Eigenvalue Estimates Using Harmonic 1−Form of Constant Length for The 𝑺𝒑𝒊𝒏𝒄 Dirac Operator

Year 2020, , 1213 - 1223, 01.06.2020

Serhan Eker

https://doi.org/10.21597/jist.644583

Cited By: 2

Abstract

In this paper, we obtain a lower bound for the eigenvalue of the 𝑆𝑝𝑖𝑛𝑐 Dirac operator on an (𝑑≥3)−dimensional compact Riemannian Spin 𝑐−manifold admitting a non−zero harmonic 1−form of constant length. Then we show that, in the limiting case, this 1−form is parallel.

Keywords

Spin and 〖Spin〗^c geometry, Dirac operator, Estimation of eigenvalues

Sabit Uzunluklu Harmonik 1−Form Kullanılarak 𝑺𝒑𝒊𝒏𝒄 Dirac Operatörünün Özdeğerlerine Tahminler

Abstract

Bu makalede, sıfır olmayan sabit uzunluklu harmonik 1-formu kabul eden (𝑑≥3)−boyutlu kompakt bir Riemann 𝑆𝑝𝑖𝑛𝑐−manifoldu üzerinde tanımlı 𝑆𝑝𝑖𝑛𝑐 Dirac operatörünün öz değeri için alt sınır elde ettik. Daha sonra, limit durumunda harmonik 1−formun paralel olduğunu gösterdik.

Keywords

Spin ve 〖Spin〗^c geometry, Dirac operatörü, Öz değer tahminleri

References

• Bär C, 1992. Lower eigenvalue estimates for Dirac operators. Math. Ann., 239: 39-46.
• Friedrich T, 1980. Der este Eigenwert des Dirac-Operators einer kompakten, Riemannschen Manningfaltigkeit nichtnegativer Skalarkrümmung. Math. Nach. 97: 117-146.
• Friedrich T, 2000. Dirac operators in Riemannian geometry. Graduate Studies in Mathematics, American Mathematical Society, 25.
• Habib G, 2007. Energy-Momentum tensor on foliations. J. Geom. Phys. 57: 2234-2248.
• Herzlich M, Moroianu A, 1999. Generalized Killing spinors and conformal eigenvalue estimates for Spin^c manifold. Ann. Global Anal. Geom., 17: 341-370.
• Hijazi O, 1986. A conformal lower bound fort he smallest eigenvalue of the Dirac operator and Killing spinors. Commun. Math. Phys. 104: 151-162.
• Hijazi O, 1991. Premire valeur propre de l’ operateur de Dirac et nombre de Yamabe. Comptes rendus de l’ Academie des sciences, Serie 1, Mathematique, 313(12): 865-868.
• Hijazi O, 1995. Lower bounds for the eigenvalues of the Dirac operator. J. Geom. Phys., 16: 27-38.
• Lawson H.B, 1989. Spin Geometry. Princeton University Press., Princeton.
• Lichnerowicz A, 1963. Spineurs harmoniques. C.R. Acad. Sci. Paris Ser. AB, 257.
• Lichnerowicz A, 1988. Killing spinors according to O. Hijazi and Applications. Spinors in Physics and Geometry (Trieste 1986), World Scientific Publishing Singapore 1-19.
• Lichnerowicz A, 1987. Spin manifolds. Killing spinors and the universality of the Hijazi inequality. Lett. Math. Phys., 3: 331-344.
• Moroianu A, Ornea L, 2004. Eigenvalue estimates for the Dirac operator and harmonic 1-forms of constant length. C.R. Math. Acad. Sci. Paris, 338(7): 561-564.
• Nakad R, 2010. Lower bounds for the eigenvalues of the Dirac operator on manifolds. J. Geom. Phys. 60(10): 1634-1642.
• Salamon D, 1995. Spin geometry and Seiberg-Witten invariants. Zurich: ETH.