geometric approach to problems in physics, many of which cannot be solved by any other methods * Text is enriched with good examples and exercises at the end of every chapter * Fine for a course or seminar directed at grad and adv. undergrad students interested in elliptic and hyperbolic differential equations, differential geometry, calculus of variations, quantum mechanics, and physics

Geometric Mechanics
Geometric Mechanics

Author: Waldyr Muniz Oliva

Publisher: Springer

Published: 2004-10-23

Total Pages: 277

ISBN-13: 354045795X

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Geometric Mechanics here means mechanics on a pseudo-riemannian manifold and the main goal is the study of some mechanical models and concepts, with emphasis on the intrinsic and geometric aspects arising in classical problems. The first seven chapters are written in the spirit of Newtonian Mechanics while the last two ones as well as two of the four appendices describe the foundations and some aspects of Special and General Relativity. All the material has a coordinate free presentation but, for the sake of motivation, many examples and exercises are included in order to exhibit the desirable flavor of physical applications.

An Introduction to Riemannian Geometry
An Introduction to Riemannian Geometry

Author: Leonor Godinho

Publisher: Springer

Published: 2014-07-26

Total Pages: 476

ISBN-13: 3319086669

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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.

Sub-Riemannian Geometry
Sub-Riemannian Geometry

Author: Ovidiu Calin

Publisher: Cambridge University Press

Published: 2009-04-20

Total Pages: 371

ISBN-13: 0521897300

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A comprehensive text and reference on sub-Riemannian and Heisenberg manifolds using a novel and robust variational approach.

On the Hypotheses Which Lie at the Bases of Geometry
On the Hypotheses Which Lie at the Bases of Geometry

Author: Bernhard Riemann

Publisher: Birkhäuser

Published: 2016-04-19

Total Pages: 181

ISBN-13: 3319260421

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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.

Geometric Dynamics
Geometric Dynamics

Author: Constantin Udriște

Publisher: Springer Science & Business Media

Published: 2000

Total Pages: 416

ISBN-13: 9780792364016

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The theme of this text is the philosophy that any particle flow generates a particle dynamics, in a suitable geometrical framework. It covers topics that include: geometrical and physical vector fields; field lines; flows; stability of equilibrium points; potential systems and catastrophe geometry; field hypersurfaces; bifurcations; distribution orthogonal to a vector field; extrema with nonholonomic constraints; thermodynamic systems; energies; geometric dynamics induced by a vector field; magnetic fields around piecewise rectilinear electric circuits; geometric magnetic dynamics; and granular materials and their mechanical behaviour. The text should be useful for first-year graduate students in mathematics, mechanics, physics, engineering, biology, chemistry, and economics. It can also be addressed to professors and researchers whose work involves mathematics, mechanics, physics, engineering, biology, chemistry, and economics.

Introduction to Riemannian Manifolds
Introduction to Riemannian Manifolds

Author: John M. Lee

Publisher: Springer

Published: 2019-01-02

Total Pages: 437

ISBN-13: 3319917552

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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.

An Introduction to the Analysis of Paths on a Riemannian Manifold
An Introduction to the Analysis of Paths on a Riemannian Manifold

Author: Daniel W. Stroock

Publisher: American Mathematical Soc.

Published: 2000

Total Pages: 290

ISBN-13: 0821838393

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Hoping to make the text more accessible to readers not schooled in the probabalistic tradition, Stroock (affiliation unspecified) emphasizes the geometric over the stochastic analysis of differential manifolds. Chapters deconstruct Brownian paths, diffusions in Euclidean space, intrinsic and extrinsic Riemannian geometry, Bocher's identity, and the bundle of orthonormal frames. The volume humbly concludes with an "admission of defeat" in regard to recovering the Li-Yau basic differential inequality. Annotation copyrighted by Book News, Inc., Portland, OR.