Chapter 1: Foundations.
The concept of a manifold, the intrinsic idea of tangent vector fields and differential 1-forms, and how these provide a framework for differential calculus with many applications for example in General Relativity. The notion of a Riemannian metric, and how it generalises the first fundamental form of surfaces in Euclidean space.
Connections or covariant derivatives , parallel transport and curvature, and how to apply them in Riemannian geometry. Riemannian metrics and the Levi-Civita connection: the fundamental theorem of Riemannian geometry. Current Department policy on feedback is available in the student handbook. Published Date: 28th June Page Count: View all volumes in this series: Pure and Applied Mathematics. For regional delivery times, please check When will I receive my book? Sorry, this product is currently out of stock.
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Riemannian Geometry | Peter Petersen | Springer
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Connect with:. Use your name:. Thank you for posting a review! We value your input. Analysis has come to the fore. Many of these results have been the product of the rapid expansion of geometric analysis , which can be roughly described as the study of partial differential equations and systems of ordinary differential equations on manifolds and submanifolds of the same dimension as the solution spaces. The solution of such equations, particularly in equations of elliptic type such as the Euler-Tricomi equation, yields geometric information on the structure of the manifold.
Conversely, aspects of the topology of the manifold yield analytic data on the PDEs. Since geometric analysis by its very nature requires differential equations expressed in local coordinates and the analytic tools needed to solve them, coordinate free concepts are of very limited use. Therefore, most of the standard introductory texts have avoided the subject altogether. The upshot of all this is that graduate students that study differential manifolds in the conventional manner will be very ill equipped to study the current literature. This not only makes the presentation more contemporary than most texts, but also covers more topics closer to the frontiers of research.
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The book is heavily computational; just about everything is expressed primarily in local coordinate tensor notation. This is differential geometry seen through the perspective of an analyst. So, in a sense, Jost has written a very modern book in an old fashioned manner.
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One would think this would result in a text that is nearly unreadable. His great triumph here is he succeeds in crafting a book that is anything but. He uses an absolute minimum of expository prose to produce the absolute maximal clarity. This results in one hell of a book, especially for serious graduate students. Chapter 1 gives — in 89 pages — a remarkable short course in differential manifolds and Lie algebras.
In the first chapter, we introduce the basic geometric concepts, like differentiable manifolds, tangent spaces, vector bundles, vector fields and one-parameter groups of diffeomorphisms, Lie algebras and groups and in particular Riemannian metrics. We also treat the existence of geodesics with two different methods, both of which are quite important in geometric analysis in general.
Thus, the reader has the opportunity to understand the basic ideas of those methods in an elementary context before moving on to more difficult versions in subsequent chapters. In this first chapter, Jost frequently uses concepts from algebra to build smooth structures on manifolds.
This is a very natural thing to do, but it requires instantaneous grasp of the notions of linear spaces and their isomorphisms. In German universities, undergraduates are expected to develop expertise in abstract algebra early. This is a lesson we can certainly take to heart for our students. He uses the concept of local coordinates to explain the Einstein summation convention. Physics majors struggling to understand this convention can do a lot worse then read this chapter. The inner product property of the Riemannian metric is emphasized; this is in keeping with the analytic flavor of the text and makes a lot of the concepts later easier to digest.
The chapter concludes with a wonderful introduction to the Lie group Spin V. The second chapter introduces de Rham cohomology and related essential tools from elliptic PDEs. He proves the existence of harmonic forms, a point not often emphasized.
The presentation of each concept leads to later ones as the discourse deepens. Restricting the elements of the cohomology classes to linear maps only allows the reader to understand some critical ideas of their general study, such as the Hodge lemma and the de Rham cohomology group on Sobolev spaces, without the technical difficulties of nonlinear problems a fully general presentation entails.
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Despite the fact that Jost is careful to define the de Rham cohomology and its relevant machinery in a completely self-contained manner, I get the distinct impression he expects the student to have at least a passing acquaintance with cohomology. He provides a separate appendix on basic topology — underscoring how adamant Jost is about making the book self contained.
The third chapter treats the general theory of linear connections and curvature. About half of this chapter is standard: parallel transport, covariant derivatives, the Bianchi identity, the Levi-Civita connection and flat connections. The remaining material is extremely nonstandard and will be of immense benefit to both analysis and physics students: an introductory discussion of the Yang-Mills operator using characteristic classes, as well as discussions of the Bochner method for harmonic forms of nonnegative Ricci curvature, connections on Spin V , and the Dirac operator.
In the fourth chapter, he gives a fairly complete development of the theory of Jacobi fields and the related machinery.