Biharmonic Hermitian vector bundles over compact Kaehlar Einstein manifolds
aa r X i v : . [ m a t h . DG ] M a y Biharmonic Hermitian vector bundles overcompact K¨ahler Einstein manifolds
Hajime Urakawa
Tohoku University, Graduate School of Information Sciences, Division of MathematicsAoba 6-3-09, Sendai 980-8579, Japan, [email protected]
May 23, 2019
Abstract
In this paper, we show that, for every Hermitian vector bundle π : ( E, g ) → ( M, h )over a compact K¨ahler Einstein manifold (
M, h ), if the projection π is biharmonic,then it is harmonic. Research of harmonic maps, which are critical points of the energy functional, is one ofthe central problems in differential geometry including minimal submanifolds. The Euler-Lagrange equation is given by the vanishing of the tension field. In 1983, Eells and Lemaire([EL]) proposed to study biharmonic maps, which are critical points of the bienergy func-tional, by definition, half of the integral of square of the norm of tension field τ ( ϕ ) for asmooth map ϕ of a Riemannian manifold ( M, g ) into another Riemannian manifold (
N, h ).After a work by G.Y. Jiang [J], several geometers have studied biharmonic maps (see [CMP],[IIU1], [IIU2], [II], [LO], [MO], [OT2], [S], etc.). Note that a harmonic maps is always bi-harmonic. One of central problems is to ask whether the converse is true.
B.-Y. Chen’sconjecture is to ask whether every biharmonic submanifold of the Euclidean space R n mustbe harmonic, i.e., minimal ([C]). There are many works supporting this conjecture ([D],[HV], [KU], [AM]). However, B.-Y. Chen’s conjecture is still open. R. Caddeo, S. Montaldo,P. Piu ([CMP]) and C. Oniciuc ([On]) raised the generalized B.-Y. Chen’s conjecture to ask1hether each biharmonic submanifold in a Riemannian manifold ( N, h ) of non-positive sec-tional curvature must be harmonic (minimal). For the generalized Chen’s conjecture, Ouand Tang gave ([OT1], [OT2]) a counter example in some Riemannian manifold of negativesectional curvature. But, it is also known (cf. [NU1], [NU2], [NUG]) that every biharmonicmap of a complete Riemannian manifold into another Riemannian manifold of non-positivesectional curvature with finite energy and finite bienergy must be harmonic. For the targetRiemannian manifold (
N, h ) of non-negative sectional curvature, theories of biharmonic mapsand biharmonic immersions seems to be quite different from the case (
N, h ) of non-positivesectional curvature. There exit biharmonic submanifolds which is not harmonic in the unitsphere. S. Ohno, T. Sakai and myself [OSU1], [OSU2] determined (1) all the biharmonichypersurfaces in irreducible symmetric spaces of compact type which are regular orbits ofcommutative Hermann actions of cohomogeneity one, and gave (2) a complete table of allthe proper biharmonic singular orbits of commutative Hermann actions of cohomogeneitytwo, and (3) a complete list of all the proper biharmonic regular orbits of ( K × K )-actionsof cohomogeneity one on G for every commutative compact symmetric triad ( G, K , K ). Wenote that recently Inoguchi and Sasahara ([IS]) also investigated biharmonic homogeneoushypersurfaces in compact symmetric spaces, and Ohno studied biharmonic orbits of isotropyrepresentations of symmetric spaces in the sphere (cf. [Oh1], [Oh2]).In this paper, we treat with an Hermitian vector bundle ( E, g ) → ( M, h ) over a compactRiemannian manifold (
M, h ). We assume (
M, h ) is a compact K¨ahler Einstein Riemannianmanifold, that is, the Ricci transform Ric h of the K¨ahler metric h on M satisfies Ric h = c Id,for some constant c . Then, we show the following: Theorem 1.1.
Let π : ( E, g ) → ( M, h ) be an Hermitian vector bundle over a compactK¨ahler Einstein Riemannian manifold ( M, h ) . If π is biharmonic, then it is harmonic. Theorem 1.1 shows the sharp contrasts on the biharmonicities between the case of vectorbundles and the one of the principle G -bundles. Indeed, we treated with the biharmonicityof the projection of the principal G -bundle over a Riemannian manifold ( M, h ) with negativedefinite Ricci tensor field (cf. Theorem 2.3 in [U4]). We also gave an example of the projectionof the principal G -bundle over a Riemannian manifold ( M, h ) which is biharmonic but notharmonic (cf. Theorem 5 in [U5]). 2
Preliminaries.
In this section, we prepare necessary materials for the first and second variational formulasfor the bienergy functional and biharmonic maps. Let us recall the definition of a harmonicmap ϕ : ( M, g ) → ( N, h ), of a compact Riemannian manifold (
M, g ) into a Riemannianmanifold (
N, h ), which is an extremal of the energy functional defined by E ( ϕ ) = Z M e ( ϕ ) v g , where e ( ϕ ) := | dϕ | is called the energy density of ϕ . That is, for any variation { ϕ t } of ϕ with ϕ = ϕ , ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 E ( ϕ t ) = − Z M h ( τ ( ϕ ) , V ) v g = 0 , (2.1)where V ∈ Γ( ϕ − T N ) is the variation vector field along ϕ which is given by V ( x ) = ddt | t =0 ϕ t ( x ) ∈ T ϕ ( x ) N , ( x ∈ M ), and the tension field is given by τ ( ϕ ) = P mi =1 B ( ϕ )( e i , e i ) ∈ Γ( ϕ − T N ), where { e i } mi =1 is a locally defined frame field on ( M, g ), and B ( ϕ ) is the secondfundamental form of ϕ defined by B ( ϕ )( X, Y ) = ( e ∇ dϕ )( X, Y )= ( e ∇ X dϕ )( Y )= ∇ X ( dϕ ( Y )) − dϕ ( ∇ X Y )= ∇ hdϕ ( X ) dϕ ( Y ) − dϕ ( ∇ X Y ) , (2.2)for all vector fields X, Y ∈ X ( M ). Furthermore, ∇ , and ∇ h , are the Levi-Civita connectionson T M , T N of (
M, g ), (
N, h ), respectively, and ∇ , and e ∇ are the induced ones on ϕ − T N ,and T ∗ M ⊗ ϕ − T N , respectively. By (2.1), ϕ is harmonic if and only if τ ( ϕ ) = 0.The second variation formula is given as follows. Assume that ϕ is harmonic. Then, d dt (cid:12)(cid:12)(cid:12)(cid:12) t =0 E ( ϕ t ) = Z M h ( J ( V ) , V ) v g , (2.3)where J is an elliptic differential operator, called Jacobi operator acting on Γ( ϕ − T N ) givenby J ( V ) = ∆ V − R ( V ) , (2.4)where ∆ V = ∇ ∗ ∇ V = − P mi =1 {∇ e i ∇ e i V − ∇ ∇ ei e i V } is the rough Laplacian and R is a linearoperator on Γ( ϕ − T N ) given by R ( V ) = P mi =1 R h ( V, dϕ ( e i )) dϕ ( e i ), and R h is the curvaturetensor of ( N, h ) given by R h ( U, V ) = ∇ hU ∇ hV − ∇ hV ∇ hU − ∇ h [ U,V ] for U, V ∈ X ( N ).3. Eells and L. Lemaire [EL] proposed polyharmonic ( k -harmonic) maps and Jiang [J]studied the first and second variation formulas of biharmonic maps. Let us consider the bienergy functional defined by E ( ϕ ) = 12 Z M | τ ( ϕ ) | v g , (2.5)where | V | = h ( V, V ), V ∈ Γ( ϕ − T N ).Then, the first variation formula of the bienergy functional is given as follows.
Theorem 2.1. ( the first variation formula ) ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 E ( ϕ t ) = − Z M h ( τ ( ϕ ) , V ) v g . (2.6) Here, τ ( ϕ ) := J ( τ ( ϕ )) = ∆ τ ( ϕ ) − R ( τ ( ϕ )) , (2.7) which is called the bitension field of ϕ , and J is given in (2 . . Definition 2.2.
A smooth map ϕ of M into N is said to be biharmonic if τ ( ϕ ) = 0. To prove Theorem 1.1, we need the following:
Proposition 3.1.
Let π : ( E, g ) → ( M, g ) be an Hermitian vector bundle over a compactK¨ahler Einstein manifold ( M, h ) . Assume that π is biharmonic. Then the following hold: (1) The tension field τ ( π ) satisfies that ∇ X τ ( π ) = 0 ( ∀ X ∈ X ( M )) . (3.1)(2) The pointwise inner product h τ ( π ) , τ ( π ) i = | τ ( π ) | is constant on ( M, g ) , say d ≥ . (3) The bitension field τ ( π ) satisfies that τ ( π ) := Z M | τ ( π ) | v h = d Vol(
M, h ) . (3.2)4y Proposition 3.1, Theorem 1.1 can be proved as follows. Assume that π : ( E, g ) → ( M, h ) is biharmonic. Due to (3.1) in Proposition 3.1, we havediv( τ ( π )) := n X i =1 ( ∇ e ′ i τ ( π ))( e ′ i ) = 0 , (3.3)where { e ′ i } ni =1 is a locally defined orthonormal frame field on ( M, h ) and we put n = dim R M .Then, for every f ∈ C ∞ ( M ), it holds that, due to Proposition (3.29) in [U1], p. 60, forexample, 0 = Z M f div( τ ( π )) v h = − Z M h ( ∇ f, τ ( π )) v h . (3.4)Therefore, we obtain τ ( π ) ≡ M, h ): Example 3.2.
A generalized flag manifold
G/H admits a unique K¨ahler Einstein metric h ([BH] and [CS]). Here, G is a compact semi-simple Lie group, and H is the centralizer of atorus S in G , i.e., G C is the complexification of G , and B is its Borel subgroup. Then, M = G/H = G C /B. The Borel subgroup B is written as B = T N , where T is a maximal torus of B and N is a nilpotent Lie subgroup of B . Every character ξ λ of a Borel subgroup B is given as ahomomorphism ξ λ : B → C ∗ = C − { } which is written as ξ λ ( tn ) = ξ λ ( t ) ( t ∈ T, n ∈ N ) . (3.5)Here ξ λ : T → U (1) is a character of T which is written as ξ λ (exp( θ H + · · · + θ ℓ H ℓ )) = e π √− k θ + ··· + k ℓ θ ℓ ) , ( θ , . . . , θ ℓ ∈ R ) , (3.6)where k , . . . , k ℓ are non-negative integers, and ℓ = dim T .Note that every character ξ λ of a nilpotent Lie group N must be ξ λ ( n ) = 1 because ξ λ ( n ) = ξ λ (exp X ) = e ξ λ ′ ( X ) where n = exp X ( X ∈ n ), and λ ′ : t → C is a homomorphism,i.e., ξ λ ′ ( X + Y ) = ξ λ ′ ( X ) + ξ λ ′ ( Y ), ( X, Y ∈ t ). Then, there exists k ∈ N which satisfies thatexp( k X ) = n k = e . Then, e k ξ λ ′ ( X ) = ξ λ ( n k ) = ξ λ ( e ) = 1. Thus, for every a ∈ R , e a k ξ λ ′ ( X ) = ( e k ξ λ ′ ( X ) ) a = 1 . k ξ λ ′ ( X ) = 0. Thus, ξ λ ′ ( X ) = 0 for all X ∈ n , i.e., ξ λ ′ ≡
0. Therefore, wehave that ξ λ ( n ) = e ( n ∈ N ). We have (3.5).For every ξ λ given by (3.5) and (3.6), we obtain the associated holomorphic vector bundle E ξ λ over G C /B as E ξ λ := { [ x, v ] | ( x, v ) ∈ G C × C } , where the equivalence relation ( x, v ) ∼ ( x ′ , v ′ ) is ( x, v ) = ( x ′ , v ′ ) if and only if there exists b ∈ B such that ( x ′ , v ′ ) = ( xb − , ξ λ ( b ) v ),denoted by [ x, v ], the equivalence class including ( x, v ) ∈ G C × C (for example, [B], [TW]). For an Hermitian vector bundle π : ( E, g ) → ( M, g ) with dim R E = m , and dim R M = n ,let us recall the definitions of the tension field τ ( π ) and the bitension field τ ( π ): τ ( π ) = m X j =1 n ∇ he j π ∗ e j − π ∗ (cid:16) ∇ ge j e j (cid:17)o ,τ ( π ) = ∆ τ ( π ) − m X j =1 R h ( τ ( π ) , π ∗ e j ) π ∗ e j . (4.1)Then, we have τ ( π ) := ∆ τ ( π ) − m X j =1 R h ( τ ( π ) , π ∗ e j ) π ∗ e j = ∆ τ ( π ) − n X j =1 R h ( τ ( π ) , e ′ j ) e ′ j (4.2)= ∆ τ ( π ) − Ric h ( τ ( π )) . (4.3)Here, recall that π : ( E, g ) → ( M, h ) is the Riemannian submersion and { e i } mi =1 and { e ′ j } nj =1 are locally defined orthonormal frame fields on ( E, g ) and (
M, h ), respectively, satisfyingthat π ∗ e j = e ′ j ( j = 1 , · · · , n ) and π ∗ ( e j ) = 0 ( j = n + 1 , · · · , m ). Therefore, we have (4 . .
3) by means of the definition of the Ricci tensor field Ric h of ( M, h ).Assume that (
M, h ) is a real n -dimensional compact K¨ahler Einstein manifold withRic h = c Id, where n is even. Then, due to (4.3), we have that π : ( E, g ) → ( M, h ) isbiharmonic if and only if ∆ τ ( π ) = c τ ( π ) . (4.4)6ince h τ ( π ) , τ ( π ) i is a C ∞ function on a Riemannian manifold ( M, h ), we have, for each j = 1 , · · · , n , e ′ j h τ ( π ) , τ ( π ) i = h∇ e ′ j τ ( π ) , τ ( π ) i + h τ ( π ) , ∇ e ′ j τ ( π ) i = 2 h∇ e ′ j τ ( π ) , τ ( π ) i , (4.5) e ′ j h τ ( π ) , τ ( π ) i = 2 e ′ j h∇ e ′ j τ ( π ) , τ ( π ) i = 2 h∇ e ′ j ( ∇ e ′ j τ ( π )) , τ ( π ) i + 2 h∇ e ′ j τ ( π ) , ∇ e ′ j τ ( π ) i , (4.6) ∇ e ′ j e ′ j h τ ( π ) , τ ( π ) i = 2 h∇ ∇ e ′ j e ′ j τ ( π ) , τ ( π ) i . (4.7)Therefore, the Laplacian ∆ h = − P nj =1 ( e ′ j − ∇ e ′ j e ′ j ) acting on C ∞ ( M ), so that∆ h h τ ( π ) , τ ( π ) i = 2 n X j =1 n −h∇ e ′ j ( ∇ e ′ j τ ( π )) , τ ( π ) i − h∇ e ′ j τ ( π ) , ∇ e ′ j τ ( π ) i + h∇ ∇ e ′ j τ ( π ) , τ ( π ) i o = 2 (cid:10) − n X j =1 (cid:8) ∇ e ′ j ∇ e ′ j − ∇ ∇ e ′ j e ′ j (cid:9) τ ( π ) , τ ( π ) (cid:11) − n X j =1 (cid:10) ∇ e ′ j τ ( π ) , ∇ e ′ j τ ( π ) (cid:11) = 2 (cid:10) ∆ τ ( π ) , τ ( π ) (cid:11) − n X j =1 h∇ e ′ j τ ( π ) , ∇ e ′ j τ ( π ) i (4.8) ≤ (cid:10) ∆ τ ( π ) , τ ( π ) (cid:11) , (4.9)because of h∇ e ′ j τ ( π ) , ∇ e ′ j τ ( π ) i ≥
0, ( j = 1 , · · · , n ).If π : ( E, g ) → ( M, h ) is biharmonic, due to (4.4), ∆ τ ( π ) = c τ ( π ), the right hand sideof (4.8) coincides with(4 .
8) = 2 c h τ ( π ) , τ ( π ) i − n X j =1 h∇ e ′ j τ ( π ) , ∇ e ′ j τ ( π ) i (4.10) ≤ c h τ ( π ) , τ ( π ) i . (4.11)Remember that due to M. Obata’s theorem, (see Proposition 4.1 below), λ ( M, h ) ≥ c, (4.12)since Ric h = c Id, And the equation in (4.11) holds, i.e., λ ( M, h ) = 2 c and∆ h h τ ( π ) , τ ( π ) i = 2 c h τ ( π ) , τ ( π ) i (4.13)7olds. Then, (4.12) implies that the equality in the inequality (4.11) holds. We have that n X j =1 h∇ e ′ j τ ( π ) , ∇ e ′ j τ ( π ) i = 0 , (4.14)which is equivalent to that ∇ X τ ( π ) = 0 ( ∀ X ∈ X ( M )) . (4.15)Due to (4.15), for every X ∈ X ( M ), X h τ ( π ) , τ ( π ) i = 2 h∇ X τ ( π ) , τ ( π ) i = 0 . (4.16)Therefore, the function h τ ( π ) , τ ( π ) i on M is a constant function on M . Therefore, it impliesthat the right hand side of (4.12) must vanish. Thus, c = 0 or τ ( π ) ≡
0. If we assume that τ ( π )
0, then by (4.12), it must hold that 2 c = 0. Then, ∆ τ ( π ) = c τ ( π ) = 0, so that τ ( π ) ≡ λ ( M, g ) be the first eigenvalue of the Laplacian ∆ of a compact Riemannian manifold(
M, g ). Recall the theorem of M. Obata:
Proposition 4.1. (cf. [U1], pp. 180, 181 ) Assume that ( M, g ) is a compact K¨ahler mani-fold, and the Ricci transform ρ of ( M, g ) satisfies that g ( ρ ( u ) , u ) ≥ α g ( u, u ) , ( ∀ u ∈ T x M ) , (4.17) for some positive constant α > . Then, it holds that λ ( M, g ) ≥ α. (4.18) If the equality holds, then M admits a non-zero holomorphic vector field. Thus, we obtain Proposition 3.1, and the following theorem (cf. Theorem 1.1):
Theorem 4.2.
Let π : ( E, g ) → ( M, g ) be an Hermitian vector over a compact K¨ahlerEinstein manifold ( M, h ) . If π is biharmonic, then it is harmonic. eferences [AM] K. Akutagawa and S. Maeta, Properly immersed biharmonic submanifolds in theEuclidean spaces , Geometriae Dedicata, (2013), 351–355.[B] R. Bott,
Homogeneous vector bundles , Ann. Math., (1957), 203–248.[BH] A. Borel and F. Hirzebruch, Characteristic classes and homogeneous spaces, I, II,III , Amer. J. Math., (1958), 458–538; (1959), 315–383; (1960), 491–504.[CMP] R. Caddeo, S. Montaldo and P. Piu, On biharmonic maps , Contemp. Math., (2001), 286–290.[C] B.-Y. Chen,
Some open problems and conjectures on submanifolds of finite type ,Soochow J. Math., (1991), 169–188.[CS] I. Chrysikos and Y. Sakane, The classification of homogeneous Einstein metrics onflag manifolds with b ( M ) = 1 . Bull. Sci. Math., (2014), 665–692.[D] F. Defever,
Hypersurfaces in E with harmonic mean curvature vector , Math. Nachr., (1998), 61–69.[EL] J. Eells and L. Lemaire, Selected Topics in Harmonic Maps , CBMS, Regional Con-ference Series in Math., Amer. Math. Soc., , 1983.[GT] O. Goertsches and G. Thorbergsson, On the geometry of the orbits of Hermannaction , Geometriae Dedicata, (2007), 101–118.[HV] T. Hasanis and T. Vlachos,
Hypersurfaces in E with harmonic mean curvaturevector field , Math. Nachr., (1995), 145–169.[HTST] D. Hirohashi, H. Tasaki, H.J. Song and R. Takagi, Minimal orbits of the isotropygroups of symmetric spaces of compact type , Differential Geom. Appl. (2000),no. 2, 167–177.[IIU1] T. Ichiyama, J. Inoguchi and H. Urakawa, Biharmonic maps and bi-Yang-Millsfields , Note di Mat., , (2009), 233–275.[IIU2] T. Ichiyama, J. Inoguchi and H. Urakawa, Classifications and isolation phenomenaof biharmonic maps and bi-Yang-Mills fields , Note di Mat., , (2010), 15–48.9I1] O. Ikawa, The geometry of symmetric triad and orbit spaces of Hermann actions ,J. Math. Soc. Japan (2011), 79–136.[I2] O. Ikawa, A note on symmetric triad and Hermann actions , Proceedings of theworkshop on differential geometry and submanifolds and its related topics, Saga,August 4–6, (2012), 220–229.[I3] O. Ikawa, σ -actions and symmetric triads , Tˆohoku Math. J., (2018), 547–565.[IST1] O. Ikawa, T. Sakai and H. Tasaki, Orbits of Hermann actions , Osaka J. Math., (2001), 923–930.[IST2] O. Ikawa, T. Sakai and H. Tasaki, Weakly reflective submanifolds and austere sub-manifolds , J. Math. Soc. Japan, (2009), 437–481.[IS] J. Inoguchi and T. Sasahara, Biharmonic hypersurfaces in Riemannian symmetricspaces I , Hiroshima Math. J. (2016), no. 1, 97–121.[II] S. Ishihara and S. Ishikawa, Notes on relatively harmonic immersions , HokkaidoMath. J., (1975), 234–246.[J] G.Y. Jiang, , ChineseAnn. Math., (1986), 388–402; Note di Mat., (2009), 209–232.[KU] N. Koiso and H. Urakawa, Biharmonic submanifolds in a Riemannian manifold ,Osaka J. Math., (2018), 325–346, arXiv: 1408.5494v1, 2014.[K] A. Kollross, A classification of hyperpolar and cohomogeneity one actions , Trans.Amer. Math. Soc. (2002), no. 2, 571–612.[LO] E. Loubeau and C. Oniciuc,
On the biharmonic and harmonic indices of the Hopfmap , Trans. Amer. Math. Soc., (2007), 5239–5256.[M] T. Matsuki,
Double coset decompositions of reductive Lie groups arising from twoinvolutions , J. Algebra, (1997), 49–91.[MO] S. Montaldo and C. Oniciuc,
A short survey on biharmonic maps between Rieman-nian manifolds , Rev. Un. Mat. Argentina (2006), 1–22.10NU1] N. Nakauchi and H. Urakawa, Biharmonic hypersurfaces in a Riemannian manifoldwith non-positive Ricci curvature , Ann. Global Anal. Geom., (2011), 125–131.[NU2] N. Nakauchi and H. Urakawa, Biharmonic submanifolds in a Riemannian manifoldwith non-positive curvature , Results in Math., (2013), 467–474.[NU3] N. Nakauchi and H. Urakawa, Polyharmonic maps into the Euclidean space , Notedi Mat., (2018), 89–100.[NUG] N. Nakauchi, H. Urakawa and S. Gudmundsson, Biharmonic maps into a Rieman-nian manifold of non-positive curvature , Geom. Dedicata (2014), 263–272.[Oh1] S. Ohno,
A sufficient condition for orbits of Hermann action to be weakly reflective ,Tokyo Journal Mathematics (2016), 537–564.[Oh2] S. Ohno, Biharmonic orbits of isotropy representations of symmetric spaces ,arXiv:1704.07541, 2017.[OSU1] S. Ohno, T. Sakai and H. Urakawa,
Biharmoic homogeneous hypersurfaces in com-pact symmetric spaces , Differential Geom. Appl. (2015), 155–179.[OSU2] S. Ohno, T. Sakai and H. Urakawa, Rigidity of transversally biharmonic maps be-tween foliated Riemannian manifolds , Hokkaido Math. J., (2018), 1–18.[OSU3] S. Ohno, T. Sakai and H. Urakawa, Biharmoic homogeneous submanifolds in com-pact symmetric spaces and compact Lie groups , Hiroshima Math. J., (2019),47–115.[On] C. Oniciuc, Biharmonic maps between Riemannian manifolds , Ann. Stiint Univ. A ℓ .I. Cuza Iasi, Mat. (N.S.), No. 2, (2002), 237–248.[OT1] Y.-L. Ou and L. Tang,
The generalized Chen’s conjecture on biharmonic submani-folds is false , arXiv: 1006.1838v1.[OT2] Y.-L. Ou and L. Tang,