This book provides a systematic presentation of the mathematical foundation of modern physics with applications particularly within classical mechanics and the theory of relativity. Written to be self-contained, this book provides complete and rigorous proofs of all the results presented within. Among the themes illustrated in the book are differentiable manifolds, differential forms, fiber bundles and differential geometry with non-trivial applications especially within the general theory of relativity. The emphasis is upon a systematic and logical construction of the mathematical foundations. It can be used as a textbook for a pure mathematics course in differential geometry, assuming the reader has a good understanding of basic analysis, linear algebra and point set topology. The book will also appeal to students of theoretical physics interested in the mathematical foundation of the theories.

In this book, I explored differential equations for operation in Lie group and for representations of group Lie in a vector space.

I tell about different mathematical tool that is important in general relativity. The text of the book includes definition of geometric object, concept of reference frame, geometry of metric\hyph affinne manifold. Using this concept I learn dynamics in general relativity. We call a manifold with torsion and nonmetricity the metric\hyph affine manifold. The nonmetricity leads to a difference between the auto parallel line and the extreme line, and to a change in the expression of the Frenet transport. The torsion leads to a change in the Killing equation. We also need to add a similar equation for the connection. The dynamics of a particle follows to the Frenet transport. The analysis of the Frenet transport leads to the concept of the Cartan connection which is compatible with the metric tensor. We need additional physical constraints to make a nonmetricity observable.

This classic monograph by a mathematician affiliated with Trinity College, Cambridge, offers a brief account of the invariant theory connected with a single quadratic differential form. Suitable for advanced undergraduates and graduate students of mathematics, it avoids unnecessary analysis and offers an accessible view of the field for readers unfamiliar with the subject.
A historical overview is followed by considerations of the methods of Christoffel and Lie as well as Maschke's symbolic method and explorations of geometrical and dynamical methods. The final chapter on applications, which draws upon developments by Ricci and Levi-Civita, presents the most successful method and can be read independently of the rest of the book.

Quantum mechanics had been started with the theory of the hydrogen atom, so when considering the quantum mechanics in Riemannian spaces it is natural to turn first to just this simplest system. A common quantum-mechanical hydrogen atom description is based materially on the assumption of the Euclidean character of the physical 3-space geometry. In this context, natural questions arise: what in the description is determined by this special assumption, and which changes will be entailed by allowing for other spatial geometries. The questions are of fundamental significance, even beyond their possible experimental testing. In the present book, detailed analytical treatment and exact solutions are given to a number of problems of quantum mechanics and field theory in simplest non-Euclidean spacetime models. The main attention is focused on new themes created by non-vanishing curvature in classical physical topics and concepts.

The book deals with some questions related to the boundary problem in complex geometry and CR geometry. After a brief introduction summarizing the main results on the extension of CR functions, it is shown in chapters 2 and 3 that, employing the classical Harvey-Lawson theorem and under suitable conditions, the boundary problem for non-compact maximally complex real submanifolds of Cn, n=3 is solvable.

In chapter 4, the regularity of Levi flat hypersurfaces Cn (n=3) with assigned boundaries is studied in the graph case, in relation to the existence theorem proved by Dolbeault, Tomassini and Zaitsev.

Finally, in the last two chapters the structure properties of non-compact Levi-flat submanifolds of Cn are discussed; in particular, using the theory of the analytic multifunctions, a Liouville theorem for Levi flat submanifolds of Cn is proved.

Information geometry provides the mathematical sciences with a new framework of analysis. It has emerged from the investigation of the natural differential geometric structure on manifolds of probability distributions, which consists of a Riemannian metric defined by the Fisher information and a one-parameter family of affine connections called the $\alpha$-connections. The duality between the $\alpha$-connection and the $(-\alpha)$-connection together with the metric play an essential role in this geometry. This kind of duality, having emerged from manifolds of probability distributions, is ubiquitous, appearing in a variety of problems which might have no explicit relation to probability theory. Through the duality, it is possible to analyze various fundamental problems in a unified perspective. The first half of this book is devoted to a comprehensive introduction to the mathematical foundation of information geometry, including preliminaries from differential geometry, the geometry of manifolds or probability distributions, and the general theory of dual affine connections.The second half of the text provides an overview of many areas of applications, such as statistics, linear systems, information theory, quantum mechanics, convex analysis, neural networks, and affine differential geometry. The book can serve as a suitable text for a topics course for advanced undergraduates and graduate students.

The Bernstein problem and the Plateau problem are central topics in the theory of minimal submanifolds. This important book presents the Douglas-Rado solution to the Plateau problem, but the main emphasis is on the Bernstein problem and its new developments in various directions: the value distribution of the Gauss image of a minimal surface in Euclidean 3-space, Simons' work for minimal graphic hypersurfaces, and author's own contributions to Bernstein type theorems for higher codimensions. The author also introduces some related topics, such as submanifolds with parallel mean curvature, Weierstrass type representation for surfaces of mean curvature 1 in hyperbolic 3-space, and special Lagrangian submanifolds.

This book is an introduction to differential geometry through differential forms, emphasizing their applications in various areas of mathematics and physics. Well-written and with plenty of examples, this textbook originated from courses on geometry and analysis and presents a widely-used mathematical technique in a lucid and very readable style. The authors introduce readers to the world of differential forms while covering relevant topics from analysis, differential geometry, and mathematical physics. The book begins with a self-contained introduction to the calculus of differential forms in Euclidean space and on manifolds. Next, the focus is on Stokes' theorem, the classical integral formulas and their applications to harmonic functions and topology. The authors then discuss the integrability conditions of a Pfaffian system (Frobenius' theorem). Chapter 5 is a thorough exposition of the theory of curves and surfaces in Euclidean space in the spirit of Cartan.The following chapter covers Lie groups and homogeneous spaces. Chapter 7 addresses symplectic geometry and classical mechanics. The basic tools for the integration of the Hamiltonian equations are the moment map and completely integrable systems (Liouville-Arnold Theorem). The authors discuss Newton, Lagrange, and Hamilton formulations of mechanics. Chapter 8 contains an introduction to statistical mechanics and thermodynamics. The final chapter deals with electrodynamics. The material in the book is carefully illustrated with figures and examples, and there are over 100 exercises. Readers should be familiar with first-year algebra and advanced calculus. The book is intended for graduate students and researchers interested in delving into geometric analysis and its applications to mathematical physics.

This book arose out of courses taught by the author. It covers the traditional topics of differential manifolds, tensor fields, Lie groups, integration on manifolds and basic differential and Riemannian geometry. The author emphasizes geometric concepts, giving the reader a working knowledge of the topic. Motivations are given, exercises are included, and illuminating nontrivial examples are discussed. Important features include the following: Geometric and conceptual treatment of differential calculus with a wealth of nontrivial examples. A thorough discussion of the much-used result on the existence, uniqueness, and smooth dependence of solutions of ODEs. Careful introduction of the concept of tangent spaces to a manifold. Early and simultaneous treatment of Lie groups and related concepts. A motivated and highly geometric proof of the Frobenius theorem. A constant reconciliation with the classical treatment and the modern approach. Simple proofs of the hairy-ball theorem and Brouwer's fixed point theorem. Construction of manifolds of constant curvature a la Chern. This text would be suitable for use as a graduate-level introduction to basic differential and Riemannian geometry.

Dieses Buch ist eine Einfuhrung in die Differentialgeometrie und ein passender Begleiter zum Differentialgeometrie-Modul (ein- und zweisemestrig). Zunachst geht es um die klassischen Aspekte wie die Geometrie von Kurven und Flachen, bevor dann hoherdimensionale Flachen sowie abstrakte Mannigfaltigkeiten betrachtet werden. Die Nahtstelle ist dabei das zentrale Kapitel "Die innere Geometrie von Flachen." Dieses fuhrt den Leser bis hin zu dem beruhmten Satz von Gauss-Bonnet, der ein entscheidendes Bindeglied zwischen lokaler und globaler Geometrie darstellt. Die zweite Halfte des Buches ist der Riemannschen Geometrie gewidmet. Den Abschluss bildet ein Kapitel uber "Einstein-Raume," die eine grosse Bedeutung sowohl in der "Reinen Mathematik" als auch in der Allgemeinen Relativitatstheorie von A. Einstein haben. Es wird grosser Wert auf Anschaulichkeit gelegt, was durch zahlreiche Abbildungen unterstutzt wird. Bei der Neuauflage wurden einige zusatzliche Losungen zu denUbungsaufgaben erganzt."

This book addresses a basic question in differential geometry that was first considered by physicists Stanley Deser and Adam Schwimmer in 1993 in their study of conformal anomalies. The question concerns conformally invariant functionals on the space of Riemannian metrics over a given manifold. These functionals act on a metric by first constructing a Riemannian scalar out of it, and then integrating this scalar over the manifold. Suppose this integral remains invariant under conformal re-scalings of the underlying metric. What information can one then deduce about the Riemannian scalar? Deser and Schwimmer asserted that the Riemannian scalar must be a linear combination of three obvious candidates, each of which clearly satisfies the required property: a local conformal invariant, a divergence of a Riemannian vector field, and the Chern-Gauss-Bonnet integrand. This book provides a proof of this conjecture.

The result itself sheds light on the algebraic structure of conformal anomalies, which appear in many settings in theoretical physics. It also clarifies the geometric significance of the renormalized volume of asymptotically hyperbolic Einstein manifolds. The methods introduced here make an interesting connection between algebraic properties of local invariants--such as the classical Riemannian invariants and the more recently studied conformal invariants--and the study of global invariants, in this case conformally invariant integrals. Key tools used to establish this connection include the Fefferman-Graham ambient metric and the author's super divergence formula.

These proceedings of the October 2005 conference held in Hanoi include survey and research articles reflecting current interest in a range of topics, including a survey on Zariski pairs and another on a categorical construction of Lie algebras, and research on elliptical parameters and defining equations for elliptic fibrations on a Kummer surface, characterization of the rational homogeneous space association with a long simple root by its variety of minimal rational tangents, maximal divisorial sets in are spaces, two non-conjugate embeddings into Cremora Group II, the Castelnuovo-Severi inequality for a double covering, a Poincare polynomial of a class of signed complete graphic arrangements, singularities of dual varieties in Characteristic 2, and Castelnuovo-Well lattices. All papers are original and have not been published elsewhere. Distributed in the US by the American Mathematical Society.

The volume contains surveys and original articles based on the talks given at the 40-th Finsler Symposium on Finsler Geometry held in the period September 9-10, 2005 at Hokkaido Tokai University, Sapporo, Japan. The Symposium's purpose was not only a meeting of the Finsler geometers from Japan and abroad, but also to commemorate the memory of the late Professor Makoto Matsumoto. The papers included in this volume contain fundamental topics of modern Riemann-Finsler geometry, interesting not only for specialists in Finsler geometry, but for researchers in Riemannian geometry or other fields of differential geometry and its applications also.Published by Mathematical Society of Japan and distributed by World Scientific Publishing Co. for all markets except North America

This volume constitutes the proceedings of the third Franco-Japanese symposium on singularities, held in Sapporo in September 2004. It contains not only research papers on the most advanced topics in the field, but also some survey articles which give broad scopes in some areas of the subject. All the articles are carefully refereed for correctness and readability.Published by Mathematical Society of Japan and distributed by World Scientific Publishing Co. for all markets except North America

This graduate-level monographic textbook treats applied differential geometry from a modern scientific perspective. Co-authored by the originator of the world's leading human motion simulator -- "Human Biodynamics Engine," a complex, 264-DOF bio-mechanical system, modeled by differential-geometric tools -- this is the first book that combines modern differential geometry with a wide spectrum of applications, from modern mechanics and physics, via nonlinear control, to biology and human sciences. The book is designed for a two-semester course, which gives mathematicians a variety of applications for their theory and physicists, as well as other scientists and engineers, a strong theory underlying their models.

Pseudo-Riemannian geometry is an active research field not only in differential geometry but also in mathematical physics where the higher signature geometries play a role in brane theory. An essential reference tool for research mathematicians and physicists, this book also serves as a useful introduction to students entering this active and rapidly growing field. The author presents a comprehensive treatment of several aspects of pseudo-Riemannian geometry, including the spectral geometry of the curvature tensor, curvature homogeneity, and Stanilov-Tsankov-Videv theory.

This book provides the first-ever systematic introduction to the theory of Riemannian submersions, which was initiated by Barrett O'Neill and Alfred Gray less than four decades ago. The authors focus their attention on classification theorems when the total space and the fibres have nice geometric properties. Particular emphasis is placed on the interrelation with almost Hermitian, almost contact and quaternionic geometry. Examples clarifying and motivating the theory are included in every chapter. Recent results on semi-Riemannian submersions are also explained. Finally, the authors point out the close connection of the subject with some areas of physics.

This volume contains the courses and lectures given during the workshop on Differential Geometry and Topology held at Alghero, Italy, in June 1992.The main goal of this meeting was to offer an introduction in attractive areas of current research and to discuss some recent important achievements in both the fields. This is reflected in the present book which contains some introductory texts together with more specialized contributions.The topics covered in this volume include circle and sphere packings, 3-manifolds invariants and combinatorial presentations of manifolds, soliton theory and its applications in differential geometry, G-manifolds of low cohomogeneity, exotic differentiable structures on R4, conformal deformation of Riemannian manifolds and Riemannian geometry of algebraic manifolds.

This book takes a historical approach to Einstein's General Theory of Relativity and shows the importance that geometry has to the theory. Starting from simpler and more general considerations, it goes on to detail the latest developments in the field and considers several cutting-edge research areas. It discusses Einstein's theory from a geometrical and a field theoretic viewpoint, before moving on to address gravitational waves, black holes and cosmology.