1980
DOI: 10.1007/bf01215090
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Classification of certain compact Riemannian manifolds with harmonic curvature and non-parallel Ricci tensor

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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%
See 1 more Smart Citation
“…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%
“…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%
“…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%