We extend a classical result by Derdzinski and Shen, on the restrictions
imposed on the Riemann tensor by the existence of a nontrivial Codazzi tensor.
The new conditions of the theorem include Codazzi tensors (i.e. closed 1-forms)
as well as tensors with gauged Codazzi condition (i.e. "recurrent 1-forms"),
typical of some well known differential structures.Comment: 5 page

Mantica, Carlo Alberto

Molinari, Luca Guido

English

arXiv

Carlo Alberto Mantica

Luca Guido Molinari

Crossref

Colloquium Mathematicum

Extended Derdziński–Shen theorem for curvature tensors

We extend a remarkable theorem of Derdzinski and Shen, on the restrictions imposed on the Riemann tensor by the existence of a nontrivial Codazzi tensor. we show that the Codazzi equation can be replaced by a more general algebraic condition. The resulting extension applies both to the Riemann tensor and to generalized curvature tensors

C. A. Mantica

L. G. Molinari

AIR Universita degli studi di Milano

.EXTENDED DERDZINSKI-SHEN THEOREMFOR CURVATURE TENSORSCARLO ALBERTO MANTICA AND LUCA GUIDO MOLINARIAbstract. We extend a remarkable theorem of Derdzinski and Shen,on the restrictions imposed on the Riemann tensor by the existence ofa nontrivial Codazzi tensor. We show that the Codazzi equation can bereplaced by a more general algebraic condition. The ensuing extensionapplies both to the Riemann and to generalized curvature tensors.1. IntroductionOn a Riemannian manifold with metric gij and Riemann connection ∇i,a Codazzi tensor is a symmetric tensor that satisfies the Codazzi equation:(1.1) ∇jbkl −∇kbjl = 0.In terms of differential forms (1.1) is the condition for closedness of the1-form bjkdxk [3, 11]. Codazzi tensors are of great interest in geometryand have been studied by several authors, such as Berger and Ebin [1],Bourguignon [3], Derdzinski [5, 6], Derdzinski and Shen [7], Ferus [8], Si-mon [15, 16]; a compendium of results is reported in Besse’s book [2] (pp.436-440). They occur in the study of Riemannian manifolds with harmoniccurvature. For example, the Ricci tensor is a Codazzi tensor if and onlyif ∇mRjklm = 0, i.e. the manifold has harmonic Riemann curvature [2](page 435). The Weyl 1-form[Rkj −R2(n−1)gkj]dxk is closed if and only if∇mCjklm = 0, i.e. the manifold has harmonic conformal curvature [2] (page435).In ref. [3] important geometric and topological consequences of the existenceof a non-trivial Codazzi tensor are examined, in particular the restrictionsimposed on the structure of the curvature operator. Derdzinski and Shenimproved them and stated the remarkable theorem (the theorem is reportedin Besse’s book [2], page 438):Theorem 1.1 (Derdzinski-Shen, [7]). Let bij be a Codazzi tensor on a Rie-mannian manifold M , x a point of M , λ and µ two eigenvalues of theDate: 20 sept 2012 - revised version.2010 Mathematics Subject Classification. Primary 53B20, Secondary 53B21.Key words and phrases. Codazzi tensor, Riemann tensor, generalized curvature tensor,recurrent 1-form.12 C.A.MANTICA AND L.G.MOLINARIoperator bij(x), with eigenspaces Vλ and Vµ in TxM . Then, the subspaceVλ ∧ Vµ is invariant under the action of the curvature operator Rx.Then, they obtained the following result: in a n-dimensional Riemannianmanifold with a Codazzi tensor having n distinct eigenvalues almost every-where, all real Pontryagin classes vanish.We point out that the Codazzi equation is a sufficient condition for thetheorem to hold. A more general one is suggested by the following lemma:Lemma 1.2. Any symmetric Codazzi tensor bkl satisfies the algebraic iden-tityRjklmbim +Rkilmbjm +Rijlmbkm = 0,(1.2)Proof. The following identity among commutators is true for a Codazzitensor:[∇j ,∇k]bil + [∇k,∇i]blj + [∇i,∇j]bkl = 0(1.3)Each commutator is evaluated: [∇i,∇j]bkl = Rijkmbml + Rijlmbkm. Cancel-lations occur by the first Bianchi identity and the result is obtained. Our first extension of the theorem states that if a symmetric tensorbkl satisfies the algebraic condition (1.2), then the same conclusions of theDerdzinski and Shen’s theorem are valid for the Riemann tensor. It turnsout that the proof of the extended theorem is much simpler than Derdzinskiand Shen’s proof.The replacement of the Codazzi equation by an algebraic condition allowsfor a further natural extension of the theorem to generalized curvature ten-sors. It includes well known tensors such as the conformal, the concircularand the conharmonic tensors [14, 9, 12].2. Extension of D-S theorem for the Riemann tensor.As in the original theorem, we need an auxiliary tensor and a lemmato prove that it is a generalized curvature tensor, a concept introduced byKobayashi and Nomizu [10] (page 198). The algebraic condition (1.2) ishere used rather than Codazzi’s equation to prove both the lemma and theextended theorem.Definition 2.1. A tensor Kijlm is a generalized curvature tensor if it hasthe symmetries of the Riemann curvature tensor:a) Kijkl = −Kjikl = −Kijlk,EXTENDED DERDZINSKI-SHEN THEOREM FOR CURVATURE TENSORS 3b) Kijkl = Kklij,c) Kijkl +Kjkil +Kkijl = 0 (first Bianchi identity).Lemma 2.2. If a symmetric tensor bkl satisfies eq.(1.2), then Kijkl =Rijrsbkrbls is a generalized curvature tensor.Proof. Properties a) are shown easily. For example: Kijlk = Rijrsblrbks =Rijsrblsbkr = −Rijrsblsbkr = −Kijkl.Property c) follows from (1.2):Kijkl+Kjkil+Kkijl = Rijrsbkrbls+Rjkrsbirbls+Rkirsbjrbls = (Rjisrbkr +Rkjsrbir +Riksrbjr)bls = 0.Property b) follows from c): Kijkl+Kjkil+Kkijl = 0. Sum the identity overcyclic permutations of all indices i, j, k, l and use the symmetries a) (thisfact was pointed out in [10]).It is easy to see that a first Bianchi identity holds also for the last threeindices: Kijkl +Kiklj +Kiljk = 0. Theorem 2.3. Let M be an n-dimensional Riemannian manifold with asymmetric tensor bkl that satisfies the algebraic equationbimRjklm + bjmRkilm + bkmRijlm = 0If X, Y and Z are three eigenvectors of the matrix brs at a point x of themanifold, with eigenvalues λ, µ and ν, then(2.1) X i Y j Zk Rijkl = 0provided that λ and µ are different from ν.Proof. Consider the first Bianchi identity for the Riemann tensor, the con-dition eq.(1.2) and the first Bianchi identity for the curvature Klijk =Rlirsbjrbks, and apply them to the three eigenvectors. The three algebraicrelations can be put in matrix form:1 1 1λ µ νµν λν λµRlijkXiY jZkRljkiXiY jZkRlkijXiY jZk =000The determinant of the matrix is (λ− µ)(λ− ν)(ν − µ). If the eigenvaluesare all different then RlijkXiY jZk = 0; by the symmetries of the Riemanntensor the statement is true for the contraction of any three indices.Suppose now that λ = µ 6= ν, i.e. X and Y belong to the same eigenspace;the system of equations implies that RlkijXiY jZk = 0. This completes the first extension of the theorem, involving the Riemanntensor. The question arises about non-Codazzi symmetric tensor fields thatfulfill (1.2), and make the theorem applicable. We give some examples.4 C.A.MANTICA AND L.G.MOLINARIDefinition 2.4. A Riemannian manifold with symmetric tensor bij is “weaklyb-symmetric” if(2.2) ∇ibkl = Aibkl +Bkbil +Dlbikwhere A, B and D are 1-forms.Weakly Ricci-symmetric manifolds [17] (see [4] and [12] for a compendium):∇iRkl = AiRkl+BkRil+DlRik, and the more general Weakly Z-symmetricmanifolds [13] are of this sort.If A−B is closed, the evaluation of the three commutators in the left handside of (1.3) yields zero, then bij is Riemann compatible.3. Extension of D-S theorem for generalized curvaturetensors.The theorem can be generalized further, and continues to hold if theRiemann tensor R is replaced by a generalized curvature tensor K, and asymmetric tensor field exists such that the condition (1.2) is valid with Rreplaced by K.Definition 3.1. Let K be a generalized curvature tensor; a symmetrictensor bij is K-compatible if(3.1) Kjklmbim +Kkilmbjm +Kijlmbkm = 0.The metric tensor is trivially K-compatible, by the first Bianchi identity forK.The following lemma is needed. Its proof is identical to that of lemma 2.2,with eq.(3.1) being used:Lemma 3.2. If K is a generalized curvature tensor and bkl is K-compatible,then Kˆijkl = Kijrsbkrbls is a generalized curvature tensor.The following fact can be proven by exactly the same argument as The-orem 2.3.Theorem 3.3. Let M be a n-dimensional Riemannian manifold with ageneralized curvature tensor K and a K-compatible tensor b. If X, Y andZ are three eigenvectors of the matrix brs at a point x of the manifold, witheigenvalues λ, µ and ν, then(3.2) X i Y j ZkKijkl = 0provided that λ and µ are different from ν.EXTENDED DERDZINSKI-SHEN THEOREM FOR CURVATURE TENSORS 5Consider the following family of curvature tensors K:Kjklm =Rjklm + ϕ(δmj Rkl − δmk Rjl +Rjmgkl −Rkmgjl)+ χ(δmj gkl − δmk gjl),(3.3)where ϕ, χ are scalar functions. Appropriate choice of the scalars give theconformal, concircular or conharmonic curvature tensors.For this family, eq.(1.2) is a sufficient condition for the extended theoremto apply:Proposition 3.4. Let K be a tensor of the form (3.3). If a symmetrictensor bkl is Riemann compatible, i.e. (1.2) holds, then it is K-compatible,and Theorem 3.3 applies.Proof. The proof is based on this identity, that holds for curvature tensorsof the form (3.3):bimKjklm + bjmKkilm + bkmKijlm = bimRjklm + bjmRkilm + bkmRijlm+ϕ [gkl(bimRjm − bjmRim) + gil(bjmRkm − bkmRjm) + gjl(bkmRim − bimRkm)]K-compatibility requires the right hand side to be zero. If bkl is Riemanncompatible, then also bimRmj − bjmRmi = 0 (this is obtained by transvecting(1.2)), and all terms in the right hand side vanish. References[1] M. Berger, D. Ebin, Some characterizations of the space of symmetric tensors on aRiemannian manifold, J. Differential Geometry 3 (1969) 379-392.[2] A. L. Besse, Einstein Manifolds, Springer (1987).[3] J. P. Bourguignon, Les varie´te´s de dimension 4 a` signature non nulle dont la courbureest harmonique sont d’Einstein, Invent. Math. 63 (2) (1981) 263-286.[4] U. C. De, On weakly symmetric structures on Riemannian manifolds, Facta Univer-sitatis, Series: Mechanics, Automatic Control and Robotics 3 (14) (2003) 805-819.[5] A. Derdzinski, Some remarks on the local structure of Codazzi tensors, in “Globaldifferential geometry and global analysis”, Lecture Notes in Math. n.838, 251-255,Springer-Verlag (1981).[6] A. Derdzinski, On compact Riemannian manifolds with harmonic curvature, Math.Ann. 259, (1982) 145-152.[7] A. Derdzinski, C. L. Shen, Codazzi tensor fields, curvature and Pontryagin forms,Proc. London Math. Soc. 47 (3) (1983) 15-26.[8] D. Ferus, A remark on Codazzi tensors in constant curvature spaces, in “GlobalDifferential Geometry and Global Analysis”, Lecture Notes in Math. n.838, 257,Springer-Verlag (1981).[9] Q. Khan, On Recurrent Riemannian manifolds, Kyungpook Math. J. 44 (2004) 269-276.[10] S. Kobayashi, K. Nomizu, Foundations of differential geometry, Vol.1, Interscience,New York (1963).[11] D. Lovelock, H. Rund, Tensors, differential forms and variational principles, reprintDover Ed. (1988).6 C.A.MANTICA AND L.G.MOLINARI[12] C. A. Mantica, L. G. Molinari, A second order identity for the Riemann tensor andapplications, Colloq. Math. 122 (1) (2011) 69-82; arXiv:0802.0650 [math.DG].[13] C. A. Mantica, L. G. Molinari, Weakly Z-symmetric manifolds, in Acta Math. Hungar135 n.1-2 (2012) 80-96; arXiv:1102.5240 [math.DG].[14] M. M. Postnikov, Geometry VI, Riemannian geometry, Encyclopaedia of Mathe-matical Sciences, Vol.91 Springer (2001).[15] U. Simon, Codazzi tensors, in “Global Differential Geometry and Global Analysis”,Lecture Notes in Math. n.838, Springer-Verlag, 289-296 (1981).[16] U. Simon, A. Schwenk-Schellschmidt, L. Vrancken, Codazzi-equivalent RiemannianMetrics, Asian J. Math. 14 (3) (2010) 291-302.[17] L. Tamassy, T. Q. Binh, On weakly symmetries of Einstein and Sasakian manifolds,Tensor (N.S) 53 (1993) 140-148.I.I.S. Lagrange, Via L. Modignani 65, 20161, Milano, Italy.E-mail address : carloalberto.mantica@libero.itPhysics Department, Universita` degli Studi di Milano, Via Celoria 16,20133, Milano, Italy.E-mail address : luca.molinari@mi.infn.it

Extended Derdzinski-Shen theorem for curvature tensors

Extended Derdzinski-Shen theorem for the Riemann tensor

Extended Derdzinski-Shen theorem for the Riemann tensor

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