On a Classification of 4-d Gradient Ricci Solitons with Harmonic Weyl Curvature

Abstract

We study a characterization of 4-dimensional (not necessarily complete) gradient Ricci solitons (M, g, f) which have harmonic Weyl curvature, i.e., $\delta W=0$. Roughly speaking, we prove that the soliton metric g is locally isometric to one of the following four types: an Einstein metric, the product $\mathbb {R}^2 \times N_{\lambda }$ of the Euclidean metric and a 2-d Riemannian manifold of constant curvature ${\lambda } \ne 0$, a certain singular metric and a locally conformally flat metric. The method here is motivated by Cao–Chen’s works (in Trans Am Math Soc 364:2377–2391, 2012; Duke Math J 162:1003–1204, 2013) and Derdziński’s study on Codazzi tensors (in Math Z 172:273–280, 1980). Combined with the previous results on locally conformally flat solitons, our characterization yields a new classification of 4-d complete steady solitons with $\delta W=0$. For the shrinking case, it re-proves the rigidity result (Fernández-López and García-Río in Math Z 269:461–466, 2011; Munteanu and Sesum in J. Geom Anal 23:539–561, 2013) in 4-d. It also helps to understand the expanding case; we now understand all 4-d non-conformally flat ones with $\delta W=0$. We also characterize locally 4-d (not necessarily complete) gradient Ricci solitons with harmonic curvature.

1 Introduction

A gradient Ricci soliton consists of a Riemannian manifold (M, g) and a smooth function f satisfying $\nabla d f = -Rc + \lambda g$, where Rc denotes the Ricci tensor of g and $\lambda$ is a constant. Gradient Ricci solitons are essential in Hamilton’s Ricci flow theory as singularity models of the flow. So it is important to understand their geometry and classify them. A gradient Ricci soliton is said to be shrinking, steady or expanding if $\lambda$ is positive, zero or negative, respectively.

Two-dimensional gradient Ricci solitons are well understood; see [2] and references therein. Any 3-d complete noncompact non-flat shrinker (shrinking Ricci soliton) is proved to be a quotient of the round cylinder $\mathbb {S}^2 \times \mathbb {R}$ in [8]; see also [22, 25, 26]. For the 3-d gradient steadiers (steady Ricci solitons), one may refer to [3, 4] and references therein.

In higher dimension, there are numerous rigidity and classification results under various geometric conditions. For the relevance to the current work, we shall focus on locally conformally flat solitons and their generalizations.

Complete locally conformally flat gradient shrinkers are classified to be a finite quotient of $\mathbb {R}^n$, $\mathbb {S}^n$, or $\mathbb {S}^{n-1} \times \mathbb {R}$, $n \ge 4$, in [9, 29, 32]; see also [18, 25]. Complete locally conformally flat gradient steadiers are classified to be either flat or isometric to the Bryant soliton [6, 10]. The 4-d half conformally flat steadiers and shrinkers are studied in [14]. More generally, Bach-flat shrinkers are classified in [7] and Bach-flat steadiers with positive Ricci curvature in [5].

A gradient soliton is said to be rigid if it is isometric to a quotient of $N \times \mathbb {R}^k$ where N is an Einstein manifold and $f = \frac{\lambda }{2} |x|^2$ on the Euclidean factor. Fernández-López and García-Río [19] showed that an n-dimensional compact Ricci soliton (M, g) is rigid if and only if it has harmonic Weyl tensor W. Then Munteanu and Sesum [24] proved that any n-dimensional complete gradient shrinker with harmonic Weyl tensor is rigid. In [31], Wu, Wu and Wylie showed that a 4-d complete gradient shrinker with $\delta W^{+}=0$ is either Einstein, or a finite quotient of $\mathbb {S}^3 \times \mathbb {R}$, $\mathbb {S}^2 \times \mathbb {R}^2$ or $\mathbb {R}^4$.

The purpose of this article is to study 4-dimensional gradient Ricci solitons (M, g, f) which have harmonic Weyl curvature. This work is most related to the above-mentioned works on locally conformally flat solitons and to [24] on shrinking solitons with $\delta W=0$. The latter needs control on geometric decay of curvature and volume from the shrinker condition, while the former resorts to the nonnegative curvedness of metrics for locally conformally flat shrinking or steady solitons, which is proved in [12, 32].

As our study includes steady and expanding solitons with $\delta W=0$, we can use neither geometric decay nor nonnegative curvedness. This work takes a different approach and is inspired by Cao and Chen’s works [6, 7] and Derdziński’s [17]. Note that the harmonicity of the Weyl tensor provides a Codazzi tensor $Rc- \frac{R}{6}g$. Riemannian metrics with a Codazzi tensor which have more than two distinct eigenvalue functions of Ricci tensor have been little understood; see Chap. 16 of [1]. In this article, combining with the soliton condition, we managed to analyze in detail the Codazzi tensor with three and four distinct eigenvalues.

Our argument is mostly local and produces a local description of soliton metrics and potential functions. So far, we worked out only in four dimensions, but we hope that our perspective might provide some way to understand the higher-dimensional case.

The main theorem of this paper is as follows.

Theorem 1.1

Any four-dimensional (not necessarily complete) connected gradient Ricci soliton (M, g, f) with harmonic Weyl curvature is one of the following four types.

• (i) g is an Einstein metric with f a constant function.

• (ii) For each point $p \in M$, there exists a neighborhood V of p such that (V, g) is isometric to a domain in the product $\mathbb {R}^2 \times N_{\lambda }$ where $\mathbb {R}^2$ has the Euclidean metric and $N_{\lambda }$ is a 2-dimensional Riemannian manifold of constant curvature ${\lambda } \ne 0$. And $f = \frac{\lambda }{2} |x|^2$ modulo a constant on the Euclidean factor.

• (iii) For each point $p \in M$, there exists a neighborhood V of p with coordinates $(s,t, x_3, x_4)$ such that (V, g) is isometric to a domain in $\mathbb {R}^4{\setminus }\{ s=0 \}$ with the Riemannian metric $ds^2 + s^{\frac{2}{3}} dt^2+ s^{\frac{4}{3}} \tilde{g}$, where $\tilde{g}$ is the Euclidean metric on the $(x_3, x_4)$-plane. Also, $\lambda =0$ and $f=\frac{2}{3} \ln (s)$ modulo a constant.

• (iv) For each point p in an open dense subset of M, there exists a neighborhood V of p with coordinates $(s,t, x_3, x_4)$ such that (V, g) is isometric to a domain in $\mathbb {R} \times W^3$ with the warped product metric $ds^2 + h(s)^2 \tilde{g}$, where $\tilde{g}$ is a constant curvature metric on a 3-manifold $W^3$ and f is not constant. And g is locally conformally flat.

For the 4-d complete shrinking soliton case, we re-prove the rigidity result in [19, 24] by a distinct method. For the 4-d complete steady case, with the result of [6, 10] on locally conformally flat solitons, we obtain the following classification.

Theorem 1.2

A 4-dimensional complete steady gradient Ricci soliton with $\delta W=0$ is either Ricci flat, or isometric to the Bryant soliton.

The expanding solitons are much less rigid, and many works have been done recently, e.g., [13, 15, 28, 30] and references therein. We prove:

Theorem 1.3

A 4-dimensional complete expanding gradient Ricci soliton with harmonic Weyl curvature is one of the following:

• (i) g is an Einstein metric with f a constant function.

• (ii) g is isometric to a finite quotient of $\mathbb {R}^2 \times N_{\lambda }$ where $\mathbb {R}^2$ has the Euclidean metric and $N_{\lambda }$ is a 2-dimensional Riemannian manifold of constant curvature ${\lambda } < 0$. And $f = \frac{\lambda }{2} |x|^2$ on the Euclidean factor.

• (iii) g is locally conformally flat.

In [28] Petersen and Wylie proved that any complete gradient Ricci soliton with harmonic curvature is rigid. But it is not clear if their argument extends to work for a local soliton. The classification of any (not necessarily complete) gradient Ricci soliton with harmonic curvature comes from Theorem 1.1; we demonstrated it as Corollary 8.3 in the final section.

To prove Theorem 1.1, from the harmonic Weyl curvature condition on gradient Ricci solitons, we observe by the arguments of [7, 19] that $\frac{\nabla f}{ | \nabla f | }$ is a Ricci-eigenvector field with its eigenvalue $\lambda _1$, there is a local function s with $\nabla s =\frac{\nabla f}{ | \nabla f | }$, and $\lambda _1$ and R are functions of s only. Next we obtain important geometric information on (Ricci-)eigenvalues, eigenvectors and eigenspaces from the Codazzi tensor $Rc- \frac{R}{6}g$ through Derdziński’s Lemma 2.4 and its extension Lemma 2.8.

Based on all the above, we show in Lemma 2.7 that the Ricci-eigenvalues $\lambda _i$, $i=1,\ldots ,4$, locally depend only on the variable s; this key lemma is crucial in the later argument. Then we divide the proof of Theorem 1.1 into several cases, depending on the distinctiveness of $\lambda _2, \lambda _3, \lambda _4$. There arise two subtle cases: when these three are pairwise distinct and when exactly two of them are equal. In the latter case we reduce the analysis to ordinary differential equations in Lemma 6.1 and resolve them to get the types (ii) and (iii). In the former we compute on the soliton equation using Codazzi tensor property, which eliminates the case, in Proposition 3.4.

The last case $\lambda _2=\lambda _3= \lambda _4$ is relatively simpler and produces the types (i) and (iv). Theorem 1.2, 1.3 and Corollary 8.3 on the harmonic curvature case can be easily deduced from Theorem 1.1.

This paper is organized as follows. In Sect. 2 we develop properties common to any gradient Ricci solitons with harmonic Weyl curvature and nonconstant f; in particular we prove that $\lambda _i$’s, $i=1,\ldots ,4$, depend only on s. In Section 3 we study the case where the three $\lambda _i$’s, $i=2,3,4$, are pairwise distinct. In Sections 4, 5 and 6, we analyze the case where two of the three $\lambda _i$’s, $i=2,3,4$, are equal. In Sect. 7, we treat the remaining case where $\lambda _2=\lambda _3= \lambda _4$. In the final Sect. 8, we summarize and prove theorems.

2 Gradient Ricci Solitons with Harmonic Weyl Curvature

We shall begin by recalling some properties of a gradient Ricci soliton with harmonic Weyl curvature in a few lemmas.

Lemma 2.1

For any gradient Ricci soliton (M, g, f), we have:

• (i) $\frac{1}{2} dR = R(\nabla f, \cdot )$, where R in the left-hand side denotes the scalar curvature, and $R(\cdot , \cdot )$ is a Ricci tensor.

• (ii) $R + |\nabla f|^2 - 2\lambda f = constant$.

Our notational convention is as follows: for orthonormal vector fields $E_i$, $i=1, \ldots , n$, on an n-dimensional Riemannian manifold, the curvature components are

$$\begin{aligned} R_{ijkl}:=R(E_i, E_j, E_k, E_l) = \langle \nabla _{E_i} \nabla _{E_j} E_k - \nabla _{E_j} \nabla _{E_i} E_k - \nabla _{[E_i, E_j]} E_k , E_l\rangle . \end{aligned}$$

We recall the formula (2.1) in [19]:

Lemma 2.2

For a gradient Ricci soliton $(M^n, g, f)$ with harmonic Weyl curvature on an n-dimensional manifold $M^n$, we have:

$$\begin{aligned} R(X, Y, Z,\nabla f )= & {} \frac{1}{n - 1} R(X,\nabla f )g(Y, Z) - \frac{1}{n - 1} R(Y,\nabla f )g(X, Z)\\= & {} \frac{1}{2(n - 1)} dR(X) g(Y, Z) - \frac{1}{2(n - 1)} dR(Y)g(X, Z). \end{aligned}$$

One may mimic arguments in [7] and get the next lemma.

Lemma 2.3

Let $(M^n, g, f)$ be a gradient Ricci soliton with harmonic Weyl curvature. Let c be a regular value of f and $\Sigma _c= \{ x | f(x) =c \}$ be the level surface of f. Then the following hold:

• (i) Where $\nabla f \ne 0$, $E_1 := \frac{\nabla f }{|\nabla f | }$ is an eigenvector field of Rc.

• (ii) R and $|\nabla f|^2$ are constant on a connected component of $\Sigma _c$.

• (iii) There is a function s locally defined with $s(x) = \int \frac{ d f}{|\nabla f|}$, so that $\ \ \ \ ds =\frac{ d f}{|\nabla f|}$ and $E_1 = \nabla s$.

• (iv) $R({E_1, E_1})$ is constant on a connected component of $\Sigma _c$.

• (v) Near a point in $\Sigma _c$, the metric g can be written as $\ \ \ g= ds^2 + \sum _{i,j > 1} g_{ij}(s, x_2, \ldots , x_n) dx_i \otimes dx_j$, where $x_2, \ldots , x_n$ is a local coordinate system on $\Sigma _c$.

• (vi) $\nabla _{E_1} E_1=0$.

Proof

Lemma 2.2 gives $R(\nabla f, X) = 0$ for $X \perp \nabla f$, hence $E_1 = \frac{\nabla f }{|\nabla f | }$ is an eigenvector of Rc.

As $d R = 2 R(\nabla f, \cdot )$ from Lemma 2.1, $d R (X) =0$ for $X \perp \nabla f$. Also, $\frac{1}{2}\nabla _X |\nabla f|^2 = - R( \nabla f, X ) + \lambda g( \nabla f, X ) = 0$ for $X \perp \nabla f$. We proved (ii). $d (\frac{ d f}{|\nabla f|}) = -\frac{1}{2 |\nabla f|^{\frac{3}{2}}} d |\nabla f|^{2} \wedge df= 0$ as $\nabla _X ( |\nabla f|^2 )=0$ for $X \perp \nabla f$. So, (iii) is proved.

Locally, R may be considered as a function of the local variable s only. We can express $dR(E_1)=\frac{dR}{ds}ds (E_1)= \frac{dR}{ds} g( \nabla s, \nabla s ) =\frac{dR}{ds}$. By Lemma 2.1, we have $dR(E_1) = 2R({E_1, E_1}) |\nabla f|$, so $R({E_1, E_1})$ is constant on a connected component of $\Sigma _c$.

As $\nabla f$ and the level surfaces of f are perpendicular, one gets (v).

For (vi), one follows the proof of Proposition 5.1 in [7]; with the local coordinates $s, x_2, \ldots , x_n$ in (v), one readily gets $\nabla s= \frac{\partial }{\partial s}$ so that $[\frac{\partial }{\partial x_i}, \nabla s]=0$. Then $\langle \nabla s, \nabla s \rangle =1$ and $\langle \frac{\partial }{\partial x_i} , \nabla s \rangle =0$ yield (vi). $\square$

A Codazzi tensor on a Riemannian manifold M is a symmetric tensor A of covariant order 2 such that $d^{\nabla } A =0$, which can be written in local coordinates as $\nabla _k A_{ij} = \nabla _i A_{kj}$. Derdziński [17] described the following: for a Codazzi tensor A and a point x in M, let $E_A(x)$ be the number of distinct eigenvalues of $A_x$, and set $M_A = \{ x \in M \ | \ E_A \mathrm{\ is \ constant \ in \ a \ neighborhood\ of \ } x \}$, so that $M_A$ is an open dense subset of M and that in each connected component of $M_A$, the eigenvalues are well-defined and differentiable functions. The next lemma is from Sect. 2 of [17].

Lemma 2.4

For a Codazzi tensor A on a Riemannian manifold M, in each connected component of $M_A$,

• (i) Given distinct eigenfunctions $\lambda , \mu$ of A and local vector fields v, u such that $A v = \lambda v$, $Au = \mu u$ with $|u|=1$, it holds that    $v(\mu ) = (\mu - \lambda ) \langle \nabla _u u, v \rangle$.

• (ii) For each eigenfunction $\lambda$, the $\lambda$-eigenspace distribution is integrable, and its leaves are totally umbilic submanifolds of M.

• (iii) Eigenspaces of A form mutually orthogonal differentiable distributions.

When a Riemannian manifold M of dimension $n \ge 4$ has harmonic Weyl curvature, i.e., $\delta W=0$, it is equivalent to $d^{\nabla } (Rc - \frac{R}{2n-2} g) =0$. So, $\mathcal {A}:=Rc - \frac{R}{2n-2} g$ is a Codazzi tensor. By Lemma 2.4, each eigenspace distribution of $\mathcal {A}$ is integrable in the open dense subset $M_{\mathcal {A}}$ of M. The leaves are totally umbilic submanifolds of M. Let $D_1, \ldots , D_k$ be all the eigenspace distributions of ${\mathcal {A}}$ in a connected component of $M_{\mathcal {A}}$. Then, the Ricci tensor also has $D_1, \ldots , D_k$ as its eigenspace distributions. Let the dimension of $D_l$ be $d_l$ for $l=1, \ldots , k$. Then in a neighborhood of each point of the connected component of $M_{\mathcal {A}}$, there exist orthonormal Ricci-eigenvector fields $E_i$, $i=1, \ldots , n$, with corresponding eigenfunctions $\lambda _i$ such that $E_1, \ldots , E_{d_1} \in D_1$, $\ \ E_{d_1 +1} , \ldots , E_{d_1+ d_2} \in D_2$, $\ldots$ , and $E_{d_1 + \cdots +d_{k-1}+1} , \ldots , E_{n} \in D_k$.

Let $(M^n,g,f)$ be a gradient Ricci soliton with harmonic Weyl curvature. As a gradient Ricci soliton, (M, g, f) is real analytic in harmonic coordinates; see [21] or argue as in [20, Prop. 2.4]. Then if f is not constant, $\{ \nabla f \ne 0 \}$ is open and dense in M. As in the above paragraph, we consider orthonormal Ricci-eigenvector fields $E_i$ in a neighborhood of each point in $M_{\mathcal {A}} \cap \{ \nabla f \ne 0 \}$. By just requiring $E_1= \frac{\nabla f}{|\nabla f| }$ to be in $D_1$ and using Lemma 2.3, we obtain:

Lemma 2.5

Let $(M^n,g,f)$ be an n-dimensional gradient Ricci soliton with harmonic Weyl curvature and non-constant f. For any point p in the open dense subset $M_{\mathcal {A}} \cap \{ \nabla f \ne 0 \}$ of $M^n$, there is a neighborhood U of p where there exist orthonormal Ricci-eigenvector fields $E_i$, $i=1, \ldots , n$ such that for all the eigenspace distributions $D_1, \ldots , D_k$ of ${\mathcal {A}}$ in U,

• (i) $E_1= \frac{\nabla f}{|\nabla f| }$ is in $D_1$,

• (ii) for $i>1$, $E_i$ is tangent to smooth level hypersurfaces of f,

• (iii) let $d_l$ be the dimension of $D_l$ for $l=1, \ldots , k$, then $E_1, \ldots , E_{d_1} \in D_1$, $\ \ E_{d_1 +1} , \ldots , E_{d_1+ d_2} \in D_2$, $\ldots$ , and $E_{d_1 + \cdots +d_{k-1}+1} , \ldots , E_{n} \in D_k$.

These local orthonormal Ricci-eigenvector fields $E_i$ of Lemma 2.5 shall be called an adapted frame field of (M, g, f).

For an adapted frame field $E_i$, $i=1, \ldots , n$, with $R_{ij}:= R(E_i, E_j)= \lambda _i \delta _{ij}$, from Lemma 2.2, for $j \in \{ 2, \ldots , n \}$ we get

$$\begin{aligned} R(E_1, E_j, E_j,\nabla f )= \frac{1}{n - 1} Ric(E_1,\nabla f )= \frac{1}{2(n - 1)} dR(E_1). \end{aligned}$$

(1)

Due to Lemma 2.3, in a neighborhood of a point $p \in M_{\mathcal {A}} \cap \{ \nabla f \ne 0 \}$, f and R may be considered as functions of the variable s only, and we write the derivative in s by a prime: $f^{\prime } = \frac{df}{ds}$ and $R^{\prime } = \frac{dR}{ds}$, etc. We recall $dR(E_1)=R^{\prime }ds (E_1)= R^{\prime } g( \nabla s, \nabla s ) =R^{\prime }$ and similarly $df(E_1) = f^{\prime }$. Also, $df(E_1) = g(\nabla f, \frac{\nabla f}{ |\nabla f| }) = |\nabla f |$. So, $|\nabla f |= f^{\prime }$. Then (1) becomes:

$$\begin{aligned} R_{1jj1} |\nabla f| = \frac{1}{n - 1} R_{11} |\nabla f| = \frac{1}{2(n - 1)} R^{\prime } . \end{aligned}$$

(2)

Lemma 2.6

For a gradient Ricci soliton (M, g, f) with harmonic Weyl curvature, and for a local adapted frame field $\{ E_i \}$ in $M_{\mathcal {A}} \cap \{ \nabla f \ne 0 \}$, setting $\zeta _i= - \langle \nabla _{E_i} E_i , E_1 \rangle$, for $i >1$, we have:

$$\begin{aligned} \nabla _{E_1} E_1= & {} 0, \ \ \ \mathrm{and} \ \ \ \nabla _{E_i} E_1 = \frac{1}{ |\nabla f|} ( \lambda - \lambda _{i})E_i. \end{aligned}$$

(3)

$$\begin{aligned} \zeta _i= & {} \frac{1}{ |\nabla f|} (\lambda - \lambda _{i}). \end{aligned}$$

(4)

Proof

From Lemma 2.3 we get $\nabla _{E_1} E_1=0$. From the gradient Ricci soliton equation, for $i >1$, $\nabla _{E_i} E_1 = \nabla _{E_i} (\frac{\nabla f}{ | \nabla f |}) = \frac{ \nabla _{E_i} \nabla f }{ | \nabla f |}= \frac{ - R({E_i}, \cdot ) + \lambda g( {E_i}, \cdot ) }{ | \nabla f |} = - \frac{1}{ |\nabla f|} ( \lambda _{i} - \lambda )E_i$. Then, $\zeta _i =- \langle \nabla _{E_i} E_i , E_1 \rangle = \langle E_i , \nabla _{E_i} E_1 \rangle = \frac{1}{ |\nabla f|} (\lambda - \lambda _{i})$. $\square$

Lemma 2.7

For a 4-dimensional gradient Ricci soliton (M, g, f) with harmonic Weyl curvature, and for a local adapted frame field $\{ E_i \}$ in $M_{\mathcal {A}} \cap \{ \nabla f \ne 0 \}$, the Ricci-eigenfunctions $\lambda _i$, $i=1, \ldots ,4$, are constant on a connected component of a regular level hypersurface $\Sigma _c$ of f, and so depend on the local variable s only. And $\zeta _i$, $i=2,3,4$, in Lemma 2.6 also depend on s only. In particular, we have $E_i (\lambda _j) = E_i (\zeta _k)= 0$ for $i,k >1$ and any j.

Proof

We write $R_{ij}:= R(E_i, E_j)$. Recall that $\lambda _i=R_{ii}$. We set $Rc^1 = Rc$ and for $k \ge 2$,

$Rc^k_{ij}= \sum _{s_1, s_2, \ldots , s_{k-1}=1}^{4} R_{i s_1} R_{s_1 s_2} \cdots R_{s_{k-1} j}$ with its trace $\mathrm{tr}(Rc^k)= \sum _{i=1}^4 (\lambda _i)^k$. We will show $\mathrm{tr}(Rc^k)$, $k=1,2,3$, depend on s only.

First, $R=\mathrm{tr}(Rc^1)$ and $\lambda _1= R_{11}$ depend on s only by Lemma 2.3. Next, for $k \ge 1$, writing the Hessian $\nabla _j \nabla _i R := \nabla _{E_j} \nabla _{ E_{i}} R$, by Lemma 2.6 we compute the following:

$$\begin{aligned} \begin{aligned}&\sum _{j, s_1, s_2, \ldots , s_{k-1}=1}^{4} ( \nabla _j \nabla _{ s_1} R) R_{s_1 s_2} \cdots R_{s_{k-1} j} \\&\quad = \sum _{j=1}^4 ( \nabla _j\nabla _{ j}R) (R_{j j})^{k-1} \\&\quad = ( \nabla _1\nabla _{ 1}R) \lambda _1^{k-1}+ \sum _{i>1} (\nabla _i \nabla _i R) \lambda _i^{k-1}\\&\quad = (R^{\prime \prime } ) \lambda _1^{k-1} + \sum _{i>1} \{ E_iE_i(R) - (\nabla _{E_i} E_i) R \} \lambda _i^{k-1}\\&\quad = ( R^{\prime \prime }) \lambda _1^{k-1}- \sum _{i>1} \frac{R^{\prime }}{ |\nabla f|} \big (\lambda _i^{k} - \lambda \cdot \lambda _i^{k-1}\big ) . \end{aligned} \end{aligned}$$

(5)

In particular, for $k = 1$, (5) shows that

$$\begin{aligned} \sum _{j=1}^4 \nabla _j \nabla _j R= R^{\prime \prime } - \sum _{i>1} \frac{R^{\prime }}{ |\nabla f|} (\lambda _{i}- \lambda ) =R^{\prime \prime } - \frac{R^{\prime }}{ | \nabla f|}(R- \lambda _{1}-3 \lambda ), \end{aligned}$$

which depends only on s. We drop summation symbols using the Einstein summation convention below.

$$\begin{aligned} \sum _{j=1}^4 \frac{1}{2}\nabla _j \nabla _{j} R= & {} \nabla _j (f_i R_{ij}) =f_{ij} R_{ij} + f_i \nabla _j R_{ij} = -(R_{ij} - \lambda g_{ij} ) R_{ij} + \frac{1}{2} f_i R_{i} \\= & {} -R_{ij}R_{ij} + \lambda R + \frac{1}{2} f^{\prime } R^{\prime }. \end{aligned}$$

So, $\mathrm{tr}(Rc^2)= R_{ij}R_{ij}$ depends only on s.

We shall use the Codazzi equation $\nabla _k R_{ij} = \nabla _i R_{kj} - \frac{R_i}{6} g_{kj} + \frac{R_k}{6} g_{ij}$.

$$\begin{aligned} \nabla _k (f_i R_{ij} R_{jk})= & {} f_{ik} R_{ij} R_{jk} + f_i (\nabla _k R_{ij}) R_{jk} + f_i R_{ij} \nabla _k R_{jk}\nonumber \\= & {} - (R_{ik} - \lambda g_{ik} ) R_{ij} R_{jk} + f_i (\nabla _i R_{kj} - \frac{R_i}{6} g_{kj} + \frac{R_k}{6} g_{ij}) R_{jk} \nonumber \\&\quad + \frac{1}{2} f_i R_{ij} R_{j} \\= & {} - \mathrm{tr}(Rc^3) + \lambda R_{ij} R_{ij} + \frac{1}{2} f_i \nabla _i ( R_{jk}R_{jk}) - f_i \frac{R_i}{6} R +\frac{ f_i R_k}{6} R_{ik}\nonumber \\&\quad + \frac{1}{2} f_i R_{ij} R_{j}\nonumber . \end{aligned}$$

(6)

All terms except $\mathrm{tr}(Rc^{3})$ in the right-hand side of (6) depend on s only. From (5) we also get

$$\begin{aligned} 2\nabla _k (f_i R_{ij} R_{jk})=\nabla _k ( R_j R_{jk}) = (\nabla _k R_j) R_{jk}+ \frac{1}{2} R_j R_{j} \\ = R^{\prime \prime } R_{11} - \sum _{i>1} \frac{R^{\prime }}{ |\nabla f|} (R_{ii}^2 - \lambda R_{ii}) + \frac{1}{2} R_j R_j, \end{aligned}$$

which depends only on s. So, we compare this with (6) to see that $\mathrm{tr}(Rc^3)$ depends only on s. Now $\lambda _1$ and $\sum _{i=1}^4 (\lambda _i)^k$, $k=1, \ldots ,3$, depend only on s. This implies that each $\lambda _i$, $i=1, \ldots ,4$, is a constant depending only on s. By (4), $\zeta _i$, $i=2, 3 ,4$ depend on s only. $\square$

We now extend Lemma 2.4 (i):

Lemma 2.8

For a Riemannian metric g of dimension $n \ge 4$ with harmonic Weyl curvature, consider orthonormal vector fields $E_i$, $i=1, \ldots , n$, such that $Rc(E_i, \cdot ) = \lambda _i g(E_i, \cdot )$. Then the following hold:

• (i) $(\lambda _j - \lambda _k ) \langle \nabla _{E_i} E_j, E_k \rangle + \nabla _{E_i} \langle E_k, {\mathcal {A}}E_j \rangle =(\lambda _i - \lambda _k ) \langle \nabla _{E_j} E_i, E_k\rangle +\nabla _{E_j} \langle E_k, {\mathcal {A}} E_i \rangle , \ \$ for any $i,j,k =1, \ldots , n$.

• (ii) If $k \ne i$ and $k \ne j$, $\ \ (\lambda _j - \lambda _k ) \langle \nabla _{E_i} E_j, E_k\rangle =(\lambda _i - \lambda _k ) \langle \nabla _{E_j} E_i, E_k\rangle$.

Proof

The tensor ${\mathcal {A}}= Rc - \frac{R}{2n-2} g$ is a Codazzi tensor with eigenfunctions $\lambda _i - \frac{R}{2n-2}$. We have

$$\begin{aligned}&\langle (\nabla _{E_i}{\mathcal {A}}) E_j , E_k \rangle = - \langle \nabla _{E_i} E_j, {\mathcal {A}} E_k\rangle - \langle \nabla _{E_i} E_k, {\mathcal {A}}E_j\rangle + \nabla _{E_i}\langle E_k, {\mathcal {A}}E_j\rangle \\&\quad = - (\lambda _k - \frac{R}{2n-2})\langle \nabla _{E_i} E_j, E_k\rangle - ( \lambda _j - \frac{R}{2n-2} )\langle \nabla _{E_i} E_k, E_j\rangle + \nabla _{E_i}\langle E_k, {\mathcal {A}}E_j\rangle \\&\quad = (\lambda _j - \lambda _k ) \langle \nabla _{E_i} E_j, E_k\rangle + \nabla _{E_i}\langle E_k, {\mathcal {A}}E_j\rangle . \end{aligned}$$

As ${\mathcal {A}}$ is a Codazzi tensor, $\langle (\nabla _{E_i}{\mathcal {A}}) E_j , E_k\rangle =\langle (\nabla _{E_j}{\mathcal {A}}) E_i , E_k\rangle$. So, we get (i). Then (ii) holds since $\nabla _{E_i} \langle E_k, {\mathcal {A}}E_j \rangle =\nabla _{E_j} \langle E_k, {\mathcal {A}}E_i \rangle =0$. $\square$

Lemma 2.9

For a gradient Ricci soliton (M, g, f) with harmonic Weyl curvature, and for a local adapted frame field $\{ E_i \}$ in $M_{\mathcal {A}} \cap \{ \nabla f \ne 0 \}$, the following holds.

For $i,j,k >1$, with $k \ne i$ and $k \ne j$, setting $\Gamma ^k_{ij}:= \langle \nabla _{E_i} E_j, E_k\rangle$, $(\zeta _k - \zeta _j ) \Gamma ^k_{ij}=(\zeta _k - \zeta _i ) \Gamma ^k_{ji}, \ \$ $(\zeta _k - \zeta _j ) \Gamma ^k_{ij}=(\zeta _i - \zeta _j ) \Gamma ^i_{kj} \$ and $\ \ \Gamma ^k_{ij} = - \Gamma ^j_{ik}$.

Proof

From (4) and Lemma 2.8, $(\zeta _k - \zeta _j ) \Gamma ^k_{ij}=(\zeta _k - \zeta _i ) \Gamma ^k_{ji}$. Others hold readily. $\square$

3 4-Dimensional Solitons with Distinct $\lambda _2, \lambda _3, \lambda _4$

Let (M, g, f) be a four-dimensional gradient Ricci soliton with harmonic Weyl curvature and non-constant f. In a neighborhood of any point in the open dense subset $M_{\mathcal {A}} \cap \{ \nabla f \ne 0 \}$ of M, there exists an adapted frame field $E_j$, $j=1,2,3,4$, of Lemma 2.5 with its eigenfunction $\lambda _j$

We may only consider three cases depending on the distinctiveness of $\lambda _2,\lambda _3,\lambda _4$: the first case is when $\lambda _i$, $i=2,3,4$, are all equal (on an open subset), and the second is when exactly two of the three are equal. And the last is when the three $\lambda _i$, $i=2,3,4$, are mutually different.

In this section we shall study the last case.

Lemma 3.1

Let (M, g, f) be a four-dimensional gradient Ricci soliton with harmonic Weyl curvature and non-constant f. Suppose that for an adapted frame field $E_j$, $j=1,2,3,4$, in an open subset W of $M_{\mathcal {A}} \cap \{ \nabla f \ne 0 \}$, the eigenfunctions $\lambda _2, \lambda _3, \lambda _4$ are distinct from each other. Then the following hold in W:

for $i, j >1$, $i \ne j$,

$$\begin{aligned}&\nabla _{E_1} E_1 =0,\quad \nabla _{E_i} E_1 = \zeta _i E_i , \quad \nabla _{E_i} E_i = -\zeta _i E_1 ,\quad \nabla _{E_1} E_i=0.\\&\nabla _{E_i} E_j= \Gamma _{ij}^k E_k \,\,\hbox {where}\,\, k \ne 1,i,j. \end{aligned}$$

Proof

From Lemma 2.6 we have $\nabla _{E_1} E_1 =0$ and $\nabla _{E_i} E_1 = \zeta _i E_i$. From Lemma 2.4 (i) and Lemma 2.7, $\langle \nabla _{E_i} E_i , E_j\rangle =0$. And $\langle \nabla _{E_i} E_i , E_1\rangle = - \langle E_i , \nabla _{E_i} E_1\rangle = - \zeta _i$. So, we get $\nabla _{E_i} E_i = -\zeta _i E_1$. Now, $-\langle \nabla _{E_i} E_j, E_i \rangle =0$, $\langle \nabla _{E_i} E_j, E_j \rangle =0$. And $\langle \nabla _{E_i} E_j, E_1 \rangle = -\langle \nabla _{E_i} E_1 , E_j\rangle =0$. So, $\nabla _{E_i} E_j= \Gamma _{ij}^k E_k$ where $k \ne 1,i,j$. Clearly $\Gamma _{ij}^k =- \Gamma _{ik}^j$.

From Lemma 2.8 (ii), $(\lambda _i - \lambda _j ) \langle \nabla _{E_1} E_i, E_j\rangle =(\lambda _1 - \lambda _j ) \langle \nabla _{E_i} E_1, E_j\rangle$. As $\langle \nabla _{E_i} E_1, E_j\rangle =0$, $\langle \nabla _{E_1} E_i, E_j\rangle =0$. This gives $\nabla _{E_1} E_i =0$. $\square$

From the above lemma, we may write

$$\begin{aligned}{}[E_2, E_3] = \alpha E_4, \ \ \ [E_3, E_4] = \beta E_2, \ \ \ [E_4, E_2] = \gamma E_3. \end{aligned}$$

(7)

Lemma 3.2

Under the hypothesis of Lemma 3.1, we have the following relation on $\zeta _i$’s and the coefficients of (7).

$$\begin{aligned} E_1(\alpha )= & {} \alpha ( \zeta _4 - \zeta _2 - \zeta _3 ), \quad E_1(\beta ) = \beta ( \zeta _2 - \zeta _3 - \zeta _4 ), \quad E_1(\gamma ) = \gamma ( \zeta _3 - \zeta _2 - \zeta _4 ) \\ \beta= & {} \frac{(\zeta _3 - \zeta _4 )^2}{(\zeta _2 - \zeta _3 )^2} \alpha , \quad \gamma = \frac{(\zeta _2 - \zeta _4 )^2}{(\zeta _2 - \zeta _3 )^2} \alpha . \end{aligned}$$

Proof

From Jacobi identity $[[X, Y], Z] + [[Y, Z], X] + [[Z, X], Y]=0$ applied to $(X, Y, Z) = (E_1, E_2, E_3)$ gives $E_1(\alpha ) = \alpha ( \zeta _4 - \zeta _2 - \zeta _3 )$. Apply it to $E_1, E_2, E_4$ and $E_1, E_3, E_4$, we get the next two.

Using $2 \langle \nabla _X Y, Z \rangle = X \langle Y, Z \rangle + Y \langle X, Z \rangle - Z \langle X,Y \rangle +\langle [X,Y], Z \rangle - \langle [X,Z], Y \rangle - \langle [Y,Z], X \rangle$ for vector fields X, Y, Z, from Lemma 2.9 we get: $\frac{-\alpha - \gamma + \beta }{2}= \Gamma _{24}^3 = \frac{( \zeta _2 - \zeta _4 )}{ \zeta _3 - \zeta _4 }\Gamma _{34}^2 =\frac{( \zeta _2 - \zeta _4 )}{ \zeta _3 - \zeta _4 }\frac{\alpha - \gamma + \beta }{2}. \ \$ So, $-\alpha - \gamma + \beta = \frac{( \zeta _2 - \zeta _4 )}{ \zeta _3 - \zeta _4 }(\alpha - \gamma + \beta )$. By symmetry, we have $-\beta - \alpha + \gamma = \frac{( \zeta _3 - \zeta _2 )}{ \zeta _4 - \zeta _2 }(\beta - \alpha + \gamma )$ and $-\gamma - \beta + \alpha = \frac{( \zeta _4 - \zeta _3 )}{ \zeta _2 - \zeta _3 }(\gamma - \beta + \alpha )$. From these, we can get the other formulas. $\square$

Lemma 3.3

Let a four-dimensional gradient Ricci soliton (M, g, f) with harmonic Weyl curvature satisfy the hypothesis of Lemma 3.1. Then the following hold in W:

For distinct $\ i, j, k>1$, $\ R_{1ii1} =-\zeta _i^{\prime } - \zeta _i^2 =R_{1jj1},$ where $\zeta _i^{\prime }= \frac{d \zeta _i}{ds}$, $\ \ R_{1ij1}= 0$.

$$\begin{aligned} R_{11}= & {} -3\zeta _2^{\prime } - 3\zeta _2^2 .\\ R_{22}= & {} - \zeta _2^{\prime } - \zeta _2^2 -\zeta _2 \zeta _3 -\zeta _2 \zeta _4 -2 \Gamma _{34}^2 \Gamma _{43}^2 .\\ R_{33}= & {} -\zeta _3^{\prime } - \zeta _3^2 -\zeta _3 \zeta _2 -\zeta _3 \zeta _4 +2 \frac{( \zeta _2 - \zeta _4 )}{ \zeta _3 - \zeta _4 }\Gamma _{34}^2 \Gamma _{43}^2 .\\ R_{44}= & {} -\zeta _4^{\prime } - \zeta _4^2 -\zeta _4 \zeta _2 -\zeta _4 \zeta _3 +2\frac{( \zeta _2 - \zeta _3 )}{ \zeta _4 - \zeta _3 } \Gamma _{34}^2 \Gamma _{43}^2 .\\ R_{1i}= & {} 0, \quad R_{ij} = E_k (\Gamma ^k_{ij}). \end{aligned}$$

Proof

One uses Lemma 3.1 and Lemma 2.7. Recall $R_{1ii1} =R_{1jj1}$ from (2). By direct computation we get $R_{1ii1} =-\zeta _i^{\prime } - \zeta _i^2$, $R_{jiij} = -\zeta _j \zeta _i - \Gamma _{ji}^k \Gamma _{ik}^j - \Gamma _{ji}^k \Gamma _{ki}^j + \Gamma _{ij}^k \Gamma _{ki}^j$ and $R_{kijk} = E_k (\Gamma ^k_{ij})$. Use Lemma 2.9 to express $R_{33}$ and $R_{44}$. $\square$

Here we set $a:= \zeta _2$, $b:= \zeta _3$ and $c:= \zeta _4$. From the soliton equation $\lambda - \zeta _i f^{\prime } = R_{ii}$, $i >1$ and Lemma 3.3,

$-(a - b)f^{\prime } =R_{22} - R_{33} = (b -a )c -2\{ 1+ \frac{( a - c )}{ b - c }\}\Gamma _{34}^2 \Gamma _{43}^2$. So,

$$\begin{aligned} f^{\prime } = c +2 \frac{( a+ b - 2c )}{ (a - b)(b - c ) }\Gamma _{34}^2 \Gamma _{43}^2. \end{aligned}$$

(8)

Similarly, $-(a - c)f^{\prime }= (c -a )b -2\{ 1+ \frac{( a - b )}{ c - b }\}\Gamma _{34}^2 \Gamma _{43}^2$. So,

$$\begin{aligned} f^{\prime }= b +2 \frac{( a + c - 2b )}{(a - c) (c - b ) }\Gamma _{34}^2 \Gamma _{43}^2. \end{aligned}$$

(9)

From (8) and (9), we get

$$\begin{aligned} 4\Gamma _{34}^2 \Gamma _{43}^2= & {} \frac{(a - b)(a - c) (b - c )^2}{ ( a^2 +b^2+ c^2 - ab - bc - ac ) }, \end{aligned}$$

(10)

$$\begin{aligned} f^{\prime }= & {} \frac{a^2 b + a^2 c + a b^2 + a c^2 + b^2 c + c^2 b - 6a b c }{ 2 ( a^2 +b^2+ c^2 - ab - bc - ac ) }. \end{aligned}$$

(11)

We are now ready to prove the following.

Proposition 3.4

Let (M, g, f) be a four-dimensional gradient Ricci soliton with harmonic Weyl curvature and non-constant f. For any adapted frame field $E_j$, $j=1,2,3,4$, in an open dense subset $M_{\mathcal {A}} \cap \{ \nabla f \ne 0 \}$ of M, the three eigenfunctions $\lambda _2, \lambda _3, \lambda _4$ cannot be pairwise distinct, i.e., at least two of the three coincide.

Proof

Suppose that $\lambda _2, \lambda _3, \lambda _4$ are pairwise distinct. We shall prove then that g should be an Einstein metric, so a contradiction.

In this proof again we set $a:= \zeta _2$, $b:= \zeta _3$ and $c:= \zeta _4$. From (10) and Lemma 2.9,

$$\begin{aligned} (\alpha - \gamma + \beta )^2= 4(\Gamma _{34}^2)^2 = 4 \Gamma _{34}^2 \Gamma _{43}^2 \frac{(a-b)}{(a-c)} =\frac{(a - b)^2 (b - c )^2}{ ( a^2 +b^2+ c^2 - ab - bc - ac ) }. \end{aligned}$$

For convenience set $P:= a^2 +b^2+ c^2 - ab - bc - ac$. From Lemma 3.2,

$$\begin{aligned} (\alpha - \gamma + \beta )^2 =\alpha ^2 \left\{ 1 - \frac{(a - c )^2}{(a - b )^2} + \frac{(b - c )^2}{(a - b )^2} \right\} ^2 = \frac{4 \alpha ^2 (b - c )^2}{(a - b )^2}. \end{aligned}$$

So, $\alpha ^2 = \frac{(a - b)^4 }{ 4P }$. Since a, b, c are all functions of s only, so is $\alpha$.

Differentiating this in s and using $b^{\prime } - a^{\prime } = a^2 - b^2$ and $c^{\prime } - a^{\prime } = a^2 - c^2$, we get

$$\begin{aligned} 2 \alpha \alpha ^{\prime }= & {} \frac{(a - b)^3 (a^{\prime } - b^{\prime }) }{ P}\\&- \frac{(a - b)^4 ( 2a a^{\prime } +2b b^{\prime } + 2c c^{\prime } - a b^{\prime } - b a^{\prime } - a c^{\prime } -c a^{\prime } - c b^{\prime }- b c^{\prime } ) }{ 4P^2 }\\= & {} \frac{-(a - b)^3 (a^2 - b^2) }{ P}\\&- \frac{(a - b)^4 \{ (a-b) (a^{\prime } -b^{\prime }) + (a-c) (a^{\prime } -c^{\prime })+ (b-c) (b^{\prime } -c^{\prime }) \} }{ 4P^2 } \\= & {} \frac{-(a - b)^4 (a+b) }{ P}\\&+ \frac{(a - b)^4 \{ (a-b) (a^2 -b^2) + (a-c) (a^2 -c^2)+ (b-c) (b^2 -c^2) \} }{ 4P^2 }\\= & {} -\frac{(a - b)^4 }{ P} \Biggr [(a+b)\\&- \frac{ \{ 2(a^3 \,{+}\,b^3 \,{+}\, c^3 \,{-}\,3abc) \,{+}\, 6abc \,{-}\,a^2b \,{-}\,ab^2 \,{-}\,a^2c \,{-}\,ac^2 \,{-}\,b^2c \,{-}\,bc^2 \} }{ 4P}\Biggr ]\\= & {} -\frac{(a - b)^4 }{ P} \Biggr [(a+b)\\&- \frac{(a+b+c)}{2} - \frac{ \{ 6abc -a^2b -ab^2 -a^2c -ac^2 -b^2c -bc^2 \} }{ 4P }\Biggr ] \\= & {} -\frac{(a - b)^4 }{ P} \Biggr [ \frac{(a+b-c)}{2}\\&- \frac{ \{ 6abc -a^2b -ab^2 -a^2c -ac^2 -b^2c -bc^2 \} }{ 4P }\Biggr ]. \end{aligned}$$

Meanwhile, from Lemma 3.2 and $\alpha ^2 = \frac{(a - b)^4 }{ 4P }$,

$$\begin{aligned} 2 \alpha \alpha ^{\prime }= 2 \alpha E_1(\alpha )=-2 \alpha ^2 ( a + b - c )= -\frac{(a - b)^4 }{ 2P }( a + b - c ). \end{aligned}$$

Equating these two expressions for $2 \alpha \alpha ^{\prime }$, we get: $6abc = a^2b + b^2a + a^2c + c^2a + b^2c + c^2b$. From (11), $f^{\prime } =0$. So, g is an Einstein metric. $\square$

4 4-Dimensional Soliton with $\lambda _2 \ne \lambda _3 =\lambda _4$

In this section we begin to study the case when exactly two of the three eigenvalues $\lambda _2, \lambda _3, \lambda _4$ are equal. We may well assume that $\lambda _2 \ne \lambda _3= \lambda _4$.

Lemma 4.1

Let (M, g, f) be a four-dimensional gradient Ricci soliton with harmonic Weyl curvature. Suppose that $\lambda _2 \ne \lambda _3= \lambda _4$ for an adapted frame field $E_j$, $j=1,2,3,4$, on an open subset of $M_{\mathcal {A}} \cap \{ \nabla f \ne 0 \}$. Then the following hold on the open subset:

• $\nabla _{E_1} E_1=0$ .

• $\nabla _{E_i} E_1= \zeta _i(s) E_i for i=2,3,4$, with $\zeta _i(s)=\frac{ 1}{|\nabla f |} (\lambda - \lambda _i )$ .

• $\nabla _{E_2} E_2 = -\zeta _2(s) E_1$ .

• $\nabla _{E_3} E_3 = -\zeta _3 E_1 - \beta _3 E_4$ , $\nabla _{E_4} E_4 = -\zeta _4 E_1 + \beta _4 E_3$ ,   for some functions $\beta _3 and \beta _4$.

• $\nabla _{E_1} E_2= 0, \nabla _{E_1} E_3= \rho E_4$ and $\nabla _{E_1} E_4= -\rho E_3$ for some function $\rho$.

• $\nabla _{E_2} E_3= q E_4$ and $\nabla _{E_2} E_4= - q E_3$ for some function q.

• $\nabla _{E_3} E_2=0$ and $\nabla _{E_4} E_2=0$.

• $\nabla _{E_3} E_4 = \beta _3 E_3$ and $\nabla _{E_4} E_3 = - \beta _4 E_4$ .

• $[E_1, E_2]= -\zeta _2 E_2$ and $[E_3, E_4]= \beta _3 E_3 + \beta _4 E_4$.

In particular, the distribution spanned by $E_1$ and $E_2$ is integrable. So is that spanned by $E_3$ and $E_4$.

Proof

The formula for $\nabla _{E_i} E_1$, $i \ge 1$, comes from (3).

Then from Lemma 2.7 and Lemma 2.4 (i): $(\lambda _2 - \lambda _i )\langle \nabla _{E_2} E_2, E_i\rangle = E_i(\lambda _2) =0$ for $i=3,4$ and $\langle \nabla _{E_2} E_2, E_1\rangle = - \langle E_2, \nabla _{E_2} E_1 \rangle = -\zeta _2(s)$. So, $\nabla _{E_2} E_2 = -\zeta _2(s) E_1$. By similar argument, $\nabla _{E_3} E_3 = -\zeta _3 E_1 - \beta _3 E_4$, $\nabla _{E_4} E_4 = -\zeta _4 E_1 + \beta _4 E_3$, for some functions $\beta _3$ and $\beta _4$.

From Lemma 2.8 (ii), $(\lambda _2 - \lambda _i ) \langle \nabla _{E_1} E_2, E_i\rangle =(\lambda _1 - \lambda _i ) \langle \nabla _{E_2} E_1, E_i\rangle =(\lambda _1 - \lambda _i) \langle \zeta _2 E_2, E_i\rangle =0$, for $i=3,4$. So, $\langle \nabla _{E_1} E_2, E_i\rangle =0$, for $i=3,4$. As $\langle \nabla _{E_1} E_2, E_1\rangle = -\langle E_2, \nabla _{E_1} E_1\rangle =0$, we have $\nabla _{E_1} E_2= 0$.

As $\langle \nabla _{E_1} E_3, E_2\rangle = -\langle E_3, \nabla _{E_1} E_2\rangle = 0$, one can readily get $\nabla _{E_1} E_3= \rho E_4$ for some function $\rho$ and $\nabla _{E_1} E_4= -\rho E_3$. And $\nabla _{E_2} E_3= q E_4$ for some function q and $\nabla _{E_2} E_4= - q E_3$.

From Lemma 2.8 (ii), $(\lambda _2 - \lambda _4 ) \langle \nabla _{E_3} E_2, E_4\rangle =(\lambda _3 - \lambda _4 ) \langle \nabla _{E_2} E_3, E_4\rangle =0$. So, $\langle \nabla _{E_3} E_2, E_4\rangle =0$. As we have $\langle \nabla _{E_3} E_2, E_a\rangle =0$ for $i=1, 3$ from above, we get $\nabla _{E_3} E_2=0$. Similarly, $\nabla _{E_4} E_2=0$.

One can easily compute $\nabla _{E_3} E_4 = \beta _3 E_3$ and $\nabla _{E_4} E_3 = - \beta _4 E_4$. From above we get $[E_1, E_2]= -\zeta _2 E_2$ and $[E_3, E_4]= \beta _3 E_3 + \beta _4 E_4$. $\square$

Lemma 4.2

Let $D^1$ and $D^2$ be both two-dimensional smooth integrable distributions on a domain $\Omega$ of a four-dimensional manifold that span the tangent space $T_p \Omega$ for each $p \in \Omega$. Let $p_0$ be a point in $\Omega$. Then there is a coordinate neighborhood $(x_1,x_2,x_3,x_4)$ near $p_0$ so that $D^1$ is tangent to the 2-dimensional level sets $\{ (x_1, x_2, x_3,x_4) | \ x_3, x_4 \ \mathrm{constants} \}$ and $D^2$ is tangent to the level sets $\{ (x_1, x_2, x_3,x_4) | \ x_1, x_2 \ \mathrm{constants} \}$.

Proof

By Frobenius’s theorem, there is a coordinate neighborhood $\mathbf{x}:=(x,y,z,w)$ near $p_0$ so that $D^2$ is tangent to the sets $\{ (x,y,z,w) | \ x,y \ \mathrm{constants} \}$. We may assume that $(x(p_0), y(p_0), w(p_0), z(p_0) )= (0, 0,0 ,0)$.

Then there are two vector fields $v_1= (a_1, b_1, c_1, d_1):= a_1\frac{\partial }{\partial x} + b_1\frac{\partial }{\partial y} + c_1\frac{\partial }{\partial z}+ d_1\frac{\partial }{\partial w}$ and $v_2=(a_2, b_2, c_2, d_2)$ for points p near $p_0$ in $D^1$, with $(a_1(p), b_1(p))$ and $(a_2(p), b_2(p))$ being linearly independent as two-dimensional vectors; if not, $D^1_{p}$ and $D^2_{p}$ will not span $T_{p} \Omega$.

By considering $X_1:= \alpha _1 v_1 + \beta _1 v_2$ and $X_2:= \alpha _2 v_1 + \beta _2 v_2$ for smooth functions $\alpha _i, \beta _i$, we have smooth vector fields $X_1, X_2 \in D^1$, of the form $X_1(p)= (1,0, a_1(p), a_2(p))$ and $X_2 = (0,1, b_1(p), b_2(p))$ for p near $p_0$ with smooth functions $a_i, b_i$, $i=1,2$.

Consider the one-parameter subgroup $\phi _t$ of $X_1$ and $\psi _s$ of $X_2$: $\frac{d }{dt} \phi _t(p) = (1,0, a_1(\phi _t(p)), a_2(\phi _t(p)) )_{\phi _t(p)}$ and $\frac{d }{ds} \psi _s(p) = (0,1, b_1(\psi _s(p)), b_2(\psi _s(p)))$.

Define a map $\Phi$ on a neighborhood of the origin in $\mathbb {R}^4= \{ (x_1, x_2, x_3, x_4) \}$ into $\mathbb {R}^4= \{ (x,y,z,w) \}$ by $\Phi (x_1, x_2, x_3, x_4) := \phi _{x_1} \psi _{x_2} (0, 0, x_3, x_4)$. This $\Phi$ gives a local coordinate system near $p_0$. From $\frac{d }{ds} \psi _s(p) = (0,1, b_1(\psi _s(p)), b_2(\psi _s(p)))$, we get $\psi _{x_2} (0, 0, x_3, x_4)=(0, x_2, *, *)$ and similarly $\phi _{x_1} \psi _{x_2} (0, 0, x_3, x_4) = \phi _{x_1}(0, x_2, *, *) =(x_1, x_2, *, *)$.

So, $\Phi (x_1, x_2, x_3, x_4) = (x_1, x_2, *, *)$. Then we get $\Phi _* ( \frac{\partial }{ \partial x_3} ), \Phi _* ( \frac{\partial }{ \partial x_4} ) \in \mathrm{span}(\frac{\partial }{ \partial z} , \frac{\partial }{ \partial w}) =D^2$. So, $D^2$ is spanned by $\Phi _* ( \frac{\partial }{ \partial x_3} )$ and $\Phi _* ( \frac{\partial }{ \partial x_4} )$.

As $D^1$ is integrable, in a neighborhood of each point $q_0:= (0, 0, c, d)$ near the origin, there is a unique surface $S_{q_0}$ containing $q_0$ which is tangent to the distribution $D^1$ at each point of $S_{q_0}$. As $X_1$ and $X_2$ are vector fields on $S_{q_0}$, at each point $q \in S_{q_0}$ we have $\{ \psi _{x_2} (q) \ | \ x_2 \in (-\epsilon , \epsilon ) \} \subset S_{q_0}$ and $\{ \phi _{x_1} (q) \ | \ x_1 \in (-\epsilon , \epsilon ) \} \subset S_{q_0}$ for small $\epsilon$. Therefore, the set $\{ \phi _{x_1} \psi _{x_2} (0, 0, c, d) \ | \ x_1, x_2 \in (-\varepsilon , \varepsilon ) \}$, for small $\varepsilon$, coincides with $S_{q_0}$ near $q_0$. So, we get $\Phi _* ( \frac{\partial }{ \partial x_1} ), \Phi _* ( \frac{\partial }{ \partial x_2} ) \in D^1$, and $D^1$ is spanned by $\Phi _* ( \frac{\partial }{ \partial x_1} ), \Phi _* ( \frac{\partial }{ \partial x_2} )$. Now we have obtained a new coordinate system $\Phi ^{-1} \circ \mathbf{x}$ with the desired property. This proves the lemma. $\square$

Using Lemma 4.1 and Lemma 4.2, we can express the metric g in the following lemma.

Lemma 4.3

Let (M, g, f) be a four-dimensional gradient Ricci soliton with harmonic Weyl curvature. Suppose that $\lambda _2 \ne \lambda _3= \lambda _4$ for an adapted frame field $E_j$, $j=1,2,3,4$, on an open subset U of $M_{\mathcal {A}} \cap \{ \nabla f \ne 0 \}$.

Then for each point $p_0$ in U, there exists a neighborhood V of $p_0$ in U with coordinates $(s,t, x_3, x_4)$ such that $\nabla s= \frac{\nabla f }{ |\nabla f |}$ and g can be written on V as

$$\begin{aligned} g= ds^2 + p(s)^2 dt^2 + h(s)^2 \tilde{g}, \end{aligned}$$

(12)

where $p:=p(s)$ and $h:=h(s)$ are smooth functions and $\tilde{g}$ is (a pull-back of) a Riemannian metric on a 2-dimensional domain with $x_3, x_4$ coordinates.

We get $E_1 =\frac{\partial }{\partial s}$ and $E_2 =\frac{1}{p} \frac{\partial }{\partial t}$.

Proof

Let $D^1$ be the 2-dimensional distribution spanned by $E_1 = \nabla s$ and $E_2$. Also let $D^2$ be the one spanned by $E_3$ and $E_4$. Then $D^1$ and $D^2$ are both integrable by Lemma 4.1. We may consider the coordinates $(x_1, x_2, x_3, x_4)$ from Lemma 4.2, so that $D^1$ is tangent to the 2-dimensional level sets $\{ (x_1, x_2, x_3,x_4) | \ x_3, x_4 \ \mathrm{constants} \}$ and $D^2$ is tangent to the level sets $\{ (x_1, x_2, x_3,x_4) | \ x_1, x_2 \ \mathrm{constants} \}$. As $D^1$ and $D^2$ are g-orthogonal, we can get the metric description for g as follows:

$g= g_{11}dx_1^2 + g_{12} dx_1 \odot dx_2 + g_{22}dx_2^2 + g_{33}dx_3^2 + g_{34} dx_3 \odot dx_4 + g_{44}dx_4^2$, where $\odot$ is the symmetric tensor product and $g_{ij}$ are functions of $(x_1, x_2, x_3, x_4)$.

As $E_1= \nabla s \in D^1$, we have $ds= g(E_1, \cdot )$. We define a 1-form $\omega _2 ( \cdot ):= g ( E_2, \cdot )$. One can readily see that $ds^2 + \omega _2^2 = g_{11}dx_1^2 + g_{12} dx_1 \odot dx_2 + g_{22}dx_2^2$. In fact, one may feed $(E_i, E_j)$ to both sides and use the fact that each of $E_1$ and $E_2$ is of the form $a\partial _1 + b\partial _2$ as they are tangent to the sets $\{ (x_1, x_2, x_3,x_4) | \ x_3, x_4 \ \mathrm{constants} \}$, while each of $E_3$ and $E_4$ is of the form $c\partial _3 + d\partial _4$ for a similar reason; here we have set $\partial _{i} := \frac{\partial }{\partial x_i}$.

Recalling $[E_1, E_2] =- \zeta _2 (s)E_2$, we define a function $p(s) = e^{\int _{s_0}^s \zeta _2(u) du}$ for a constant $s_0$ so that $\zeta _2= \frac{p^{\prime }}{p}$. Then, the 2-form $d (\frac{\omega _2}{p})$ satisfies $d (\frac{\omega _2}{p})(E_1, E_2) = -\frac{dp \wedge \omega _2 }{p^2}(E_1, E_2) + \frac{1}{p} d \omega _2(E_1, E_2) =- \frac{p^{\prime }}{p^2}+ \frac{p^{\prime }}{p^2} = 0$. And for $i \in \{3,4 \}$ and for any $j \in \{1,2,3,4 \}$, $d (\frac{\omega _2}{p})(E_i, E_j) = -\frac{dp \wedge \omega _2 }{p^2}(E_i, E_j) + \frac{1}{p} d \omega _2(E_i, E_j) = \frac{1}{p} d\omega _2(E_i, E_j)= -\frac{1}{p} \omega _2([E_i, E_j]) = 0$ by Lemma 4.1.

So, $d (\frac{\omega _2}{p})=0$ and $\frac{\omega _2}{p} = d t$ for some function t modulo a constant in a neighborhood of $p_0$. The metric g can now be written as

$$\begin{aligned} g= ds^2 + p(s)^2 dt^2 + g_{33}dx_3^2 + g_{34} dx_3 \odot dx_4 + g_{44}dx_4^2, \end{aligned}$$

(13)

where $g_{ij}$ are functions of $(x_1, x_2, x_3, x_4)$. In the coordinate system $(s,t, x_3, x_4)$, one easily gets $E_1 =\frac{\partial }{\partial s}$ and $E_2 =\frac{1}{p} \frac{\partial }{\partial t}$.

Now we use new coordinates $(s,t, x_3, x_4)$ in computations below, so that $\partial _{1}=\frac{\partial }{\partial s}$ and $\partial _{2} =\frac{\partial }{\partial t}$, etc. From Lemma 4.1, we have $\langle \nabla _{E_i} E_{j} , E_2\rangle =0$ for $i, j \in \{ 3,4 \}$. As $\partial _{3}$ and $\partial _{4}$ are both of the form $\gamma E_3 + \delta E_4$, we have that $\langle \nabla _{\partial _{i}} \partial _{j} , \partial _2\rangle = 0$ for $i,j \in \{ 3,4 \}$.

We set $g_{ij}= g( \partial _i , \partial _{j} )$. Due to (13), for $i,j \in \{ 3,4 \}$:

$$\begin{aligned} \begin{aligned} 0 =&\langle \nabla _{\partial _{i}} \partial _{j} , \partial _{2}\rangle = \sum _{k=1}^4 \langle \Gamma ^{k}_{ij} \partial _k , \partial _{2} \rangle \\ =&\sum _{k, l=1}^4 \langle \frac{1}{2} g^{kl}( \partial _i g_{l{j}}+ \partial _j g_{l{i}} - \partial _l g_{ij} )\partial _k , \partial _{2} \rangle \\ =&- \sum _{k, l=1}^4 \frac{1}{2} g^{kl} \partial _l g_{ij} \langle \partial _k , \partial _{2} \rangle = -\frac{1}{2} \partial _{2} g_{ij}. \end{aligned} \end{aligned}$$

(14)

We have shown:

$$\begin{aligned} \frac{\partial g_{33}}{\partial t} = \frac{\partial g_{34}}{\partial t} =\frac{\partial g_{44}}{\partial t}=0. \end{aligned}$$

(15)

We consider the second fundamental form of a leaf for $D^2$ with respect to $E_1$: $H^{E_1} ( u , u ) = - \langle \nabla _{u} u , E_1\rangle$. As $D^2$ is totally umbilic by Lemma 2.4 (ii), $H^{E_1} ( u , u ) = \zeta \cdot g( u , u)$ for some function $\zeta$ and any u tangent to $D^2$. Then, $H^{E_1} (E_3, E_3 ) = - \langle \nabla _{E_3} E_3 , E_1\rangle = \zeta _3$ So, $\zeta = \zeta _3$, which is a function of s only by Lemma 2.7.

For $i, j \in \{ 3,4 \}$, we compute similarly as in (14),

$$\begin{aligned} \zeta _3 g_{ij}= & {} H^{E_1} ( \partial _i , \partial _{j} ) = - \left\langle \nabla _{\partial _i} \partial _{j} , \frac{\partial }{\partial s}\right\rangle = - \left\langle \sum _k \Gamma ^{k}_{i{j}} \partial _k , \frac{\partial }{\partial s} \right\rangle \\= & {} - \sum _k \left\langle \frac{1}{2} g^{kl}( \partial _i g_{lj} +\partial _{j} g_{li} - \partial _l g_{ij} )\partial _k , \frac{\partial }{\partial s} \right\rangle = \frac{1}{2} \frac{\partial }{\partial s} g_{i{j}}. \end{aligned}$$

So, $\frac{1}{2} \frac{\partial }{\partial s} g_{i{j}} = \zeta _3 g_{ij}$. Integrating it, for $i, j \in \{ 3,4 \}$, we get $g_{ij} = e^{C_{ij}} h(s)^2$. Here the function $h(s)>0$ is independent of i, j, and each function $C_{ij}$ depends only on $x_3, x_4$ by (15).

Now g can be written as $g= ds^2 + p(s)^2 dt^2 + h(s)^2 \tilde{g}$, where $\tilde{g}$ can be viewed as a Riemannian metric in a domain of the $(x_3, x_4)$-plane. $\square$

5 Analysis of the Metric When $\lambda _2 \ne \lambda _3 =\lambda _4$

We shall study more about the metric $g = ds^2 + p(s)^2 dt^2 + h(s)^2 \tilde{g}$ of (12) obtained in Lemma 4.3.

Lemma 5.1

Let (M, g, f) be a four-dimensional gradient Ricci soliton with harmonic Weyl curvature which satisfies the hypothesis of Lemma 4.3. For the metric $g = ds^2 + p(s)^2 dt^2 + h(s)^2 \tilde{g}$ of (12), the two-dimensional metric $\tilde{g}$ has constant curvature, say k.

Proof

In local coordinates $(x_1:=s, \ x_2:=t, \ x_3, x_4)$ of Lemma 4.3, we write some Christoffel symbols $\Gamma _{ij}^k$ and Ricci curvature of g. In this proof, for any (0, 2)-tensor P, $P( \frac{\partial }{\partial _{x_i}}, \frac{\partial }{\partial _{x_j}} )$ shall be denoted by $P_{ij}$. We let $\tilde{\nabla }$, $\tilde{\Gamma }_{ij}^k$ and $R^{\tilde{g}}_{ij}$ be the Levi-Civita connection, Christoffel symbols and Ricci curvature of $\tilde{g}$, respectively. For $i,j,k \in \{ 3,4 \}$, we get:

$$\begin{aligned} \Gamma _{ij}^k= & {} \tilde{\Gamma }_{ij}^k \nonumber \\ R_{ij}= & {} - \tilde{g}_{ij} \{ h h^{\prime \prime } + \frac{p^{\prime }}{p} h h^{\prime } + { h^{\prime }}^2 \} +R^{\tilde{g}}_{ij}. \end{aligned}$$

(16)

From (16), for $i,j,k \in \{ 3,4 \}$, we have $\nabla _k \tilde{g}_{ij}=\tilde{\nabla }_k \tilde{g}_{ij}=0$ and $\nabla _k R^{\tilde{g}}_{ij}=\tilde{\nabla }_k R^{\tilde{g}}_{ij}$ so that $\nabla _k R_{ij} = \tilde{\nabla }_k R^{\tilde{g}}_{ij}$. The condition $\delta W=0$ gives $\nabla _k R_{ij} -\nabla _j R_{ik} = - \frac{R_j}{6} g_{ki} + \frac{R_k}{6} g_{ij}$. For $i, j, k \in \{ 3,4 \}$, $R_j=R_k=0$, so $\nabla _k R_{ij} =\nabla _j R_{ik}$.

Then, we get $\tilde{\nabla }_k R^{\tilde{g}}_{ij} = \tilde{\nabla }_j R^{\tilde{g}}_{ik}$. By the contracted second Bianchi identity the 2-dimensional metric $\tilde{g}$ then has constant curvature. $\square$

The metric $\tilde{g}$ of Lemma 5.1 is locally isometric to the Riemannian metric $g_0= dr^2 + u(r)^2 d \theta ^2$ on a domain in $\mathbb {R}^2$ with polar coordinates $(r, \theta )$, where $u(r)=r$ when $k=0$, $u(r) =\sin (\sqrt{k} \cdot r)$ when $k>0$ or $u(r)=\sinh (\sqrt{-k} \cdot r)$ when $k<0$. We may identify $\tilde{g}$ with $g_0$ locally and set $e_3 = \frac{\partial }{\partial r}$ and $e_4 = \frac{1}{u(r)} \frac{\partial }{ \partial \theta }$, which then form an orthonormal basis of $\tilde{g}$.

Lemma 5.2

For the local soliton metric $g = ds^2 + p(s)^2 dt^2 + h(s)^2 \tilde{g}$ of (12) obtained in Lemma 4.3 with the metric $\tilde{g}$ of constant curvature k, if we set $E_1 = \frac{\partial }{\partial s}$, $E_2 = \frac{1}{p(s)}\frac{\partial }{ \partial t}$, $E_3 = \frac{1}{h(s)} e_3$ and $E_4 = \frac{1}{h(s)} e_4$, where $e_3$ and $e_4$ are as in the above paragraph, then the connection form, Ricci and scalar curvature of g are as below. Here $R_{ij} = R(E_i, E_j)$ and $R_{ijkl} = R(E_i, E_j, E_k, E_l)$.

• $\nabla _{E_1}E_i = 0$, for $i=1,2,3,4$.

• $\nabla _{E_i}E_1 = \zeta _iE_i$, for $i=2,3,4$ with $\zeta _2= \frac{p^{\prime }}{p}$, $\zeta _3=\zeta _4= \frac{h^{\prime }}{h}$.

• $\nabla _{E_2} E_2 = -\zeta _2 E_1$, $\ \ \ \ \nabla _{E_3} E_3 = -\zeta _3 E_1$, $\ \ \ \ \nabla _{E_4} E_4 = -\zeta _4 E_1 + \beta _4 E_3$.

• $\nabla _{E_2}E_3 =\nabla _{E_3}E_2 = \nabla _{E_4}E_2 = \nabla _{E_2}E_4 = 0$.

• $\nabla _{E_3}E_4 = 0$, $\ \ \ \ \nabla _{E_4}E_3 = -\beta _4 E_4, \$ where $\ \beta _4 = \frac{u^{\prime }(r)}{h(s)u(r)}$.

$$\begin{aligned} R_{1221}= & {} - \frac{p^{\prime \prime }}{p}=-\zeta _2^{\prime } - \zeta _2^2 \ = \ R_{1ii1} =-\zeta _i^{\prime } - \zeta _i^2= - \frac{h^{\prime \prime }}{h}, \ \ \mathrm{for} \ i \ge 3. \\ R_{11}= & {} -3\zeta _2^{\prime } - 3\zeta _2^2 = - 3 \frac{h^{\prime \prime }}{h}. \\ R_{22}= & {} -\zeta _2^{\prime } - \zeta _2^2 -2\zeta _2 \zeta _3 = -\frac{h^{\prime \prime }}{h} -2 \frac{p^{\prime }}{p} \frac{h^{\prime }}{h} .\\ R_{33}= & {} R_{44} = -\zeta _3^{\prime } - \zeta _3^2 -\zeta _3 \zeta _2 -(\zeta _3)^2 + \frac{k}{h^2}= -\frac{h^{\prime \prime }}{h} - \frac{p^{\prime }}{p} \frac{h^{\prime }}{h} - \frac{(h^{\prime })^2}{h^2} + \frac{k}{h^2}. \\ R_{ij}= & {} 0, \,\,\, \mathrm{if} \, \, i \ne j. \\ R= & {} -6\zeta _3^{\prime } - 6\zeta _3^2 -4\zeta _3 \zeta _2 -2(\zeta _3)^2 + 2\frac{k}{h^2}= - 6 \frac{h^{\prime \prime }}{h} -4 \frac{p^{\prime }}{p} \frac{h^{\prime }}{h} - 2\frac{(h^{\prime })^2}{h^2} + 2\frac{k}{h^2}. \end{aligned}$$

Proof

One may verify all the formulas by direct computation. In particular, $\zeta _2= \frac{p^{\prime }}{p}$ and $\zeta _3=\zeta _4= \frac{h^{\prime }}{h}$. We get $\frac{p^{\prime \prime }}{p} = \frac{h^{\prime \prime }}{h}$ from (2). $\square$

What emerges from the above discussions can be highlighted as the following soliton on an open set, which results from Lemmas 4.3 and 5.1:

A four-dimensional gradient Ricci soliton (M, g, f) with harmonic Weyl curvature has a connected coordinate neighborhood $(V, (s,t, x_3, x_4)) \subset M_{\mathcal {A}} \cap \{ \nabla f \ne 0 \}$, in which

$$\begin{aligned} g= ds^2 + p(s)^2 dt^2 + h(s)^2 \tilde{g}\ \ \ \ \mathrm{on} \ V, \end{aligned}$$

(17)

where $\tilde{g}$ is a 2-dimensional Riemannian metric of constant curvature k on an $(x_3, x_4)$-domain. We have the adapted frame field

$$\begin{aligned}&E_1 = \frac{\nabla f }{ |\nabla f |}=\frac{\partial }{\partial s}, \quad E_2 =\frac{1}{p} \frac{\partial }{\partial t}, \quad E_3 =\frac{1}{h} e_3, \quad E_4 =\frac{1}{h} e_4 \,\,\mathrm{on} \ V, \nonumber \\&\qquad \mathrm{and} \,\,\,\lambda _2 \ne \lambda _3= \lambda _4, \end{aligned}$$

(18)

where $e_3$ and $e_4$ are an orthonormal frame field of $\tilde{g}$ as in Lemma 5.2.

Remark 5.3

As mentioned in Sect. 2, g and f are real analytic (in harmonic coordinates), so is $|\nabla f|$ where $\nabla f \ne 0$. The Ricci eigenvalues $\lambda _i$ are real analytic in $M_{\mathcal {A}} \cap \{ \nabla f \ne 0 \}$. So are $\zeta _i(s)=\frac{ 1}{|\nabla f |} (\lambda - \lambda _i )$.

Also $R^{\prime }= dR(E_1)$ is real analytic since it equals $dR( \frac{\nabla f}{|\nabla f|} )$. From (2) $R(E_1, E_2, E_2, E_1)$ is real analytic. As $-\zeta _2^{\prime } - \zeta _2^2=-\zeta _3^{\prime } - \zeta _3^2 =R(E_1, E_2, E_2, E_1)$, $\zeta _2^{\prime }$ as well as $\zeta _3^{\prime }$ are real analytic.

To exploit the real analyticity, we shall use the following simple fact: if $P \cdot Q$ equals zero (identically) on an open connected set W for two real analytic functions P and Q, then either P equals zero on W or Q equals zero on W.

For the rest of this section we denote $a:=\zeta _2$ and $b:=\zeta _3$ for convenience.

In the adapted frame field $\{ E_i \}$ of (18), we can write components of the soliton equation $\nabla d f(E_i, E_i) = -(Rc- \lambda g)(E_i, E_i)$, $i=1,2,3$ as follows:

$$\begin{aligned} f^{\prime \prime }= & {} 3a^{\prime } + 3a^2 + \lambda . \end{aligned}$$

(19)

$$\begin{aligned} f^{\prime } a= & {} a^{\prime } + a^2 +2b a + \lambda . \end{aligned}$$

(20)

$$\begin{aligned} f^{\prime } b= & {} b^{\prime } + b^2 +b a +b^2 -\frac{k}{h^2} + \lambda . \end{aligned}$$

(21)

In the next section we are going to deduce several linear or quadratic equations in a and b from (19)–(21) and $\delta W=0$. But before we get to it, in the next three lemmas we shall understand three linear cases (when $a=0$, $b=0$ and $a+b=0$ on a domain).

Lemma 5.4

For the soliton metric g of (17) with harmonic Weyl curvature and with the adapted frame field (18), the function a cannot vanish on V.

Proof

If $a=0$, then $b^{\prime } + b^2 = a^{\prime } + a^2=0$. Integrate for $b= \frac{h^{\prime }}{h}$ to get $\frac{h^{\prime }}{h}=\frac{1}{s-c}$ for a constant c, as $b \ne a=0$. So, $h = c_h (s-c)$, for a constant $c_h \ne 0$. From (20), $\lambda =0$. From (19), $f^{\prime \prime } = 0$ and $f^{\prime }$ is constant. From (21) we get $f^{\prime }= \frac{1}{s-c} (1 - \frac{k}{c_h^2} )$. Then, $c_h^2=k>0$ and $f^{\prime } =0$. So, g is Einstein, a contradiction to the hypothesis $\lambda _2 \ne \lambda _3$. $\square$

Lemma 5.5

For the soliton metric g of (17) with harmonic Weyl curvature and with the adapted frame field (18), assume that $b=0$ on V. Then g is locally isometric to a domain in $\mathbb {R}^2 \times (N, \tilde{g})$ with $g= ds^2 + s^2 dt^2+ \tilde{g}$, where $\tilde{g}$ is a Riemannian metric of constant curvature $\lambda \ne 0$ on a two-dimensional manifold N. And $f = \frac{\lambda }{2}s^2 +C_1$, for a nonzero constant $C_1$.

Proof

If $b=0$, then $a^{\prime } + a^2=0$. Integrate for $a= \frac{p^{\prime }}{p}$ to get $\frac{p^{\prime }}{p}=\frac{1}{s-c_1}$ for a constant $c_1$, as $a \ne b=0$. So, $p = c_p (s-c_1)$, for a constant $c_p \ne 0$. As h is constant, we set $h=h_0>0$.

From (20), $f^{\prime } = \lambda (s-c_1)$. We get $f(s) = \frac{1}{2} \lambda (s-c_1)^2+C_1$. If $\lambda =0$, then f is constant and g is Einstein, which violates the $\lambda _2 \ne \lambda _3$ hypothesis. So, $\lambda \ne 0$. From (21), we have $\frac{k}{h_0^2}=\lambda$. And by absorbing a constant to the variable t, we can write the metric $g= ds^2 + (s-c_1)^2 dt^2+ h_0^2 \tilde{g}$, where $h_0^2\tilde{g}$ is a Riemannian metric of constant curvature $\frac{k}{h_0^2}=\lambda$. The metric g is isometric to $ds^2 + s^2 dt^2+ h_0^2 \tilde{g}$. This proves the lemma. $\square$

Lemma 5.6

For the soliton metric g of (17) with harmonic Weyl curvature and with the adapted frame field (18), the function $a+b$ cannot vanish on V.

Proof

Suppose $a+b=0$ on V. Then $a^{\prime } -b^{\prime } = b^2 -a^2 =0$. So, $a-b = C$, a constant. Then $a= \frac{p^{\prime }}{p}= \frac{C}{2}, b= \frac{h^{\prime }}{h}= -\frac{C}{2}$. As $a \ne b$, $C \ne 0$. Then $h= c_h e^{ -\frac{C}{2}s}$ for a constant $c_h>0$. Put it into (20) and (21), and we have $k = \lambda =0$ and $f^{\prime }$ is a constant. Then (19) gives $C^2=0$, which is a contradiction. $\square$

6 Characterization of the Metric When $\lambda _2 \ne \lambda _3 =\lambda _4$

In this section we shall characterize the soliton metric g of (17) with harmonic Weyl curvature and with the adapted frame field (18).

From (20) and (21),

$$\begin{aligned} (a -b ) f^{\prime } = b (a - b) + \frac{k}{h^2}. \end{aligned}$$

(22)

Differentiating, $(a -b )^{\prime } f^{\prime } + (a -b ) f^{\prime \prime }= b^{\prime } (a - b) +b (a - b)^{\prime } -2\frac{k h^{\prime }}{h^3}.$

Meanwhile, from (19), (22) and $a^{\prime } -b^{\prime }=-a^{2} +b^{2}$,

$$\begin{aligned} (a -b )^{\prime } f^{\prime } + (a -b ) f^{\prime \prime }= & {} -(a^{2} -b^{2} ) f^{\prime } + (a -b ) ( - \lambda _1 + \lambda ) \\= & {} (a +b ) \Big \{ -b (a - b) - \frac{k}{h^2}\Big \} + (a -b ) ( - \lambda _1 + \lambda ). \end{aligned}$$

So, we get $b^{\prime } (a - b) +b (b^2 - a^2) -2\frac{k h^{\prime }}{h^3}= (a +b ) \{ -b (a - b) - \frac{k}{h^2} \} + (a -b ) ( - \lambda _1 + \lambda )$. Then, as $b = \frac{h^{\prime }}{h}$,

$$\begin{aligned} b^{\prime } (a - b)= & {} (a +b ) \Big \{ - \frac{k}{h^2}\Big \} + 2\frac{k h^{\prime }}{h^3} + (a -b ) ( - \lambda _1 + \lambda ) \\= & {} (a -b ) \Big \{ - \frac{k}{h^2}\Big \} + (a -b ) ( - \lambda _1 + \lambda ). \end{aligned}$$

As $\lambda _2 \ne \lambda _3$, we have $a -b \ne 0$. We then have:

$$\begin{aligned} 2(b^{\prime } + b^2) + b^2 - \frac{k}{h^2} + \lambda =0. \end{aligned}$$

(23)

From (20), (21) and $b^{\prime } = a^{\prime } +a^2 - b^2$, we have

$$\begin{aligned} b( a^{\prime } + a^2 +2b a + \lambda ) = a\Big (b^{\prime } + b^2 +b a +b^2 -\frac{k}{h^2} + \lambda \Big ), \end{aligned}$$

and so

$$\begin{aligned} -(a-b)a^{\prime } -a^3 +ab^2 + \lambda (b-a)= -a\frac{k}{h^2}. \end{aligned}$$

(24)

Next, we shall exploit the harmonic Weyl curvature condition. In $\{ E_i \}$, we have $\nabla _k R_{ij} -\nabla _j R_{ik} = - \frac{R_j}{6} g_{ki} + \frac{R_k}{6} g_{ij}$. Then as $\nabla _{E_1}E_2 = \nabla _{E_1}E_3=0$,

$$\begin{aligned} 0= & {} \nabla _1 R_{22} - \nabla _2 R_{12} - \frac{R^{\prime }}{6} \nonumber \\= & {} \nabla _1 (R_{22}) + R ( \nabla _{E_{2} } E_{1}, E_{2}) + R ( \nabla _{E_2 }E_2, E_1) - \frac{R^{\prime }}{6} \\= & {} ( R_{22})^{\prime } + a R_{22} -a R_{11} - \frac{R^{\prime }}{6}.\nonumber \end{aligned}$$

(25)

$$\begin{aligned} 0= & {} \nabla _1 R_{33} - \nabla _3 R_{13} - \frac{R^{\prime }}{6}\nonumber \\= & {} \nabla _1 (R_{33}) + R ( \nabla _{E_3 } E_1, E_3) + R ( \nabla _{E_3 }E_3, E_1) - \frac{R^{\prime }}{6}\\= & {} ( R_{33})^{\prime } +b R_{33} -b R_{11} - \frac{R^{\prime }}{6}.\nonumber \end{aligned}$$

(26)

Subtracting (26) from (25), with Lemma 5.2 we get $( -a b +b^2 - \frac{k}{h^2} )^{\prime } + a(- a^{\prime } - a^2 -2a b ) -(a -b)(-3a^{\prime } - 3a^2 ) -b ( - b^{\prime } - b^2 -b a -b^2 + \frac{k}{h^2} ) =0$, from which we obtain

$$\begin{aligned} -(a-b)a^{\prime } -a^3 + b^3 + 2a^2b -2ab^2 = b\frac{k}{h^2}. \end{aligned}$$

(27)

Subtracting (24) from (27),

$$\begin{aligned} (a-b) ( 2ab -b^2+\lambda )= (a+b)\frac{k}{h^2}. \end{aligned}$$

(28)

Lemma 6.1

For the soliton metric g of (17) with harmonic Weyl curvature and with the adapted frame field (18), assume that $k \ne 0$. Then the following holds:

$$\begin{aligned} b(\lambda + 3ab )(\lambda -2a^2 +ab )=0. \end{aligned}$$

(29)

Proof

We start from (28). Our hypothesis $k \ne 0$ and Lemma 5.6 implies that $2ab -b^2+\lambda$ does not vanish. So, we may take the natural log of (28) and differentiate it:

$$\begin{aligned} -a-b + \frac{2a^{\prime }b+ 2ab^{\prime } - 2bb^{\prime }}{2ab - b^2 + \lambda } = \frac{a^{\prime } + b^{\prime }}{a+b} -2b. \end{aligned}$$

Then put $b^{\prime } = a^{\prime } +a^2 - b^2$ into it:

$$\begin{aligned} \frac{ a a^{\prime }+ (a - b )(a^2 -b^2) }{2ab - b^2 + \lambda }= \frac{a^{\prime }+a^2 -b^2}{a+b}. \end{aligned}$$

Arranging terms, we obtain:

$$\begin{aligned} -a^{\prime } {(a^2+ b^2 -ab -\lambda ) } = ( a^2-b^2)(a^2 -2ab -\lambda ). \end{aligned}$$

(30)

Meanwhile, using that $a-b \ne 0$, from $b \times (24) + a \times (27)=0$ we have

$$\begin{aligned} -(a+b) a^{\prime } = a^3 +2ab^2 +\lambda b. \end{aligned}$$

(31)

Removing $a^{\prime }$ in (30) and (31) and simplifying, we can get:

$$\begin{aligned} b(\lambda + 3ab )(\lambda -2a^2 +ab )=0. \end{aligned}$$

$\square$

We need to characterize the two equalities appearing in (29): $\lambda + 3ab=0$ and $\lambda -2a^2 +ab=0$.

Lemma 6.2

For the soliton metric g of (17) with harmonic Weyl curvature and with the adapted frame field (18), assume that $k \ne 0$ and that h is not constant. Then $\lambda + 3ab$ does not vanish on V.

Proof

If $\lambda + 3ab=0$ vanishes, we have $0=a^{\prime }b + ab^{\prime }=(b^{\prime } +b^2 - a^2)b+ ab^{\prime } = (a+b) (b^{\prime }+b^2 -ab )$. By Lemma 5.6, we have $b^{\prime }+b^2 =ab =-\frac{\lambda }{3}$. Due to (23), $b^2 - \frac{k}{h^2} =-\frac{\lambda }{3}$. From (21), $f^{\prime } b = - \frac{\lambda }{3} - \frac{\lambda }{3} - \frac{\lambda }{3} + \lambda =0.$ As h is not constant, we have $f^{\prime }=0$, a contradiction. $\square$

We study the equation $\lambda -2a^2 +ab =0$:

Lemma 6.3

For the soliton metric g of (17) with harmonic Weyl curvature and with the adapted frame field (18), assume that $k \ne 0$ and that h is not constant. Then $\lambda -2a^2 +ab$ does not vanish on V.

Proof

If $\lambda -2a^2 +ab =0$ on V, put $\lambda =2a^2 -ab$ into (31) to get: $-a^{\prime } = \frac{a^3 +2a^2b +ab^2 }{ a+b}= a(a+b)$. So, $a^{\prime } +a^2+ ab=0$, i.e., $p^{\prime \prime }h + p^{\prime }h^{\prime } =0$. Integrating this, we get $p^{\prime } h =c_1$ for a constant $c_1$. As $\frac{h^{\prime \prime }}{h} = \frac{p^{\prime \prime }}{p}$, we have $h^{\prime \prime }p+p^{\prime }h^{\prime } =0$, which integrates to $h^{\prime } p =c_2$ for a constant $c_2$. As a does not vanish by Lemma 5.5 and $b \ne 0$ from the hypothesis, $c_1c_2$ is not zero. So $\frac{h^{\prime }}{h} = \frac{c_2}{c_1} \frac{p^{\prime }}{p}$, i.e., $b= ca$, for $c \ne 0$. So, $0=\lambda -2a^2 +ab = \lambda + (c-2)a^2$.

If $c \ne 2$, then a is a nonzero constant. $a^{\prime } +a^2+ ab=0$ yields $a+b = 0$, which is not possible by Lemma 5.6.

If $c=2$, then $\lambda =0$ and $2a = b$. Put these and $a^{\prime } +a^2+ ab=0$ into (20) to get $f^{\prime } =2a$. Then from (21), we get $k=0$, a contradiction. $\square$

Lemma 6.4

For the soliton metric g of (17) with harmonic Weyl curvature and with the adapted frame field (18), assume that $k=0$.

Then g is locally isometric to the metric $ds^2 + s^{\frac{2}{3}} dt^2+ s^{\frac{4}{3}} \tilde{g}$ on a domain of $\mathbb {R}^4$, where $\tilde{g}$ is flat. Also, $\lambda =0$ and $f=\frac{2}{3} \ln s + C_2$, for a constant $C_2$.

Furthermore, the Ricci curvature components and scalar curvature of g are as follows: $R_{11} = \frac{2}{3s^2}$, $R_{22} = -\frac{2}{9s^2}$, $R_{33}=R_{44} = -\frac{4}{9s^2}$, $R_{ij} =0$, $i \ne j$, and $R= -\frac{4}{9s^2}$. And the Weyl curvature of g is not zero.

Proof

As $k=0$ and $a \ne b$, $2ab -b^2 + \lambda =0$ from (28). From the computation in Lemma 5.2, we get $\ R = -6 ( a^{\prime } + a^2) - 8ab-2 \lambda$. (25) becomes:

$$\begin{aligned} 0= & {} -\{ a^{\prime } +a^2 + 2 ab \}^{\prime } - a \{a^{\prime } + a^2 + 2ab \} +3a ( a^{\prime } + a^2) \\&- \frac{ 1}{6} \{ -6 ( a^{\prime } + a^2) - 8ab-2 \lambda \}^{\prime } \\= & {} -\frac{2}{3} (ab)^{\prime } + 2a( a^{\prime }+ a^2-a b) \\= & {} -\frac{2}{3} \{ a^{\prime }b + a(a^{\prime } + a^2 -b^2) \} + 2a( a^{\prime }+ a^2-a b) \\= & {} -\frac{2}{3} a^{\prime }b + \frac{4}{3} aa^{\prime } + \frac{4}{3} a^3 + \frac{2}{3} ab^2 -2 a^2 b. \end{aligned}$$

We get:

$$\begin{aligned} (2a-b) (a^{\prime } + a^2 -ab)=0. \end{aligned}$$

If $a^{\prime }+ a^2 -ab=0$, we get $p^{\prime \prime } = \frac{p^{\prime } h^{\prime }}{h}$. Then $\frac{p^{\prime }}{h}=c_1$, a constant. From $\frac{ h^{\prime \prime }}{h}=\frac{p^{\prime \prime }}{p}= \frac{p^{\prime } h^{\prime }}{ph}$, we also get $\frac{h^{\prime }}{p}=c_2$, a constant. So, $ab =\frac{p^{\prime }h^{\prime }}{ph} =c_1c_2$. And $2ab - b^2 + \lambda =0$ tells that b is a constant. If $b=0$, then $\lambda =k=0$ and from (20) $f^{\prime }a=0$. So, $f^{\prime }=0$ and g is Einstein, a contradiction to the hypothesis. Now b is a nonzero constant. Then $b^{\prime } + b^2 =a^{\prime }+ a^2=ab$ gives $a=b$, a contradiction to the hypothesis.

If $2a=b$, then $0= 2ab - b^2 + \lambda = \lambda$. From $a^{\prime } +a^2 = b^{\prime } +b^2 = 2a^{\prime } + 4a^2$, we get $a^{\prime } +3a^2 =0$. Integrating it to get $a=\frac{p^{\prime }}{p}= \frac{1}{3s -c_2}$ for a constant $c_2$. (20) gives $f^{\prime } a = 2a^2$, so that $f^{\prime } = 2a=\frac{2}{3s -c_2}$. As $2 \frac{p^{\prime }}{p} = \frac{ h^{\prime }}{h}$, we have $p^2 = e^c h$ for a constant c. We get $p = e^{c_3} (3s- c_2)^{\frac{1}{3}}$ and $h= e^{c_4} (3s- c_2)^{\frac{2}{3}}$.

So, g is locally isometric to the metric $ds^2 + s^{\frac{2}{3}} dt^2+ s^{\frac{4}{3}} \tilde{g}$ on a domain of $\mathbb {R}^4$, where $\tilde{g}$ is flat. And $f=\frac{2}{3} \ln s + C_2$, for a constant $C_2$.

One can check that the above (g, f) satisfy the soliton equation including (19), (20), (21) and the harmonicity of the Weyl curvature, and so is a steady Ricci soliton. One can easily compute the curvature components of g. $\square$

Based on the real analyticity of a, b, $a^{\prime }$ and $b^{\prime }$ from Remark 5.3, we combine the previous lemmas to obtain the next proposition.

Proposition 6.5

Let (M, g, f) be a four-dimensional gradient Ricci soliton with harmonic Weyl curvature. Suppose that $\lambda _2 \ne \lambda _3= \lambda _4$ for an adapted frame field $E_j$, $j=1,2,3,4$, in an open subset U of $M_{\mathcal {A}} \cap \{ \nabla f \ne 0 \}$.

Then for each point $p_0$ in U, there exists a neighborhood V of $p_0$ in U with coordinates $(s,t, x_3, x_4)$ in which (V, g, f) can be one of the following:

• (i) (V, g) is isometric to a domain in $\mathbb {R}^2 \times N$ with $g= ds^2 + s^2 dt^2+ \tilde{g}$, where $(N, \tilde{g})$ is a Riemannian manifold of constant curvature $\lambda \ne 0$. And $f = \frac{\lambda }{2} s^2+C_1$, for a constant $C_1$.

• (ii) (V, g) is isometric to a domain in $\mathbb {R}^4$ with the Riemannian metric $ds^2 + s^{\frac{2}{3}} dt^2+ s^{\frac{4}{3}} \tilde{g}$, where $\tilde{g}$ is flat. Also, $\lambda =0$ and $f=\frac{2}{3} \ln s+C_2$, for a constant $C_2$. The metric g is not locally conformally flat.

Proof

We exploit the real analyticity. Lemma 6.4 settles the $k=0$ case. Lemma 6.1 divides the $k \ne 0$ case into three subcases $b=0$, $\lambda + 3ab=0$ and $\lambda -2a^2 +ab=0$ which are treated in Lemmas 5.5, 6.2 and 6.3, respectively. $\square$

7 4-Dimensional Soliton with $\lambda _2 = \lambda _3 =\lambda _4$.

In this section we treat the remaining case of $\lambda _2 = \lambda _3 =\lambda _4$ for an adapted frame field.

Proposition 7.1

Suppose that (M, g, f) is a four-dimensional gradient Ricci soliton with harmonic Weyl curvature and non-constant f and that $\lambda _2 = \lambda _3 =\lambda _4 \ne \lambda _1$ for an adapted frame field in an open subset U of $M_{\mathcal {A}} \cap \{ \nabla f \ne 0 \}$.

Then for each point $p_0$ in U, there exists a neighborhood V of $p_0$ in U where g is a warped product:

$$\begin{aligned} g= ds^2 + h(s)^2 \tilde{g}, \end{aligned}$$

(32)

for a positive function h, where the Riemannian metric $\tilde{g}$ has constant curvature, say k. In particular, g is locally conformally flat.

Proof

Near $p_0$ in U, we use a local coordinate system $(x_1 := s, \ x_2, x_3, x_4)$ from Lemma 2.3 (v) in which the metric $g = ds^2 + \sum _{i, j \ge 2}^{4}g_{ij} dx_i dx_j$ with $g_{ij} =g_{ij}(x_1, \ldots , x_4)$.

By Lemma 2.7, near $p_0$, each $\lambda _i$, $i = 1,2,3,4$ is a function of s only. We consider the second fundamental form of the level hypersurfaces $\Sigma _c$ of f with respect to $E_1$: $H^{E_1} ( u , u ) = - \langle \nabla _{u} u , E_1\rangle$. As $\Sigma _c$ is totally umbilic by Lemma 2.4 (ii), $H^{E_1} ( u , u ) = G \cdot g( u , u)$ for any u tangent to $\Sigma _c$ and some function G. Then, by Lemma 2.4 (i) $\langle \nabla _{E_2} E_2 , E_1\rangle = \frac{ \lambda _2^{\prime } -\frac{1}{6}R^{\prime } }{\lambda _2 - \lambda _1 }$ So, $G= - \frac{ \lambda _2^{\prime } -\frac{1}{6}R^{\prime } }{\lambda _2 - \lambda _1 }$ is a function of s only.

For $i, j \in \{2, 3,4 \}$, setting $\partial _i:= \frac{\partial }{\partial x_i}$, we compute,

$$\begin{aligned} G(s) \cdot g_{ij}= & {} H^{E_1} ( \partial _i , \partial _{j} ) = - \left\langle \nabla _{\partial _i} \partial _{j} , \frac{\partial }{\partial s}\right\rangle = - \left\langle \sum _{k=1}^4 \Gamma ^{k}_{i{j}} \partial _k , \frac{\partial }{\partial s} \right\rangle \\= & {} - \sum _k \left\langle \frac{1}{2} g^{kl}( \partial _i g_{lj} +\partial _{j} g_{li} - \partial _l g_{ij} )\partial _k , \frac{\partial }{\partial s} \right\rangle = \frac{1}{2} \frac{\partial }{\partial s} g_{i{j}}. \end{aligned}$$

So, $\frac{1}{2} \frac{\partial }{\partial s} g_{i{j}} = G(s) g_{ij}$. Integrating it, we get $g_{ij} = e^{C_{ij}} w(s)$. Here the function w(s) is independent of i, j and each $C_{ij}$ depends only on $x_2, x_3, x_4$.

Now g can be written as $g= ds^2 + h(s)^2 \tilde{g}$, where $\tilde{g}$ can be viewed as a Riemannian metric in a domain of the $(x_2, x_3, x_4)$-plane.

To prove that $\tilde{g}$ has constant curvature, we modify the proof of Derdziński’s Lemma 4 in [17], which is stated for the harmonic curvature case.

For $i,j \in \{2, 3,4 \}$, we compute the Christoffel symbols and Ricci curvature of g:

$$\begin{aligned} \Gamma _{ij}^1= & {} - h h^{\prime } \tilde{g}_{ij}, \quad \Gamma _{1j}^i = \frac{h^{\prime }}{h} \delta _{ij}, \nonumber \\ \quad R_{1i}= & {} 0,\quad R_{11}= -3 \frac{h^{\prime \prime }}{h}, \quad R_{ij} = - \tilde{g}_{ij} ( h h^{\prime \prime } + 2 { h^{\prime }}^2 ) +R^{\tilde{g}}_{ij}. \end{aligned}$$

(33)

The condition $\delta W=0$ gives $\nabla _k R_{ij} - \nabla _j R_{ik} = - \frac{R_j}{6} g_{ki} + \frac{R_k}{6} g_{ij}$. In particular, for $i, j \in \{2, 3,4 \}$, $\nabla _1 R_{ij} -\nabla _j R_{i1} = \frac{R_1}{6} g_{ij}$. From (33),

$$\begin{aligned} \frac{\partial _1 R}{6} h^2 \tilde{g}_{ij}= & {} \frac{\partial _1 R}{6} g_{ij}=\nabla _1 R_{ij} -\nabla _{i} R_{1 j}\\= & {} \partial _1 R_{ij} - R ( \nabla _{\partial _{1} }\partial _{j}, \partial _{i}) + R ( \nabla _{\partial _{i} }\partial _{j}, \partial _{1}) \\= & {} \partial _1 R_{ij} - \frac{ h^{\prime } }{h} R ( {\partial _{j} }, \partial _{i}) - h h^{\prime } R ( \partial _{1 }, \partial _{1}) \tilde{g}( \partial _{i } , \partial _{j } )\\= & {} - \tilde{g}_{ij} \partial _1 ( h^{\prime \prime } h + 2 {h^{\prime }}^2 ) - \frac{ h^{\prime } }{h} \left[ - \tilde{g}_{ij} \big ( h h^{\prime \prime } +2 { h^{\prime }}^2 \big ) +R^{\tilde{g }}_{ij} \right] - h h^{\prime } R_{11} \tilde{g}_{ij}. \end{aligned}$$

As R depends only on s, so does $\partial _1 R= \frac{\partial R }{\partial s}$. Therefore, we get $R^{\tilde{g }}_{ij}=H(s) \cdot \tilde{g}_{ij}$ for a function H(s) of s only. So, $\tilde{g}$ is a 3-dimensional Einstein metric. $\square$

For the metric in (32), h and f satisfy the following equations from $\nabla \nabla f + Rc = \lambda g$:

$$\begin{aligned}&f^{\prime \prime } - 3\frac{h^{\prime \prime }}{h} = \lambda , \end{aligned}$$

(34)

$$\begin{aligned}&\frac{h^{\prime }}{h} f^{\prime } + \frac{2k}{h^2} - \frac{h^{\prime \prime }}{h} - 2\frac{(h^{\prime })^2}{h^2} = \lambda . \end{aligned}$$

(35)

Remark 7.2

If all $\lambda _i$’s, $i=1, \ldots , 4$, are equal, then the metric is Einstein. And if f is not constant, then the conclusion of Proposition 7.1 still holds. In fact, from Sect. 1 of [11], the Einstein metric g becomes locally of the form $g= ds^2 + (f^{\prime }(s))^2 \tilde{g}$ where $\tilde{g}$ has constant curvature. Then, the soliton can be seen to be either Gaussian or a flat metric with $\nabla d f=0$; see also Proposition 2 of [28].

8 Classification of Gradient Ricci Solitons with Harmonic Weyl Curvature

We are going to combine Propositions 3.4, 6.5 and 7.1 to prove Theorem 1.1 after we settle the next lemma:

Lemma 8.1

No two of the local four types of solitons (i)–(iv) in the statement of Theorem 1.1 can exist on a connected soliton.

Proof

When the real analytic function f is constant in an open subset, then it is constant on M as M is connected. So, if a soliton is the type (i) in an open subset, it will be so on M.

If g is a locally conformally flat metric on an open subset U with non-constant f, then $|W|^2=0$ on U and the real analytic function $|W|^2=0$ everywhere on M. So, g is locally conformally flat on M and f is nowhere constant on M. The types (ii) and (iii) do not satisfy $|W|^2=0$.

If g is isometric, on an open subset V, to a domain in $\mathbb {R}^2 \times N_{\lambda }$, then $R=2 \lambda$ on V and by real analyticity $R=2 \lambda$ on M. But if g is isometric, on another open subset W, to the metric $ds^2 + s^{\frac{2}{3}} dt^2+ s^{\frac{4}{3}} \tilde{g}$, then the scalar curvature $R= -\frac{4}{9s^2}$ is not locally constant. This proves the lemma. $\square$

Proof of Theorem 1.1

Due to Lemma 8.1 we may consider only one type on M. When f is constant, it corresponds to the type (i).

So, suppose that f is not constant. Note that the statement (iv) holds by Proposition 7.1 and Remark 7.2. We denote the open dense subset $M_{\mathcal {A}} \cap \{ \nabla f \ne 0 \}$ by K. If $K=M$, then the statements for (ii) and (iii) also hold from Proposition 6.5..

For the rest of proof we assume that there is a point $p_0 \in M{\setminus }K$.

When (K, g) is of the type (ii), (K, g) is locally isometric to $\mathbb {R}^2 \times N_{\lambda }$, where the Ricci tensor is parallel. As K is dense in M, the Ricci tensor is parallel near $p_0$ with eigenvalues $\lambda$ and 0 of both multiplicity two by continuity. We can decompose the tangent bundle over a neighborhood of $p_0$: $TM = \eta _1 \oplus \eta _2$, where $\eta _1, \eta _2$ are 2-dimensional parallel distributions with $Rc_{| \eta _1} = \lambda \cdot \mathrm{Id}$ and $Rc_{| \eta _2} = 0 \cdot \mathrm{Id}$. By de Rham’s decomposition theorem [27, Sect. 8.3.1], $p_0$ has an open ball $B \subset M$ with $p_0$ as the center, where B is isometric to (to be identified with) a ball in $\mathbb {R}^2 \times N_{\lambda }$. Now we can just solve for f from the gradient soliton equation $\nabla d f = -Rc + \lambda g$ to get: $f= \frac{\lambda s^2}{2} + C$ where $s(\cdot ) := d_{\mathbb {R}^2}( p_0,\cdot )$ is the Euclidean distance function from $p_0$. So, a neighborhood of $p_0$ is of type (ii).

Suppose that (K, g) is of the type (iii). Let $\gamma _1: [0,1] \rightarrow M$ be a smooth path with $\gamma _1(0) =p_0$ and $\gamma _1(1) \in K$. Let $c \in [0,1)$ be the largest element in $\{ t \in [0,1) \ | \ \gamma _1(t) \in M{\setminus }K \}$. Define $\gamma$ to be the restriction of $\gamma _1$ on [c, 1]. Set $p:= \gamma (c)$ which is in $M{\setminus }K$. Then $\gamma ((c,1]) \subset K$. Near any point $q \in \gamma ((c,1])$, by Proposition 6.5 we have local coordinates neighborhood $B_q \subset K$ with $(s_q, t, x_3, x_4)$ in which $f= \frac{2}{3} \ln (s_q) +C_q$ with the function $s_q$ and constant $C_q$ depending on q. In a neighborhood $B_r \subset K$ of another point $r \in \gamma ((c,1])$, we have a similar expression of $f= \frac{2}{3} \ln (s_r) +C_r$. On a possible overlap region $B_q \cap B_r$, $\frac{2}{3} \ln (s_q) +C_q = \frac{2}{3} \ln (s_r) +C_r$. By taking its gradient, we have $\frac{\nabla s_q}{s_q} = \frac{\nabla s_r}{s_r}$. As $\nabla s_q= \frac{\nabla f}{|\nabla f|} =\nabla s_r$, we get $s_q =s_r$ and then $C_q= C_r$.

We may set $s:=s_q$ and $C:=C_q$ which are independent of q and $f= \frac{2}{3} \ln (s) + C$ near $\gamma ((c,1])$. As $|\nabla s| \equiv 1$, the oscillation of s along $\gamma$ is less than or equal to the length of $\gamma$, which is finite. So, $|\nabla f| = \frac{2}{3s}$ cannot be zero at p. From Lemma 6.3, the Ricci-eigenfunctions of g are $\lambda _{1} = \frac{2}{3s^2}$, $\lambda _{2} = -\frac{2}{9s^2}$, $\lambda _{3}=\lambda _{4} = -\frac{4}{9s^2}$. So, p shall stay in $M_{\mathcal {A}}$ by definition. Then $p \in K$. This contradiction implies that $M{\setminus }K$ is an empty set.

Proposition 3.4 shows that there are no other types than (i)–(iv). This proves the theorem. $\square$

We remark that the incomplete steady gradient soliton in Theorem 1.1 (iii) has negative scalar curvature, in contrast to the fact that complete steady gradient solitons should have nonnegative scalar curvature.

As a Corollary to Theorem 1.1, we state a classification of 4-dimensional complete gradient Ricci solitons with harmonic Weyl curvature. The case Theorem 1.1 (iii) can only yield an incomplete soliton. And for case (ii), when g is complete and locally isometric to $\mathbb {R}^2 \times N_{\lambda }$, its universal cover is isometric to $\mathbb {R}^2 \times N_{\lambda }$.

Theorem 8.2

Let (M, g, f) be a complete four-dimensional gradient Ricci soliton $\nabla df = -Rc + \lambda g$ with harmonic Weyl curvature. Then it is one of the following:

• (i) g is an Einstein metric with f a constant function.

• (ii) g is isometric to a finite quotient of $\mathbb {R}^2 \times N_{\lambda }$ where $\mathbb {R}^2$ has the Euclidean metric and $N_{\lambda }$ is a 2-dimensional Riemannian manifold of constant curvature ${\lambda } \ne 0$. And $f = \frac{\lambda }{2} |x|^2$ modulo a constant on the Euclidean factor.

• (iii) g is locally conformally flat.

Complete locally conformally flat steady gradient Ricci solitons are classified to be either flat or isometric to the Bryant soliton, in [6, 10]. This result and Theorem 8.2 yield Theorem 1.2. We also understand better complete expanding gradient Ricci solitons with harmonic Weyl curvature as in Theorem 1.3.

As mentioned in the Introduction, we can show the local classification of gradient Ricci soliton with harmonic curvature as a corollary of Theorem 1.1.

Corollary 8.3

Let (M, g, f) be a (not necessarily complete) four-dimensional gradient Ricci soliton satisfying $\nabla df = -Rc + \lambda g$ with harmonic curvature. Then it is locally one of the three types (i)–(iii) below; for each point p, there exists a neighborhood V of p such that (V, g, f) can be one of the following:

• (i) g is an Einstein metric and f is constant.

• (ii) g is isometric to a domain in $\mathbb {R}^2 \times N_{\lambda }$ where $\mathbb {R}^2$ has the Euclidean metric and $N_{\lambda }$ is a 2-dimensional Riemannian manifold of constant curvature ${\lambda } \ne 0$. And $f = \frac{\lambda }{2} |x|^2$ modulo a constant on the Euclidean factor.

• (iii) g is isometric either to a domain in the Gaussian soliton or to a domain in $\mathbb {R} \times M_{\lambda }$ with the product metric, where $M_{\lambda }$ is a 3-dimensional Riemannian manifold of constant curvature $\frac{ \lambda }{2} \ne 0$, and $f = \frac{\lambda }{2} |x|^2$ modulo a constant on the Euclidean factor.

Proof

In this proof we do not rely on Theorem 1.2 of [28] as it works for a complete soliton.

The soliton metric $ds^2 + s^{\frac{2}{3}} dt^2+ s^{\frac{4}{3}} \tilde{g}$ in Theorem 1.1 (iii) does not have constant scalar curvature, so does not have harmonic curvature.

Note that the above (iii) should come from Theorem 1.1 (iv), in which the metric is of the form $g=ds^2 + h(s)^2 \tilde{g}$, where $\tilde{g}$ has constant curvature. Lemma 2.1 (ii) gives $R + |\nabla f|^2 - 2\lambda f = \mathrm{constant}$. We differentiate with the local variable s where $|\nabla f| \ne 0$, and get $2 f^{\prime } f^{\prime \prime } = 2\lambda f^{\prime }$ since R is constant. So, $f^{\prime \prime } = \lambda$. From (34), $h^{\prime \prime } =0$. Either $h=a$ or $h=bs$ for constants $a, b \ne 0$ after shifting s by a constant.

When $h=a$, from (35) we get $\frac{k}{a^2} = \frac{ \lambda }{2}$. We have $g = ds^2 + \tilde{g}$ where $\tilde{g}$ has constant curvature $\frac{ \lambda }{2}$. And we may set $f= \frac{\lambda }{2} s^2+C$ by shifting s. As f is not constant, $\lambda \ne 0$.

When $h=bs$, using (35) and $f^{\prime \prime } = \lambda$ we obtain that $f^{\prime } = \lambda s$ and $k= b^2$. We get $f = \frac{1}{2}\lambda s^2 +C$ so that $\lambda \ne 0$. And $g = ds^2 + s^2 \tilde{g},$ where $\tilde{g}$ has constant curvature $+1$. This yields the Gaussian soliton.

(As an alternative to settle (iii), Sect. 2.2 of [10] may be cited. But that section is based on the existence of a self-similar solution, which exists if the soliton metric is complete [33]. Here the metric may be incomplete.) $\square$

Remark 8.4

In Theorem 1.1 (iii) we got a four-dimensional incomplete soliton. One may ask if there exist complete non-conformally flat gradient Ricci solitons of dimension ${\ge }5$ with harmonic Weyl curvature and $\lambda \le 0$.

There are a number of objects to study by extending our method; it would be interesting to characterize the higher-dimensional gradient Ricci solitons with harmonic Weyl curvature as well as other Ricci solitons. Of course, other geometric structures than solitons can also be approached by the method here.

Remark 8.5

There is much literature on orbifolds in the theory of Ricci flow, for instance, [16, 23]. As our result is a local description, it is possible to state an orbifold version of Theorem 8.2.

Remark 8.6

B.L. Chen proved a local version of a Hamilton–Ivey type estimate for three dimensions in [12], which has been extended to the $W=0$ case by Zhang [32]. From Theorem 1.1, one may ask if such a local version still holds when $\delta W=0$.