Tensor Geometry

The Geometric Viewpoint and its Uses
Author: Christopher T. J. Dodson,Timothy Poston
Publisher: Springer Science & Business Media
ISBN: 3642105149
Category: Mathematics
Page: 434
View: 5235

Continue Reading →

This treatment of differential geometry and the mathematics required for general relativity makes the subject accessible, for the first time, to anyone familiar with elementary calculus in one variable and with some knowledge of vector algebra. The emphasis throughout is on the geometry of the mathematics, which is greatly enhanced by the many illustrations presenting figures of three and more dimensions as closely as the book form will allow.

Tensor Geometry

The Geometric Viewpoint and Its Uses
Author: C. T. J. Dodson,T. Poston
Publisher: Springer
ISBN: N.A
Category: Mathematics
Page: 432
View: 2422

Continue Reading →

Foundations of Differentiable Manifolds and Lie Groups


Author: Frank W. Warner
Publisher: Springer Science & Business Media
ISBN: 1475717997
Category: Mathematics
Page: 276
View: 6437

Continue Reading →

Foundations of Differentiable Manifolds and Lie Groups gives a clear, detailed, and careful development of the basic facts on manifold theory and Lie Groups. Coverage includes differentiable manifolds, tensors and differentiable forms, Lie groups and homogenous spaces, and integration on manifolds. The book also provides a proof of the de Rham theorem via sheaf cohomology theory and develops the local theory of elliptic operators culminating in a proof of the Hodge theorem.

Manifolds, Tensors and Forms


Author: Paul Renteln
Publisher: Cambridge University Press
ISBN: 1107042194
Category: Science
Page: 340
View: 9085

Continue Reading →

Comprehensive treatment of the essentials of modern differential geometry and topology for graduate students in mathematics and the physical sciences.

Geometrical Methods of Mathematical Physics


Author: Bernard F. Schutz
Publisher: Cambridge University Press
ISBN: 1107268141
Category: Science
Page: N.A
View: 3692

Continue Reading →

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.

Introduction to Tensor Analysis and the Calculus of Moving Surfaces


Author: Pavel Grinfeld
Publisher: Springer Science & Business Media
ISBN: 1461478677
Category: Mathematics
Page: 302
View: 9894

Continue Reading →

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.

Differential Geometry and Topology

With a View to Dynamical Systems
Author: Keith Burns,Marian Gidea
Publisher: CRC Press
ISBN: 9781584882534
Category: Mathematics
Page: 400
View: 8329

Continue Reading →

Accessible, concise, and self-contained, this book offers an outstanding introduction to three related subjects: differential geometry, differential topology, and dynamical systems. Topics of special interest addressed in the book include Brouwer's fixed point theorem, Morse Theory, and the geodesic flow. Smooth manifolds, Riemannian metrics, affine connections, the curvature tensor, differential forms, and integration on manifolds provide the foundation for many applications in dynamical systems and mechanics. The authors also discuss the Gauss-Bonnet theorem and its implications in non-Euclidean geometry models. The differential topology aspect of the book centers on classical, transversality theory, Sard's theorem, intersection theory, and fixed-point theorems. The construction of the de Rham cohomology builds further arguments for the strong connection between the differential structure and the topological structure. It also furnishes some of the tools necessary for a complete understanding of the Morse theory. These discussions are followed by an introduction to the theory of hyperbolic systems, with emphasis on the quintessential role of the geodesic flow. The integration of geometric theory, topological theory, and concrete applications to dynamical systems set this book apart. With clean, clear prose and effective examples, the authors' intuitive approach creates a treatment that is comprehensible to relative beginners, yet rigorous enough for those with more background and experience in the field.

Differential Geometric Structures


Author: Walter A. Poor
Publisher: Courier Corporation
ISBN: 0486151913
Category: Mathematics
Page: 352
View: 371

Continue Reading →

This introductory text defines geometric structure by specifying parallel transport in an appropriate fiber bundle and focusing on simplest cases of linear parallel transport in a vector bundle. 1981 edition.

Séminaire de Probabilités XLII


Author: Catherine Donati-Martin,Michel Émery,Alain Rouault,Christophe Stricker
Publisher: Springer
ISBN: 3642017630
Category: Mathematics
Page: 449
View: 9373

Continue Reading →

This book offers an introduction to rough paths. Coverage also includes the interface between analysis and probability to special processes, Lévy processes and Lévy systems, representation of Gaussian processes, filtrations and quantum probability.

Complexity Dichotomies for Counting Problems: Volume 1, Boolean Domain


Author: Jin-Yi Cai,Xi Chen
Publisher: Cambridge University Press
ISBN: 1108508820
Category: Computers
Page: N.A
View: 5966

Continue Reading →

Complexity theory aims to understand and classify computational problems, especially decision problems, according to their inherent complexity. This book uses new techniques to expand the theory for use with counting problems. The authors present dichotomy classifications for broad classes of counting problems in the realm of P and NP. Classifications are proved for partition functions of spin systems, graph homomorphisms, constraint satisfaction problems, and Holant problems. The book assumes minimal prior knowledge of computational complexity theory, developing proof techniques as needed and gradually increasing the generality and abstraction of the theory. This volume presents the theory on the Boolean domain, and includes a thorough presentation of holographic algorithms, culminating in classifications of computational problems studied in exactly solvable models from statistical mechanics.

Clifford (Geometric) Algebras

with applications to physics, mathematics, and engineering
Author: William Baylis
Publisher: Springer Science & Business Media
ISBN: 1461241049
Category: Science
Page: 517
View: 7897

Continue Reading →

This volume is an outgrowth of the 1995 Summer School on Theoretical Physics of the Canadian Association of Physicists (CAP), held in Banff, Alberta, in the Canadian Rockies, from July 30 to August 12,1995. The chapters, based on lectures given at the School, are designed to be tutorial in nature, and many include exercises to assist the learning process. Most lecturers gave three or four fifty-minute lectures aimed at relative novices in the field. More emphasis is therefore placed on pedagogy and establishing comprehension than on erudition and superior scholarship. Of course, new and exciting results are presented in applications of Clifford algebras, but in a coherent and user-friendly way to the nonspecialist. The subject area of the volume is Clifford algebra and its applications. Through the geometric language of the Clifford-algebra approach, many concepts in physics are clarified, united, and extended in new and sometimes surprising directions. In particular, the approach eliminates the formal gaps that traditionally separate clas sical, quantum, and relativistic physics. It thereby makes the study of physics more efficient and the research more penetrating, and it suggests resolutions to a major physics problem of the twentieth century, namely how to unite quantum theory and gravity. The term "geometric algebra" was used by Clifford himself, and David Hestenes has suggested its use in order to emphasize its wide applicability, and b& cause the developments by Clifford were themselves based heavily on previous work by Grassmann, Hamilton, Rodrigues, Gauss, and others.

A New Approach to Differential Geometry using Clifford's Geometric Algebra


Author: John Snygg
Publisher: Springer Science & Business Media
ISBN: 081768283X
Category: Mathematics
Page: 465
View: 8669

Continue Reading →

Differential geometry is the study of the curvature and calculus of curves and surfaces. A New Approach to Differential Geometry using Clifford's Geometric Algebra simplifies the discussion to an accessible level of differential geometry by introducing Clifford algebra. This presentation is relevant because Clifford algebra is an effective tool for dealing with the rotations intrinsic to the study of curved space. Complete with chapter-by-chapter exercises, an overview of general relativity, and brief biographies of historical figures, this comprehensive textbook presents a valuable introduction to differential geometry. It will serve as a useful resource for upper-level undergraduates, beginning-level graduate students, and researchers in the algebra and physics communities.

Geometry of Manifolds


Author: Richard L. Bishop,Richard J. Crittenden
Publisher: American Mathematical Soc.
ISBN: 0821829238
Category: Mathematics
Page: 273
View: 8903

Continue Reading →

First published in 1964, this book served as a text on differential geometry to several generations of graduate students all over the world. The first half of the book (Chapters 1-6) presents basics of the theory of manifolds, vector bundles, differential forms, and Lie groups, with a special emphasis on the theory of linear and affine connections. The second half of the book (Chapters 7-11) is devoted to Riemannian geometry. Following the definition and main properties of Riemannian manifolds, the authors discuss the theory of geodesics, complete Riemannian manifolds, and curvature. Next, they introduce the theory of immersion of manifolds and the second fundamental form. The concluding Chapter 11 contains more complicated results on which much of the research in Riemannian geometry is based: the Morse index theorem, Synge's theorem on closed geodesics, Rauch's comparision theorem, and Bishop's volume-comparision theorem. Clear, concise writing as well as many exercises and examples make this classic an excellent text for a first-year graduate course on differential geometry.

Tensors

Geometry and Applications
Author: J. M. Landsberg
Publisher: American Mathematical Soc.
ISBN: 0821869078
Category: Mathematics
Page: 439
View: 9399

Continue Reading →

Tensors are ubiquitous in the sciences. The geometry of tensors is both a powerful tool for extracting information from data sets, and a beautiful subject in its own right. This book has three intended uses: a classroom textbook, a reference work for researchers in the sciences, and an account of classical and modern results in (aspects of) the theory that will be of interest to researchers in geometry. For classroom use, there is a modern introduction to multilinear algebra and to the geometry and representation theory needed to study tensors, including a large number of exercises. For researchers in the sciences, there is information on tensors in table format for easy reference and a summary of the state of the art in elementary language. This is the first book containing many classical results regarding tensors. Particular applications treated in the book include the complexity of matrix multiplication, P versus NP, signal processing, phylogenetics, and algebraic statistics. For geometers, there is material on secant varieties, G-varieties, spaces with finitely many orbits and how these objects arise in applications, discussions of numerous open questions in geometry arising in applications, and expositions of advanced topics such as the proof of the Alexander-Hirschowitz theorem and of the Weyman-Kempf method for computing syzygies.

Differential Geometry

Connections, Curvature, and Characteristic Classes
Author: Loring W. Tu
Publisher: Springer
ISBN: 3319550845
Category: Mathematics
Page: 347
View: 3118

Continue Reading →

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.

Geometry: Euclid and Beyond


Author: Robin Hartshorne
Publisher: Springer Science & Business Media
ISBN: 0387226761
Category: Mathematics
Page: 528
View: 1045

Continue Reading →

This book offers a unique opportunity to understand the essence of one of the great thinkers of western civilization. A guided reading of Euclid's Elements leads to a critical discussion and rigorous modern treatment of Euclid's geometry and its more recent descendants, with complete proofs. Topics include the introduction of coordinates, the theory of area, history of the parallel postulate, the various non-Euclidean geometries, and the regular and semi-regular polyhedra.

Geometric Control of Mechanical Systems

Modeling, Analysis, and Design for Simple Mechanical Control Systems
Author: Francesco Bullo,Andrew D. Lewis
Publisher: Springer Science & Business Media
ISBN: 9780387221953
Category: Science
Page: 727
View: 2581

Continue Reading →

The area of analysis and control of mechanical systems using differential geometry is flourishing. This book collects many results over the last decade and provides a comprehensive introduction to the area.

Quantum Groups


Author: Christian Kassel
Publisher: Springer Science & Business Media
ISBN: 1461207835
Category: Mathematics
Page: 534
View: 4492

Continue Reading →

Here is an introduction to the theory of quantum groups with emphasis on the spectacular connections with knot theory and Drinfeld's recent fundamental contributions. It presents the quantum groups attached to SL2 as well as the basic concepts of the theory of Hopf algebras. Coverage also focuses on Hopf algebras that produce solutions of the Yang-Baxter equation and provides an account of Drinfeld's elegant treatment of the monodromy of the Knizhnik-Zamolodchikov equations.

Geometric Realizations of Curvature


Author: Miguel Brozos Vázquez,Peter B Gilkey,Stana Nikcevic
Publisher: World Scientific
ISBN: 1908977744
Category: Mathematics
Page: 264
View: 5385

Continue Reading →

A central area of study in Differential Geometry is the examination of the relationship between the purely algebraic properties of the Riemann curvature tensor and the underlying geometric properties of the manifold. In this book, the findings of numerous investigations in this field of study are reviewed and presented in a clear, coherent form, including the latest developments and proofs. Even though many authors have worked in this area in recent years, many fundamental questions still remain unanswered. Many studies begin by first working purely algebraically and then later progressing onto the geometric setting and it has been found that many questions in differential geometry can be phrased as problems involving the geometric realization of curvature. Curvature decompositions are central to all investigations in this area. The authors present numerous results including the Singer–Thorpe decomposition, the Bokan decomposition, the Nikcevic decomposition, the Tricerri–Vanhecke decomposition, the Gray–Hervella decomposition and the De Smedt decomposition. They then proceed to draw appropriate geometric conclusions from these decompositions. The book organizes, in one coherent volume, the results of research completed by many different investigators over the past 30 years. Complete proofs are given of results that are often only outlined in the original publications. Whereas the original results are usually in the positive definite (Riemannian setting), here the authors extend the results to the pseudo-Riemannian setting and then further, in a complex framework, to para-Hermitian geometry as well. In addition to that, new results are obtained as well, making this an ideal text for anyone wishing to further their knowledge of the science of curvature. Contents:Introduction and Statement of ResultsRepresentation TheoryConnections, Curvature, and Differential GeometryReal Affine GeometryAffine Kähler GeometryRiemannian GeometryComplex Riemannian Geometry Readership: Graduate students, researchers, mathematicians and physicist interested in the study of curvature. Keywords:Affine Geometry;Riemannian Geometry;Pseudo-Riemannian Geometry;Kähler Geometry;Para-Kähler Geometry;Hermitian Geometry;Para-Hermitian Geometry;Hyper-Hermitian Geometry;Curvature Tensor;Weyl Geometry;Curvature Decompositions;Almost Complex GeometryKey Features:There are no competing titlesComplete proofs are given that are often only sketched in other literature. Curvature decompositions are presented in parallel for many different structure groups. Full computations of spaces of quadratic invariants appear - this is central to the subject and missing in previously published literature. New results in the pseudo-Riemannian, pseudo-Hermitian and para-Hermitian contexts are included. Geometric realization results are clearly organized and discussedThe relevant “background material” concerning differential geometry and representation theory is introduced, developed and presented in detail, therefore the book is self-contained

Geometry from a Differentiable Viewpoint


Author: John McCleary
Publisher: Cambridge University Press
ISBN: 0521116074
Category: Mathematics
Page: 357
View: 8609

Continue Reading →

A thoroughly revised second edition of a textbook for a first course in differential/modern geometry that introduces methods within a historical context.