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The Weyl Tensor of Gradient Ricci Solitons

This paper derives new identities for the Weyl tensor on a gradient Ricci soliton, particularly in dimension four. First, we prove a Bochner-Weitzenb\"ock type formula for the norm of the self-dual Weyl tensor and discuss its applications, including connections between geometry and topology. In the second part, we are concerned with the interaction of different components of Riemannian curvature and (gradient and Hessian of) the soliton potential function. The Weyl tensor arises naturally in these investigations. Applications here are rigidity results.


Introduction
The Ricci flow, which was first introduced by R. Hamilton in [30], describes a oneparameter family of smooth metrics g(t), 0 ≤ t < T ≤ ∞, on a closed n-dimensional manifold M n , by the equation The subject has been studied intensively, particularly in the last decade thanks to seminal contributions by G. Perelman in his proof of the Poincaré conjecture (cf. [39,40]). It also gains more popularity after playing a key role in the proofs of the classification theorem for manifolds with 2-positive curvature operators by C. Böhm and B. Wilking [6], and the Differentiable Sphere Theorem of S. Brendle and R. Schoen [8,10].
As a weakly parabolic system, the Ricci flow can develop finite-time singularities and, consequently, the study of singularity models becomes essentially crucial. In this paper, we are concerned with gradient Ricci solitons (GRS), which are selfsimilar solutions of Hamilton's Ricci flow (1.1) and arise naturally in the analysis of singularities. A GRS (M, g, f, λ) is a Riemannian manifold endowed with a special structure given by a (soliton) potential function f , a constant λ, and the equation (1.2) Rc + Hessf = λg.
Depending on the sign of λ, a GRS is called shrinking (positive), steady (zero), or expanding (negative). In particular an Einstein manifold N can be considered as a special case of a GRS where f is a constant and λ becomes the Einstein constant. A less trivial example is a Gaussian soliton (R k , g sd , λ |x| 2 2 , λ) with g sd being the standard metric on Euclidean space. It is interesting to note that λ can be an arbitrary real number and that the Gaussian soliton can be either shrinking, steady or expanding. Furthermore, a combination of those two above, by the notation of P. Petersen and W. Wylie [42], is called a rank k rigid GRS, namely a quotient of N × R k . Other nontrivial examples of GRS are rare and mostly Kähler, see [12,25].
In recent years, following the interest in the Ricci flow, there have been various efforts to study the geometry and classification of GRS's; for example, see [13] and the citations therein. In particular, the low-dimensional cases (n = 2, 3) are relatively well-understood. For n = 2, Hamilton [32] completely classified shrinking gradient solitons with bounded curvature and showed that they must be either the round sphere, projective space, or Euclidean space with standard metric. For n = 3, utilizing the Hamilton-Ivey estimate, Perelman [40] proved an analogous theorem. Other significant results include recent development of Brendle [7] showing that a non-collapsed steady GRS must be rotationally symmetric and is, therefore, isometric to the Bryant soliton.
In higher dimensions, the situation is more subtle mainly due to the non-triviality of the Weyl tensor (W) which is vacuously zero for dimension less than four. One general approach to the classification problem so far has been imposing certain restrictions on the curvature operator. An analogue of Hamilton-Perelman results was obtained by A. Naber proving that a four dimensional complete non-compact GRS with bounded nonnegative curvature operator must be a finite quotient of R 4 , S 2 × R 2 or S 3 × R [37]. In [33], B. Kotschwar classified all rotationally symmetric GRS's with given diffeomorphic types on R n , S n−1 × R or S n . Note that any rotationally symmetric Riemannian manifold has vanishing Weyl tensor.
Thus, a natural development is to impose certain conditions on that Weyl tensor. If the dimension is at least four, then a complete shrinking GRS with vanishing Weyl tensor must be a finite quotient of R n , or S n−1 × R or S n following the works of [38,50,18,43]; a steady GRS is flat or rotationally symmetric (that is, a Bryant Soliton) by [14]. The assumption W ≡ 0 can be weakened to δW ≡ 0, a closed or non-compact shrinking GRS must be rigid [18,26,36]; or in dimension four, to the vanishing of self-dual Weyl tensor only, a shrinking GRS with bounded curvature must be a finite quotient of R 4 , S 3 × R, S n , or CP 2 , and steady GRS must be a Bryant soliton or flat [21]. There are some other classifications based on, for instance, Bach flatness [15] or assumptions on the radial sectional curvature [43].
As the major obstruction to understand GRS in higher dimensions is the nontriviality of the Weyl tensor, this paper is devoted to studying the delicate role of the Weyl tensor within a gradient soliton structure. Our perspective here is to view a GRS as both generalization of an Einstein manifold as well as a self-similar solution to the Ricci flow. In particular, this paper derives several new identities on the Weyl tensor of GRS in dimension four. In the first part, we prove the following Bochner-Weitzenböck type formula for the norm of the self-dual Weyl tensor using flow equations and some ideas related to Einstein manifolds. Theorem 1.1. Let (M, g, f, λ) be a four-dimensional GRS. Then we have the following Bochner-Weitzenböck formula: For the relevant notation, see Section 2. Identity (1.3) potentially has several applications and we will present a couple of them in this paper including a gap theorem. More precisely, if the GRS is not locally conformally flat and the divergence of the Weyl tensor is relatively small, then the L 2 -norm of the Weyl tensor is bounded below by a topological constant (cf. Theorem 4.1). The proof, in a similar manner to that of [28], uses some ideas from the solution to the Yamabe problem.
In the second part, we are mostly concerned with the interaction of different curvature components, gradient and Hessian of the potential function. In particular, an interesting connection is illustrated by the following integration by parts formula. Theorem 1.2. Let (M, g, f, λ) be a closed GRS. Then we have the following identity: In particular, in dimension four, the identity becomes  Remark 1.2. This result exposes the intriguing interaction between the Weyl tensor and the potential function f on a GRS. It will be interesting to extend those identities to a (possibly non-compact) smooth metric measure space or generalized Einstein manifold. Remark 1.3. In dimension four, the statement also holds if replacing W by W ± , see Corollary 5.8.
The interactions of various curvature components and the soliton potential function can be applied to study the classification problem. For example, Theorem 6.1 asserts rigidity of the Ricci curvature tensor in dimension four. More precisely, if the Ricci tensor at each point has at most two eigenvalues with multiplicity one and three, then any such closed GRS must be rigid. It is interesting to compare this result with classical classification results of the Codazzi tensor, which requires both distribution of eigenvalues and information on the first derivative (see [5,Chapter 16,Section C]). This paper is organized as follows. In Section 2, we fix our notation and collect some preliminary results. Section 3 provides a proof of Theorem 1.1 using the Ricci flow technique. Section 4 gives some immediate applications of the new Bochner-Weitzenböck type formula including the aforementioned gap theorem. In Section 5, we first discuss a general framework to study the interaction of different components of the curvature with the potential function, and then prove Theorem 1.2. In Section 6, we apply our framework to obtain various rigidity results. Finally, in the Appendix, we collect a few related formulas.

Notation and Preliminaries
In this section, we will fix the notation and convention that will be used throughout the paper. R, W, Rc, S, E will stand for the Riemannian curvature operator, Weyl tensor, Ricci curvature, scalar curvature, and the traceless part of the Ricci respectively.
For a finite dimensional real vector space (bundle), Λ 2 (V ) denotes the space of bi-vectors or two-forms. In our case, the space of interest is normally the tangent bundle and when the context is clear, the dependence on V is omitted.
Given an orthonormal basis {E i } n i=1 of T p M, it is well-known that we can construct an orthonormal frame about p such that e i (p) = E i and ∇e i | p = 0. Such a frame is called normal at p. Also e 12 is the shorthand notation for e 1 ∧ e 2 ∈ Λ 2 .
The modified Laplacian is defined as For any (m, 0)-tensor T , its divergence operator is defined as while its interior product by a vector field X is defined as Furthermore, we will interchange the perspective of a vector and a covector freely, i.e., a (2, 0) tensor will also be seen as a (1, 1) tensor. Similarly, a (4, 0) tensor such as R, W can be interpreted as an operator on bi-vectors, that is, a map from Λ 2 (T M) → Λ 2 (T M). Consequently, the norm of these operators is agreed to be sum of all eigenvalues squared (this agrees with the tensor norm defined in [22] for (2, 0) tensors but differs by 1/4-factor for (4, 0) tensors). More precisely, In addition, the norm of covariant derivative and divergence on these tensors can be defined accordingly, For a tensor T : Finally, when the context is clear, we will omit the measure when integrating.
2.1. Gradient Ricci Solitons. In this subsection, we recall some well-known identities for GRS's. A GRS is characterized by the Ricci soliton equation Algebraic manipulation of (2.3) and application of the Bianchi identities lead to following formulas (for a proof see [22]), △S + 2|Rc| 2 = ∇f, ∇S + 2λS. Moreover, a complete GRS has positive scalar curvature unless it is isometric to the flat Euclidean space [44].

2.2.
Four-Manifolds. In this subsection, we give a brief review of the algebraic structure of curvature and geometry on an oriented four-manifold (M, g).
First we recall the Kurkani-Nomizhu product for (2, 0) symmetric tensors A and B, Then we have the following decomposition of curvature, In dimension four this becomes, An important feature in dimension four is that the Hodge star operator decomposes the space of bi-vectors (Λ 2 ) orthogonally according to the eigenvalues ±1. The Riemannian curvature inherits this decomposition and, consequently, has a special structure. To be more precise, let {e i } 4 i=1 be an orthonormal basis of the tangent space at any arbitrary point on M, then one pair of orthonormal bases of bi-vectors is given by Accordingly, the curvature now is with C essentially the traceless part. It is easy to observe that W (Λ ± 2 ) ∈ Λ ± 2 , so it is unambiguous to define W ± := W |Λ ± . In particular, with α ± and β ± the projection of α, β onto Λ ± 2 .
Furthermore, as W is traceless and satisfies the first Bianchi identity, there is a normal form by M. Berger [4] (see also [46]). That is, there exists an orthonormal basis Then, by (2.12), Using the basis given in (2.10), we get Hence we obtain the following well-known identities.
Lemma 2.1. Let (M 4 , g) be a four-dimensional Riemannian manifold, then the following tensor equations hold, Proof. Note that these identities only depend on the decomposition of these tensors. In particular, it suffices to prove for the Weyl tensor. Using the normal form discussed above, we calculate that, Similar calculation can be done for all other pair of indexes to verify the statements.
Remark 2.2. The first identity can also be found in [23, 2.31].
In addition, in reference to the decomposition of curvature in (2.11), we have the following relations.
If the manifold is closed, then the Gauss-Bonnet-Chern formula for the Euler characteristic and Hirzebauch formulas for the signature (cf. [5] for more details) are given by, The Hodge operator in dimension four is related to a certain decomposition on the tangent bundle. Let {α i } 3 i=1 be a positive-oriented orthogonal basis of Λ + 2 with |α i | = √ 2, according to [1], if sign(i, j, k) = 1, then we have Here sign(i, j, k) is the sign-um of the permutation of {1, 2, 3}. The positive-orientation is just to agree with the sign convention. An example of such a basis is given by multiplying √ 2 the basis given in (2.10). Consequently, we have the following result.
Lemma 2.2. Suppose (M, g) is a four-dimensional Riemannian manifold and X is a vector field on M. At any point p such that X p = 0, Pick an orthogonal basis of Λ + 2 as above then it follows that {α i (X p )} 3 i=1 are three orthogonal vectors and each is perpendicular to X p . So the statement follows.
Remark 2.4. By symmetry, the statement also holds for Λ − 2 . When the context is clear, we normally omit the sub-index of the point.
We further define a new "curvature" tensor R by Thus, it follows immediately that, with respect to (2.10), Id. Furthermore, following the argument in [4], we obtain, Proposition 2.4. There exists a normal form for R. More precisely, at each point, there exits an orthonormal base {e i } 4 i=1 , such that with respect to the corresponding base {e 12 , e 13 , e 14 , e 34 , e 42 , e 23 } for Λ 2 and as an operator on 2-forms, where K is the "sectional curvature" of R, i.e., K(e 1 , e 2 ) = R 1212 for any orthonormal vectors e 1 and e 2 .
Remark 2.6. Can a GRS be characterized by the existence of such a function f with R constructed as above having the normal form?
Next, we investigate the assumption of having a lower bound on this new sectional curvature similar to [29]. For ǫ < 1/3, suppose that Equivalently, for any orthonormal pairs e i and e j , that is Then we have the following lemma.
Lemma 2.5. Let (M, g, f, λ) be a GRS, then assumption (2.20) implies the following: The equality happens in the last formula if and only if W ± has the form a ± diag(−1, −1, 2), with a ± ≥ 0 and Proof. All inequalities follow from tracing equation (2.21) and the soliton equation S + ∆f = 4λ except the last one. For the last inequality, first note that any two form φ can be written as a simple wedge product of 1-forms iff φ ∧ φ = 0. In dimension four, with respect to (2.10), that is equivalent to φ = φ + + φ − and |φ + | = |φ − . Therefore, in light of Proposition 2.13, assumption (2.20) is equivalent to with a + , a − are the smallest eigenvalues of W ± . Using the algebraic inequalities we obtain: Equality happens if and only if the equality happens in (2.22) and (2.23) (or (2.24)). The result then follows immediately.
Lemma 2.6. Let (M, g, f, λ) be a closed GRS with assumption (2.20), then Again equality holds if W ± has the form a ± diag(−1, −1, 2) with a ± ≥ 0 and Proof. Using equation (2.8), we compute: The above results lead to the following estimate on the Euler characteristic.
We now claim that the equality case can not happen. Suppose otherwise then |W + ||W − | = 0 and equality also happens in Lemma 2.6. By the regularity theory for solitons [3], we can choose an orientation such that In the first case, by the classification of locally conformally flat four-dimensional closed GRS's as discussed in Introduction, (M, g) is flat, this is a contradiction.
In the second case, Rc = 0 implies S = 0 = λ, and since equality happens in Lemma 2.6, W + = 0. Hence the above argument applies.
Remark 2.8. The Euler characteristic of a closed Ricci soliton has been studied by [24]. If the manifold is Einstein and ǫ = 0, we recover some results of [29].

A Bochner-Weitzenböck Formula
In this section, we prove Theorem 1.1, a new Bochner-Weitzenböck formula for the Weyl tensor of GRS's, which generalizes the one for Einstein manifolds. Bochner-Weitzenböck formulas have been proven a powerful tool to find connections between topology and geometry with certain curvature conditions (for example, see [27,41,47]).
Particularly, in dimension four, if δW + = 0 (this contains all Einstein manifolds), we have the following well-known formula (see [5,16.73]), This equation plays a crucial role to obtain a L 2 -gap theorem of the Weyl tensor and to study the classification problem of Einstein manifolds (cf. [28,29,49]).
Our first technical lemma gives a formula of ∆ f W in a local frame. Also it is noticed that the Einstein summation convention is used repeatedly here.
be a local normal frame, then the following holds, Proof. First, we recall how a GRS can be realized as a self-similar solution to the Ricci flow (1.1), as in Section 2.1. Let ) is a solution to the Ricci flow. Furthermore, W(t) = τ φ * W. Let p be a point in M and {e i } n i=1 be a basis of T p M, and we obtain a local normal frame via extending e i to a neighborhood by parallel translation along geodesics with respect to g(0). First, we observe, where L X is the Lie derivative with respect to X. Furthermore, by definition, We calculate that W([∇f, e i ], e j , e k , e l ) =W(∇ ∇f e i − ∇ e i ∇f, e j , e k , e l ) = −W(∇ e i ∇f, e j , e k , e l ).
By the soliton structure, Along the Ricci flow, the Weyl tensor is evolving under the following (for example, The result then follows.
Furthermore, in dimension four, we are able to obtain significant simplification due to the special structure given by the Hodge operator. That gives the proof of our first main theorem.
Proof. (Theorem 1.1) We observe that, Therefore, To calculate the first term of the right hand side, we use the normal form of the Weyl tensor (2.13). As usual, a local normal frame is obtained by parallel translation along geodesic lines. Then (2.10) gives a basis of eigenvectors In order to use Lemma 3.1, it is necessary to calculate the C ijkl terms. By the normal form, we have Thus, Therefore, Similar calculations hold when replacing α 1 by α 2 , α 3 , Combining (3.7), (3.8), (3.9), (3.10) yields, The first equality then follows. The second equality comes from the soliton equation, the property that W + is trace-free and Remark 2.5.

4.1.
A Gap Theorem for the Weyl Tensor. In [28], under the assumptions W + = 0, δW + = 0, and the positivity of the Yamabe constant, M. Gursky proves the following inequality, relating ||W + || L 2 with topological invariants of a closed fourmanifold, Our main result in this section is to prove an analog for GRS's. It is noted that the particular structure of GRS allows us to relax the harmonic self-dual condition above with the cost of a worse coefficient due to the absence of an improved Kato's inequality.
Thus, it is clearly weaker than the assumption of harmonic self-dual.
To prove Theorem 4.1, we follow an idea of [28] and introduce a Yamabe-type conformal invariant. First, the conformal Laplacian is given by, where a and b are constants to be determined later. Under a conformal transformation as described in (7.1), for any function Φ, we havẽ The Yamabe problem is, for a given Riemannian manifold (M, g), to find a constant scalar curvature metric in its conformal class [g]. That is equivalent to find a critical point of the following functional, for any C 2 positive function u, letg = u 2 g, define Then the conformal invariant Y is defined as : u is a positive C 2 function on M}.
For an expository account on the Yamabe problem, see [34].
As F a,b conformally transforms like the scalar curvature, in analogy with the discussion above, we can define the following conformal invariant.
For the case of interest, we shall denote   Proof. The first inequality follows from the definition and the following observation. Given a metric g, a positive function u and b ≥ 0, then The second inequality is a result of T. Aubin [2] and R. Schoen [45]. The last inequality is an immediate consequence of the fact that the standard metric on S n is locally conformally flat (W = 0).
On a complete gradient shrinking soliton, the scalar curvature is positive unless the soliton is isometric to the flat Euclidean space [44]. Therefore, if the GRS is not flat then the existence of a solution to the Yamabe problem [34] implies that Y g > 0. This observation is essential because of the following result.   Proof. By Theorem 1.1, we have

Integrating both sides and applying (4.2) yield
Therefore, we arrive at We also have the following pointwise estimates, The first one is the classical Kato's inequality while the second one is purely algebraic. Thus, for u = |W + |, Hence if | W + | > 0 everywhere then the statement follows.
Now, equality holds only if M ( 1 3 F u 2 + 2|∇u| 2 )dµ = 0 and the equality happens in each point-wise estimate above. The result then follows.
We are now ready to prove the main result of this section.
Now the equality holds only if all equalities hold in (4.7), (4.6) and (4.2). The first one implies thatg is Einstein. Therefore, by [29, Theorem 1], inequality (4.6) is strict unless S ≡ 0. But this is a contradiction to Y (M) > 0. Thus the inequality is strict.

A Framework Approach
In this section, we shall propose a framework to study interactions between components of curvature operator and the potential function on a GRS (M, g, f, λ). In particular, we represent the divergence and the interior product i ∇f on each curvature component as linear combinations of four operators P, Q, M, N. The geometry of these operators, in turn, gives us information about the original objects. It should be noted that some identities here have already appeared elsewhere.
Let (M n , g) be an n-dimensional oriented Riemannian manifold. Using the pointwise induced inner product, any anti-symmetric (2,0) tensor α (a two-form) can be seen as an operator on the tangent space by In particular, a bi-vector acts on a vector X as follows For instance, in dimension four, the complete description is given by the table below. In a similar manner, any symmetric (2, 0) tensor b can be seen as an operator on the tangent space, Consequently, when b is viewed as a 1-form valued 1-form, d ∇ b denotes the exterior derivative (a 1-form valued 2-form). That is, Now we can define the fundamental tensors of our interest here, first via a local frame and then using the operator language. Let α ∈ Λ 2 , X, Y, Z ∈ T M, and {e i } n i=1 be a local normal orthonormal frame on a GRS (M n , g, f, λ).
Remark 5.1. The tensors P ± , Q ± , M ± , N ± : Λ ± 2 T M ⊗ T M → R are defined by restricting α ∈ Λ ± 2 T M. They can be seen as operators on Λ 2 by standard projection.

Decomposition Lemmas.
Using the framework above, we now can represent the interior product i ∇f on components of the curvature tensor as follows. Again the Einstein summation convention is used here.
Lemma 5.2. Let (M, g, f, λ) be a GRS, for P, Q, M, N as in Definition 5.1, in a local normal orthonormal frame, we have W ijkp ∇ p f = W(e i , e j , e k , ∇f ) (5.10) .
Proof. The first formula is well-known (cf. [16]), following from the soliton equation and Bianchi identities. For the second, we compute, For the third, we use (2.6) to calculate The next formula is a consequence of the above formulas, definition of H (2.18) and the soliton equation (2.3). Finally, the last one comes from decomposition of the curvature operator (2.9) and previous formulas; it appeared, for example, in [21].
In addition, the divergence on these components can be written as linear combinations of P, Q, M, N.
Proof. The first formula is well-known and comes from the second Bianchi identity [16]. For the second, we compute, For the next one, we use (2.5) to calculate, Finally, the last one comes from decomposition of curvature (2.9) and previous formulas; it also appeared in, for example, [23,Eq. (9)]. Remark 5.3. C defined in (5.14) is also called the Cotton tensor in literature.
The following observation is an immediate consequence of Lemma 5.3.
Proposition 5.4. Let (M n , g, f, λ), n > 2, be a GRS and H given by (2.18). Then the tensor Remark 5.5. The result can be viewed as a generalization of the harmonicity of the Weyl tensor on an Einstein manifold.

Norm Calculations.
Lemma 5.5. Let (M, g, f, λ) be a GRS, then the following identities hold: Proof. The main technique is to compute under a normal orthonormal local frame. For example, Other equations follow from similar calculation.
When M is closed, we can integrate by parts. In particular, the first equation was first derived in [16]. For the second, we compute that Hence, the statement follows.
Remark 5.6. The factor of 2 is due to our convention of calculating norm. Some special cases of dimension four also appeared in [9,Proposition 4].
An interesting consequence of the above calculation is the following corollary, which exposes the orthogonality of Q, N versus i ∇f W, δW. b. If M is closed, then, Proof. Part a) follows immediately from Lemmas 5.2, 5.3, 5.5, and our convention (2.1). For example, Other formulas follow from similar calculations. For part b) we observe that, , P .
Notice that we apply part a) in the last step. Consequently, applying Lemma 5.5 again yields , )e −f .
Remark 5.7. Part b) recovers the well-known fact that harmonic curvature implies harmonic Weyl tensor and constant scalar curvature.
Now we are ready to prove Theorem 1.2.
Proof. (Theorem 1.2) First, we observe, Proof. It suffices to show the statements is true for the self-dual part. Let {e i } 4 i=1 be a normal orthonormal local frame and let {α i } 4 i=1 be an orthonormal basis for Λ + 2 . Then Furthermore, we can choose a special basis, namely the normal form as in (2.13).
Then α i 's diagonalize W + with eigenvalues λ i 's. Consequently, Thus, Now by (5.1), it is easy to see that each η k = 0 because W + is traceless.
The statements involved N follow from analogous calculations as By manipulation as in the proof of Theorem 1.2, using Remark 5.4 (replacing Lemmas 5.2 and 5.3) and Lemma 5.7 (replacing Lemma 5.6), we immediately obtain the following result.

Rigidity Results
In this section, we present conditions that imply the rigidity of a GRS using the analysis on the framework discussed in the previous section.
First, Proposition 6.10 provides a geometrical way to understand tensor D defined in (5.15). In particular, it says that D ≡ 0 is equivalent to a special condition, namely, the normalization of ∇f (if not trivial) is an eigenvector of the Ricci tensor, and all other eigenvectors have the same eigenvalue. Such a structure will imply rigidity as the geometry of the level surface (of f ) being well-described.
On the other hand, Theorem 1.2 reveals an interesting connection between the Ricci tensor and the Weyl tensor in dimension four. That allows us to obtain rigidity results using only the structure of the Ricci curvature for a GRS. Theorem 6.1. Let (M 4 , g, f, λ) be a closed four-dimensional GRS. Assume that at each point the Ricci curvature has one eigenvalue of multiplicity one and another of multiplicity three, then the GRS is rigid, hence Einstein.
We also find conditions that imply the vanishing of tensor D.
In dimension four, the result can be improved significantly. As discussed in the last section, there are some similarities between taking the divergence and interior product i ∇f of the Weyl tensor, for example, see Corollary 5.6. The following theorem is inspired by condition (1) of Theorem 6.2. An immediate consequence of the results above (plus known classifications discussed in the Introduction) is to obtain rigidity results. Corollary 6.5. Let (M n , g, f, λ), n ≥ 4, be a complete shrinking GRS.
In particular, when the dimension is four, we have the following result. The general strategy to prove aforementioned statements is to use the framework to study the structure of the Ricci tensor. 6.1. Eigenvectors of the Ricci curvature. Here we study various interconnections between the eigenvectors of the Ricci curvature, the Weyl tensor, and the potential function. First, we observe the following lemma. Lemma 6.7. Let (M, g) be a Riemannian manifold. Assume that, at each point, the Ricci curvature has one eigenvalue of multiplicity one and another of multiplicity n − 1. Then we have, W, Rc • Rc = 0.
Proof. Without loss of generality, we can choose a basis {e i } n i=1 of T p M consisting of eigenvectors of Rc, namely Rc 11 = η and Rc ii = ζ for i = 2, ..., n. Then, We observe that, The result then follows.
Next, a consequence of our previous framework (on P, Q, M, and N) is the following characterization about the condition Rc(∇f ) = µ∇f . (1) being equivalent to (3) follows from an identical argument as above.
(2) being equivalent to (4) follows from Q(Y, Z, X), (2) being equivalent to (5) follows from Furthermore, the rigidity of these operators Q, M, N is captured by the following result. ii. In dimension four, fix a unit vector e i and note that T (α, e i ) = 0 for any α ∈ Λ + 2 . By Lemma 2.2 and Remark 2.4, T (β, e i ) = 0 for all β ∈ Λ − 2 . As e i is arbitrary the result then follows.
Recall that tensor D is a special linear combination of M, N, Q. Therefore, we obtain the following geometric characterization.
Thus, D ≡ 0 is equivalent to For the tensor D, .
Remark 6.4. Under that conformal change of the metric, the Ricci tensor is given by Therefore, at each point, Rc has at most two eigenvalues. Furthermore, sinceg has harmonic Weyl tensor, its Schouten tensor is a Codazzi tensor with at most two eigenvalues. Using the splitting results for Riemannian manifolds admitting such a tensor gives another proof of results in [15]. This method is inspired by [19].
Now we investigate several conditions which will imply that Rc(∇f ) = µ∇f .
Proof. The idea is to find a connection of each condition with Lemma 6.8.
Assuming (1): We observe that Therefore, the result follows from Lemma 6.9 and Proposition 6.10.
The proof of Theorem 6.4 follows from a similar argument.

Proof. (Theorem 6.3)
Using a normal local frame, we can rewrite the assumption as, We pick an arbitrary index a and multiply both sides with W ajkl to arrive at, Applying identity (2.14) yields, Since index a is arbitrary, we have ∇f = 0 or |W + | = 0.
Proof. (Corollary 6.5) By Theorem 6.2 and Theorem 6.4, each condition implies D ≡ 0. Then, [15,Lemma 4.2] further implies that δW = 0. It follows, from classification results for harmonic Weyl tensor as discussed in the Introduction, that the manifold must be rigid. We now look at each case closely and observe that not all ranks can arise.
i. In this case, Lemma 6.9 reveals that λ 0 − λ i = 0 with Rc(∇f ) = λ 0 ∇ f , and λ i is any other eigenvalue of Rc. Therefore, the manifold structure must be Einstein.
ii. In this case, since D ≡ 0 implies Rc has at most two eigenvalues with one of multiplicity 1 and another of n − 1. So k can only be 0, 1, n.
iii. In this case, there is no obvious obstruction, so all rank can arise.

Appendix
In this appendix, we collect a few formulas that are related to this paper, they follow from direct computation. 7.1. Conformal Change Calculation. In this subsection, we state the change of covariant derivative of the Weyl tensor and Bochner-Weitzenböck type formula, with respect to the conformal transformation of a metric. We first fix our notation. Let (M n , g) be a smooth Riemannian manifold and u = e f be a smooth positive function on M. A conformal change is defined by: Then, for any tensor D with respect to g, the corresponding forg is denoted by D.
We can calculate the transformation of the covariant derivative. For fixed X, Y, Z, Thus, Consequently, with the convention of a ∇ 2 f − df ⊗ df + 1 2 |∇f | 2 g, we have R =e 2f R − e 2f a • g, R l ijk = R l ijk − a l i g jk − a jk δ l i + a ik δ l j + a l j g ik , d µ =e nf dµ, Next, we calculate the conformal change of the norm of the covariant derivative of the Weyl tensor. Lemma 7.2. Let (M, g) be a four-dimensional Riemmanian manifold, andg = u 2 g, for some positive smooth function u. Then we have, (7.3) | ∇ W| 2 = u −6 |∇W| 2 + 18u −8 |∇u| 2 |W| 2 − 10u −7 ∇u∇|W| 2 + 16 δW, i ∇u W .
The result then follows immediately.
We now can calculate the conformal change of Bochner-Weitzenböck's formula.
The result then follows by combining above equations with Lemma 7.2.

7.2.
Along the Ricci Flow. Inspired by the simplification in Bochner-Weitzenböck formula in Theorem 1.1, we carry out a similar calculation on a Ricci solution. As a consequence, we obtain several interesting evolution equations involving the self-dual part and other components of the curvature operator. First, we state some useful lemmas.
An immediate application of the computation above and the maximum principle is the result below.
Proposition 7.8. Let (M, g(t)), 0 ≤ t < T ≤ ∞, be a closed solution to the Ricci flow (1.1). If det Λ 2 + W + is nonpositive along the Ricci flow then there exists a constant C = C(g(0)), such that |W + | S < C is preserved along the flow.