Dirac-Operatoren in der Riemannschen Geometrie
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Home > Mathematics and Science Textbooks > Mathematics > Geometry > Dirac-Operatoren in der Riemannschen Geometrie: Mit einem Ausblick auf die Seiberg-Witten-Theorie(Advanced Lectures in Mathematics)
Dirac-Operatoren in der Riemannschen Geometrie: Mit einem Ausblick auf die Seiberg-Witten-Theorie(Advanced Lectures in Mathematics)

Dirac-Operatoren in der Riemannschen Geometrie: Mit einem Ausblick auf die Seiberg-Witten-Theorie(Advanced Lectures in Mathematics)


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About the Book

Dieses Buch entstand nach einer einsemestrigen Vorlesung an der Humboldt-Universitat Berlin im Studienjahr 1996/ 97 und ist eine Einfuhrung in die Theorie der Spinoren und Dirac-Operatoren uber Riemannschen Mannigfaltigkeiten. Vom Leser werden nur die grundlegenden Kenntnisse der Algebra und Geometrie im Umfang von zwei bis drei Jahren eines Mathematik- oder Physikstudiums erwartet. Ein Anhang gibt eine Einfuhrung in das aktuelle Gebiet der Seiberg-Witten-Theorie.

Table of Contents:
1 Clifford-Algebren und Spin-Darstellung.- 1.1 Lineare Algebra quadratischer Formen.- 1.2 Die Clifford-Algebra einer quadratischen Form.- 1.3 Clifford-Algebren reeller, negativ-definiter quadratischer Formen.- 1.4 Die Pin- und die Spin-Gruppe.- 1.5 Die Spin-Darstellung.- 1.6 Die Gruppe SpinC.- 1.7 Reelle und quaternionische Strukturen im Raum der n-Spinoren.- 1.8 Literatur und Aufgaben.- 2 Spin-Strukturen.- 2.1 Existenz und Klassifikation von Spin-Strukturen eines SO(n)-Hauptfaserbündels.- 2.2 Beschreibung von Spin-Strukturen in Überlagerungen.- 2.3 Spin-Strukturen von G-Hauptfaserbündeln.- 2.4 Existenz von SpinC-Strukturen.- 2.5 Assoziierte Spinorbündel.- 2.6 Literatur und Aufgaben.- 3 Dirac-Operatoren.- 3.1 Zusammenhänge in Spinorbündeln.- 3.2 Der Dirac- und der Laplace-Operator im Spinorbündel.- 3.3 Die Lichnerowicz-Formel.- 3.4 Hermitesche Mannigfaltigkeiten und Spinoren.- 3.5 Der Dirac-Operator eines Riemannsch-symmetrischen Raumes.- 3.6 Literatur und Aufgaben.- 4 Analytische Eigenschaften der Dirac-Operatoren.- 4.1 Die wesentliche Selbstadjungiertheit von Dirac-Operatoren in L2.- 4.2 Das Spektrum von Dirac-Operatoren über kompakten Mannigfaltigkeiten.- 4.3 Dirac-Operatoren sind Fredholm-Operatoren.- 4.4 Literatur und Aufgaben.- 5 Abschätzungen der Eigenwerte des Dirac-Operators und Lösungen der Twistorgleichung.- 5.1 Abschätzungen von unten der Eigenwerte des Dirac-Operators.- 5.2 Riemannsche Mannigfaltigkeiten mit Killing-Spinoren.- 5.3 Die Twistorgleichung auf Riemannschen Mannigfaltigkeiten.- 5.4 Abschätzungen von oben der Eigenwerte des Dirac-Operators.- 5.5 Literatur und Aufgaben.- 6 Anhang 1: Seiberg-Witten-Invarianten.- 6.1 Zur Topologie 4-dimensionaler Mannigfaltigkeiten.- 6.2 Die Seiberg-Witten-Gleichung.- 6.3 Die Seiberg-Witten-Invariante.-6.4 Verschwindungssätze.- 6.5 Der Fall dim mL(g) = 0.- 6.6 Der Kähler-Fall.- 6.7 Literatur.- 7 Anhang 2: Hauptfaserbündel und Zusammenhänge.- 7.1 Lokal-triviale Faserungen, Hauptfaserbündel, assoziierte Bündel und Vektorbündel — Beispiele und Definitionen.- 7.2 Der Klassifizierungsraum einer topologischen Gruppe und die Homotopieklassifikation der Hauptfaserbündel.- 7.3 Zusammenhänge in Hauptfaserbündeln.- 7.4 Absolutes Differential und Krümmung eines Zusammenhangs.- 7.5 Zusammenhänge in U(1)-Hauptfaserbündeln und der Satz von Weyl.- 7.6 Induzierte Zusammenhänge und Reduktion eines Zusammenhangs.- 7.7 Die globale Variante des Frobenius-Theorems.- 7.8 Der Satz von Freudenthal-Yamabe.- 7.9 Holonomietheorie.- Literatur.- Namens- und Sachverzeichnis.

About the Author :
Prof. Dr. sc. Thomas Friedrich lehrt Mathematik an der Humboldt-Universität zu Berlin.


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Product Details
  • ISBN-13: 9783528069261
  • Publisher: Springer Fachmedien Wiesbaden
  • Publisher Imprint: Vieweg+Teubner Verlag
  • Height: 229 mm
  • No of Pages: 207
  • Series Title: Advanced Lectures in Mathematics
  • Width: 162 mm
  • ISBN-10: 3528069260
  • Publisher Date: 13 Jun 1997
  • Binding: Paperback
  • Language: German
  • Returnable: Y
  • Sub Title: Mit einem Ausblick auf die Seiberg-Witten-Theorie


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Dirac-Operatoren in der Riemannschen Geometrie: Mit einem Ausblick auf die Seiberg-Witten-Theorie(Advanced Lectures in Mathematics)
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Dirac-Operatoren in der Riemannschen Geometrie: Mit einem Ausblick auf die Seiberg-Witten-Theorie(Advanced Lectures in Mathematics)
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