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Eigenvalues of the Dirac Operator, Twistors and Killing Spinors on Riemannian Manifolds

to the memory of Albert Crumeyrolle

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Clifford Algebras and Spinor Structures

Part of the book series: Mathematics and Its Applications ((MAIA,volume 321))

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Abstract

Let M n be an n-dimensional oriented Riemannian manifold with a fixed spin structure. We understand the spin structure as a reduction P of the SO(n)-principal bundle of M n to the universal covering Spin(n) SO(n) (n ≥ 3) of the special orthogonal group.

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References

  1. Anghel, M.: Extrinsic upper bounds for eigenvalues of Dirac type operators. Proc. AMS, 117, Nr. 2 (1993), 501–509

    Article  MathSciNet  MATH  Google Scholar 

  2. Atiyah, M.F.; Hitchin, N.J.; Singer, I.M.: Self-duality in four dimensional Riemannian geometry. Proc. R. Soc. London Serie A, 362 (1978), 425–461

    Article  MathSciNet  MATH  Google Scholar 

  3. Bär, Ch.: Upper Eigenvalue Estimates for Dirac operators. Ann. of Global Anal. and Geom. 10 (1992), 171–177

    Article  MATH  Google Scholar 

  4. Bär, Ch.: Real Killing spinors and holonomy. To appear in Comm. Math. Phys.

    Google Scholar 

  5. Baum, H.: Spin Strukturen und Dirac Operatoren über Pseudo-Riemannschen Mannigfaltigkeiten, Teubner-Texte zur Mathematik, Band 41, Teubner Verlag Leipzig 1981

    Google Scholar 

  6. Baum, H.: Variétés riemanniennes admettant des spineurs de Killing imaginaires. C.R. Acad. Sci. Paris Ser. I, t. 309 (1989), 47–49

    MathSciNet  MATH  Google Scholar 

  7. Baum, H.: Odd-dimensional Riemannian manifolds with imaginary Killing spinors. Ann. Global Anal. Geom. 7 (1989), 141–154

    Article  MathSciNet  MATH  Google Scholar 

  8. Baum, H.: Complete Riemannian manifolds with imaginary Killing spinors. Ann. Global Anal. Geom. 7 (1989), 205–226

    Article  MathSciNet  MATH  Google Scholar 

  9. Baum, H.: An upper bound for the first eigenvalue of the Dirac operator on compact spin manifolds. Math. Zeitschrift 206 (1991), 409–422

    Article  MathSciNet  MATH  Google Scholar 

  10. Baum, H.; Friedrich, Th.; Grunewald, R.; Kath, I.: Twistors and Killing Spinors on Riemannian Manifolds. Teubner-Verlag Leipzig-Stuttgart, 1991

    MATH  Google Scholar 

  11. Boyer, C.P.; Galicki, K.; Mann, B.M.: Quaternionic reduction and Einstein manifolds. Comm. in Analysis and Geometry, Vol. i (1993), 1–51

    MathSciNet  Google Scholar 

  12. Boyer, C.P.; Galicki, K.; Mann, B.M.: The geometry and topology of 3-Sasakian manifolds, Preprint 1993

    Google Scholar 

  13. Bunke, U.: Upper bounds of small eigenvalues of the Dirac operator and isometric immersions. Ann. of Global Anal. and Geom. 9 (1991), 211–243

    Article  Google Scholar 

  14. Cahen, M.; Gutt, S.; Lemaire, L.; Spindel, T.: Killing spinors. Bull. Soc. Math. Belg. Ser. A 38 (1986), 75–102

    MathSciNet  MATH  Google Scholar 

  15. Friedrich, Th.: Der erste Eigenwert des Dirac-Operators einer kompakten Riemannschen Mannigfaltigkeit nichtnegativer Skalarkrümmung. Math. Nachrichten 97 (1980), 117–146

    Article  MathSciNet  MATH  Google Scholar 

  16. Friedrich, Th.: Zur Existenz paralleler Spinorfelder der Riemannschen Mannigfaltigkeiten. Colloq. Math. 44 (1981) 2, 277–290

    MathSciNet  Google Scholar 

  17. Friedrich, Th.: A remark on the first eigenvalue of the Dirac operator on 4-dimensional manifolds, Math. Nachrichten 102 (1981), 53–56

    Article  MathSciNet  MATH  Google Scholar 

  18. Friedrich, Th.: On the conformal relation between twistors and Killing spinors. Suppl. Rend. Circ. Mat. Palermo 22 (1989), 59–75

    Google Scholar 

  19. Friedrich, Th.: The classification of 4-dimensional Kähler manifolds with small eigenvalue of the Dirac operator. Math. Ann. 295 (1993), 565–574

    Article  MathSciNet  MATH  Google Scholar 

  20. Friedrich, Th.; Grunewald, R.: On the first eigenvalue of the Dirac operator on 6-dimensional manifolds. Ann. Global Anal. Geom. 3 (1985), 265–273

    Article  MathSciNet  MATH  Google Scholar 

  21. Friedrich, Th.; Kath, I.: Varietés riemanniennes compactes de dimension 7 admettant des spineurs de Killing. C.R. Acad. Sci. Paris Serie I, 307 (1988), 967–969

    MathSciNet  MATH  Google Scholar 

  22. Friedrich, Th.; Kath, I.: Einstein manifolds of dimension five with small eigenvalue of the Dirac operator. J. Diff. Geom. 29 (1989), 263–279

    MathSciNet  MATH  Google Scholar 

  23. Friedrich, Th.; Kath, I.: Compact seven-dimensional manifolds with Killing spinors. Comm. Math. Phys. 133 (1990), 543–561

    Article  MathSciNet  MATH  Google Scholar 

  24. Friedrich, Th.; Kath. I.: Compact 5-dimensional Riemannian manifolds with parallel spinors. Math. Nachrichten 147 (1990), 161–165

    Article  MathSciNet  MATH  Google Scholar 

  25. Friedrich, Th.; Kurke, H.: Compact four-dimensional self-dual Einstein manifolds with positive scalar curvature. Math. Nachrichten 106 (1982), 271–299

    Article  MathSciNet  MATH  Google Scholar 

  26. Grunewald, R.: Six-dimensional Riemannian manifolds with a real Killing spinor. Ann. Global Anal. Geom. 8 (1990), 43–59

    Article  MathSciNet  MATH  Google Scholar 

  27. Grunewald, R.: On the relation between real Killing spinors and almost hermitian structures. Preprint Nr. 271, Fachbereich Mathematik, Humboldt-Universität Berlin, 1991

    Google Scholar 

  28. Habermann, K.: The twistor equation on Riemannian manifolds. JGP Vol. 7 (1990), 469–488

    MathSciNet  MATH  Google Scholar 

  29. Habermann, K.: Twistor spinor and their zeroes, to appear. (Preprint SFB 288 No. 30, Berlin)

    Google Scholar 

  30. Hijazi, O.: A conformal lower bound for the smallest eigenvalue of the Dirac operator and Killing spinors. Comm. Math. Phys. 104 (1986), 151–162

    Article  MathSciNet  MATH  Google Scholar 

  31. Hijazi, O.: Caractérisation de la sphere par les l’operator de Dirac en dimension 3, 4, 7 et 8. C.R. Acad. Sci. Paris Serie I, 303 (1986), 417–419

    MathSciNet  MATH  Google Scholar 

  32. Hijazi, O.; Lichnerowicz, A.: Spineurs harmonique, spineurs-twisteurs et géométrie conforme. C.R. Acad. Sci. Paris Serie I, 307 (1988), 833–838

    MathSciNet  MATH  Google Scholar 

  33. Hijazi, O.; Milhorat, J.-L.: Spineurs et valeurs propres de l’opérateur de Dirac sur le variétés Kählerian-Quaternioniennes. Preprint Université de Nantes 1993

    Google Scholar 

  34. Hitchin, N.: Harmonic spinors. Adv. in Mathem. 14 (1974), 1–55

    Article  MathSciNet  MATH  Google Scholar 

  35. Hitchin, N.: Kählerian twistor spaces. Proc. London Math. Soc. 43 (1981), 133–150

    Article  MathSciNet  MATH  Google Scholar 

  36. Kath, I.: Varietés riemanniennes de dimension 7 admettant un spineur de Killing réel. C.R. Acad. Sci. Paris Serie I, 311 (1990), 553–555

    MathSciNet  MATH  Google Scholar 

  37. Kirchberg, K.-D.: An estimation for the first eigenvalue of the Dirac operator on closed Kähler manifolds of positive scalar curvature. Ann. Global Anal. Geom. 4 (1986), 291–325

    Article  MathSciNet  MATH  Google Scholar 

  38. Kirchberg, K.-D.: Compact six-dimensional Kähler spin manifolds of positive scalar curvature with the smallest possible first eigenvalue of the Dirac operator. Math. Ann. 282 (1988), 157–176

    Article  MathSciNet  MATH  Google Scholar 

  39. Kirchberg, K.-D.: Twistor-spinors on Kähler manifolds and the first eigenvalue of the Dirac operator. J. Geom. Phys. 7 (1990), 449–468

    Article  MathSciNet  MATH  Google Scholar 

  40. Kirchberg, K.-D.: Properties of Kählerian twistor-spinors and vanishing theorems. Math. Ann. 293 (1992), 349–369

    Article  MathSciNet  MATH  Google Scholar 

  41. Kirchberg, K.-D.: Some further properties of Kählerian twistor spinors. Math. Nachrichten 163 (1993), 229–255

    Article  MathSciNet  MATH  Google Scholar 

  42. Kühnel, W.; Rademacher, H.-B.: Twistor spinors with zeroes. Preprint 1993

    Google Scholar 

  43. Lichnerowicz, A.: Varietés spinorielles et universalité de l’inegalité de Hijazi. C.R. Acad. Sci. Paris Serie I, 304 (1987), 227–231

    MathSciNet  MATH  Google Scholar 

  44. Lichnerowicz, A.: Spin manifolds, Killing spinors and universality of the Hijazi inequality. Lett. Math. Phys. 13 (1987), 331–344

    Article  MathSciNet  MATH  Google Scholar 

  45. Lichnerowicz, A.: Les spineurs-twisteurs sur une varieté spinorielle compacte. C.r. Acad. Sci. Paris Serie I, 306 (1988), 381–385

    MathSciNet  MATH  Google Scholar 

  46. Lichnerowicz, A.: Sur les résultates de H. Baum et Th. Friedrich concernant les spineurs de Killing a valeur propre imaginary. C.R. Acad. Sci. Paris Serie I, 306 (1989), 41– 45

    MathSciNet  Google Scholar 

  47. Lichnerowicz, A.: On the twistor spinors. Lett. Math. Physics 18 (1989), 333–345

    Article  MathSciNet  MATH  Google Scholar 

  48. Lichnerowicz, A.: Sur les zéros des spineurs-twisteurs. C.R. Acad. Sci. Paris Serie I, 310 (1990), 19–22

    MathSciNet  MATH  Google Scholar 

  49. Lichnerowicz, A.: La premier value propre de l’operateur de Dirac pour une varieté Kählerienne et sone case limite. C.R. Acad. Sci. Paris Serie I, 311 (1990), 883–887

    MathSciNet  MATH  Google Scholar 

  50. Lott: Eigenvalue bounds for the Dirac operator. Pac. Journ. of Math. 125 (1986)

    Google Scholar 

  51. Penrose, P.’ Rindler, W.: Spinors and Space Time, Vol. 2, Cambr. Mono. in Math. Physics, 1986

    Google Scholar 

  52. Rademacher, H.-B.: Generalized Killing Spinors with Imaginary Killing Function and Conformal Killing Fields. in: Proceedings of the Conference on Global Analysis and Global Differential Geometry, 193–198. Lecture Notes in Math. Nr. 1481, Springer-Verl. Berlin 1990

    Google Scholar 

  53. Wang, ; McKenzie, Y.: Parallel spinors and parallel forms. Ann. Global Anal. Geom. 7 (1989), 59–68

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Fachbereich Mathematik, Humboldt-Universität zu Berlin, 10099, Berlin, Germany

    Helga Baum & Thomas Friedrich

Authors
  1. Helga Baum
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  2. Thomas Friedrich
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Editor information

Editors and Affiliations

  1. Gannon University, Erie, Pennsylvania, USA

    Rafał Ablamowicz

  2. Helsinki University of Technology, Espoo, Finland

    Pertti Lounesto

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Baum, H., Friedrich, T. (1995). Eigenvalues of the Dirac Operator, Twistors and Killing Spinors on Riemannian Manifolds. In: Ablamowicz, R., Lounesto, P. (eds) Clifford Algebras and Spinor Structures. Mathematics and Its Applications, vol 321. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-8422-7_14

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  • DOI: https://doi.org/10.1007/978-94-015-8422-7_14

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-90-481-4525-6

  • Online ISBN: 978-94-015-8422-7

  • eBook Packages: Springer Book Archive

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