Biharmonic Hermitian vector bundles over compact Kaehlar Einstein manifolds

TL;DR

This paper proves that on compact Kaehler Einstein manifolds, biharmonic projections of Hermitian vector bundles are necessarily harmonic, revealing a rigidity property of such geometric structures.

Contribution

It establishes a new result linking biharmonic and harmonic projections in Hermitian vector bundles over compact Kaehler Einstein manifolds.

Findings

Biharmonic projections are harmonic in this setting

The result applies to all Hermitian vector bundles over the given manifolds

Provides insight into the geometric structure of vector bundles on Einstein manifolds

Abstract

In this paper, we show that, for every Hermitian vector bundle over a compact Kaehler Einstein manifold, if the projection is biharmonic, then it is harmonic.

Equations

$$E(\phi )={\int }_{M}e(\phi )v_g,$$

$$\frac{d}{dt}\bigg{|}_{t=0}E(\varphi_{t})=-\int_{M}h(\tau(\varphi),V)v_{g}=0,$$

$$B(\phi )(X,Y)$$

$$=({\tilde{\nabla }}_{X}d\phi )(Y)$$

$$={\overline{\nabla }}_X(d\phi (Y))-d\phi ({\nabla }_{X}Y)$$

$$={\nabla }_{d\phi (X)}^{h}d\phi (Y)-d\phi ({\nabla }_{X}Y),$$

$$\frac{d^{2}}{dt^{2}}\bigg{|}_{t=0}E(\varphi_{t})=\int_{M}h(J(V),V)v_{g},$$

$$J(V)=\overline{\Delta }V-\mathcal{R}(V),$$

$$E_2(\phi )=\frac{1}{2}{\int }_M|\tau (\phi ){|}^2{v}_g,$$

$$\frac{d}{dt}\bigg{|}_{t=0}E_{2}(\varphi_{t})=-\int_{M}h(\tau_{2}(\varphi),V)v_{g}.$$

$${\tau }_2(\phi ):=J(\tau (\phi ))=\overline{\Delta }\tau (\phi )-\mathcal{R}(\tau (\phi )),$$

$${\overline{\nabla }}_X\tau (\pi )=0(\forall X\in \mathfrak{X}(M)).$$

$${\tau }_2(\pi ):={\int }_M|\tau (\pi ){|}^2{v}_h=d\text{mbox}\mathrm{Vol}(M,h).$$

$$\text{mbox}\mathrm{div}(\tau (\pi )):=\sum _{i=1}^n({\overline{\nabla }}_{e_i^{\prime }}\tau (\pi ))(e_i^{\prime })=0,$$

$$0={\int }_{M}f\text{mbox}\mathrm{div}(\tau (\pi ))v_h=-{\int }_{M}h(\nabla f,\tau (\pi ))v_h.$$

$$M=G/H=G^{\mathbb{C}}/B.$$

$${\xi }_{\lambda }(tn)={\xi }_{\lambda }(t)(t\in T,n\in N).$$

$${\xi }_{\lambda }(\exp ({\theta }_1{H}_1+\cdots +{\theta }_{\ell }{H}_{\ell }))=e^{2\pi \sqrt{-1}(k_1{\theta }_1+\cdots +k_{\ell }{\theta }_{\ell })},({\theta }_1,\dots ,{\theta }_{\ell }\in \mathbb{R}),$$

$$e^{ak{\xi }_{{\lambda }^{\prime }}(X)}=(e^{k{\xi }_{{\lambda }^{\prime }}(X)}{)}^a=1.$$

$$\{\begin{array}{rl}\tau (\pi )& =\sum _{j=1}^m\left\{{\overline{\nabla }}_{e_j}^h{\pi }_{\ast }{e}_j-{\pi }_{\ast }\left({\nabla }_{e_j}^g{e}_j\right)\right\},\\ {\tau }_2(\pi )& =\overline{\Delta }\tau (\pi )-\sum _{j=1}^m{R}^h(\tau (\pi ),{\pi }_{\ast }{e}_j){\pi }_{\ast }{e}_j.\end{array}$$

$${\tau }_2(\pi )$$

$$=\overline{\Delta }\tau (\pi )-\sum _{j=1}^n{R}^h(\tau (\pi ),e_j^{\prime })e_j^{\prime }$$

$$=\overline{\Delta }\tau (\pi )-\text{mbox}{\mathrm{Ric}}^h(\tau (\pi )).$$

$$\overline{\Delta }\tau (\pi )=c\tau (\pi ).$$

$$e_j^{\prime }⟨\tau (\pi ),\tau (\pi )⟩$$

$$=2⟨{\overline{\nabla }}_{e_j^{\prime }}\tau (\pi ),\tau (\pi )⟩,$$

$${e_j^{\prime }}^2⟨\tau (\pi ),\tau (\pi )⟩$$

$$=2⟨{\overline{\nabla }}_{e_j^{\prime }}({\overline{\nabla }}_{e_j^{\prime }}\tau (\pi )),\tau (\pi )⟩+2⟨{\overline{\nabla }}_{e_j^{\prime }}\tau (\pi ),{\overline{\nabla }}_{e_j^{\prime }}\tau (\pi )⟩,$$

$${\nabla }_{e_j^{\prime }}{e}_j^{\prime }⟨\tau (\pi ),\tau (\pi )⟩$$

$${\Delta }_h⟨\tau (\pi ),\tau (\pi )⟩$$

$$\displaystyle=2\big{\langle}-\sum_{j=1}^{n}\big{\{}\overline{\nabla}_{e^{\prime}_{j}}\overline{\nabla}_{e^{\prime}_{j}}-\overline{\nabla}_{\nabla_{e^{\prime}_{j}}e^{\prime}_{j}}\big{\}}\tau(\pi),\tau(\pi)\big{\rangle}-2\sum_{j=1}^{n}\big{\langle}\overline{\nabla}_{e^{\prime}_{j}}\tau(\pi),\overline{\nabla}_{e^{\prime}_{j}}\tau(\pi)\big{\rangle}$$

$$\displaystyle=2\big{\langle}\overline{\Delta}\tau(\pi),\tau(\pi)\big{\rangle}-2\sum_{j=1}^{n}\langle\overline{\nabla}_{e^{\prime}_{j}}\tau(\pi),\overline{\nabla}_{e^{\prime}_{j}}\tau(\pi)\rangle$$

$$\displaystyle\leq 2\big{\langle}\overline{\Delta}\tau(\pi),\tau(\pi)\big{\rangle},$$

$$(4.8)$$

$$\le 2c⟨\tau (\pi ),\tau (\pi )⟩.$$

$${\lambda }_1(M,h)\ge 2c,$$

$${\Delta }_h⟨\tau (\pi ),\tau (\pi )⟩=2c⟨\tau (\pi ),\tau (\pi )⟩$$

Full text

1 Introduction.

Research of harmonic maps, which are critical points of the energy functional, is one of the 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 functional, by definition, half of the integral of square of the norm of tension field $\tau(\varphi)$ for a smooth map $\varphi$ 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 biharmonic. One of central problems is to ask whether the converse is true. B.-Y. Chen’s conjecture is to ask whether every biharmonic submanifold of the Euclidean space ${\mathbb{R}}^{n}$ must be 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 ask whether each biharmonic submanifold in a Riemannian manifold $(N,h)$ of non-positive sectional curvature must be harmonic (minimal). For the generalized Chen’s conjecture, Ou and Tang gave ([OT1], [OT2]) a counter example in some Riemannian manifold of negative sectional curvature. But, it is also known (cf. [NU1], [NU2], [NUG]) that every biharmonic map of a complete Riemannian manifold into another Riemannian manifold of non-positive sectional curvature with finite energy and finite bienergy must be harmonic. For the target Riemannian manifold $(N,h)$ of non-negative sectional curvature, theories of biharmonic maps and biharmonic immersions seems to be quite different from the case $(N,h)$ of non-positive sectional curvature. There exit biharmonic submanifolds which is not harmonic in the unit sphere. S. Ohno, T. Sakai and myself [OSU1], [OSU2] determined (1) all the biharmonic hypersurfaces in irreducible symmetric spaces of compact type which are regular orbits of commutative Hermann actions of cohomogeneity one, and gave (2) a complete table of all the proper biharmonic singular orbits of commutative Hermann actions of cohomogeneity two, and (3) a complete list of all the proper biharmonic regular orbits of $(K_{2}\times K_{1})$-actions of cohomogeneity one on $G$ for every commutative compact symmetric triad $(G,K_{1},K_{2})$. We note that recently Inoguchi and Sasahara ([IS]) also investigated biharmonic homogeneous hypersurfaces in compact symmetric spaces, and Ohno studied biharmonic orbits of isotropy representations of symmetric spaces in the sphere (cf. [Oh1], [Oh2]).

In this paper, we treat with an Hermitian vector bundle $(E,g)\rightarrow(M,h)$ over a compact Riemannian manifold $(M,h)$. We assume $(M,h)$ is a compact Kähler Einstein Riemannian manifold, that is, the Ricci transform ${\rm Ric}^{h}$ of the Kähler metric $h$ on $M$ satisfies ${\rm Ric}^{h}=c\,\,{\rm Id}$, for some constant $c$. Then, we show the following:

Theorem 1.1**.**

Let $\pi:\,\,(E,g)\rightarrow(M,h)$ be an Hermitian vector bundle over a compact Kähler Einstein Riemannian manifold $(M,h)$. If $\pi$ is biharmonic, then it is harmonic.

Theorem 1.1 shows the sharp contrasts on the biharmonicities between the case of vector bundles and the one of the principle $G$-bundles. Indeed, we treated with the biharmonicity of the projection of the principal $G$-bundle over a Riemannian manifold $(M,h)$ with negative definite Ricci tensor field (cf. Theorem 2.3 in [U4]). We also gave an example of the projection of the principal $G$-bundle over a Riemannian manifold $(M,h)$ which is biharmonic but not harmonic (cf. Theorem 5 in [U5]).

2 Preliminaries.

In this section, we prepare necessary materials for the first and second variational formulas for the bienergy functional and biharmonic maps. Let us recall the definition of a harmonic map $\varphi:\,(M,g)\rightarrow(N,h)$, of a compact Riemannian manifold $(M,g)$ into a Riemannian manifold $(N,h)$, which is an extremal of the energy functional defined by

[TABLE]

where $e(\varphi):=\frac{1}{2}|d\varphi|^{2}$ is called the energy density of $\varphi$. That is, for any variation $\{\varphi_{t}\}$ of $\varphi$ with $\varphi_{0}=\varphi$,

[TABLE]

where $V\in\Gamma(\varphi^{-1}TN)$ is the variation vector field along $\varphi$ which is given by $V(x)=\frac{d}{dt}|_{t=0}\varphi_{t}(x)\in T_{\varphi(x)}N$, $(x\in M)$, and the tension field is given by $\tau(\varphi)=\sum_{i=1}^{m}B(\varphi)(e_{i},e_{i})\in\Gamma(\varphi^{-1}TN)$, where $\{e_{i}\}_{i=1}^{m}$ is a locally defined frame field on $(M,g)$, and $B(\varphi)$ is the second fundamental form of $\varphi$ defined by

[TABLE]

for all vector fields $X,Y\in{\mathfrak{X}}(M)$. Furthermore, $\nabla$, and $\nabla^{h}$, are the Levi-Civita connections on $TM$, $TN$ of $(M,g)$, $(N,h)$, respectively, and $\overline{\nabla}$, and $\widetilde{\nabla}$ are the induced ones on $\varphi^{-1}TN$, and $T^{\ast}M\otimes\varphi^{-1}TN$, respectively. By (2.1), $\varphi$ is harmonic if and only if $\tau(\varphi)=0$.

The second variation formula is given as follows. Assume that $\varphi$ is harmonic. Then,

[TABLE]

where $J$ is an elliptic differential operator, called Jacobi operator acting on $\Gamma(\varphi^{-1}TN)$ given by

[TABLE]

where $\overline{\Delta}V=\overline{\nabla}^{\ast}\overline{\nabla}V=-\sum_{i=1}^{m}\{\overline{\nabla}_{e_{i}}\overline{\nabla}_{e_{i}}V-\overline{\nabla}_{\nabla_{e_{i}}e_{i}}V\}$ is the rough Laplacian and ${\mathcal{R}}$ is a linear operator on $\Gamma(\varphi^{-1}TN)$ given by ${\mathcal{R}}(V)=\sum_{i=1}^{m}R^{h}(V,d\varphi(e_{i}))d\varphi(e_{i})$, and $R^{h}$ is the curvature tensor of $(N,h)$ given by $R^{h}(U,V)=\nabla^{h}{}_{U}\nabla^{h}{}_{V}-\nabla^{h}{}_{V}\nabla^{h}{}_{U}-\nabla^{h}{}_{[U,V]}$ for $U,\,V\in{\mathfrak{X}}(N)$.

J. 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

[TABLE]

where $|V|^{2}=h(V,V)$, $V\in\Gamma(\varphi^{-1}TN)$.

Then, the first variation formula of the bienergy functional is given as follows.

Theorem 2.1**.**

$($the first variation formula$)$

[TABLE]

Here,

[TABLE]

which is called the bitension field of $\varphi$, and $J$ is given in $(2.4)$.

Definition 2.2**.**

A smooth map $\varphi$ of $M$ into $N$ is said to be biharmonic if $\tau_{2}(\varphi)=0$.

3 Proof of Theorem 1.1.

To prove Theorem 1.1, we need the following:

Proposition 3.1**.**

Let $\pi:\,\,(E,g)\rightarrow(M,g)$ be an Hermitian vector bundle over a compact Kähler Einstein manifold $(M,h)$. Assume that $\pi$ is biharmonic. Then the following hold:

$(1)$* The tension field $\tau(\pi)$ satisfies that*

[TABLE]

$(2)$* The pointwise inner product $\langle\tau(\pi),\tau(\pi)\rangle=|\,\tau(\pi)\,|^{2}$ is constant on $(M,g)$, say $d\geq 0$.*

$(3)$* The bitension field $\tau_{2}(\pi)$ satisfies that*

[TABLE]

By Proposition 3.1, Theorem 1.1 can be proved as follows. Assume that $\pi:\,(E,g)\rightarrow(M,h)$ is biharmonic. Due to (3.1) in Proposition 3.1, we have

[TABLE]

where $\{e_{i}^{\prime}\}_{i=1}^{n}$ is a locally defined orthonormal frame field on $(M,h)$ and we put $n=\dim_{\mathbb{R}}M$. Then, for every $f\in C^{\infty}(M)$, it holds that, due to Proposition (3.29) in [U1], p. 60, for example,

[TABLE]

Therefore, we obtain $\tau(\pi)\equiv 0$. ∎

We will prove Proposition 3.1, later. Here, we give examples of the line bundles over some compact homogeneous Kähler Einstein manifolds $(M,h)$:

Example 3.2**.**

A generalized flag manifold $G/H$ admits a unique Kähler Einstein metric $h$ ([BH] and [CS]). Here, $G$ is a compact semi-simple Lie group, and $H$ is the centralizer of a torus $S$ in $G$, i.e., $G^{\mathbb{C}}$ is the complexification of $G$, and $B$ is its Borel subgroup. Then,

[TABLE]

The Borel subgroup $B$ is written as $B=TN$, where $T$ is a maximal torus of $B$ and $N$ is a nilpotent Lie subgroup of $B$. Every character $\xi_{\lambda}$ of a Borel subgroup $B$ is given as a homomorphism $\xi_{\lambda}:\,B\rightarrow{\mathbb{C}}^{\ast}={\mathbb{C}}-\{0\}$ which is written as

[TABLE]

Here $\xi_{\lambda}:\,T\rightarrow U(1)$ is a character of $T$ which is written as

[TABLE]

where $k_{1},\,\ldots,\,k_{\ell}$ are non-negative integers, and $\ell=\dim T$.

Note that every character $\xi_{\lambda}$ of a nilpotent Lie group $N$ must be $\xi_{\lambda}(n)=1$ because $\xi_{\lambda}(n)=\xi_{\lambda}(\exp X)=e^{\xi_{\lambda^{\prime}}(X)}$ where $n=\exp X$ $(X\in{\mathfrak{n}})$, and $\lambda^{\prime}:\,{\mathfrak{t}}\rightarrow{\mathbb{C}}$ is a homomorphism, i.e., $\xi_{\lambda^{\prime}}(X+Y)=\xi_{\lambda^{\prime}}(X)+\xi_{\lambda^{\prime}}(Y)$, $(X,\,Y\in{\mathfrak{t}})$. Then, there exists $k\in{\mathbb{N}}$ which satisfies that $\exp(k\,X)=n^{k}=e$. Then, $e^{k\,\xi_{\lambda^{\prime}}(X)}=\xi_{\lambda}(n^{k})=\xi_{\lambda}(e)=1$. Thus, for every $a\in{\mathbb{R}}$,

[TABLE]

This implies that $k\,\xi_{\lambda^{\prime}}(X)=0$. Thus, $\xi_{\lambda^{\prime}}(X)=0$ for all $X\in\mathfrak{n}$, i.e., $\xi_{\lambda^{\prime}}\equiv 0$. Therefore, we have that $\xi_{\lambda}(n)=e$ ($n\in N$). We have (3.5).

For every $\xi_{\lambda}$ given by (3.5) and (3.6), we obtain the associated holomorphic vector bundle $E_{\xi_{\lambda}}$ over $G^{\mathbb{C}}/B$ as $E_{\xi_{\lambda}}:=\{[x,v]|(x,v)\in G^{\mathbb{C}}\times{\mathbb{C}}\}$, where the equivalence relation $(x,v)\sim(x^{\prime},v^{\prime})$ is $(x,v)=(x^{\prime},v^{\prime})$ if and only if there exists $b\in B$ such that $(x^{\prime},v^{\prime})=(xb^{-1},\xi_{\lambda}(b)v)$, denoted by $[x,v]$, the equivalence class including $(x,v)\in G^{\mathbb{C}}\times{\mathbb{C}}$ (for example, [B], [TW]).

4 Proof of Proposition 3.1.

For an Hermitian vector bundle $\pi:\,(E,g)\rightarrow(M,g)$ with $\dim_{\mathbb{R}}E=m$, and $\dim_{\mathbb{R}}M=n$, let us recall the definitions of the tension field $\tau(\pi)$ and the bitension field $\tau_{2}(\pi)$:

[TABLE]

Then, we have

[TABLE]

Here, recall that $\pi:\,(E,g)\rightarrow(M,h)$ is the Riemannian submersion and $\{e_{i}\}_{i=1}^{m}$ and $\{e^{\prime}_{j}\}_{j=1}^{n}$ are locally defined orthonormal frame fields on $(E,g)$ and $(M,h)$, respectively, satisfying that $\pi_{\ast}e_{j}=e^{\prime}_{j}$ $(j=1,\cdots,n)$ and $\pi_{\ast}(e_{j})=0$ $(j=n+1,\cdots,m)$. Therefore, we have $(4.2)$ and $(4.3)$ by means of the definition of the Ricci tensor field ${\rm Ric}^{h}$ of $(M,h)$.

Assume that $(M,h)$ is a real $n$-dimensional compact Kähler Einstein manifold with ${\mbox{\rm Ric}}^{h}=c\,\mbox{\rm Id}$, where $n$ is even. Then, due to (4.3), we have that $\pi:\,(E,g)\rightarrow(M,h)$ is biharmonic if and only if

[TABLE]

Since $\langle\tau(\pi),\tau(\pi)\rangle$ is a $C^{\infty}$ function on a Riemannian manifold $(M,h)$, we have, for each $j=1,\cdots,n$,

[TABLE]

Therefore, the Laplacian $\Delta_{h}=-\sum_{j=1}^{n}(e^{\prime}_{j}{}^{2}-\nabla_{{}_{e^{\prime}_{j}}}e^{\prime}_{j})$ acting on $C^{\infty}(M)$, so that

[TABLE]

because of $\langle\overline{\nabla}_{e^{\prime}_{j}}\tau(\pi),\overline{\nabla}_{e^{\prime}_{j}}\tau(\pi)\rangle\geq 0$, $(j=1,\cdots,n)$.

If $\pi:\,(E,g)\rightarrow(M,h)$ is biharmonic, due to (4.4), $\overline{\Delta}\tau(\pi)=c\,\tau(\pi)$, the right hand side of (4.8) coincides with

[TABLE]

Remember that due to M. Obata’s theorem, (see Proposition 4.1 below),

[TABLE]

since $\mbox{\rm Ric}_{h}=c\,\mbox{\rm Id}$, And the equation in (4.11) holds, i.e., $\lambda_{1}(M,h)=2c$ and

[TABLE]

holds. Then, (4.12) implies that the equality in the inequality (4.11) holds. We have that

[TABLE]

which is equivalent to that

[TABLE]

Due to (4.15), for every $X\in{\mathfrak{X}}(M)$,

[TABLE]

Therefore, the function $\langle\tau(\pi),\tau(\pi)\rangle$ on $M$ is a constant function on $M$. Therefore, it implies that the right hand side of (4.12) must vanish. Thus, $c=0$ or $\tau(\pi)\equiv 0$. If we assume that $\tau(\pi)\not\equiv 0$, then by (4.12), it must hold that $2c=0$. Then, $\overline{\Delta}\tau(\pi)=c\,\tau(\pi)=0$, so that $\tau(\pi)\equiv 0$ due to (4.4).

Let $\lambda_{1}(M,g)$ be the first eigenvalue of the Laplacian $\Delta$ 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 Kähler manifold, and the Ricci transform $\rho$ of $(M,g)$ satisfies that

[TABLE]

for some positive constant $\alpha>0$. Then, it holds that

[TABLE]

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 $\pi:\,(E,g)\rightarrow(M,g)$ be an Hermitian vector over a compact Kähler Einstein manifold $(M,h)$. If $\pi$ is biharmonic, then it is harmonic.