Conformal Killing L 2 − forms on complete Riemannian manifolds with nonpositive curvature operator
CConformal Killing -forms on complete Riemannian manifolds with nonpositive curvature operator L S ERGEY S TEPANOV
AND I RINA T SYGANOK
We give a classification for connected complete locally irreducible Riemannian manifolds with nonpositive curvature operator, which admit a nonzero closed or co-closed conformal Killing -form. Moreover, we L prove vanishing theorems for closed and co-closed conformal Killing L -forms on some complete Riemannian manifolds.
1. Introduction and results
Conformal Killing forms (also called as conformal Killing-Yano tensors) have been defined on Riemannian manifolds more than forty-five years ago by S. Tachibana and T. Kashiwada (see [24] and [9]) as a natural generalization of conformal Killing vector fields. We also know from the literature about closed conformal Killing forms or, otherwise, closed conformal Killing-Yano tensors and co-closed conformal Killing forms or, otherwise, Killing-Yano tensors (see, for example, [7, pp. 426-427]; [18; pp. 559-564]). We remark here that the Hodge dual of a co-closed Killing form is a closed conformal Killing form. Moreover, the converse is also true (see [14]). Surveys of the publications on conformal Killing, co-closed and closed conformal Killing forms and their numerous applications can be found in the introductions to our papers [19] and [20]. In addition, it should be taken into account the list of recent papers on these forms: [4]; [6]; [13]; [21]; [22] and [27]. In the present paper we consider conformal Killing, co-closed and closed conformal Killing L -forms of degree p for −≤≤ np on a simply connected and complete Riemannian manifold ( ) gM, of dimension for . n ≥ n Here we regard the Riemannian curvature tensor Rm of as a symmetric ( gM, ) algebraic operator : R ( ) ( ) MTMT xx Λ→Λ on the vector space ( ) MT x Λ of -2 forms over tangent space at an arbitrary MT x Mx ∈ (see [11, pp. 36-37]). We say that the manifold has a nonpositive (respectively nonnegative) curvature ( gM, ) perator R if ( ) ( ) ≤θθ ,Rg (respectively ( ) ( ) ≥θθ ,Rg ) for all two-forms 0 ≠ θ . There have been many papers on the relationship between the curvature operator R of a Riemannian manifold ( ) gM, and some global characterization of it, such as its homotopy type, topological types and etc. In connection with above, the first our result on conformal Killing forms will be proved by the most important analytic method of differential geometry “in the large” which derived by S. Bochner for proving so-called vanishing theorems under appropriate curvature conditions on compact Riemannian manifolds (see [28]). S.-T. Yau generalized this method of proving vanishing theorems for the case of complete noncompact Riemannian manifolds (see, for example, [29]). We use a generalization of the “Bochner technique” to prove the following statement. Lemma 1.
Suppose the curvature operator R is negative semi-define at every point of a complete non-compact Riemannian manifold ( ) gM, . Then every closed or co-closed conformal Killing L -form on ( ) gM, is a parallel form. If either the volume of ( is infinite, then every closed or co-closed conformal Killing ) gM, L -form on is identically zero. ( gM, ) If ω is a parallel form on ( ) gM, then x ω is invariant under the holonomy representation ( ) x Hol at each point Mx ∈ . Therefore, finding on a Riemannian manifold ( parallel forms is equivalent to the algebraic problem of finding the ) gM, invariant forms of the holonomy representation ( ) g Hol . We give an example of the implementation of this principle, which is the main result of the present paper.
Theorem 1 . Let a simply connected complete non-compact Riemannian manifold ( gM, ) with nonpositive curvature operator admits a non-zero not decomposable closed or co-closed conformal Killing L -form. If in addition, we assume that ( gM, ) is a locally irreducible Riemannian manifold, then it is a manifold of the following list: globally symmetric manifolds of non-compact type, Kähler and quaternionic-Kähler manifolds. emark 1 . In addition, we recall that an arbitrary co-closed conformal Killing p -form ω on an -dimension compact Kählerian manifold is parallel for n −≤≤ np (see [25]). From this statement follows that an arbitrary closed conformal Killing -form ( pn − ) ω on an -dimension compact Kählerian n manifold is parallel as well. In particular, for pM = we prove the following lemma using a generalization of the “Bochner technique”. Lemma 2.
Suppose the curvature operator R is negative semi-define at every point of a -p dimensional complete non-compact Riemannian manifold ( ) gM, . Then every closed or co-closed conformal Killing L -form of degree p is parallel form on ( . If either the volume of ) gM, ( ) gM, is infinite, then every closed or co-closed conformal Killing L -form of degree p is identically zero on . ( ) gM,
Remark 2 . In turn, we have proved that an arbitrary conformal Killing p -form ( −≤≤ np ) is parallel on an -dimension compact Riemannian manifold with n nonpositive curvature operator (see [17]). The following theorem is an analogue of Theorem 1. Theorem 2 . Let a -p dimensional simply connected complete non-compact Riemannian manifold with nonpositive curvature operator admits a non- ( gM, ) zero not decomposable conformal Killing L -form of degree p . If in addition, we assume that is a locally irreducible, then it is a manifold of the following ( gM, ) list: globally symmetric manifolds of non-compact type, Kähler and quaternionic-Kähler manifolds. We can formulate the converse statement in one particular case.
Corollary 1.
A globally symmetric space of non-compact type ( with infinite ) gM, volume does not admit a nonzero conformal Killing -L form. Remark 3 . In addition, we know that if a symmetric space ( gM, ) of compact type carries a non-parallel co-closed conformal Killing p -form for 2 ≥ p , then its niversal cover M~ is either a round sphere, or has a factor isometric to a round sphere in its de Rham decomposition (see [12]).An -dimensional Riemannian manifold п ( ≥ n ) ( ) gM, is generic (see [1, p. 291]) if each point Mx ∈ admits the orthonormal base { } i е of the tangent space T M x such that the ( ) − nn endomorphisms ( ) ji e,eR for the curvature tensor R of ( gM, ) will be linearly independent in the Lie algebra of the orthogonal group ( ) nO . In this case, the corollary is true. Corollary 2 . A generic complete Riemannian manifold with nonpositive curvature operator does not admit nonzero closed and co-closed conformal Killing L -forms.
In [1, p. 419] was proved the statement on Kähler manifold with parallel form that is not proportional to any degree of the Kähler 2-form. Here we interpret this statement as the following
Corollary 3 . Let a complete non-compact locally irreducible Kähler manifold admits a nonzero irreducible closed or co-closed conformal Killing L -form ω . If the curvature operator ≤ R everywhere on the Kähler manifold and ω is not some power of the Kähler form then the Kähler manifold is an Einstein manifold.
2. Preliminary information
Let M p Λ be the bundle of differential p -forms over a connected complete Riemannian manifold ( of dimensions ) g,M ≥ n and ( ) MC p Λ ∞ be the space of ∞ C -sections of the bundle M p Λ for 11 −≤≤ np . By ∇ we will denote the Levi-Civita connection on ( g,M ) and by ∗ ∇ the canonical formal adjoint of ∇ . In addition, let ( ) ( ) MCMCd pp : +∞∞ Λ→Λ be the well known operator of exterior derivative and ( ) ( )
MCMCd pp : −∞∞∗ Λ→Λ be the codifferentiation operator which defined as the canonical formal adjoint of d . We recall that if 0 = ω d , then the p -form ω ( ) MC p Λ∈ ∞ is said to be closed . In tern, if , then the 0 = ∗ ω d p -form ω ( ) MC p Λ∈ ∞ is said to be co-closed . Using these operators, one constructs he well-known Hodge-de Rham Laplacian which admits a ∗∗ +=Δ dddd Weitzenböck decomposition (see [1, p. 53]; [8, p. 144]; [11, p. 211]) (2.1) ( ) ωωω E +∇∇=Δ ∗ for any ( ) MC p Λ∈ ∞ ω and an algebraic symmetric operator MME pp Λ→Λ : that depends linearly in a known way (see, for example, [23]; [26]) on the Riemannian curvature tensor Rm and the Ricci tensor Ric of ( ) gM, . Further, a direct calculation based on (2.1) yields the well-known Bochner-Weitzenböck formula (see, for example, [26]) (2.2) ( ) ( )( ωωωωωω ,Eg,g −∇−Δ=Δ ) . Let be locally defined orthonormal frame fields of the tangent bundle { n e,...,e } TM and { n ,..., } ωω be its dual coframe fields. Then in terms of the basis, we can write pp iii...i ...a ωωω ∧∧= where the summation is being performed over the multi-index ( ) p i,...,iI = for ni,...,i p ≤≤ ( )( ) =ωω ,Eg ( ( ) ) ωω ,Rg for the -form ααα ωωω iji...j...i p a ∧= . J.-P. Bourguignon in [2] proved the existence of the basis { } D,dd, ∗ in the space of natural (with respect to isometric diffeomorphisms) first-order differential operators on the space ( ) MC p Λ ∞ with value in the space of homogeneous tensor on ( gM, ) . The third basis operator was not given explicitly. In our papers [14] D and [15] we used some classical theorems of H. Weyl about the representation theory of the orthogonal group ( ) nO and shown that the third operator has the D form ∗ ∧−−∇= +−+ dgdD pnp here ( ) ( ) ( ) ( ) ( ) ( ) paapa aap X,...,X,X,...,XdX,XgX,...,X,Xdg +−= ∗∗ ∑ −=∧ ω for any ( ) MC p Λ∈ ∞ ω and ( ) TMCX,...,X p ∞ ∈ . We also proved that the kernel of consists of conformal Killing D p -forms. Therefore, a closed conformal Killing form ω is determined by the conditions 0 = ω D and 0 = ω d . In turn, the conditions 0 = ω D and define a co-closed conformal Killing form 0 = ∗ ω d ω . In addition, we proved in [14] that if ω is a closed conformal Killing -p form then ω∗ must be a co-closed conformal Killing ( ) -pn − form for the Hodge star operator MM pnp − Λ→Λ∗ : , which is a familiar isomorphism of vector bundles M p Λ and M p-n Λ (see, for example, [1, p. 33]). Moreover, the converse is also true (see also [14]). In [16] we found the operator formally adjoint to and then constructed the ∗ D D second-order differential operator (2.4) ⎟⎠⎞⎜⎝⎛ −−∇∇= ∗∗∗∗ +−++ ddddDD pnpp p with 11 −≤≤ np . Properties of the operator were studied in the following DD ∗ papers [16]; [19]; [20] and [21]. In particular, we proved in [16] that i DD ∗ s a second-order self-adjoint elliptic differential operator acting on ( ) MC p Λ ∞ .
3. Proofs of the statements
A direct calculation yields the second inequality of Kato (see [2, p. 380]) (3.1) ( ) ωωωω ,g ∇∇≤Δ ∗ for an arbitrary form ( ) MC p Λ∈ ∞ ω . By virtue of the Weitzenböck decomposition (2.1), the Kato inequality (3.1) can be written in the form (3.2) ( ) ( )( ) ωωωωωω ,Eg,g −Δ≤Δ . At the same time, by the Weitzenböck decomposition (2.1), the operator can DD ∗ be written in the following form 3.3) ( ) ( )( ) ⎟⎠⎞⎜⎝⎛ −−Δ= ∗∗ +−+ −+− −+ ωωωω ddEDD pnp pnpn pnp p
11 211 for an arbitrary -p form ( ) MC p Λ∈ ∞ ω . Then from (3.3) we obtain equation (3.4) ( ) ωω E pn pn − +− =Δ for a closed conformal Killing p -form ω . Using (3.4) we write the inequality (3.2) in the form (3.5) ( )( ) ωωωω ,Eg pn − ≤Δ for a closed conformal Killing p -form ω . If the curvature operator is nonpositive, then based on (2.3) we obtain from (3.5) the inequality ( ) ≥Δ− ωω . For a Riemannian manifold ( ) gM, we have the “natural” Hilbert space ( ) ( ) g d,MLML Vol = where g Vol is the Riemannian volume form associated to the metric g . In our case, assume that ω is a L -form on a complete , then ( gM, ) by Yau theorem (see [29, p. 664]) the function ω is constant for an arbitrary closed conformal Killing form ω . Then it follows from (3.5) that ( )( ) =ωω ,Eg . In this case, we obtain from (3.4) that ( ) =Δ ωω ,g . Finally, we obtain from the Weitzenböck formula (2.2) that 0 =∇ ω for an arbitrary closed conformal Killing L -form ω on a complete manifold ( ) gM, with nonpositive curvature operator. Then ω∗ is a co-closed conformal Killing form whose covariant derivative vanishes. In particular, it means that const =∗ ω . In this case, we have ( ) ∞< M g Vol for the volume ( ) M g Vol of ( ) gM, because ∫ ∞< M g d Vol ω for an arbitrary L -form ω . Therefore, if the volume ( ) M g Vol of a complete manifold ( gM, ) with nonpositive curvature operator is infinite then ( ) gM, does not admit a onzero closed (resp. co-closed) conformal Killing − L form ω . This completes the proof of Lemma 1. We recall that the description of all the connected complete locally irreducible Riemannian manifolds with a form whose covariant derivative vanishes is given in Corollary 10.110 from the monograph [1, p. 306-307]. Namely, the following statement holds. Let ( gM, ) be a connected complete locally irreducible Riemannian manifold admitting a nonzero and not decomposable form ω whose covariant derivative is vanishing. Then ( ) gM, should fall in one of the following categories: locally symmetric manifolds, Kähler manifolds, quaternionic-Kähler manifolds, manifolds with holonomy group or G ( ) Spin . We recall that any complete, simply-connected locally symmetric space is globally symmetric. In addition, a Riemannian globally symmetric space of non-compact type has a nonpositive sectional curvature. Note that symmetric spaces of non-compact type are non-compact. Now the assertion about symmetric spaces in our theorem becomes obvious. Next, we note here that manifolds with holonomy group or G ( ) Spin are Ricci-flat manifolds. On the other hand, every non-compact complete Riemannian manifold with non-negative Ricci curvature has infinite volume (see [29, p. 667]). Therefore, any non-compact complete Riemannian manifolds with holonomy group or G ( ) Spin do not admit a not decomposable parallel L -form. These remarks complete the proof of our Theorem 1. Remark 4 . S.-T. Yau used the Laplacian Δ in the form without the ∇=Δ g trace minus sing (see [29]) that we took into account in our proof. In particular, if pn = we obtain from (3.4) that (3.7) ( ) ⎟⎠⎞⎜⎝⎛ −Δ= ++ ∗ ωωω EDD p pp p . Suppose that ω is a conformal Killing -p form, then it follows from (3.7) that . T ( ) ( ωω Epp − +=Δ ) hen based on (2.3) we obtain the inequality ( ) ≥Δ− ωω if ≤ R everywhere on ( ) gM, . Further, it can be shown that =∇ ω if ω is a conformal Killing L -form on a complete manifold with ( gM, ) nonpositive curvature operator. In this case, the assertions of Lemma 2 and Theorem 2 become obvious. A Riemannian globally symmetric space ( ) gM, is complete. We also know that a Riemannian symmetric space has nonpositive curvature operator if and only if it has nonpositive sectional curvature (see [5]). After the above remarks, the assertion of Corollary 1 becomes obvious. Let be the restricted holonomy group of ( ) x Hol ( ) g,M at an arbitrary point Mx ∈ (see [1, p. 280]). Note that ( ) x Hol is always contained in special orthogonal group . But for a generic Riemannian manifold ( ) nSO Hol ( ) nSO = at all its points (see [1, p. 291]). In this case, there is no a nontrivial parallel form ω , i.e. there is no a nontrivial form ω with zero covariant derivative (see [1, p. 306]). This implies the validity of Corollary 2. In [1, p. 307]) was proved the following statement: “Let ( ) g,M be a Riemannian manifold admitting an exterior form ω whose covariant derivative is vanishing. Assume now that ω is not zero, not decomposable and not some power of the Kähler form if is Kähler. Then ( g,M ) ( ) g,M is automatically an Einstein manifold”. Our Corollary 3 is a corollary of Theorem 1 and this statement.
4. Appendix
A form ω ( ) MC p Λ∈ ∞ is said to be harmonic if 0 =Δ ω . According to Yau proposition (see [29, p. 663]) an L -form ω on a complete Riemannian manifold ( g,M ) is harmonic if and only if it is closed and co-closed. The Bochner-Weitzenböck formula (2.2) implies that any harmonic p -form ω satisfies the inequality (4.1) ( )( ) ≥+∇=Δ− ωωωω ,Eg . n ( g,M ) with the nonnegative curvature operator R . Then by Yau theorem (see [29, p. 663]) the function ω is constant for an arbitrary harmonic form ω on a complete Riemannian manifold ( ) g,M with 0 ≥ R . An elementary linear algebra argument show that if the curvature operator in non-negative, then all the sectional curvatures of ( ) g,M are also non-negative. Then the Ricci curvature of ( ) g,M is non-negative as well. In this case, the volume ( ) M g Vol is infinite if ( is a ) gM, non-compact complete manifold (see [29, p. 667]). As a result, a complete non-compact Riemannian manifold ( ) gM, with non-negative curvature operator does not admit a nonzero harmonic − L form ω . Therefore, we can not formulate a theorem on harmonic forms which is an analog of our Theorem 1. But the following statements hold. Corollary 4.
A globally symmetric space of compact type ( ) gM, does not admit a non-parallel harmonic form.
Proof.
It is well known that a Riemannian globally symmetric space of compact type ( gM, ) has a nonpositive sectional curvature and also ( ) gM, is compact. Then from (4.1) we obtain const = ω for an arbitrary harmonic form ω on a Riemannian globally symmetric space of compact type ( ) gM, . In this case, the covariant derivative of ω is vanishing. Corollary 5 . A generic complete Riemannian manifold with non-negative curvature operator does not admit nonzero harmonic L -forms. Proof.
It is well known that for a generic Riemannian manifold Hol ( ) nSO = at all its points (see [1, p. 291]). In this case, there is no a nontrivial form ω with zero covariant derivative (see [1, p. 306]). On the other hand, from (4.1) we obtain 0 =∇ ω for an arbitrary harmonic L - forms ω on a complete Riemannian manifold with non-negative curvature operator. This contradiction proves Corollary 5. Acknowledgments ur work was supported by RBRF grant 16-01-00053- а (Russia). References [1] Becce A., Einstein manifolds, Springer-Verlag, Berline-Heidelberg (1987). [2] Bérard P.H., From vanishing theorems to estimating theorems: the Bochner technique revisited, Bulletin of the American Mathematical Society (1988) no. 2, 371-406. [3] Bourguignonj J.-P., Formules de Weitzenbock en dimension 4, Geometrie riemannienne en dimension 4, Semin. Arthur Besse 1978/79, Paris, Cedic 3 (1981), 308-333. [4] David L., The conformal-Killing equation on - and -structures, Journal of Geometry and Physics (2011), no. 6, 1070-1078. G Spin [5] Duchesne B., Infinite dimensional Riemannian symmetric spaces with fixed-sing curvature operator, Ann. Inst. Fourier, Grenoble (2015), no. 1, 211-244. [6] Ertem U., Lie algebra of conformal Killing-Yano forms, Classical and Quantum Gravity (2016), no. 12, 125033 [13 pp.]. [7] Frolov V.P., Zelenkov A., Introduction to Black Hole Physics, Oxford, OUP Oxford (2011). [8] Jost J., Riemannian geometry and Geometric Analysis, Berlin, Springer-Verlag (2011). [9] Kashiwada T., The curvature operator of the second kind, Natural Science Report, Ochanomizu University (1993), no. 2, 69-73. [10] Li P., Geometric Analysis, Cambridge, Cambridge University Press (2012). [11] Petersen P., Riemannian Geometry, NY, Springer (2006). [12] Semmelmann U., Belgum F., Moroianu A., Killing forms on symmetric spaces, Differential Geometry and its Applications (2006), 215-222. [13] Slesar V., Visinescu M., Vilcu G.E., Special Killing forms on toric Sasaki-Einstein manifolds, Physica Scripta (2014), no. 12, 125205 [15 pp.].
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