Classification of certain compact Riemannian manifolds with harmonic curvature and non-parallel Ricci tensor

Derdziński, Andrzej

doi:10.1007/bf01215090

Cited by 64 publications

(71 citation statements)

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“…As r is a Codazzi tensor with respect to g and has constant ^-trace (LEMMA 1), it follows from [8,Lemma 3] that Tg is p-parallel. This proves that if g has harmonic curvature and non-parallel Ricci tensor, K\ or K^ is constant, which implies (*) in view of (11).…”

Section: By a Codazzi Tensor On A Riemannian Manifold (Mgmentioning

confidence: 99%

“…Applying contractions to the second Bianchi identity dR = 0 (in local coordinates, qRijk£ + ^iRjqkt + V' jRqiki = 0), we obtain the following relations : Here R, W, r and u are the curvature tensor, Weyl conformal tensor, Ricci tensor and scalar curvature of g, respectively, with the sign conventions such that rij = Risj 8 1 u = g^rij and (…”

Section: Preliminariesmentioning

confidence: 99%

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Riemannian metrics with harmonic curvature on 2-sphere bundles over compact surfaces

Derdziński¹

1988

Bul. Soc. Math. France

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Section: By a Codazzi Tensor On A Riemannian Manifold (Mgmentioning

confidence: 99%

Section: Preliminariesmentioning

confidence: 99%

Riemannian metrics with harmonic curvature on 2-sphere bundles over compact surfaces

Derdziński¹

1988

Bul. Soc. Math. France

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“…Proof: Using the formulas (3), (4) and (7) we have (8) is a preliminary version of the Weitzenböck formula, which we will apply in the proof of our main result. Our next aim is to express the uncontrollable last term on the right-hand side by terms that are controllable.…”

Section: The Weitzenböck Formula For the Operator Q Tmentioning

confidence: 99%

“…Local products of Einstein manifolds; 2. conformally flat manifolds with constant scalar curvature; 3. warped products S 1 × f 2 N n−1 of an Einstein manifold with positive scalar curvature R = 4(n − 1)/n by S 1 (see [4], [5], [6]), where the function F := f n/2 is a positive, periodic solution of the differential equation…”

Section: A Mini-max Principle For the Estimate Of The Eigenvaluesmentioning

confidence: 99%

Eigenvalue estimates of the Dirac operator depending on the Ricci tensor

Friedrich

Kirchberg

2002

Mathematische Annalen

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We prove a new lower bound for the first eigenvalue of the Dirac operator on a compact Riemannian spin manifold by refined Weitzenböck techniques. It applies to manifolds with harmonic curvature tensor and depends on the Ricci tensor. Examples show how it behaves compared to other known bounds.Subj. Class.: Differential Geometry. 2000 MSC: 53C27, 53C25. Keywords: Dirac operator, eigenvalues, harmonic Weyl tensor. IntroductionIf M n is a compact Riemannian spin manifold with positive scalar curvature R, then each eigenvalue λ of the Dirac operator D satisfies the inequalitywhere R 0 is the minimum of R on M n . The estimate (1) is sharp in the sense that there exist manifolds for which (1) is an equality for the first eigenvalue λ 1 of D. If this is the case, then each eigenspinor ψ corresponding to λ 1 is a Killing spinor with the Killing number λ 1 /n, i.e., ψ is a solution of the field equationand M n must be an Einstein space (see [7]). A generalization of this inequality was proved in the paper [10], where a conformal lower bound for the spectrum of the Dirac operator occured. Moreover, for special Riemannian manifolds better estimates for the eigenvalues of the Dirac operator are known, see [11], [12]. However, all these estimates of the spectrum of the Dirac operators depend only on the scalar curvature of the underlying * Supported by the SFB 288 of the DFG.

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“…They are Gmanifolds of cohomogeneity one and Riemannian manifolds with harmonic curvature, non-parallel Ricci tensor and such that the operator Ric has less than three distinct eigenvalues. The last class of manifolds was studied by Derdzisnki in [7] and [8]. Such manifolds were the first examples of compact manifolds with harmonic curvature and non-parallel Ricci tensor and hence not Einstein.…”

Section: Introductionmentioning

confidence: 99%