Extended Derdzinski-Shen theorem for the Riemann tensor
aa r X i v : . [ m a t h . DG ] J a n EXTENDED DERDZINSKI-SHEN THEOREMFOR THE RIEMANN TENSOR
CARLO ALBERTO MANTICA AND LUCA GUIDO MOLINARI
Abstract.
We extend a classical result by Derdzinski and Shen, on the re-strictions imposed on the Riemann tensor by the existence of a nontrivialCodazzi 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. Introduction
Codazzi tensors are of great interest in the geometric literature and have beenstudied by several authors, as Berger and Ebin [1], Bourguignon [3], Derdzinski[5, 6], Derdzinski and Shen [7], Ferus [8], Simon [14]; a compendium of results isreported in Besse’s book [2].In the following, M is a n ≥ g ij and Riemannian connection ∇ ; the Ricci tensor is R kl = − R mklm and the scalarcurvature is R = g ij R ij [16]. A (0 ,
2) symmetric tensor is a
Codazzi tensor if itsatisfies the Codazzi equation:(1) ∇ j b kl − ∇ k b jl = 0 . A Codazzi tensor is trivial if it is a constant multiple of the metric tensor [7]. Interms of differential forms, the Codazzi equation is the condition for closedness ofthe 1-form B j = b jk dx k , with covariant exterior differential DB j = ∇ i b jk dx i ∧ dx k [11, 3].Codazzi tensors occur naturally in the study of harmonic Riemannian manifolds.For example, the Ricci tensor is a Codazzi tensor if and only if ∇ m R jklm = 0 (bythe contracted second Bianchi identity), i.e. the manifold has harmonic Riemanncurvature [2]. The other way, the Weyl 1-form Σ j = (cid:16) R kj − R n − g kj (cid:17) dx k isclosed if and only if ∇ m C jklm = 0, where C jklm is the conformal curvature tensor[13], i.e. the manifold has harmonic conformal curvature [2].Berger and Ebin [1] proved that in a compact Riemannian manifold the Codazzitensors are parallel if the sectional curvature is non-negative and if there is a pointwhere it is positive. In ref.[3] the important geometric and topological consequencesof the existence of a non-trivial Codazzi tensor are examined, particularly the re-strictions imposed on the structure of the curvature operator. Derdzinki and Shenimproved these results and proved a remarkable theorem on the consequences ofthe existence of a non trivial Codazzi tensor [7]. The theorem is reported also inBesse’s book [2]. Mathematics Subject Classification.
Primary 53B20, Secondary 53B21.
Key words and phrases.
Codazzi tensor, Riemann tensor, generalized curvature tensor, recur-rent 1-forms.
Theorem 1.1 (Derdzinski-Shen) . Let b ij be a Codazzi tensor on a Riemannianmanifold M , x a point of M , λ and µ two eigenvalues of the operator b ij ( x ) , witheigenspaces V λ and V µ in T x M . Then, the subspace V λ ∧ V µ is invariant under theaction of the curvature operator R x . The theorem can be rephrased as follows: given eigenvalues λ , µ , ν of b ij ( x ) andcorresponding eigenvectors X , Y , Z in T x M , it is R ( X, Y ) Z = 0, provided that λ and µ are different from ν .We point out that the Codazzi equation is a sufficient condition for the theoremto hold. A more general condition is suggested by the lemma: Lemma 1.2.
Any (0 , symmetric Codazzi tensor b kl satisfies the algebraic iden-tity: (2) b im R jklm + b jm R kilm + b km R ijlm = 0 . Proof.
The following condition involving commutators (indices are cyclically per-muted) is true for a Codazzi tensor:(3) [ ∇ j , ∇ k ] b il + [ ∇ k , ∇ i ] b lj + [ ∇ i , ∇ j ] b kl = 0Each commutator is evaluated: [ ∇ i , ∇ j ] b kl = R ijkm b ml + R ijlm b km . Three termscancel by the first Bianchi identity, and the result is obtained. (cid:3) We note in passing that also the following equation is solved by Codazzi tensors:(4) [ ∇ i , ∇ j ] b kl + [ ∇ j , ∇ k ] b li + [ ∇ k , ∇ l ] b ij + [ ∇ l , ∇ i ] b jk = 0By evaluating the commutators and simplifying the result by means of the firstBianchi identity and the Codazzi property, the equation gives:(5) R kij m b ml + R jlim b mk + R ljkm b mi + R iklm b mj = 0We are ready to formulate the main theorem of the paper: it states that if asymmetric tensor b kl satisfies the algebraic identity (2), then the same conclusionsof the Derdzinski-Shen theorem are valid. Moreover, the proof is much simpler.2. An extension of the Derdzinski-Shen theorem.
Definition 2.1.
A (0 ,
4) tensor K ijlm is a generalized curvature tensor [10] if ithas the symmetries of the Riemann curvature tensor:a) K ijkl = − K jikl = − K ijlk ,b) K ijkl = K klij ,c) K ijkl + K jkil + K kijl = 0 (first Bianchi identity). Lemma 2.2.
If a symmetric tensor b kl satisfies eq.(2), then K ijkl = R ijrs b kr b ls isa generalized curvature tensor.Proof. Properties a) are shown easily. For example: K ijlk = R ijrs b lr b ks = R ijsr b ls b kr = − R ijrs b ls b kr = − K ijkl .Property c) follows from the condition (2): K ijkl + K jkil + K kijl = R ijrs b kr b ls + R jkrs b ir b ls + R kirs b jr b ls = ( R jisr b kr + R kjsr b ir + R iksr b jr ) b ls = 0.Property b) follows from c): K ijkl + K jkil + K kijl = 0. Sum the identity over cyclicpermutations of all indices i, j, k, l and use the symmetries a) (this fact was pointedout in [10]). XTENDED DERDZINSKI-SHEN THEOREM FOR THE RIEMANN TENSOR 3
It is easy to see that a first Bianchi identity holds also for the last three indices: K ijkl + K iklj + K iljk = 0. (cid:3) Theorem 2.3 (main theorem) . Let M be a n -dimensional Riemannian manifold.If a (0 , symmetric tensor b kl satisfies the algebraic equation b im R jklm + b jm R kilm + b km R ijlm = 0 then, for any x ∈ M , arbitrary eigenvalues λ and µ of b ij ( x ) , with eigenspaces V λ and V µ in T x M , the subspace V λ ∧ V µ is invariant under the action of the curvatureoperator R x . The theorem can be rephrased as follows: If b rs is a symmetric tensor with the property (2) and X , Y and Z are three eigen-vectors of the matrix b rs at a point x of the manifold, with eigenvalues λ , µ and ν ,then (6) X i Y j Z k R ijkl = 0 provided that λ and µ are different from ν .Proof. Consider the first Bianchi identity for the Riemann tensor, the conditioneq.(2) and the third Bianchi identity for the curvature K ℓijk = R ℓirs b jr b ks , andapply them to the three eigenvectors. The resulting algebraic relations can be putin matrix form:(7) λ µ νµν λν λµ R ℓijk X i Y j Z k R ℓjki X i Y j Z k R ℓkij X i Y j Z z = The determinant of the matrix is ( λ − µ )( λ − ν )( ν − µ ). If the eigenvalues areall different then R ℓijk X i Y j Z k = 0; by the symmetries of the Riemann tensor thestatement is true for the contraction of any three indices.Suppose now that λ = µ = ν , i.e. X and Y belong to the same eigenspace; thesystem of equations implies that R ℓkij X i Y j Z k = 0. (cid:3) Proposition 2.4.
The hypothesis (2) of the main theorem is fulfilled if the symmet-ric tensor b ij is a Codazzi tensor or, more generally, if it solves a gauged Codazziequation(8) ( ∇ k − β k ) b ij = ( ∇ i − β i ) b kj where the covariant gauge field β k is closed: ∇ k β j − ∇ j β k = 0 .Proof. The proof is straightforward and may start from eq.(3) by noting that theclosedness condition for the gauge field ensures that [ ∇ i − β i , ∇ j − β j ] = [ ∇ i , ∇ j ].Then eq.(3) is again true and eq.(2) follows. (cid:3) The gauged Codazzi equation (8) can be interpreted in terms of differentialforms.
CARLO ALBERTO MANTICA AND LUCA GUIDO MOLINARI Recurrent tensors forms
Definition 3.1.
A 1-form B j = b kj dx k is recurrent if there is a nonzero scalar1-form β = β i dx i such that(9) DB j = β ∧ B j . In local components it is ( ∇ i − β i ) b kl ( dx i ∧ dx k ) = 0. Therefore, the 1-form B j = b kj dx k is recurrent if and only if [11](10) ( ∇ i − β i ) b kl = ( ∇ k − β k ) b il . If β = 0 the closedness of the 1-form B j = b kj dx k and the Codazzi equation arerecovered. Eq.(10) enlarges the ordinary notion of recurrence, by which a tensoris recurrent if ∇ i b kl = β i b kl . An example of ordinary recurrence are the Riccirecurrent spaces, where ∇ i R kl = β i R kl [9].The extended recurrence eq.(10) includes well known differential structures, as the Weakly b symmetric manifolds, defined by the condition(11) ∇ i b kl = A i b kl + B k b il + D l b ik . For these manifolds, eq.(10) is satisfied with the choice β i = A i − B i and, if thecovector β i is a closed 1-form, the main theorem applies. Weakly Ricci symmetricmanifolds (see [4] and [12] for a compendium) are of this sort:(12) ∇ i R kl = A i R kl + B k R il + D l R ik . They were introduced by Tamassy and Binh [15].
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XTENDED DERDZINSKI-SHEN THEOREM FOR THE RIEMANN TENSOR 5
Physics Department, Universit´a degli Studi di Milano and I.N.F.N. sez. Milano, ViaCeloria 16, 20133 Milano, Italy., Present address of C. A. Mantica: I.I.S. Lagrange,Via L. Modignani 65, 20161, Milano, Italy
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