1 Introduction

In this paper, we study complete Riemannian manifolds whose Ricci tensor \(\rho \) satisfies the relation

$$\begin{aligned} \nabla _X\rho (X,X) =\frac{2X \text {Scal}}{n+2} g(X,X) \end{aligned}$$
(1.1)

where \( {\text {Scal}}= \text {tr}_g \rho \) is the scalar curvature of (Mg) and \(n=\dim M\). This property was studied by A. Gray [4] and by A. Besse [1]. A. Gray called Riemannian manifolds satisfying (1.1) \({\mathcal {AC}}^{\perp }\) manifolds; we shall also call them Gray manifolds. We describe all simply connected and real analytic Gray \({\mathcal {AC}}^{\perp }\) manifolds whose Ricci tensor has two eigenvalues \(\lambda ,\mu \) of multiplicity \(\dim M-1\) and 1 .

If ([g], D) is an Einstein–Weyl structure on a compact manifold M, and \(g_0\) is the Gauduchon metric in the conformal class [g], then \((M,g_0)\) is an \(\mathcal {AC}^{\perp }\) manifold (see [6]) with the above property. If \(\omega _0\) is the Lee form of \(g_0\), i.e., \(Dg_0=\omega _0\otimes g_0\), then the Weyl structure determined by the pair \((g_0,-\omega _0)\) is also Einstein–Weyl (see [6]). Hence if ([g], D) is an Einstein–Weyl structure on a compact manifold M, and \(g_0\) is the Gauduchon metric of ([g], D), then on M there is another Einstein–Weyl structure \(([g],D^-)\) such that \(D^-g_0=-\omega _0\otimes g_0\). Thus, \((M,g_0)\) is then a Riemannian manifold which admits a pair of Einstein–Weyl structures determined by the pairs \((g_0,\omega _0)\) and \((g_0,-\omega _0)\). More generally, a Riemannian manifold (Mg) which admits two Einstein–Weyl structures determined by pairs \((g,\omega )\) and \((g,-\omega )\) is an \(\mathcal {AC}^{\perp }\) manifold whose Ricci tensor has two eigenvalues \(\lambda ,\mu \) of multiplicity \(\dim M-1\) and 1. We prove that Gray manifolds with two eigenvalues \(\lambda ,\mu \) of multiplicity \(\dim M-1\) and 1 always admit a conformal vector field \(\xi \) which is an eigenfield of the Ricci tensor of (Mg). The field \(\xi \) is a Killing vector field or a conformal closed gradient vector field.

In particular, we give a description of all complete, real analytic, simply connected Riemannian manifolds (Mg) admitting a pair of Einstein–Weyl structures \((g,\omega ),(g,-\omega )\). We show that on such manifolds \(\omega (X)=g(\xi ,X)\) where \(\xi \) is either Killing vector field or a closed gradient conformal field. In the second case, we give a complete classification of such manifolds. We prove that in this case the two Einstein–Weyl structures are conformally Einstein.

For general \(\mathcal {AC}^{\perp }\) manifolds, we prove that the eigenvalues \(\lambda ,\mu \) of the Ricci tensor satisfy the equations: \( n\mu -2(n-1)\lambda =C_0=\mathrm{const}\) in the case of a non-Killing conformal vector field and \((n-4)\lambda +2\mu =C_0=\mathrm{const}\) in the case of a Killing vector field.

We also give new examples of compact \(\mathcal {AC}^{\perp }\) manifolds diffeomorphic to the sphere \(S^{n},n>2\).

2 \({\mathcal {AC}}^{\perp }\) manifolds and Einstein–Weyl manifolds

By an \({\mathcal {AC}}^{\perp }\) manifold (see [1, 4]), we mean a Riemannian manifold (Mg) satisfying the condition

$$\begin{aligned} \mathfrak {C}_{X Y Z}\nabla _X\rho (Y,Z)=\frac{2}{\dim M+2}\mathfrak {C}_{X Y Z}X \text {Scal} g(Y,Z), \end{aligned}$$
(2.1)

where \(\rho \) is the Ricci tensor of (Mg), \(\text {Scal}\) is the scalar curvature and \(\mathfrak {C}\) means the cyclic sum.

In [6], it is proved that a Riemannian manifold (Mg) is an \({\mathcal {AC}}^{\perp }\) manifold if and only if the Ricci endomorphism Ric of (Mg) is of the form \({Ric}=S+\frac{2}{n+2} \text {Scal} Id\) where S is a Killing tensor and \(n=\,\)dimM. Let us recall that a (1, 1) tensor S on a Riemannian manifold (Mg) is called a Killing tensor if \(g(\nabla S(X,X),X)=0\) for all \(X\in \mathrm{TM}\).

Define the integer-valued function \(E_S(x)=(\)the number of distinct eigenvalues of \(S_x\)) and set \(M_S=\{ x\in M:E_S\) is constant in a neighborhood of \(x\}\). The set \(M_S\) is open and dense in M, and the eigenvalues \(\lambda _i\) of S are distinct and smooth in each component U of \(M_S\). Let us denote by \(D_{\lambda _i}\) the eigendistribution corresponding to \(\lambda _i\). We have (see [6]).

Proposition

Let S be a Killing tensor on M and U be a component of \(M_S\) and \(\lambda _1,\ldots ,\lambda _k \in C^{\infty }(U)\) be eigenfunctions of S. Then for all \(X\in D_{\lambda _i}\) we have

$$\begin{aligned} \nabla S(X,X)=-\frac{1}{2} \nabla \lambda _i\Vert X \Vert ^2 \end{aligned}$$
(2.2)

and \(D_{\lambda _i}\subset \ker d\lambda _i\). If \(i\ne j\) and \(X\in \Gamma (D_{\lambda _i}),Y \in \Gamma (D_{\lambda _j})\) then

$$\begin{aligned} g(\nabla _X X ,Y)=\frac{1}{2}\frac{Y\lambda _i}{\lambda _j-\lambda _i}\Vert X \Vert ^2. \end{aligned}$$
(2.3)

If \(T(X,Y)=g(SX,Y)\) is a Killing tensor on (Mg) and c is a geodesic on M, then the function \(\phi (t)=T(\dot{c}(t),\dot{c}(t))\) is constant on the domain of c. In fact, \(\phi '(t)=\nabla _{\dot{c}(t)}T((\dot{c}(t),\dot{c}(t))=0\).

A conformal vector field \(\xi \) on a Riemannian manifold satisfies the relation \(L_{\xi }g=\alpha g\). This is equivalent to \(\nabla _X\omega (Y)+\nabla _Y\omega (X)=\alpha g(X,Y)\) where \(\omega =g(\xi ,.)\).

Finally, let us recall that a Riemannian manifold with Killing Ricci tensor is called after A. Gray an \(\mathcal A\) manifold ( it is an \({\mathcal {AC}}^{\perp }\) manifold with constant scalar curvature).

We start with some basic facts concerning Einstein–Weyl geometry. For more details, see [10, 12], [11].

Let M be an n-dimensional manifold with a conformal structure [g] and a torsion-free affine connection D. This defines an Einstein–Weyl (E–W) structure if D preserves the conformal structure, i.e., there exists a 1-form \(\omega \) on M such that

$$\begin{aligned} Dg=\omega \otimes g \end{aligned}$$
(2.4)

and the Ricci tensor \(\rho ^D(X,Y)=\text {tr}\{Z\rightarrow R^D(Z,X)Y\}\) of D, where \(R^D\) is the curvature tensor of (MD), satisfies the condition

$$\begin{aligned} \rho ^D(X,Y)+\rho ^D(Y,X)=\bar{\Lambda }g(X,Y) \text { for every } X,Y\in \mathrm{TM} \end{aligned}$$

for some function \(\bar{\Lambda }\in C^{\infty }(M)\). P. Gauduchon ([2]) proved the fundamental theorem that if M is compact, then there exists a Riemannian metric \(g_0\in [g]\) for which the Lee form \(\omega _0\) associated with \(g_0\) is co-closed-\(\delta \omega _0=0\) where \(Dg_0=\omega _0\otimes g_0\), and \(g_0\) is unique up to homothety. The metric \(g_0\) is called Gauduchon or the standard metric of the E–W structure ([g], D) on M. Note that if M is not compact, a standard metric may not exist.

Let \(\rho \) be the Ricci tensor of (Mg) and let us denote by Ric the Ricci endomorphism of (Mg), i.e., \(\rho (X,Y)=g(X, Ric Y)\). We recall an important theorem (see [10]):

Theorem

A metric g and a 1-form \(\omega \) determine an E–W structure if and only if there exists a function \(\Lambda \in C^{\infty }(M)\) such that

$$\begin{aligned} \rho +\frac{1}{4}(n-2) \mathcal D\omega =\Lambda g \end{aligned}$$
(2.5)

where \(\mathcal D\omega (X,Y)=(\nabla _X\omega )Y+(\nabla _Y\omega )X+\omega (X)\omega (Y)\) and \(n=\dim \ M\). If (1.5) holds then

$$\begin{aligned} \bar{\Lambda }=2\Lambda +\text {div}\omega -\frac{1}{2} (n-2)\parallel \omega ^{\sharp }\parallel ^2 \end{aligned}$$
(2.6)

Compact Einstein–Weyl manifolds with the Gauduchon metric are Gray manifolds. By a Riemannian manifold with a pair of Einstein–Weyl structures, we mean a manifold (Mg) for which there exists a 1-form \(\omega \) such that \((g,\omega ),(g,-\omega )\) determine two Einstein Weyl structures \(([g],D),([g],D^-)\) with \(Dg=\omega \otimes g, D^-g=-\omega \otimes g\). Note that every Einstein–Weyl structure (D, [g]) on a compact manifold M determines a Riemannian manifold (Mg) with two Einstein–Weyl structures, namely \(([g], D),([g], D^-)\) where g is the Gauduchon metric for the conformal manifold (M, [g]) and \(D^-g=-\omega _0\otimes g\) where \(\omega _0\) is the Lee form of g with respect to D (see [6]).

It is not difficult to prove that any Riemannian manifold admitting a pair of E–W structures is an \(\mathcal {AC}^{\perp }\) manifold with two eigenvalues \(\lambda ,\mu \) of the Ricci tensor of multiplicities \(n-1,1\) and such that \(\lambda \ge \mu \) (see [3]). In this case, \(\omega (X)=g(\xi ,X)\) where \(\xi \) is a conformal vector field. We will prove the converse: if (Mg) is a simply connected, real analytic \(\mathcal {AC}^{\perp }\) manifold with two eigenvalues \(\lambda ,\mu \) of multiplicities \(n-1,1\) and \(\lambda >\mu \) at least at one point, then (Mg) admits a pair of Einstein–Weyl structures \(( g, \omega )\) and \(( g,-\omega )\).

3 \({\mathcal {AC}}^{\perp }\) manifolds (Mg) whose Ricci tensor has two eigenvalues of multiplicity \(\dim M-1\) and 1

We start with:

Theorem 1

Let (Mg) be a real analytic complete, simply connected \({\mathcal {AC}}^{\perp }\) manifold whose Ricci tensor has two eigenvalues \(\lambda ,\mu \) of multiplicity \(n-1,1\) in the set \(V=\{x\in M:\lambda (x)\ne \mu (x)\}\). Then, the set \(N=\{x\in M:\lambda (x)=\mu (x)\}\) has an empty interior and there exists a conformal vector field \(\xi \) such that \(g(\xi ,\xi )=|\lambda -\mu |\) and \((Ric-\mu Id)\xi =0\). If \(\xi \) is a Killing vector field, then \(\lambda '\) is constant, while if \(\xi \) is conformal not Killing, then \(\mu '\) is constant. Here \(\lambda ',\mu '\) are eigenvalues of the Killing tensor \(S=Ric-\frac{2\text {Scal}}{n+2 }Id\) corresponding to \(\lambda ,\mu \) respectively.

Proof

Let \(V_+=\{x\in M:\lambda (x)>\mu (x)\}\), \(V_-=\{x\in M:\lambda (x)<\mu (x)\}\). Note that \(M=V_+\cup V_-\cup N\). Let \(\mathcal D\) be the distribution on V defined by \(D_{\mu }=\mathcal D=\ker (Ric-\mu I)\). Let us define on V the tensor \(m=g(p_{\mathcal D}.,p_{\mathcal D}.)\) where \(p_{\mathcal D}:\mathrm{TM}\rightarrow \mathcal D\) is the orthogonal projection onto \(\mathcal D\). Then, \(\rho =\lambda g+(\mu -\lambda )m\). Let us assume that \(V_+\ne \emptyset \). The distribution \(\mathcal D_{|V_+}\) is locally spanned by a unit vector field \(\xi _0\). We have \(Ric\xi _0=\mu \xi _0\). Hence on \(V_+\),

$$\begin{aligned} \rho =\lambda g- \omega \otimes \omega \end{aligned}$$

where \(\omega =g(\xi , .)\) and \(\xi =\sqrt{\lambda -\mu }\xi _0\).

Now we shall check when the tensor \(S=Ric-\frac{2}{n+2}\text {Scal} g\) is a Killing tensor. Note that for \(T(X,Y)=g(SX,Y)\) we have \(T=\lambda 'g-\omega \otimes \omega \) where \(\lambda '=\lambda -\frac{2}{n+2}\text {Scal}\). Hence

$$\begin{aligned} \nabla _XT(X,X)=X\lambda 'g(X,X)-2\nabla _X\omega (X)\omega (X)=X\lambda 'g(X,X)-h(X,X)\omega (X) \end{aligned}$$

where h is the symmetric (2, 0) tensor defined by \(h(X,Y)=\nabla _X\omega (Y)+\nabla _Y\omega (X)\). Consequently, T is a Killing tensor if and only if

$$\begin{aligned} X\lambda 'g(X,X)-h(X,X)\omega (X)=0. \end{aligned}$$
(3.1)

If \(\omega (X)=0\), then from (3.1) it follows that \(d\lambda '(X)=0\). Thus, \(d\lambda '=\alpha \omega \) for some \(\alpha \in C^{\infty }(V_+)\). Consequently

$$\begin{aligned} \omega (X)(\alpha g(X,X)-h(X,X))=0 \end{aligned}$$

for every \(X\in \mathrm{TM}_{|V_+}\).

Let \(F(X,Y)=\alpha g(X,Y)-h(X,Y)\). Then, F is a symmetric (2, 0) tensor and

$$\begin{aligned} \omega (X)F(X,X)=0 \end{aligned}$$
(3.2)

for every \(X\in \mathrm{TM}_{|V_+}\). Let us write \(X=tY+\xi \) where \(Y\in D_{\lambda }\), \(\omega (Y)=0\), \(t\in {\mathbb {R}}\). Then, \(\omega (X)=g(\xi ,\xi )\ne 0\), and from (2.2) we get \(F(tY+\xi ,tY+\xi )=0\). Consequently, \(t^2F(Y,Y)+2tF(Y,\xi )+F(\xi ,\xi )=0\). Since \(t\in {\mathbb {R}}\) is arbitrary, we obtain \(F(Y,Y)=F(\xi ,Y)=F(\xi ,\xi )=0\). Thus, \(F=0\). Hence, \(h(X,Y)=\alpha g(X,Y)\) and \(\xi \) is a local conformal field. Since \(h(X,Y)=\nabla _X\omega (Y)+\nabla _Y\omega (X)\), we get \(n\alpha =-2\delta \omega \). We also have

$$\begin{aligned} d\lambda '=\alpha \omega \end{aligned}$$
(3.3)

and \(0=d\alpha \wedge \omega +\alpha d\omega \). Hence in the set \(U_+=\{x\in V_+:\alpha (x)\ne 0\}\), we obtain \(d\omega =-d\ln |\alpha |\wedge \omega \). It follows that in \(U_+\) the distribution \(D_{\lambda }=\ker \omega \) is integrable. Thus for \(X,Y\in D_{\lambda }\), we get \(\nabla _X\omega (Y)=g(\nabla _X\xi ,Y)=-g(\xi ,\nabla _XY)=-g(\xi ,\nabla _YX)=g(\nabla _Y\xi ,X)=\nabla _Y\omega (X)\). Consequently, \(\nabla _X\omega (Y)=\frac{1}{2}\alpha g(X,Y)\) and \(p_{D_{\lambda }}(\nabla _X\xi )=\frac{1}{2}\alpha X\). Hence for a vector field \(\xi \) and \(X\in D_{\lambda }\), we get \(\nabla _X\xi =\frac{\alpha }{2}X+\phi (X)\xi \) for \(\phi =\frac{1}{2}d\ln g(\xi ,\xi )\). Now for \(X,Y\in D_{\lambda }\), we have

$$\begin{aligned} 2\nabla _X\nabla _Y\xi= & {} X\alpha Y+\alpha \nabla _XY+2X\phi (Y)\xi +\phi (Y)(\alpha X+2\phi (X)\xi ),\\ 2\nabla _Y\nabla _X\xi= & {} Y\alpha X+\alpha \nabla _YX+2Y\phi (X)\xi +\phi (X)(\alpha Y+2\phi (Y)\xi ),\\ 2\nabla _{[X,Y]}\xi= & {} \alpha [X,Y]+2\phi ([X,Y])\xi \end{aligned}$$

Thus, \(2R(X,Y)\xi =X\alpha Y-Y\alpha X-\phi (X)\alpha Y+\phi (Y)\alpha X \) for \(X,Y\in \ker \omega \). Let \(\beta =d\alpha -\alpha \phi \). Consequently, taking the trace over vectors perpendicular to \(\xi \) in the previous formula we get \(0=\rho (Y,\xi )=\frac{1}{2}(\beta (Y)-(n-1)\beta (Y))=-\frac{n-2}{2}\beta (Y)\). Hence, \(\beta (X)=0\) for \(X\in D_{\lambda }\) and

$$\begin{aligned} R(X,Y)\xi =0 \end{aligned}$$

for \(X,Y\in D_{\lambda }\). We have \( d\ln |\alpha |(X)=\phi (X)\) for \(X\in D_{\lambda }\). Hence, for \(X\in D_{\lambda }\), \(\nabla _X\xi =\frac{\alpha }{2}X+d\ln |\alpha |(X)\xi \) and \(\nabla _X\omega (\xi )=d\ln |\alpha |(X)\omega (\xi )\). Thus, \(\nabla _{\xi }\omega (X)=-d\ln |\alpha |(X)\omega (\xi )\) and \(d\omega (X,\xi )=2d\ln |\alpha |(X)\omega (\xi )\). On the other hand, \(d\omega =-d\ln |\alpha |\wedge \omega \) and \(d\omega (X,\xi )=-d\ln |\alpha |(X)\omega (\xi )\). Consequently \(d\ln |\alpha |(X)=0\) for \(X\in D_{\lambda }\). It follows that \(d\omega =0\) and \(\nabla _X\xi =\frac{1}{2}\alpha X\) for all \(X\in \mathrm{TM}\). The formula \(\nabla _X\xi =\frac{1}{2}\alpha X\) is clear if \(g(X,\xi )=0\). We will show that it also holds for \(X=\xi \). In fact, if X is any vector perpendicular to \(\xi \) then \(0=\alpha g(X,\xi )=g(\nabla _X\xi ,\xi )+g(\nabla _{\xi }\xi ,X)=g(\nabla _{\xi }\xi ,X)\). Hence \(\nabla _{\xi }\xi =\gamma \xi \) for some function \(\gamma \). On the other hand, we have \(\alpha g(\xi ,\xi )=g(\nabla _{\xi }\xi ,\xi )+g(\nabla _{\xi }\xi ,\xi )=2\gamma g(\xi ,\xi )\). It follows that \(\gamma =\frac{1}{2}\alpha \). Thus, \( R(X,Y)\xi =\frac{1}{2} d\alpha (X)Y-\frac{1}{2}d\alpha (Y)X\), and consequently, \( \rho (Y,\xi )=-\frac{n-1}{2}d\alpha (Y)\). Hence, \(d\alpha =-\frac{2\mu }{n-1}\omega \).

We also have \(d g(\xi ,\xi )(X)=2g(\nabla _X\xi ,\xi )=\alpha \omega (X)=d\lambda '(X)\). Note that \(d\mu '=0\) in \(M-N\) where \(\mu '\) is an eigenvalue of the Killing tensor associated with Ric corresponding to the one-dimensional distribution \(D_{\mu }\) [see (2.3)]. Note that \(g(\xi ,\xi )=\lambda '-\mu '\).

Analogously, we construct a conformal local vector field \(\xi \) in \(V_-\). In particular in \(V_-\), we have \(d\lambda '=-\alpha \omega \) where \(n\alpha =-2\delta \omega \) and \(\nabla _X\xi =\frac{\alpha }{2}X\). Note that if \(\xi ,\xi '\) are two local fields constructed as above, then \(\xi =\pm \xi '\) in the intersection of their domains. Now let us define \(M_-=\text {int}\{x\in M-N:\alpha (x)=0\}\) and \(M_+=\{x\in M-N:\alpha (x)\ne 0\}\).

We now show that if \(M_-\ne \emptyset \), then \(M_+=\emptyset \) hence \(\xi \) is a Killing vector field (see [6]). Let \(x_0\in M_-\) and \(x_1\in M_+\). If there exists a smooth curve c joining \(x_0\) and \(x_1\) and such that \(\text {im }c\cap N=\emptyset \), then by analytic continuation we see that \(\lambda '\) is constant along this curve, i.e., \(\lambda '\) is constant on a certain neighborhood \(U_x\) of any \(x\in \text {im} c\). Analogously, \(\mu '\) is constant, and hence, \(\alpha (x_1)=0\), a contradiction. A curve as above indeed exists: if U is a connected component of \(x_1\) in \(M_+\), then there exists a geodesic c with \(c(0)=x_0 \) and \(c(1)\in U\) which doesn’t meet N (see [6]).

If \(N=\emptyset \), then \(\xi \) exists and is smooth in a neighborhood of any point \(x\in M\). Note that in the case of a Killing vector field the constructed vector field is smooth also if \(N\ne \emptyset \), and then one of the sets \(V_+,V_-\) vanishes. Next we show that a similar situation holds for a conformal vector field.

Let us assume now that \(\text{ int }\{x:\alpha (x)=0\}=\emptyset \). In every component of the set \(M-N\) and in the interior of N, the function \(\mu '\) is constant. Since \(\mu '\) is continuous, it is constant on M. The set \(\{x:\alpha (x)\ne 0\}\) is dense in \(M-N\). In this set, \(\nabla _X\xi =\frac{\alpha }{2}X\). If \(x_0\in M- N\), then in a neighborhood of \(x_0\) the local field \(\xi \) is smooth and \(\nabla _X\xi =\frac{\alpha }{2}X\) also if \(\alpha (x_0)=0\) (we take the limit). Hence in \(M-N\), we have \(d\alpha =-\frac{2\mu }{n-1}\omega \). Note that the function \(\alpha ^2\) is independent of the choice of the local field \(\xi \), and \(\alpha ^2=\frac{||\nabla \lambda '||^2}{|\lambda -\mu |}\); hence,\(\alpha \) is continuous in \(M-N\). Note that \(\alpha d\alpha =-\frac{2\mu }{n-1}\alpha \omega =-\frac{2\mu }{n-1}d\lambda '\). On the other hand, since \(\mu '\) is constant, it follows that \(n\mu -2(n-1)\lambda =C_0=\mathrm{const}\) and \(d\lambda =\frac{n}{2(n-1)}d\mu \). Since \(\lambda -\mu =\lambda '-\mu '\) and \(\mu '\) is constant, we get \(d\lambda '=d\lambda -d\mu \) and \((n-1) \alpha d\alpha =-2\mu ( d\lambda - d\mu )=\frac{2-n}{n-1}\mu d\mu \). Hence in any component of \(M-N\), we have \(\pm (n-1)^2\alpha ^2+(n-2)\mu ^2=C_1=\mathrm{const}\) (the minus sign is in the components contained in \(V_-\)). Since \(\mu '\) is constant, it follows that \(\lambda ',\lambda ,\mu \) are real analytic functions on the whole of M (we have \((n-1)\lambda +\mu =\text {Scal}, n\mu -2(n-1)\lambda =C_0=\mathrm{const}\) and the scalar curvature \(\text {Scal}\) of (Mg) is real analytic).

Now we show that \(\nabla \lambda '=0\) in N. It is clear that \(\nabla \lambda '=0\) in the interior of N. Let \(x_0\in \partial N\). Then, there exists a sequence of points \(x_m\) from one component of \(M-N\) such that \(lim x_m=x_0\). This component, for example, is contained in \(V_+\). In that component, \((n-1)^2\alpha ^2+(n-2)\mu ^2=C_0=\mathrm{const}\). In particular, \(lim_{m\rightarrow \infty }\alpha ^2(x_m)=\frac{1}{(n-1)^2}(C_0-(n-2)\mu (x_0)^2)\) is finite which means that \(\nabla \lambda '=0\) at \(x_0\).

Now let assume that \(N\ne \emptyset \) and let \(x_0\in N\). Let c(t) be a geodesic with \(c(0)=x_0,\Vert \dot{c}(0)\Vert =1\). If \(S=Ric-\frac{2\text {Scal}}{n+2}g\), then \(T(.,.)=g(S.,.)\) is a Killing tensor and \(T-\lambda 'g=\pm \omega \otimes \omega \). Note that \(\lim _{t\rightarrow 0}T(\dot{c}(t),\dot{c}(t))=\mu '\), and since \(T(\dot{c}(t),\dot{c}(t))\) is constant, it follows that \(g(\xi (c(t)),\dot{c}(t))^2=\pm (\lambda '-\mu ')\). Hence, \(g(\frac{1}{\sqrt{|\lambda '-\mu '|}}\xi (c(t)),\dot{c}(t))=\pm 1\). Since both vectors have length 1, it follows that \(\xi (c(t))=\pm \sqrt{|\lambda '-\mu '|}\dot{c}(t)\). If we introduce geodesic polar normal coordinates centered at \(x_0\), then we can assume that \(\xi =\sqrt{|\lambda '-\mu '|}\frac{\partial }{\partial t}\) where t is a radial coordinate. In particular, \(\nabla \lambda '=\alpha \sqrt{|\lambda -\mu |}\frac{\partial }{\partial t}\) on \(M-N\) and \(\nabla \lambda '=0\) on N. Hence, \(d\lambda '(X)=0\) for any X tangent to the geodesic sphere and \(\lambda '\) is constant on geodesic spheres with center \(x_0\). Thus, \(\lambda '\) is a function of the radial coordinate t only.

If there existed a hypersurface \(\Sigma \subset B(x_0,r)\) with \(x_0\in \Sigma \) such that the intersection of \(\Sigma \) with every geodesic sphere \(S(x_0,s),s<r,\) is not empty, and on which \(\lambda '-\mu '=0\), then we would have \(\lambda '=\mu '\) on the whole of B. In fact, if for \(x_1\ne x_0\) we have \(\lambda '(x_1)=\mu '\), then \(\lambda '(x)=\mu '\) on the sphere \(S(x_0,s)\) containing the point \(x_1\). Hence, the sphere \(S(x_0,s)\) would be a hypersurface \(\Sigma \) in a ball \(B(x_1,t)\), where t is the injectivity radius at the point \(x_1\), which has the above property. It follows that the set N has a non-empty interior U. Since \(\lambda ,\mu \) are real analytic and \(\lambda =\mu \) on U, it follows that \(\lambda =\mu \) on M, a contradiction. It follows that in a geodesic ball \(B(x_0,r)\) where r is the injectivity radius of M at \(x_0\), the set \(\{x\in B(x_0,r): \lambda '(x)=\mu '\}\) consists of only the point \(x_0\). Hence, the set N consists of isolated points. This implies that \(M-N\) has only one component. It follows that \(\lambda '-\mu '\) has a constant sign and one of the sets \(V_+,V_-\) is empty.

For \(x_0\in N\), let us define a local field \(\xi \) on the geodesic ball \(B(x_0,r)\) where r is the injectivity radius at \(x_0\) by \(\xi =\sqrt{|\lambda '-\mu '|}\frac{\partial }{\partial t}\) where t is the radial coordinate. Then since M is simply connected, there exists a continuous vector field \(\xi \) on M such that \(\rho =\lambda g -\omega \otimes \omega \) if \(V_-=\emptyset \) or \(\rho =\lambda g +\omega \otimes \omega \) if \(V_+=\emptyset \) and \(\omega =g(\xi ,.)\) (we take the germs of local fields \(\xi \) as points of a two-sheeted covering of M). The field \(\xi \) is smooth on \(M-N\) and defines a function \(\alpha \) on \(M-N\). The function \(\alpha \) is smooth on \(M-N\). We shall show that \(\xi \) is in fact smooth.

First we show that the function \(\alpha \) has a continuous extension to the points \(x_0\in N\). We assume, for example, that \(V_-=\emptyset \). Then, the function \(\alpha ^2\) is real analytic and \((n-1)^2\alpha ^2+(n-2)\mu ^2=C_1=\mathrm{const}\) in \(M-N\); hence, \(\alpha ^2\) has a real analytic extension on the whole of M. If \(\alpha (x_0)^2>0\), then \(\alpha \) has a constant sign in a certain neighborhood of \(x_0\) and \(\alpha =\pm \frac{1}{n-1}\sqrt{C_1-(n-2)\mu ^2}\) is a smooth extension of \(\alpha \) on the neighborhood of \(x_0\). If an extension of \(\alpha ^2\) satisfies \(\alpha ^2(x_0)=0\), then certainly \(\mathrm{lim}_{x\rightarrow x_0}\alpha (x)=0\) and taking \(\alpha (x_0)=0\) we get a function continuous at \(x_0\).

Let c(t) be a unit speed geodesic with \(c(0)=x_0\in N\). We take \(\xi \) which in geodesic coordinates around \(x_0\) is \(\xi =\sqrt{|\lambda '-\mu '|}\frac{\partial }{\partial t}\). Note that \(\sqrt{|\lambda '-\mu '|}=\pm g(\dot{c},\xi )\). The function \(\phi (t)=\sqrt{|\lambda '-\mu '|}\circ c(t) \) if \(t\ge 0\) and \(\phi (t)=-\sqrt{|\lambda '-\mu '|}\circ c(t)\) if \(t<0\) is differentiable at 0 and smooth in a neighborhood of 0. Indeed note that \(\phi (t)=g(\dot{c}(t),\xi (c(t)))\) and \(\phi '(t)=g(\dot{c},\nabla _{\dot{c}}\xi )=\frac{1}{2}\alpha (c(t)\) for \(t\ne 0\). Since \(\alpha \) is continuous, it follows that \(\phi \) is differentiable at 0 and has continuous derivative. The function \(\alpha (t)=\alpha (c(t))\) is differentiable at 0 and \(\alpha '(0)=0\). In fact, \(d\alpha =-\frac{2}{n-1}\mu \omega \) and \(d\alpha (\dot{c})=-\frac{2}{n-1}\mu \phi (t)\). Hence, \(\alpha '(t)=-\frac{2}{n-1}\mu \circ c(t)\phi (t)\). Note that \(\frac{1}{2}\alpha (t)=\phi '(t)\). The last equation holds also for \(t=0\) since \(\alpha \) is continuous. Hence, we get an equation \(\phi ''(t)=-\frac{\mu (c(t))}{n-1}\phi (t)\) with initial conditions \(\phi (0)=0,\phi '(0)=\frac{1}{2}\alpha (x_0)\) where the function \(\mu (c(t))\) is real analytic. If \(\alpha (x_0)=0\), then we get \(\phi (t)=0\), a contradiction. Hence, \(\alpha (x_0)\ne 0\) and \(\xi =\frac{1}{\alpha }\nabla \lambda '\) is a smooth vector field in a neighborhood of \(x_0\). Hence, \(\xi \) is a smooth vector field.\(\diamondsuit \)

Remark

\(\mathcal {AC}^{\perp }\) manifolds for which \(\xi \) is a Killing vector field are described in [6]. In the rest of the paper, we shall assume that \(\dim M=n+1\).

Theorem 2

Let us assume that (Mg) is a real analytic, simply connected \(\mathcal {AC}^{\perp }\) manifold, whose Ricci tensor has two eigenvalues \(\lambda ,\mu \) of multiplicities n, 1 respectively, and such that \(\mu -\frac{2\text {Scal}}{n+3}\) is constant and \(\lambda -\frac{2\text {Scal}}{n+3}\) is non-constant. Then, \(\lambda -\mu \ge 0\) on the whole of M or \(\lambda -\mu \le 0\) on the whole of M. If \(|\lambda -\mu |>0\) on M, then M is a warped product \(M={\mathbb {R}}\times _{f^2} M_*\) where \(M_*\) is an Einstein manifold. If \(N\ne \emptyset \), then \(N=\{x_0\}\) or \(N=\{x_0,x_1\}\) and \(M={\mathbb {R}}_+\times _{f^2} M_*\) in the first case, while \(M=(0,\epsilon )\times _{f^2} M_*\) in the second case, where \(M_*=(S^n,g_\mathrm{can})\) is a sphere with the standard metric of constant sectional curvature. In all cases, the function f satisfies an equation

$$\begin{aligned} -(n-1)\left( \frac{f''}{f}-\frac{(f')^2}{f^2}\right) =\tau f^{-2}-Cf^2 \end{aligned}$$
(3.4)

where \(C\in {\mathbb {R}}-\{0\}\) and \(Ric_{M_*}=\tau g_{M_*}\). If \(N\ne \emptyset \) and \(M_*\) is a sphere with sectional curvature 1, then the function f satisfies the initial conditions \(f(0)=0,f'(0)=1\) in the first case and in the second case \(f(0)=0,f'(0)=1, f(\epsilon )=0,f'(\epsilon )=-1\).

Proof

If \(N=\emptyset \), the proof is given in [7] (see also [5]). Let us assume first that \(\sharp N\ge 2\). Let \(x_0\in N\) and \(r=d(x_0,N-\{x_0\})=d(x_0,x_1)\) where \(x_1\in N\). We show that the map \(\mathrm{exp}_{x_0}:B(0,r)\rightarrow M\), where \(B(0,r)=\{X\in T_{x_0}M:||X||<r\}\), is a diffeomorphism onto its image.

If \(\mathrm{exp}_{x_0}X=\mathrm{exp}_{x_0}Y\), then we consider two cases \(X=0,Y\ne 0\) and \(X,Y\ne 0\). In the first case, the geodesic \(c(t)=\mathrm{exp}_{x_0}t\frac{Y}{||Y||}\) is a loop of length \(s=||Y||<r\). In the second case, let \(c(t)=\mathrm{exp}_{x_0}t\frac{X}{||X||}, c_1(t)=\mathrm{exp}_{x_0}t\frac{Y}{||Y||}\). Let \(t_0=||X||,t_1=||Y||\). Then, \(c(t_0)=c_1(t_1)\) and \(\dot{c}(t_0)=\dot{c}_1(t_1)\) since \(\xi (c(t_0))=\sqrt{|\lambda -\mu |}\dot{c}(t_0)=\sqrt{|\lambda -\mu |}\dot{c}_1(t_0)\). It follows that \(c(t_0+s)=c_1(t_1+s)\). Let us assume that \(t_1>t_0\). Then, \(c(0)=x_0=c_1(t_1-t_0)\). Hence, \(c_1\) is a geodesic loop at \(x_0\) of length \(t_1-t_0<r\). This contradicts \(\xi (c_1(t))=\sqrt{|\lambda -\mu |}\dot{c}_1(t)\) which is valid up to the first value of \(t>0\) for which \(\lambda (c_1(t))-\mu (c_1(t))=0\), if we start from \(t=0\) (note that \(|\lambda (c_1(t))-\mu (c_1(t))|=0\) only if \(c_1(t)=x_0\) for \(t\in (0,r)\)). Thus, \(t_0=t_1\) and \(c=c_1\), which means that \(X=Y\). Similarly, we obtain \(Y=0\) if \(X=0\). The mapping \(\mathrm{exp}_{x_0}:B(0,r)\rightarrow M\) is a diffeomorphism. In fact, otherwise we would have a conjugate point on a geodesic \(c(t)=\mathrm{exp} tX\). Hence, we would have two different geodesics starting at \(x_0\) and joining \(x_0\) to \(x_1=\mathrm{exp} sX\), where \(s>1, s||X||<r\). It leads to a contradiction as above.

Note that \(\lambda -\mu \) depends only on the distance from \(x_0\). Hence, \(\mathrm{exp}_{x_0}rS\subset N\) where S is a unit sphere in \(T_{x_0}M\). Since \(\mathrm{exp}_{x_0}\) is a continuous function, \(\mathrm{exp}_{x_0}rS\) is connected and N consists of isolated points, it follows that \(\mathrm{exp}_{x_0}rS=\{x_1\}\). Hence, \(N=\{x_0,x_1\}\) and every unit geodesic c starting from \(x_0\) satisfies \(c(r)=x_1\). It follows that \(\mathrm{exp}_{x_0}:B(0,r)\rightarrow M-\{x_1\}\) is a diffeomorphism. In the same way, \(\mathrm{exp}_{x_1}:B(0,r)\rightarrow M-\{x_0\}\) is a diffeomorphism onto \(M-\{x_0\}\).

Note that by the Gauss lemma, the leaves of \(D_{\lambda }\) are compact spheres \(S^n\) via the diffeomorphism \(\mathrm{exp}_{x_0}:S(0,t)\rightarrow M\) where \(S(0,t)=\{X\in T_{x_0}M:\Vert X\Vert =t\) where \(t\in (0,r)\). Hence, \(M=[0,r]\times _{f^2}S^n\). From [B, Lemma 9.114, page 269], it follows that on \(S^n\) there is a metric of constant sectional curvature \(\lambda ^2g_\mathrm{can}\) and \(f(0)=0=f(r),f'(0)=\frac{1}{\lambda }=-f'(r)\). Replacing f by a function \(\lambda f\), we see that we can assume that \(f(0)=0,f'(0)=1,f(r)=0,f'(r)=-1\).

Now consider the case \(N=\{x_0\}\). Then, as above one can prove that \(\mathrm{exp}_{x_0}:T_{x_0}M\rightarrow M\) is a diffeomorphism. Hence in this case, \(M={\mathbb {R}}_+\times _{f^2} S^n\) where \(S^n\) has the a canonical metric. The function f satisfies the initial conditions \(f(0)=0,f'(0)=1\).\(\diamondsuit \)

Now we consider the equation (see [7, p.27])

$$\begin{aligned} -(n-1)(\frac{f''}{f}-\frac{(f')^2}{f^2})=\tau f^{-2}-Cf^2 \end{aligned}$$

with initial conditions \(f(0)=0,f'(0)=1\) ; hence, we consider the warped product \(M=(0,\epsilon )\times _{f}S^{n}\) or \(M={\mathbb {R}}_+\times _{f}S^{n}\) and \(\tau =n-1\).

If we write \(g=\ln f\), then \((g)''=-\frac{\tau }{n-1}e^{-2g}+\frac{C}{n-1}e^{2g}\). We get \(\frac{1}{2}((g')^2)'=(\frac{\tau }{2(n-1)}e^{-2g}+\frac{C}{2(n-1)}e^{2g})'\) and \((g')^2=\frac{\tau }{(n-1)}e^{-2g}+\frac{C}{(n-1)}e^{2g}+A\) for some constant \(A\in {\mathbb {R}}\). Consequently, if we assume that \(\tau =n-1\), we get an equation for f

$$\begin{aligned} (f')^2=1+Af^2+\frac{C}{n-1}f^4. \end{aligned}$$

Using homothety, we can assume that \(C=\pm (n-1)\). Thus, we get \((f')^2=1+Af^2+f^4\) or \((f')^2=1+Af^2-f^4\).

First let us assume that the quadratic polynomial \(1+At+\epsilon t^2\) where \(\epsilon \in \{-1,1\}\) has real roots. This happens for \(\epsilon =-1\), and for \(|A|\ge 2\) if \(\epsilon =1\). Hence, if \(\epsilon =1\) and \(A<-2\), we get \((f')^2=(a^2-f^2)(\frac{1}{a^2}-f^2)\) for some \(a\in (0,1)\). Let f be the solution of this equation with initial conditions \(f(0)=0,f'(0)=1\).

We shall show that f increases until it attains its maximum \(a=f(t_0)\) and then decreases, and \(f(2t_0)=0,f'(2t_0)=-1\). In fact, f satisfies an equation \(f'=\sqrt{\left( a^2-f^2\right) \left( \frac{1}{a^2}-f^2\right) }\) if \(f<a\). It follows that \(\arcsin (\frac{f}{a})'=\sqrt{ \frac{1}{a^2}-f^2}\ge \sqrt{\frac{1}{a^2}-a^2}\) if \(f<a\). Consequently, \(\arcsin (\frac{f}{a})\ge t\sqrt{\frac{1}{a^2}-a^2}\) and f increases until it attains the value \(f(t_0)=a\). The solution exists on an interval \([0,t_0]\). Now we define \(f(t_0+s)=f(t_0-s)\) for \(s\in (0,t_0]\). Then, f is defined on an interval \([0,2t_0]\) and is a smooth function on an open neighborhood of this interval.

Similarly if \(\epsilon =-1\), then we get an equation \(f'=\sqrt{\left( a^2-f^2\right) \left( \frac{1}{a^2}+f^2\right) }\) where \(a>0\). Thus, \(\arcsin (\frac{f}{a})\ge t\sqrt{\frac{1}{a^2}}\), and again there exists \(t_0>0\) such that \(f(t_0)=a\). We extend the solution on the interval \([0,2t_0]\) and obtain a solution satisfying \(f(2t_0)=0,f'(2t_0)=-1\).

We have to show that f is an odd function at \(t=0\) and \(t=2t_0\). Note that f satisfies the equation \(f''=Af+2\epsilon f^3\). Since f is real analytic, it suffices to show that \(f^{(2k)}(0)=0\) for \(k\in {\mathbb {N}}\). We use induction. We have \(f(0)=0,f''(0)=0\). Let us assume that \(f^{(2k)}(0)=0\) for \(k\le n\) where \(n\ge 1\). We will show that \(f^{(2n+2)}(0)=0\). Note that

$$\begin{aligned} f^{(2n+2)}(0)=Af^{(2n)}(0)+2\epsilon \sum _{r=0}^{2n}\sum _{s=0}^rC^{2n}_rC^r_sf^{(s)}f^{(r-s)}f^{(2n-r)}, \end{aligned}$$
(3.5)

where \(C^n_k=\frac{n!}{(n-k)!k!}\). If s and \(r-s\) are odd numbers, then \(r=r-s+s\) is an even number and \(f^{(2n-r)}(0)=0\). Hence in all cases, the left-hand side of (3.5) equals 0. It follows that in both cases considered above we get a compact \(\mathcal {AC}^{\perp }\) manifold diffeomorphic to \(S^{n+1}\). If \(\epsilon =1\) and \(A=-2\), then one can easily check that \(f(t)=\tanh t\). In that case, we obtain a complete non-compact \(\mathcal {AC}^{\perp }\) manifold diffeomorphic to \({\mathbb {R}}^{n+1}\) (see [7]). If \(A>-2\), then we can obtain a solution which tends to infinity which may not be defined on the whole of \({\mathbb {R}}_+\). For example, for \(A=2\) we get \(f(t)=\tan t\). Summarizing we have

Theorem 3

On the sphere \(S^{n+1}\), there exists a one-parameter family of \(\mathcal {AC}^{\perp }\) metrics \(g_A\) such that \(g_A=dt^2+f^2g_\mathrm{can}\) and \((S^{n+1},g_A)=[0,2t_0]\times _{f^2}S^{n}\) where f satisfies the equation

$$\begin{aligned} (f')^2=1+Af^2+f^4 \end{aligned}$$
(3.6)

with \(A\in (-\infty ,-2)\), or

$$\begin{aligned} (f')^2=1+Af^2-f^4 \end{aligned}$$
(3.7)

for \(A\in {\mathbb {R}}\). In the case of Eq. (3.6), the corresponding \(\mathcal {AC}^{\perp }\) manifold admits a pair of Einstein–Weyl structures both conformally Einstein and conformal to the standard sphere with constant sectional curvature.

Proof

Note that in the case (3.6) the corresponding \(\mathcal {AC}^{\perp }\) manifold has a Ricci tensor with eigenvalues \(\lambda ,\mu \) such that \(\lambda -\mu =Cf^2\ge 0\). Hence from (2.5), it determines a pair of Einstein–Weyl structures \((g,\omega '), (g,-\omega ')\) where \(\omega '=\frac{2}{\sqrt{n-1}}\omega \) and \(\omega (X)=g(\xi ,X)\) was defined in the first part of the paper. Since \(d\omega =0\), both Einstein–Weyl structures are conformally Einstein. Let \(\omega =d\phi \), then the metrics \(g_1=\exp (\phi )g, g_2=\exp (-\phi )g\) are Einstein metrics on \(S^{n+1}\). They are conformally equivalent to each other \(g_1=\exp (2\phi )g_2\). Thus, it follows from [8, 9] that \(g_i\) are standard metrics of constant sectional curvature on \(S^{n+1}\). \(\square \)

Remark

In the case of Eq. (3.6) with \(A=-2\), we obtain a solution \(f(t)=\tanh t\). The corresponding \(\mathcal {AC}^{\perp }\) manifold \(M={\mathbb {R}}_+\times _{f^2} S^n\) is complete (see [7]) and diffeomorphic to \({\mathbb {R}}^{n+1}\). It also admits a pair of Einstein–Weyl structures which are conformally Einstein. At least one of them has to be non-complete since any complete Einstein manifold which is conformal to a complete Einstein manifold has to be a sphere with standard metric of constant sectional curvature (see [8]).

Theorem 4

On a manifold \(M={\mathbb {R}}\times _{f^2} M_*\) where \((M_*,g_{M_*})\) is an Einstein manifold such that \(Ric_{M_*}=\tau g_{M_*}\) with \(\tau <0\) there exists a one-parameter family of complete \(\mathcal {AC}^{\perp }\) metrics \(g_A\) such that \(g_A=dt^2+f^2g_{M_*}\), where f satisfies the equation

$$\begin{aligned} (f')^2=\frac{\tau }{n-1}+Af^2-f^4 \end{aligned}$$
(3.8)

with \(A\in {\mathbb {R}}, A^2>-4\tau ,\tau <0\). The eigenvalues \(\lambda ,\mu \) of the Ricci tensor of \((M,g_A)\) satisfy \(\lambda \le \mu \); hence, these manifolds do not admit a pair of Einstein–Weyl structures.

Proof

Equation (3.8) is equivalent to the equation

$$\begin{aligned} f''=Af-2f^3 \end{aligned}$$
(3.9)

with appropriate initial conditions. Hence, we can apply [B,Lemma 16.37 p.445 ]. Note that \((f')^2=-(f^2-a^2)(f^2+\frac{\tau '}{a^2})\) if \(A^2>-4\tau ',A>0,\tau '<0\) where \(\tau '=\frac{\tau }{n-1}, a>0\). We can assume that \(a<\frac{\sqrt{-\tau '}}{a}\). Hence for \(P(f)=-(f^2-a^2)(f^2+\frac{\tau '}{a^2})\), we have \(P(a)=P(\frac{\sqrt{-\tau '}}{a})=0\) and \(P'(a)P'(\frac{\sqrt{-\tau '}}{a})\ne 0\). Consequently, the equation \((f')^2=P(f)\) admits a periodic solution \(f:{\mathbb {R}}\rightarrow [a,\frac{\sqrt{-\tau '}}{a}]\). In that way, we get complete manifolds \((M,g_A)\). Note that these manifolds admit compact quotients.\(\diamondsuit \)

Remark

Note that an equation of type (3.9) ( for a function \(\frac{1}{f}\)) was studied in [1] and complete and compact manifolds constructed in Theorem 4 are known (as manifolds with \(Dr\in C(Q)\) in [1]). However, the \(\mathcal {AC}^{\perp }\) metrics on spheres \(S^{n+1}\) were not found by A. Besse.