University Press Scholarship Online. Sign in. Not registered? Sign up. Publications Pages Publications Pages. Recently viewed 0 Save Search. Users without a subscription are not able to see the full content. Authors Affiliations are at time of print publication. Your current browser may not support copying via this button. This book will supply a graduate student in mathematics or theoretical physics with the fundamentals of these objects.
Many of the tools used in differential topology are introduced and the basic results about differentiable manifolds, smooth maps, differential forms, vector fields, Lie groups, and Grassmanians are all presented here. Differential Geometry uses many of the classical examples from, and applications of, the subjects it covers, in particular those where closed form expressions are available, to bring abstract ideas to life.
Helpfully, proofs are offered for almost all assertions throughout. All of the introductory material is presented in full and this is the only such source with the classical examples presented in detail. An excellent reference for anyone needing to examine properties of harmonic vector fields to help them solve research problems. The book provides the main results of harmonic vector fields with an emphasis on Riemannian manifolds using past and existing problems to assist you in analyzing and furnishing your own conclusion for further research.
It emphasizes a combination of theoretical development with practical applications for a solid treatment of the subject useful to those new to research using differential geometric methods in extensive detail. A useful tool for any scientist conducting research in the field of harmonic analysis Provides applications and modern techniques to problem solving A clear and concise exposition of differential geometry of harmonic vector fields on Reimannian manifolds Physical Applications of Geometric Methods.
Given a mathematical structure, one of the basic associated mathematical objects is its automorphism group. The object of this book is to give a biased account of automorphism groups of differential geometric struc tures. Amongst the former, Riemannian and complex structures stand out for their beauty and wealth. A major portion of this book is therefore devoted to these two structures. Chapter I describes a general theory of automorphisms of geometric structures with emphasis on the question of when the automorphism group can be given a Lie group structure.
The concept of G-structure or that of pseudo-group structure enables us to treat most of the interesting geo metric structures in a unified manner. This chapter is partly based on lec tures I gave in Tokyo and Berkeley in Book Detail: Author : Loring W. 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.
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. ISBN: Size : Differential geometry began as the study of curves and surfaces using the methods of calculus. In time, the notions of curve and surface were generalized along with associated notions such as length, volume, and curvature.
At the same time the topic has become closely allied with developments in topology. The basic object is a smooth manifold, to which some extra structure has been attached, such as a Riemannian metric, a symplectic form, a distinguished group of symmetries, or a connection on the tangent bundle.
This book is a graduate-level introduction to the tools and structures of modern differential geometry. Included are the topics usually found in a course on differentiable manifolds, such as vector bundles, tensors, differential forms, de Rham cohomology, the Frobenius theorem and basic Lie group theory.
The book also contains material on the general theory of connections on vector bundles and an in-depth chapter on semi-Riemannian geometry that covers basic material about Riemannian manifolds and Lorentz manifolds. An unusual feature of the book is the inclusion of an early chapter on the differential geometry of hyper-surfaces in Euclidean space. There is also a section that derives the exterior calculus version of Maxwell's equations.
This book will supply a graduate student in mathematics or theoretical physics with the fundamentals of these objects. Introduction to vector bundles 4. Home Questions Tags Users Unanswered. Differential Clifofrd uses many of the classical examples from, and applications of, the subjects it covers, in particular those where closed form expressions are available, to bring abstract ideas to life.
Every geometer should read it, it will blow your mind, and it will change your life. Many of the tools used in differential topology are introduced Nonequilibrium Statistical Physics Noelle Pottier. Mathematics Stack Exchange works best with JavaScript enabled.
By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service. My library Help Advanced Book Search. Bundles, Connections, Metrics and Curvature. I would certainly second the suggestion of taking a look at any book by John Lee. An Introduction to Curvature is great for everyone!
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