The booklet offers an basic and finished advent to Spin Geometry, with specific emphasis at the Dirac operator, which performs a basic position in differential geometry and mathematical physics. After a self-contained presentation of the fundamental algebraic, geometrical, analytical and topological parts, a scientific examine of the spectral houses of the Dirac operator on compact spin manifolds is performed. The classical estimates on eigenvalues and their restricting circumstances are mentioned subsequent, highlighting the delicate interaction of spinors and precise geometric buildings. a number of functions of those rules are awarded, together with spinorial proofs of the optimistic Mass Theorem or the type of confident Kähler-Einstein touch manifolds. illustration thought is used to explicitly compute the Dirac spectrum of compact symmetric areas. The distinct beneficial properties of the publication contain a unified therapy of and conformal spin geometry (with precise emphasis at the conformal covariance of the Dirac operator), an outline with proofs of the speculation of elliptic differential operators on compact manifolds in keeping with pseudodifferential calculus, a spinorial characterization of specific geometries, and a self-contained presentation of the representation-theoretical instruments wanted on the way to understand spinors. This e-book can help complex graduate scholars and researchers to get extra acquainted with this pretty, even though no longer sufficiently recognized, area of arithmetic with nice relevance to either theoretical physics and geometry. A e-book of the ecu Mathematical Society (EMS). disbursed in the Americas via the yank Mathematical Society.
By H. S. M. Coxeter
From the experiences of the 1st edition:"... right here ... a wealth of fabric is displayed for us, an excessive amount of to even point out in a assessment. ... Your reviewer was once very inspired by way of the contents of either volumes (EMS 2 and 4), recommending them with none restriction." Mededelingen van het Wiskundig genootshap 1992
Fibre bundles, now an essential component of differential geometry, also are of significant value in sleek physics--such as in gauge thought. This ebook, a succinct advent to the topic through renown mathematician Norman Steenrod, was once the 1st to offer the topic systematically.
It starts off with a basic advent to bundles, together with such themes as differentiable manifolds and protecting areas. the writer then offers short surveys of complicated subject matters, equivalent to homotopy concept and cohomology thought, earlier than utilizing them to review additional homes of fibre bundles. the result's a vintage and undying paintings of serious application that might entice severe mathematicians and theoretical physicists alike.
This e-book begins with the fundamental thought of topological teams, harmonic research, and unitary representations. It then concentrates on geometric constitution, harmonic research, and unitary illustration idea in commutative areas. these areas shape a simultaneous generalization of compact teams, in the neighborhood compact abelian teams, and riemannian symmetric areas. Their geometry and serve as idea is an more and more lively subject in mathematical study, and this e-book brings the reader as much as the frontiers of that study sector with the new classifications of weakly symmetric areas and of Gelfand pairs. half 1, "General conception of Topological Groups", is an creation with many examples, together with all the commonplace semisimple linear Lie teams and the Heisenberg teams. It offers the development of Haar degree, the invariant vital, the convolution product, and the Lebesgue areas. half 2, "Representation idea and Compact Groups", presents history at a marginally greater point. along with the fundamentals, it comprises the Mackey Little-Group approach and its program to Heisenberg teams, the Peter-Weyl Theorem, Cartan's optimum weight idea, the Borel-Weil Theorem, and invariant functionality algebras. half three, "Introduction to Commutative Spaces", describes that quarter as much as its fresh resurgence. round services and linked unitary representations are constructed and utilized to harmonic research on $G/K$ and to uncertainty ideas. half four, "Structure and research for Commutative Spaces", summarizes riemannian symmetric house idea as a rôle version, and with that orientation delves into fresh learn on commutative areas. the implications are particular for areas $G/K$ of nilpotent or reductive style, and the hot constitution and type idea will depend on these circumstances. elements 1 and a couple of are available to first-year graduate scholars. half three takes just a little analytic sophistication yet ordinarily is obtainable to graduate scholars. half four is meant for mathematicians starting their study careers in addition to mathematicians drawn to seeing simply how a long way you may opt for this unified view of algebra, geometry, and research.
Some of the earliest books, quite these relationship again to the 1900s and ahead of, at the moment are tremendous scarce and more and more dear. we're republishing those vintage works in reasonable, top of the range, sleek variations, utilizing the unique textual content and art.
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