Then, we extend these estimates in terms of the Yamabe number and the area of the submanifold under the conformal change of the metric.
📌 3 Key Takeaways
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Objective & Hypothesis:In this paper, we construct two modified spinorial Levi−Civita connection based on the Energy−Momentum tensor and its trace to bring an optimal lower bound to the eigenvalues of the compact Riemannian Spin−submanifold Dirac operator in terms of the the Energy−Momentum tensor and its trace.
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Key Findings & Data:In this paper, we construct two modified spinorial Levi−Civita connection based on the Energy−Momentum tensor and its trace to bring an optimal lower bound to the eigenvalues of the compact Riemannian Spin−submanifold Dirac operator in terms of the the Energy−Momentum tensor and its trace.
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Conclusion & Impact:Then, we extend these estimates in terms of the Yamabe number and the area of the submanifold under the conformal change of the metric.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Domain Parameters
Abstract
In this paper, we construct two modified spinorial Levi−Civita connection based on the Energy−Momentum tensor and its trace to bring an optimal lower bound to the eigenvalues of the compact Riemannian Spin−submanifold Dirac operator in terms of the the Energy−Momentum tensor and its trace. Then, we extend these estimates in terms of the Yamabe number and the area of the submanifold under the conformal change of the metric.
Özet
Bu makalede, kompakt Riemannian Spin altmanifold Dirac operatörünün özdeğerlerine Energy−Momentum tensörü ve onun izine bağlı olarak optimal bir alt sınır getirmek için spinorial Levi−Civita konneksiyonunu Energy−Momentum tensörü ve onun izine bağlı olarak deforme ettik. Daha sonra bu tahminleri konformal değişim metriği altında Yamabe sayısı ve altmanifoldun alanına bağlı olarak genişlettik.
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Cite This Article
EKER, S. (2022). Lower Bound Eigenvalue Problems of the Compact Riemannian Spin−Submanifold Dirac Operator. Erzincan Üniversitesi Fen Bilimleri Enstitüsü Dergisi, Cilt: 13 - Sayı: Özel Sayı - 1, 63–69. https://ajindex.com/articles/650188
Friedrich T. (2000). “Dirac operators in Riemannian geometry”, American Mathematical Society, 25. Friedrich T., Kim E.C. (2001). “Some remarks on the Hijazi inequality and generalizations of the Killing equation for spinors”, J. Geom. Phys., 37 (1-2),1-14.
Hijazi O. (1986). “A conformal lower bound for the smallest eigenvalue of the Dirac operator and Killing spinors”,Comm.Math. Phys., 104, 151–162.
Hijazi O. (1995). “Lower bounds for the eigenvalues of the Dirac operator”, J.Geom. Phys. 16, 27-38.
Hijazi O. (1991). “Première valeur propre de l'opérateur de Dirac et nombre de Yamabe”, Comptes rendus de l'Aca-démie, des sciences. Série 1, Mathé-matique, 313 (12), 865–868.
Hijazi O., Zhang X. (2001). “ Lower bounds for the eigenvalues of the Dirac operator: part I. The hypersurface Dirac operator”, Ann. Global Anal. Geom.,19 (4), 355–376.
Hijazi O., Zhang X. (2001). “ Lower bounds for the eigenvalues of the Dirac operator: part II. The Submanifold Dirac operator”, Ann. Global Anal. Geom. 20 (2), 163–181.
Hijazi O., Montiel S., Zhang X. (2001). “Eigenvalues of the Dirac Operator on Manifolds with Boundary”, Comm. Math. Phys., 221 (2), 255–265.
Lawson H., Michelsohn M. (1989). “Spin Geometry”, Princeton university press. Nakad R., Roth J. (2013). “The Spin 𝑐 Dirac operator on hypersurfaces and applications”, Differential Geom. Appl. 31 (1), 93-103.
Lichnerowicz A. (1963). “ “Spineurs harmoniques”, C.R. Acad. Sci. Paris Ser. AB, 257, 1963.
Naber G.L. (1997). “Topology, geometry, and gauge fields”, Springer, New York. Zhang X. (1998). “ Lower bounds for eigenvalues hypersurface Dirac operators, Math. Res. Lett., 5 (2), 199–210.
Zhang X. (1998) “ Angular momentum and positive mass theorem”, Comm. Math. Phys., 206 (1), 137–155.