Matematik / Mathematics DOI: 10.21597/jist.644583 Araştırma Makalesi / Research Article Iğdır Üniversitesi Fen Bilimleri Enstitüsü Dergisi, 10(2): 1213-1223, 2020 Journal of the Institute of Science and Technology, 10(2): 1213-1223, 2020 ISSN: 2146-0574, eISSN: 2536-4618 Eigenvalue Estimates Using Harmonic 1−Form of Constant Length for The 푺풑풊풏풄 Dirac Operator Serhan EKER1* ABSTRACT: In this paper, we obtain a lower bound for the eigenvalue of the 푆푝푖푛푐 Dirac operator on an (푑 ≥ 3) −dimensional compact Riemannian Spin 푐 −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 푆푝푖푛푐 geometry, Dirac operator, Estimation of eigenvalues. Sabit Uzunluklu Harmonik 1−Form Kullanılarak 푺풑풊풏풄 Dirac Operatörünün Özdeğerlerine Tahminler ÖZET: Bu makalede, sıfır olmayan sabit uzunluklu harmonik 1-formu kabul eden (푑 ≥ 3) −boyutlu kompakt bir Riemann 푆푝푖푛푐 −manifoldu üzerinde tanımlı 푆푝푖푛푐 Dirac operatörünün öz değeri için alt sınır elde ettik. Daha sonra, limit durumunda harmonik 1 −formun paralel olduğunu gösterdik. Anahtar Kelimeler: Spin ve 푆푝푖푛푐 geometry, Dirac operatörü, Öz değer tahminleri. 1 Serhan EKER (Orcid ID: 0000-0003-1039-0551), Ağrı İbrahim Çeçen Üniversitesi, Fen Edebiyat Fakültesi, Matematik Bölümü, Ağrı, Türkiye *Sorumlu Yazar/Corresponding Author: Serhan EKER, e-mail: [email protected] Geliş tarihi / Received: 08-11-2019 Kabul tarihi / Accepted: 05-01-2020 1213 Serhan EKER 10(2): 1213-1223, 2020 Eigenvalue Estimates Using Harmonic 1−Form of Constant Length for The 푺풑풊풏풄 Dirac Operator INTRODUCTION The Dirac operator is an important tool that provides information about the topology and geometry of the compact Riemannian 푆푝푖푛푐 −manifold and compact Riemannian 푆푝푖푛 manifold. Due to this feature of the Dirac operator, many authors have been systematically worked on it. One of these studies is to give a lower bound to the the square of the eigenvalue of the Dirac operator. In 1963 A. Lichnerowicz (Lichnerowicz, 1963) presented the following formula called Schrödinger−Lichnerwicz formula 푅 퐷2 = Δ + (1) 4 where Δ is the Laplacian acting on any spinor field and 푅 is the scalar curvature of (푀, 푔). By using (1) A. Lichnerowicz obtained the following estimates for the eigenvalue of the Dirac operator 퐷, 1 휆2 ≥ 푖푛푓푅. (2) 4 푀 In (Friedrich, 1980) T. Friedrich proved that on a Spin manifold (푀, 푔) of dimension 푑 ≥ 2, any eigenvalue 휆 of the Dirac operator satisfies 푑 휆2 ≥ 푖푛푓푅. (3) 4(푑−1) 푀 The proof is based on the modified spinorial Levi−Civita connection 푓 ∇푉휑 = ∇푉휑 + 푓푉 ⋅ 휑, (4) where 푓 ∈ 퐶∞(푀, ℝ). The limiting case of (3) implies that the existence of Killing spinor, i.e a spinor 휑 satisfying the equation: ∇푉φ + 푓푉 ⋅ φ = 0, ∀ 푉 ∈ 휒(푀). (5) In dimensions 2, C. Bär (Bär, 1992) obtained a bound to the eigenvalue 휆 of the Dirac operator according to the Euler−Poincare characteristic 풳(푀) of 푀 as follows: 2휋풳(푀) 휆2 ≥ . (6) 퐴푟푒푎(푀,푔) Later on, by using the conformal covariance of the Dirac operator, O. Hijazi improved the inequality (3), on a Spin manifold (푀, 푔) of dimension 푑 ≥ 3, 푑 휆2 ≥ 휇 , (7) 4(푑−1) 1 here 휇1 denotes the first eigenvalue of the Yamabe operator 퐿 given by 푑−1 퐿: = 4 Δ + 푅 (8) 푑−2 푔 and Δ푔 denotes the positive Laplacian acting on functions. In 1995, O. Hijazi (Hijazi, 1995) modified the spinorial Levi−Civita connection in the direction of symmetric endomorphism 푙φ as 1214 Serhan EKER 10(2): 1213-1223, 2020 Eigenvalue Estimates Using Harmonic 1−Form of Constant Length for The 푺풑풊풏풄 Dirac Operator 푙φ ∇푉 φ = ∇푉φ + 푙φ(푉) ⋅ φ (9) Then, he obtained that 2 푅 2 휆 ≥ 푖푛푓 ( + |푙φ| ). (10) 푀 4 Also, O. Hijazi has shown that using the modified spinorial Levi−Civita connection given in (9) and the conformal covariance of the Dirac operator on a Spin manifold (푀, 푔), any eigenvalue of the Dirac operator is satisfied 1 휇 + 푖푛푓|푙 |2, 푖푓 푑 ≥ 3 4 1 휑 휆2 ≥ { 푀 (11) 휋풳(푀) 2 + 푖푛푓|푙φ| , 푖푓 푑 = 2, 퐴푟푒푎(푀,푔) 푀 where 휇1 is the first eigenvalue of the Wamabe operator 퐿. In the limiting case of (10), O. Hijazi obtained the following relations: 푅 (푡푟푙 )2 = + |푙 |2, φ 4 φ 푔푟푎푑(푡푟푙φ) = −푑푖푣(푡푟푙φ), (12) where (푡푟푙φ) is the trace part of 푙φ. Subsequently, G. Habib (Habib, 2007) modified the spinorial Levi- Civita connection (9) in the direction of the skew−symmetric endomorphism 푞φ of the 2 −tensor 퐸, as 푙φ ∇푉 φ = ∇푉φ + 푙φ(푉) ⋅ φ + 푞φ(푉) ⋅ φ. (13) By using (13), he improved (10) as follows: 2 푅 2 2 휆 ≥ 푖푛푓 ( + |푙φ| + |푞φ| ). (14) 푀 4 As mentioned above, many studies have been done to improve lower bound (3), but the fundemantal question is: When is the equalitW in (3) hold? Accordingly, O. Hijazi (Hijazi, 1986) and A. Lichnerowicz (Lichnerowicz, 1988; Lichnerowicz 1987) noticed that the equalitW in (3) cannot hold on the Spin manifolds admitting a non−zero parallel 푟 −form for some 푟 ∈ {1,2, . , 푑 − 1}. Under this assumption, A. Moroianu and L. Ornea (Moroianu et al., 2004) enhanced the lower bound obtained in (3) on a 푑 −dimensional Spin manifolds admitting a non−trivial harmonic 1 −form of constant length as follows: 푑−1 휆2 ≥ 푖푛푓푅. (15) 푑−2 푀 In the limiting case, they show that, the universal cover of the manifold is isometric to the ℝ × 푁 where 푁 is a manifold admitting Killing spinors. In this paper, we consider the same assumption for the compact 푑 −dimensional 푆푝푖푛 푐 −manifold admitting a non−trivial harmonic 1 −form of constant length. Before mentioning to this assumption, we briefly touch on what kind of studies are done to obtain lower bound estimates for the eigenvalue of the 푆푝푖푛푐 Dirac operator defined . 1215 Serhan EKER 10(2): 1213-1223, 2020 Eigenvalue Estimates Using Harmonic 1−Form of Constant Length for The 푺풑풊풏풄 Dirac Operator All the inequalities mentioned above is obtained on the Spin manifold. This paper deals onlW with the eigenvalue of the Dirac operator defined on the 푆푝푖푛푐 − manifold. In 1999, A. Moroianu and M. Herzlich (Moroianu et al., 1999) proved that on a compact Riemannian 푆푝푖푛푐 manifold of dimension 푛 ≥ 3, any eigenvalue 휆 of the Dirac operator satisfies 푑 휆2 ≥ 휇 , (16) 4(푑−1) 1 Ω where 휇1 denotes the first eigenvalue of the perturbed Yamabe operator 퐿 given by Ω 퐿 : = 퐿 − 푐푑|Ω|푔, (17) and Ω is the curvature form of the line bundle ℒ. In (Herzlich et al., 1999), theW showed that there are no generalized Killing spinors on a Spin 푐 − manifold of dimension 푑 ≥ 4, except the usual Killing spinors. Using the modified spinorial Levi−Civita connection in the direction of 푙휑 + 푞φ, R. Nakad (Nakad, 2010) proved that, on a compact 푆푝푖푛푐 −manifold of dimension 푑 ≥ 2 any eigenvalue of the Dirac operator satisfies 2 푅 푐푑 2 2 휆 ≥ 푖푛푓 ( − |Ω|푔 + |푙φ| + |푞φ| ), (18) 푀 4 4 푑 1 where 푐 = 2[ ]2. Also, by considering the deformation of the spinorial Levi−Civita conection in the 푑 2 direction of the symmetric endomorphism 푙Φ given in (9), he obtained 2 푅 푐푑 2 휆 ≥ 푖푛푓 ( − |Ω|푔 + |푙φ| ). (19) 푀 4 4 Furthermore, using the modified spinorial Levi−Civita connecion in the direction of 푙Φ and conformal covariance of the Dirac operator, he has shown that on a on a 푆푝푖푛푐 −manifold, any eigenvalue of the Dirac operator satisfies 1 휇 + 푖푛푓|푙 |2, 푖푓 푑 ≥ 3 4 1 φ 휆2 ≥ { 푀 (20) 휋휒(푀) 2 + 푖푛푓|푙φ| , 푖푓 푑 = 2, 퐴푟푒푎(푀,푔) 푀 Ω where 휇1 denotes first eigenvalue of the perturbed Yamabe operator 퐿 . Then, in the limiting case, he obtained the following relations 푅 푑 1 (푡푟푙 )2 = − [ ] ⁄2|Ω| + |푙 |2, φ 4 2 푔 φ 푔푟푎푑(푡푟푙φ) = −푑푖푣(푡푟푙휑), (21) where (푡푟푙φ) is the trace part of 푙φ. 1216 Serhan EKER 10(2): 1213-1223, 2020 Eigenvalue Estimates Using Harmonic 1−Form of Constant Length for The 푺풑풊풏풄 Dirac Operator In this paper, we show that any eigenvalue 휆 of 퐷 on an (푑 ≥ 3) −dimensional 푆푝푖푛 푐 −manifold admitting a non−zero harmonic 1 −form of constant lenght satisfies 2 푑−1 휆 ≥ 푖푛푓(푅 − 푐푑|Ω|푔). (22) 4(푑−2) 푀 Furthermore, in the limiting case, this 1 −form is parallel. In the following section, some basic notions concerning Riemannian 푆푝푖푛푐-manifold and Dirac operator is introduced. MATERIALS AND METHODS 푺풑풊풏풄 Geometry and the Dirac operator Definitions of 푆푝푖푛푐 −structures on (푀, 푔) are obtained as follows: The structure group of 푑 −dimensional compact Riemannian manifold (푀, 푔) is 푆푂(푑) and there is an open covering {푈훼}훼∈퐴 with the transition functions 푔훼훽: 푈훼 ∩ 푈훽 ⟶ 푆푂(푑) for (푀, 푔). Accordingly, if there exists another collection of transition functions 푐 푔̃훼훽: 푈훼 ∩ 푈훽 ⟶ 푆푝푖푛 (푑) such that the following diagram commutes that is, 휆 ∘ 푔̃훼훽 = 푔훼훽 and the cocycle condition 푔̃훼훽(푥) ∘ 푔̃훽훾(푥) = 푔̃훼훾(푥) on 푈훼 ∩ 푈훽 ∩ 푈훾 is satisfied, then 푀 is called 푆푝푖푛푐 manifold. Along with the 푆푝푖푛푐 manifold (푀, 푔), one can construct two principal bundles such as 푃푆푂(푑), 푃푆푝푖푛푐 (Friedrich, 2000). 푐 On a Riemannian 푆푝푖푛 manifold, an associated spinor bundle 핊 = 푃푆푝푖푛푐 ×푑 Δ푑 can be constructed by using spinor representations 푐 휅푛: 푆푝푖푛 (푑) ↦ 퐴푢푡(Δ푑) where Δ푑 is the irreducible representation of Clifford algebra (Friedrich, 2000). The sections of 핊 are called spinor fields. The spinor bundle 핊 carries a natural Hermitian product, denoted by ( , ) and satisfies (푉 ⋅ 휑, ψ) = −(φ, 푉 ⋅ ψ) for every 푉 ∈ χ(M) and φ, ψ ∈ Γ(핊).