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AuthorCnops, Jan. author
TitleAn Introduction to Dirac Operators on Manifolds [electronic resource] / by Jan Cnops
ImprintBoston, MA : Birkhรคuser Boston : Imprint: Birkhรคuser, 2002
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Descript XI, 211 p. online resource


Dirac operators play an important role in several domains of mathematics and physics, for example: index theory, elliptic pseudodifferential operators, electromagnetism, particle physics, and the representation theory of Lie groups. In this essentially self-contained work, the basic ideas underlying the concept of Dirac operators are explored. Starting with Clifford algebras and the fundamentals of differential geometry, the text focuses on two main properties, namely, conformal invariance, which determines the local behavior of the operator, and the unique continuation property dominating its global behavior. Spin groups and spinor bundles are covered, as well as the relations with their classical counterparts, orthogonal groups and Clifford bundles. The chapters on Clifford algebras and the fundamentals of differential geometry can be used as an introduction to the above topics, and are suitable for senior undergraduate and graduate students. The other chapters are also accessible at this level so that this text requires very little previous knowledge of the domains covered. The reader will benefit, however, from some knowledge of complex analysis, which gives the simplest example of a Dirac operator. More advanced readers---mathematical physicists, physicists and mathematicians from diverse areas---will appreciate the fresh approach to the theory as well as the new results on boundary value theory


1 Clifford Algebras -- 1 Definition and basic properties -- 2 Dot and wedge products -- 3 Examples of Clifford algebras -- 4 Modules over Clifford algebras -- 5 Subgroups -- 2 Manifolds -- 1 Manifolds -- 2 Derivatives and differentials -- 3 The Spin group as a Lie group -- 4 Exterior derivatives and curvature -- 5 Spinors -- 6 Spinor fields -- 3 Dirac Operators -- 1 The vector derivative -- 2 The spinor Dirac operator -- 3 The Hodgeโ{128}{148}Dirac operator -- 4 Gradient, divergence and Laplace operators -- 4 Conformal Maps -- 1 Mรถbius transformations -- 2 Liouvilleโ{128}{153}s Theorem -- 3 Conformal embeddings -- 4 Maps between manifolds -- 5 Unique Continuation and the Cauchy Kernel -- 1 The unique continuation property -- 2 Sobolev spaces -- 3 The Cauchy kernel -- 4 The case of Euclidean space -- 6 Boundary Values -- 1 The Cauchy transform -- 2 Boundary values and boundary spinors -- 3 Boundary spinors and integral operators -- Appendix. General manifolds -- 1 Vector bundles -- 2 Connections -- 3 Connections on SO(M) -- 4 Spinor bundles -- List of Symbols

Mathematics Group theory Functions of complex variables Operator theory Differential geometry Physics Mathematics Functions of a Complex Variable Group Theory and Generalizations Differential Geometry Operator Theory Theoretical Mathematical and Computational Physics


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