An Introduction to Dirac Operators on Manifolds

1. Front Matter

   Pages i-xi

   PDF

2. Clifford Algebras

   • Jan Cnops

   Pages 1-24

3. Manifolds

   • Jan Cnops

   Pages 25-60

4. Dirac Operators

   • Jan Cnops

   Pages 61-89

5. Conformal Maps

   • Jan Cnops

   Pages 91-121

6. Unique Continuation and the Cauchy Kernel

   • Jan Cnops

   Pages 123-144

7. Boundary Values

   • Jan Cnops

   Pages 145-169

8. Back Matter

   Pages 171-211

   PDF

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.

• Complex analysis
• applications of mathematics
• curvature
• differential geometry group
• lie groups
• manifold
• mathematical physics

"The text should be accessible for senior undergraduate and graduate students. It requires very little previous knowledge of the domains covered. More advanced readers could perhaps appreciate the new approach to the theory as well as some new results on boundary value theory."

—Mathematical Reviews

"This book gives an introduction to Dirac operators on manifolds for readers with little knowledge in differential geometry and analysis.... Compared to other books treating similar subjects...the present book is considerably more elementary and is mostly restricted to results that can easily be obtained out of the definitions."

—Zentralblatt Math

"The extraordinary importance of Dirac operators in variuos domains of mathematics and physics is well known. So, although there are some remakrable monographs on Dirac operators, the high number of recent papers covering several subjects needs periodical surveys...

The book is excellent for beginners offering several ideas of research and a global picture of a fascinating theory!"  ---Memoriile Sectiilor Stiintifice

• Department of Computer Sciences, Gent Polytechnic, Gent, Belgium

  Jan Cnops

Policies and ethics