Tachibana-type Theorems and special Holonomy

We prove rigidity results for compact Riemannian manifolds in the spirit of Tachibana. For example, we observe that manifolds with divergence free Weyl tensors and $\lfloor \frac{n-1}{2} \rfloor$-nonnegative curvature operators are locally symmetric or conformally equivalent to a quotient of the sphere. The main focus of the paper is to prove similar results for manifolds with special holonomy. In particular, we consider K\"ahler manifolds with divergence free Bochner tensor. For quaternion K\"ahler manifolds we obtain a partial result towards the LeBrun-Salamon conjecture.


Introduction
In this paper we establish rigidity theorems for compact Riemannian manifolds. According to a famous theorem of Tachibana [Tac74], manifolds with harmonic curvature tensors and positive curvature operators have constant sectional curvature. If the curvature operator is nonnegative, then the manifold is locally symmetric.
In dimension n = 4, Micallef-Wang [MW93] proved that a Riemannian manifold with harmonic curvature tensor and nonnegative isotropic curvature is locally symmetric or locally conformally flat. In particular, if the metric is Einstein, then the manifold is locally symmetric.
In the case of Einstein manifolds, the convergence theorems for the Ricci flow due to Hamilton [Ham82,Ham86], Chen [Che91], Böhm-Wilking [BW08], Ni-Wu [NW07], Brendle-Schoen [BS09,BS08], Brendle [Bre08] and Seshadri [Ses09] imply Tachibana-type theorems. Moreover, Brendle [Bre10] proved that Einstein manifolds with nonnegative isotropic curvature are locally symmetric. These results rely on the fact that, e.g., nonnegative curvature operator or nonnegative isotropic curvature are Ricci flow invariant curvature conditions. We recall that the curvature operator of a Riemannian manifold is k-nonnegative if the sum of its lowest k eigenvalues is nonnegative.
In [PW21a], the authors proved that Einstein manifolds with ⌊ n−1 2 ⌋-nonnegative curvature operators are locally symmetric. In contrast to the previously mentioned curvature conditions, ⌊ n−1 2 ⌋-nonnegative curvature operator is not preserved by the Ricci flow for n ≥ 7. Note that a Riemannian manifold has harmonic curvature tensor if and only if it has constant scalar curvature and the Weyl tensor is divergence free. The main theme of this paper is to show that the assumption of divergence free Weyl tensor is sufficient to prove Tachibana-type theorems.
In fact, Tran [Tra17] observed that manifolds with divergence free Weyl tensors and nonnegative curvature operators are locally symmetric or locally conformally flat. Based on the work of Schoen-Yau [SY88], Noronha [Nor93] classified compact locally conformally flat 2010 Mathematics Subject Classification. 32Q10, 32Q15, 32Q20, 53C20, 53C26 MW funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy EXC 2044-390685587, Mathematics Münster: Dynamics-Geometry-Structure. manifolds with nonnegative Ricci curvature. Their universal cover is either conformally equivalent to S n or isometric to S n−1 × R or R n .
Theorem A. Let (M, g) be a compact n-dimensional Riemannian manifold with divergence free Weyl tensor.
If (M, g) has ⌊ n−1 2 ⌋-nonnegative curvature operator, then (M, g) is locally symmetric or conformally equivalent to a quotient of the standard sphere.
For manifolds with special holonomy, the assumption on the eigenvalues of the curvature operator reduces to nonnegative curvature operator. This is immediate from the observation that the curvature operator of a Riemannian manifold vanishes on the complement of the holonomy algebra.
In order to establish Tachibana-type results for manifolds with special holonomy, it is therefore natural to study the restriction of curvature operator to the holonomy algebra, R |hol : hol → hol.
The analogue of the Weyl tensor for Kähler manifolds is the Bochner tensor, identified by Bochner in [Boc49]. In analogy to the generic holonomy case, a Kähler manifold has harmonic curvature tensor if and only if it has constant scalar curvature and the Bochner tensor is divergence free.
Bryant classified compact Kähler manifolds with vanishing Bochner tensors in [Bry01,Corollary 4.17]. In particular, a compact Bochner flat Kähler manifold with nonnegative Ricci curvature is isometric to CP m or its universal cover is isometric to C m . Therefore, we have the following generalization of [PW21b, Theorem E] on Kähler-Einstein manifolds: Theorem B. Let (M, g) be a compact Kähler manifold of real dimension 2m with divergence free Bochner tensor.
A Riemannian manifold of real dimension 4m ≥ 8 with holonomy contained in Sp(m) · Sp(1) is called quaternion Kähler manifold. If the scalar curvature is positive, then the manifold is called positive quaternion Kähler manifold.
The LeBrun-Salamon conjecture asserts that every positive quaternion Kähler manifold is symmetric. In real dimension 8 this was proven by Poon-Salamon [PS91] and with different techniques by LeBrun-Salamon [LS94]. If the real dimension of the manifold is 12 or 16, then the conjecture follows from the work of Buczyński-Wiśniewski [BW19]. A quaternion Kähler manifold of real dimension 4 is by definition a half conformally flat Einstein manifold. In this case the LeBrun-Salamon conjecture follows from Hitchin's work [Hit87].
In analogy to the Kähler case, for a quaternion Kähler manifold we consider the corresponding quaternion Kähler curvature operator by restricting the Riemannian curvature operator to the holonomy algebra sp(m) ⊕ sp(1).
Notice that in real dimension 4m ≥ 8 quaternion Kähler manifolds are necessarily Einstein. In particular, the curvature tensor is automatically harmonic.
Therefore, the analogue of Theorems A and B for quaternion Kähler manifolds is Theorem C. Let (M, g) be a compact quaternion Kähler manifold of real dimension 4m ≥ 8.
Positive quaternion Kähler manifolds are necessarily compact and due to a result of Salamon, [Sal82, Theorem 6.6], also simply connected. Hence we have the following partial result towards the LeBrun-Salamon conjecture: Corollary. Let (M, g) be a positive quaternion Kähler manifold of real dimension 4m ≥ 8. If (M, g) has ⌊ m+1 2 ⌋-nonnegative quaternion Kähler curvature operator, then (M, g) is a symmetric space.
Symmetric quaternion Kähler manifolds with positive scalar curvature are classified by Wolf [Wol65]. In particular, to identify (M, g) isometrically as HP m , in addition to being ⌊ m+1 2 ⌋-nonnegative, the quaternion Kähler curvature operator only needs to be k(m)-positive for a function k(m) ∼ m 2 .
We note that Amann [Ama12] proved that positive quaternion Kähler manifolds are symmetric provided the dimension of the isometry group is large. Other partial results towards the LeBrun-Salamon conjecture have been obtained by, e.g., Amann [ Recall that the remaining holonomy groups in Berger's list of irreducible holomony groups force the metric to be either locally symmetric or Ricci flat. Notice that a Ricci flat manifold whose curvature operator satisfies one of the nonnegativity assumptions in Theorems A -C is flat.
The proofs of Theorems A -C rely on the Bochner technique and the fact that the Lichnerowicz Laplacian preserves tensor bundles which are invariant under the holonomy representation. In particular, if R is a harmonic curvature tensor on (M, g) and R is the curvature operator of (M, g), then R satisfies the Bochner identity where the curvature term g(R(R hol ), R hol ) is adapted to the holonomy algebra hol. Thus, if g(R(R hol ), R hol ) ≥ 0, then the manifold is locally symmetric. For example, if R |hol : hol → hol is 2-nonnegative, then g(R(R hol ), R hol ) ≥ 0 according to proposition 1.3. This is a useful observation in low dimensions.
Furthermore, corollaries 2.3, 3.3 and proposition 4.6 show that g(R(R hol ), R hol ) ≥ 0 provided a weighted sum of eigenvalues of the curvature operator is nonnegative. In particular, Theorems A -C generalize to these weighted curvature conditions. The paper is structured as follows: Section 1 briefly reviews the relevant details of the Bochner technique. Section 2 proves Theorem A by combining corollary 2.3 with results from the literature. In section 3 we show that Kähler manifolds with harmonic Bochner tensors are Bochner flat or have constant scalar curvature, and deduce Theorem B. Finally, in section 4 we compute the curvature term of the Lichnerowicz Laplacian for quaternion Kähler curvature tensors and prove Theorem C.

Preliminaries
We summarize the relevant material from [PW21b, Section 1] and focus on the Bochner technique for curvature tensors.
1.1. Tensors. Let (V, g) be an n-dimensional Euclidean vector space. The metric g induces a metric on k V * and k V . In particular, if {e i } i=1,...,n is an orthonormal basis for V , Notice that 2 V inherits a Lie algebra structure from so(V ). The induced Lie algebra action on V is given by In particular, for Ξ α , Ξ β ∈ 2 V we have Similarly, for T ∈ k V * and L ∈ so(V ) set T (X 1 , . . . , LX i , . . . , X k ).
A tensor Rm ∈ 4 V * is an algebraic curvature tensor if In particular, it induces the curvature operator R :

The associated symmetric bilinear form is denoted by
In particular, g g is the curvature tensor of the sphere of radius 1/ √ 2.
Remark 1.2. The curvature operator R of a Riemannian manifold (M, g) vanishes on the complement of the holonomy algebra hol. In particular, it induces R |hol : hol → hol and the corresponding curvature tensor R ∈ Sym 2 B (hol) . (1)) is the quaternion Kähler curvature tensor.
In particular, if {λ α } denote the corresponding eigenvalues, then be an algebraic curvature tensor and let R : g → g be the corresponding curvature operator.
The divergence of T is given by In this paper we will focus on algebraic curvature tensors on Riemannian manifolds. Notice that the proof of [Pet16, Theorem 9.4.2] also shows Proposition 1.4. Let (M, g) be a Riemannian manifold. Suppose that T is an algebraic curvature tensor on M, i.e. T satisfies If in addition T satisfies the second Bianchi identity and T is divergence free, then T is harmonic, A curvature tensor R ∈ Sym 2 B (T M) is called harmonic if the corresponding (0, 4)-curvature tensor Rm is harmonic.
Corollary 1.5. Let (M, g) be a Riemannian manifold. Let R : 2 T M → 2 T M denote its curvature operator and hol its holonomy algebra. If T is a harmonic curvature tensor on M, then In particular, if in addition M is compact and g(R(T hol ), T hol ) ≥ 0, then T is parallel.
Proof. According to [PW21b, Proposition 1.6] the curvature term in the Bochner formula can be computed by Thus the claim follows from proposition 1.4 and the maximum principle.
A general criterion to show g(R(T hol ), T hol ) ≥ 0 based on the eigenvalues of the curvature operator R |hol : hol → hol is established in [PW21b, Lemma 1.8]. As an application thereof, proposition 1.3 shows that if R |hol : hol → hol is 2-nonnegative and R ∈ Sym 2 B (hol) is the associated curvature tensor, then g(R(R hol ), R hol ) ≥ 0. Thus we have Corollary 1.6. Let (M, g) be a compact Riemannian manifold with holonomy algebra hol. If the curvature operator R |hol : hol → hol is 2-nonnegative, then (M, g) is locally symmetric.

Manifolds with divergence free Weyl tensors
Let (M, g) be a compact n-dimensional Riemannian manifold. The decomposition of the space of curvature tensors into orthogonal, irreducible, O(n)-invariant modules yields whereRic = Ric − scal n g denotes the trace-free Ricci tensor and W the Weyl tensor. Remark 2.1. Recall that the curvature tensor of a Riemannian manifold is divergence free if and only if the Weyl tensor is divergence free and the scalar curvature is constant, since Proposition 2.2. Let (M, g) be a Riemannian manifold. If the Weyl curvature W is divergence free, then W satisfies the second Bianchi identity and Proof. The fact that divergence free Weyl tensors satisfy the second Bianchi identity is explained in [Eis50, section 28]. The Bochner formula follows from proposition 1.4.
Corollary 2.3. Let (M, g) be a compact n-dimensional Riemannian manifold. Suppose that the Weyl tensor is divergence free. If the eigenvalues λ 1 ≤ . . . ≤ λ ( n 2 ) of the curvature operator satisfy   Proof. This was established by G lodek in [G lo71]. We include a modified proof to illustrate the idea behind the proof of proposition 3.4, the Kähler analogue of proposition 2.4.
It follows from remark 2.1 and ∇W = 0 that Z)) .
Consider the Lie algebra action of the curvature tensor R(X, Y ) ∈ so(T M) on the Weyl tensor. Since ∇W = 0 we have (R(X, Y ))W = 0 and consequently ((∇ Z R) (X, Y )) W = 0.
Overall we obtain Contraction of E 1 with Z yields since W is totally trace free and satisfies the algebraic Bianchi identity. Inserting this equation back into the equation above and setting E 1 = ∇ scal implies |∇ scal | 2 · W = 0 and the claim follows.
Proof of Theorem A. The assumptions in Theorem A and corollary 2.3 imply that the Weyl tensor is parallel. Thus G lodek's work [G lo71] shows that (M, g) is conformally flat or has constant scalar curvature.
If the scalar curvature is constant, then a result of Derdziński-Roter [DR77], see also Roter [Rot76], shows that the Ricci tensor is parallel. Hence the curvature tensor is parallel and (M, g) is locally symmetric.
If the manifold is conformally flat, then the classification of compact conformally flat manifolds with nonnegative Ricci curvature due to Noronha [Nor93] implies that (M, g) is locally symmetric or conformally equivalent to a quotient of the sphere.
The curvature tensor decomposes into a Kähler curvature tensor with constant holomorphic sectional curvature, a Kähler curvature tensor with trace-free Ricci curvature and the Bochner tensor, Rm = scal 4m(m + 1) The tensor B was introduced by Bochner in [Boc49] as the analogue of the Weyl tensor. Alekseevski [Ale68] observed that this is indeed the decomposition of a Kähler curvature tensor according to decomposition of the space of Kähler curvature tensors into orthogonal, U(m)-invariant, irreducible subspaces.
Hence the Bochner tensor satisfies In addition, Tachibana [Tac67] proved that the Bochner tensor is totally trace-free. That is, if e 1 , . . . , e 2m is an orthonormal basis of T M, then It is straightforward to compute that  Proof. The fact that divergence free Bochner tensors satisfy the second Bianchi identity is a result of Omachi [Oma03]. Proposition 1.4 shows that hence the Bochner tensor is harmonic.
Recall from remark 1.2 that the Kähler curvature operator is the restriction of the Riemannian curvature operator to the holonomy algebra u(m). Proof. If div B = 0, then Note that and hence the terms with factors of 2d scal(JY ) cancel. Therefore we obtain In view of remark 3.1, contraction of E 1 with Y yields where we used the algebraic Bianchi identity in the penultimate step. Therefore we conclude that B(∇ scal, ·, ·, ·) = B(J∇ scal, ·, ·, ·) = 0 since B(J·, J·, ·, ·) = B(·, ·, ·, ·) = 0. Inserting this back into the above equation we find Finally, set E 1 = ∇ scal and note that d scal(J∇ scal) = g(J∇ scal, ∇ scal) = 0 to conclude that |∇ scal | 2 · B = 0 as required.
Proof of Theorem B. It follows from corollary 3.3 that the Bochner tensor is parallel. Proposition 3.4 shows that hence the scalar curvature is constant or the Bochner tensor vanishes.
According to a theorem of Kim [Kim09], a Kähler manifold with divergence free Bochner tensor and constant scalar curvature has parallel Ricci tensor. Therefore, in this case, the curvature tensor is in fact parallel and (M, g) is locally symmetric.
On the other hand, if (M, g) is Bochner flat, then it is locally symmetric due to Bryant's classification of compact Bochner flat Kähler manifolds in [Bry01,Corollary 4.17].
Locally there exist almost complex structures I, J, K such that IJ = −JI = K. For a local orthonormal frame {e i , Ie i , Je i , Ke i } i=1,...,m consider It is straightforward to check that The curvature operator of quaternionic projective space is given by Remark 4.1. In this normalization of the metric, the curvature operator has eigenvalues 4m and 4. The eigenspace for the eigenvalue 4m is isomorphic to sp(1) and spanned by ω I , ω J , ω K . The eigenspace for the eigenvalue 4 is isomorphic to sp(m) and spanned by W ij = 1 2 (e i ∧ e j + Ie i ∧ Ie j + Je i ∧ Je j + Ke i ∧ Ke j ) for 1 ≤ i < j ≤ m, Recall that dim sp(m) = m(2m + 1). In particular, |R HP m | 2 = 16m(5m + 1) and scal(R HP m ) = 16m(m + 2).
The curvature operator R ∈ Sym 2 B (T M) of a quaternion Kähler manifold satisfies where R 0 is the hyper-Kähler component. Recall that hyper-Kähler manifolds have holonomy contained in Sp(m) and are necessarily Ricci flat. Due to a result of Alekseevski [Ale68], see also [Sal82], this is indeed the decomposition of the curvature tensor of a quaternion Kähler manifold according to the decomposition of the space of quaternion Kähler curvature tensors into orthogonal, Sp(m) · Sp(1)-invariant, irreducible subspaces.
The key ingredient in the proof of Theorem C is the computation of |R sp(m)⊕sp(1) | 2 for any quaternion Kähler curvature tensor R ∈ Sym 2 B (sp(m) ⊕ sp(1)) in corollary 4.5. The curvature tensor of quaternionic projective space satisfies |R sp(m)⊕sp(1) HP m | 2 = 0 due to the following observation.
Proposition 4.2. Let (M, g) be an isotropy irreducible symmetric space with holonomy algebra hol.
The computation of |R sp(m)⊕sp(1) | 2 in terms of the hyper-Kähler component |R 0 | 2 in corollary 4.5 is based on the computation of |R sp(m)⊕sp(1) | 2 for the Wolf spaces SO(m+4) S(O(m)×O(4)) . Example 4.3. Let denote the Grassmannian of p-planes in R p+q . Under the identification of the tangent space with R q×p the metric is given by g(X, Y ) = tr(X T Y ) and the curvature tensor by cf. [Bal06,Example B.42]. In particular, if E ij denotes the standard orthonormal basis of R q×p , then the eigenspaces of the curvature operator are given by Proof. Example 4.3 exhibits the geometry of the Grassmannians. To emphasize the quaternion Kähler structure we use the identification It is straightforward to describe the eigenspaces Eig(m) and Eig(4) in terms of the quaternion Kähler geometry. Moreover, using the eigenbasis of the curvature operator of HP m in remark 4.1 as a basis for sp(m) ⊕ sp(1), it is easy to find an orthonormal eigenbasis of the curvature operator

Specifically, let
and for L = I, J , K define Recall that I i , J i , K i and W ij are defined in remark 4.1. It follows that the eigenspaces for R W : sp(m) ⊕ sp(1) → sp(m) ⊕ sp(1) are given by In particular, In the following, we will consider the orthonormal eigenbasis We will compute the overall sum by separately evaluating for orthonormal bases B i for suitable subspaces of Eig(m), Eig(4), ker(R W ).
Recall from the proof of proposition 4.2 that the Lie algebra action of Eig(m) on Eig(4) is trivial.
Next we consider the action of Eig(m) on ker(R W ). Clearly, is an orthonormal eigenbasis for Eig(m). Remark 4.1 shows that sp(1) = span ω + I , ω + J , ω + K is part of the isotropy of HP m . Thus it acts trivially on ker (R W ) ⊂ sp(m). On the other hand, for the Lie algebra sp(1) = span ω − I , ω − J , ω − K , we obtain In fact, a similar diagram is valid for then B 2 ∪ B 3 is an orthonormal eigenbasis for ker(R W ). Observe that and similarly Finally consider the action of Eig(4) on ker(R W ). Let denote an orthonormal eigenbasis of Eig(4). Firstly, we compute the action of B 4 on B 2 . For two sets A, B let A∆B = (A ∪ B) \ (A ∩ B) denote the symmetric difference. It is straightforward to check that for i < j, k < l and where L k is defined in remark 4.1 for L = I, J , K.
is orthogonal to all basis elements in B 0 except possibly L k . Note that Thus the only non-zero inner products of (W ij )L ij with elements in B 0 are given by for i < j < k + 1 for L = I, J , K.
In particular, R sp(m)⊕sp(1) = 0 if and only of R is a multiple of R HP m .
Hence the maximum principle shows that R is parallel.