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# ðŸ“™ Dirac operators in Riemannian geometry by Thomas Friedrich â€” epub download

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For a Riemannian manifold $M$, the geometry, topology and analysis are interrelated in ways that are widely explored in modern mathematics. Bounds on the curvature can have significant implications for the topology of the manifold. The eigenvalues of the Laplacian are naturally linked to the geometry of the manifold. For manifolds that admit spin (or $\textrm{spin}^\mathbb{C}$) structures, one obtains further information from equations involving Dirac operators and spinor fields. In the case of four-manifolds, for example, one has the remarkable Seiberg-Witten invariants. In this text, Friedrich examines the Dirac operator on Riemannian manifolds, especially its connection with the underlying geometry and topology of the manifold. The presentation includes a review of Clifford algebras, spin groups and the spin representation, as well as a review of spin structures and $\textrm{spin}^\mathbb{C}$ structures. With this foundation established, the Dirac operator is defined and studied, with special attention to the cases of Hermitian manifolds and symmetric spaces. Then, certain analytic properties are established, including self-adjointness and the Fredholm property. An important link between the geometry and the analysis is provided by estimates for the eigenvalues of the Dirac operator in terms of the scalar curvature and the sectional curvature. Considerations of Killing spinors and solutions of the twistor equation on $M$ lead to results about whether $M$ is an Einstein manifold or conformally equivalent to one. Finally, in an appendix, Friedrich gives a concise introduction to the Seiberg-Witten invariants, which are a powerful tool for the study of four-manifolds. There is also an appendix reviewing principal bundles and connections. This detailed book with elegant proofs is suitable as a text for courses in advanced differential geometry and global analysis, and can serve as an introduction for further study in these areas. This edition is translated from the German edition published by Vieweg Verlag

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- Series:
**Graduate Studies in Mathematics 025** - Author:
**Thomas Friedrich** - Year:
**2000** - Publisher:
**American Mathematical Society** - Language:
**English** - ISBN:
**0-8218-2055-9,9780821820551,18-1996-131-1,12-1975-173-1,310-1998-55-7,43-1998-615-6,93-1984-227-2**

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**600** - File size:
**1 563 505** - Format:
**djvu**

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For a Riemannian manifold $M$, the geometry, topology and analysis are interrelated in ways that are widely explored in modern mathematics. Bounds on the curvature can have significant implications for the topology of the manifold. The eigenvalues of the ...

The Dirac operator has many useful applications in theoretical physics and mathematics. This book provides a clear, concise and self-contained introduction to the global theory of the Dirac operator and to the analysis of spectral asymptotics with local o...

One of the basic ideas in differential geometry is that the study of analytic properties of certain differential operators acting on sections of vector bundles yields geometric and topological properties of the underlying base manifold. Symplectic spinor ...

This monograph presents a comprehensive treatment of important new ideas on Dirac operators and Dirac cohomology. Dirac operators are widely used in physics, differential geometry, and group-theoretic settings (particularly, the geometric construction of ...

This book presents a comprehensive treatment of important new ideas on Dirac operators and Dirac cohomology. Using Dirac operators as a unifying theme, the authors demonstrate how some of the most important results in representation theory fit together wh...

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 se...

This book is an exposition of "Singular Semi-Riemannian Geometry"- the study of a smooth manifold furnished with a degenerate (singular) metric tensor of arbitrary signature. The main topic of interest is those cases where the metric tensor is assumed to ...

In his latest book A Panoramic View of Riemannian Geometry, Marcel Berger does a remarkable job giving an in-depth survey ranging over almost the full spectrum of Riemannian geometry, furnishing the reader with some of the most exciting and elegant topics...

These lecture notes grew out of an M.Sc. course on differential geometry which I gave at the University of Leeds 1992. Their main purpose is to introduce the beautiful theory of Riemannian Geometry a still very active research area of mathematics. This is...

A study which uniquely integrates the analytical and policy aspects of transportation planning, providing an up-to-date review of transport policy. This is a basic text for students of planning, transport and public administration and an important referen...

During the past decade there has been a rapid growth in literature on traumatic experience, but what remains missing from the expanding field of commentary is any sustained consideration of how those who are outsiders to the experience deal with the chall...

This study expands on Reynolds' 'transversal poetics' - the theory, methodology, and aesthetics developed in response to the need for an approach that fosters agency, creativity and conscientious scholarship and pedagogy. It offers new readings of plays b...