2.1 The Chern connection

Let $(M,\,\mathbb {J},\, {\mathbb g})$ be a connected, complete, Hermitian manifold of real dimension $\dim _{\mathbb {R}}M=2\,m$. Let us denote by $\omega := {\mathbb g}(\mathbb {J} \,\cdot \,,\, \cdot \,)$ its fundamental $2$-form, by $D$ its Levi-Civita connection and by $\nabla$ its Chern connection, which is defined by

(2.1)\begin{equation} {\mathbb g}(\nabla_AB,C) := {\mathbb g}(D_AB,C)-\frac 12\operatorname{d\!}\omega(\mathbb{J} A,B,C) \end{equation}

for any $A,\,B,\,C \in \Gamma (TM)$. Since, by the Koszul formula,

(2.2)\begin{align} 2 {\mathbb g}(D_AB,C) & = \mathcal{L}_A( {\mathbb g}(B,C))+\mathcal{L}_B( {\mathbb g}(A,C))-\mathcal{L}_C( {\mathbb g}(A,B)) \nonumber\\ & \quad+ {\mathbb g}([A,B],C)- {\mathbb g}([A,C],B)- {\mathbb g}([B,C],A) \end{align}

and

(2.3)\begin{align} \operatorname{d\!}\omega(A,B,C) & = \mathcal{L}_A( {\mathbb g}(\mathbb{J} B,C))-\mathcal{L}_B( {\mathbb g}(\mathbb{J} A,C))+\mathcal{L}_C( {\mathbb g}(\mathbb{J} A,B)) \nonumber\\ & \quad+ {\mathbb g}([A,B],\mathbb{J} C)+ {\mathbb g}([B,C],\mathbb{J} A)+ {\mathbb g}([C,A],\mathbb{J} B), \end{align}

it follows that

(2.4)\begin{align} 2 {\mathbb g}(\nabla_AB,C)& = \mathcal{L}_A( {\mathbb g}(B,C)) - \mathcal{L}_{\mathbb{J} A}( {\mathbb g}(\mathbb{J} B,C)) + {\mathbb g}([A,B],C) \nonumber\\ & \quad- {\mathbb g}([\mathbb{J} A,B],\mathbb{J} C) - {\mathbb g}([A,C],B) + {\mathbb g}([\mathbb{J} A,C],\mathbb{J} B). \end{align}

Moreover, it is well-known that the Chern connection is characterized by the following properties

(2.5)\begin{equation} \nabla {\mathbb g} = \nabla \mathbb{J} =0, \quad \mathbb{J} \tau(A,B)=\tau(\mathbb{J} A,B)=\tau(A,\mathbb{J} B), \end{equation}

where $\tau (A,\,B) := \nabla _AB -\nabla _BA - [A,\,B]$ is the torsion tensor of $\nabla$.

For later use, we observe the following straightforward

Lemma 2.1 For any $A \in \Gamma (TM)$ it holds that

(2.6)\begin{align} \mathcal{L}_A\mathbb{J} & =-[\nabla A, \mathbb{J}], \end{align}

(2.7)\begin{align} \mathcal{L}_{\mathbb{J} A}\mathbb{J} & = \mathbb{J} \circ \mathcal{L}_A\mathbb{J}. \end{align}

Proof. If $E \in \Gamma (\operatorname {End}(TM))$, then

\begin{align*} (\mathcal{L}_AE)B & = \mathcal{L}_A(EB)-E(\mathcal{L}_AB) \\ & = \nabla_A(EB) -\nabla_{EB}A -\tau(A,EB) -E(\nabla_AB) +E(\nabla_BA) +E(\tau(A,B)) \\ & =-[\nabla A, E]B +(\nabla_AE)B +E(\tau(A,B)) -\tau(A,EB) \,. \end{align*}

Then, equation (2.6) follows by setting $E=\mathbb {J}$ and using equation (2.5). On the other hand, from equations (2.5) and (2.6) we get

\[ \mathcal{L}_{\mathbb{J} A}\mathbb{J} =-[\nabla \mathbb{J} A, \mathbb{J}] =-[\mathbb{J}\nabla A, \mathbb{J}] =-\mathbb{J} [\nabla A, \mathbb{J}] = \mathbb{J} \circ \mathcal{L}_A\mathbb{J} \]

and so the thesis follows.

A real vector field $A \in \Gamma (TM)$ is holomorphic if $\mathcal {L}_A\mathbb {J}=0$. Hence, we get

Finally, we set $\operatorname {d\!}^{\,c} := \mathbb {J}^{-1} \! \circ \operatorname {d\!}\, \circ \mathbb {J}$, so that

\[ \operatorname{d\!} = \partial+\bar{\partial}, \quad \operatorname{d\!}^{\,c} =-\sqrt{-1}(\partial-\bar{\partial}), \ \operatorname{d\!}\operatorname{d\!}^{\,c}=2\sqrt{-1}\partial\bar{\partial} \]

and, for any smooth function $f: M \to \mathbb {R}$, we denote by $\Delta _{ {\mathbb g}}^{\rm Ch}f := {\mathbb g}(\operatorname {d\!}\operatorname {d\!}^{\,c}\!f,\,\omega )$ the Chern–Laplacian of $f$.

2.2 Second-Chern–Einstein metrics

Let $(M,\,\mathbb {J},\, {\mathbb g})$ be a connected, complete, Hermitian manifold of real dimension $\dim _{\mathbb {R}}M=2\,m$. We recall that, by the lack of symmetries of the Chern-curvature

\[ R^{\rm Ch}( {\mathbb g})(A,B) := \nabla_{[A,B]} - [\nabla_A,\nabla_B], \]

we have (at least) two ways to trace the Ricci tensor. We call first Chern–Ricci curvature the tensor defined by

\[ \operatorname{Ric}^{\rm Ch[1]}( {\mathbb g})(A,B)_x := \sum_{e_\alpha} {\mathbb g}_x\Big(R^{\rm Ch}( {\mathbb g})_x(A_x,\mathbb{J} B_x)\,e_{\alpha},\,\mathbb{J} e_\alpha\Big), \]

where $(e_\alpha,\, \mathbb {J} e_\alpha )$ is a $(\mathbb {J},\, {\mathbb g})$-unitary frame for the tangent space at $x$. Similarly, we call second Chern–Ricci curvature the tensor defined by

\[ \operatorname{Ric}^{\rm Ch[2]}( {\mathbb g})(A,B)_x := \sum_{e_\alpha} {\mathbb g}_x(R^{\rm Ch}( {\mathbb g})_x(e_\alpha,\mathbb{J} e_\alpha)\,A_x,\,\mathbb{J} B_x). \]

Finally, the Chern-scalar curvature is the function given by

\[ \operatorname{scal}^{\rm Ch}( {\mathbb g})(x) := 2 \sum_{e_\alpha}\operatorname{Ric}^{\rm Ch[i]}( {\mathbb g})_x(e_\alpha,e_\alpha), \quad i=1,2. \]

We remark that, according to our notation, when $ {\mathbb g}$ is Kähler it holds that $\operatorname {Ric}^{\rm Ch[1]}( {\mathbb g})=\operatorname {Ric}^{\rm Ch[2]}( {\mathbb g})=\operatorname {Ric}( {\mathbb g})$ and $\operatorname {scal}^{\rm Ch}( {\mathbb g})=\operatorname {scal}( {\mathbb g})$, where $\operatorname {Ric}( {\mathbb g})$ and $\operatorname {scal}( {\mathbb g})$ denote the Riemannian Ricci tensor and the Riemannian scalar curvature of $ {\mathbb g}$, respectively.

This yields to the following

Definition 2.3 Let $i \in \{1,\,2\}$. The metric $ {\mathbb g}$ is said to be weakly (respectively, strongly) $i{\rm th}$-Chern–Einstein if there exists $\lambda \in \mathcal {C}^{\infty }(M,\, \mathbb {R})$ (respectively, $\lambda \in \mathbb {R}$) such that

\[ \operatorname{Ric}^{{\rm Ch}[i]}( {\mathbb g})=\frac{\lambda}{2m} {\mathbb g}. \]

We stress that the first-Chern–Einstein problem is basically understood for compact complex manifolds $X=(M,\,\mathbb {J})$, see [Reference Angella, Calamai and Spotti7, Reference Tosatti52]. Indeed, strongly first-Chern–Einstein metrics with non-zero Chern-scalar curvature are Kähler–Einstein. Moreover, by conformal methods, if a Hermitian metric $ {\mathbb g}$ is weakly first-Chern–Einstein with non-identically zero Chern-scalar curvature, then it is conformal to a Kähler metric in the class $\pm c_1(X)$ (see [Reference Angella, Calamai and Spotti7, theorem A]). Finally, compact complex manifolds with first-Bott–Chern class $c_1^{BC}(X)=0$ are the so-called non-Kähler Calabi–Yau manifolds [Reference Tosatti52] and always admit Chern–Ricci flat metrics [Reference Tosatti52, theorem 1.2], see also [Reference Tosatti and Weinkove53, corollary 2].

On the other hand, the second-Chern–Einstein problem seems to be geometrically more appealing, see e.g. [Reference Angella, Calamai and Spotti7, Reference Gauduchon and Ivanov24, Reference Podestá47, Reference Streets and Tian51]. Note that a Hermitian metric $ {\mathbb g}$ on $X=(M,\,\mathbb {J})$ is second-Chern–Einstein according to definition 2.3 if and only if the induced Hermitian metric $h(V,\,W) := {\mathbb g}(V,\,\overline {W})$ on the holomorphic tangent bundle $T^{1,0}X$ is Hermite–Einstein by taking trace with itself [Reference Gauduchon22].

Remark 2.4 We observe that the second-Chern–Einstein condition is satisfied by a Hermitian metric $ {\mathbb g}$ if and only if it is satisfied by all the metrics in its conformal class (see [Reference Gauduchon22]), since for any smooth function $f: M \to \mathbb {R}$ it holds that

\[ \operatorname{Ric}^{{\rm Ch}[2]}({\rm e}^{f} {\mathbb g}) = \operatorname{Ric}^{{\rm Ch}[2]}( {\mathbb g})-(\Delta_{ {\mathbb g}}^{\rm Ch}f) {\mathbb g}. \]

We remark that this is strongly different from the Riemannian analogue, i.e. the Einstein condition. On the other hand, we stress that a Kähler metric is second-Chern–Einstein if and only if it is Einstein.

Note that, up to our knowledge, a non-Kähler example of second-Chern–Einstein metric on a compact complex manifold of complex dimension $m > 2$ with negative scalar curvature is still missing.

2.3 Cohomogeneity one group actions on Hermitian manifolds

Let us consider a compact, connected real Lie group $\mathsf {G}$ which acts effectively by holomorphic isometries on $(M,\,\mathbb {J},\, {\mathbb g})$ with cohomogeneity one [Reference Bérard-Bergery9, Reference Besse10, Reference Mostert41, Reference Podestà and Spiro48]. Then, the orbit space $\Omega := \mathsf {G} \backslash M$ is homeomorphic to one of the following:

\[ \text{(i)}\ \Omega \simeq \mathbb{R}, \quad \text{(ii)}\ \Omega \simeq [0,+\infty), \text{(iii)}\ \Omega \simeq S^{1}, \text{(iv)}\ \Omega \simeq [0,\pi]. \]

Up to homothety, we can choose a unit speed geodesic $\gamma :\overline {I} \to M$ which intersects orthogonally any $\mathsf {G}$-orbit [Reference Bérard-Bergery9, § 2.7], where the interval $I \subset \mathbb {R}$ is defined as

\[ \text{(i) } I := \mathbb{R}, \quad \text{(ii) } I := (0,+\infty), \ \text{(iii) } I := (-\pi,\pi), \ \text{(iv) } I := (0,\pi). \]

Then, for $r \in I$, the orbit $\mathcal {S}_r := \mathsf {G} \cdot \gamma (r)$ is principal and can be identified with a fixed homogeneous space by means of the $1$-parameter family of $\mathsf {G}$-equivariant diffeomorphisms

\[ \phi_r: \mathsf{G}/\mathsf{H} \to \mathcal{S}_r , \quad \phi_r(a\mathsf{H}) := a \cdot \gamma(r), \]

where $\mathsf {H} \subset \mathsf {G}$ is a closed subgroup. For $r \in \partial I$, the following cases occur:

1. (i) $\partial I = \emptyset$ and all the orbits are principal;

2. (ii) $\partial I = \{0\}$, the orbit $\mathcal {S}_0 := \mathsf {G} \cdot \gamma (0)$ is non-principal and $\mathsf {G}$-equivariantly diffeomorphic to a homogeneous space $\mathsf {G}/\mathsf {L}$, where $\mathsf {H} \subset \mathsf {L} \subset \mathsf {G}$ is an intermediate subgroup and $\mathsf {L} / \mathsf {H}$ is a sphere, see e.g. [Reference Bérard-Bergery9, § 2.12];

3. (iii) $\partial I = \{\pm \pi \}$, the orbits $\mathcal {S}_{\pm \pi } := \mathsf {G} \cdot \gamma (\pm \pi )$ are principal but in general $\gamma (-\pi ) \neq \gamma (\pi )$, see e.g. [Reference Bérard-Bergery9, § 2.10];

4. (iv) $\partial I = \{0,\,\pi \}$, the orbits $\mathcal {S}_0 := \mathsf {G} \cdot \gamma (0)$, $\mathcal {S}_{\pi } := \mathsf {G} \cdot \gamma (\pi )$ are non-principal and $\mathsf {G}$-equivariantly diffeomorphic to two homogeneous space $\mathsf {G}/\mathsf {L}_{\pm }$, with $\mathsf {H} \subset \mathsf {L}_- \cap \mathsf {L}_+$ and $\mathsf {L}_{\pm } / \mathsf {H}$ are spheres, see e.g. [Reference Bérard-Bergery9, § 2.13].

The subset $M^{\rm reg} := \bigcup _{r \in I} \mathcal {S}_r$ of regular points is an open, dense submanifold of $M$ which projects onto $I$ by means of the canonical projection $M \to \Omega$ and the restricted Riemannian metric splits as

\[ {\mathbb g}|_{M^{\rm reg}}=\operatorname{d\!} r^{2} + {\mathbb g}|_{\mathcal{S}_r}, \]

where $r$ is the coordinate on $I$. We fix an $\operatorname {Ad}(\mathsf {G})$-invariant inner product $Q$ on the Lie algebra $\mathfrak {g} := \operatorname {Lie}(\mathsf {G})$ and a $Q$-orthogonal decomposition $\mathfrak {g}=\mathfrak {h}+\mathfrak {m}$, with $\mathfrak {h} := \operatorname {Lie}(\mathsf {H})$. We consider identified $\mathfrak {m} \simeq T_{e\mathsf {H}}\mathsf {G}/\mathsf {H}$ by means of the evaluation map $X \mapsto \frac {\rm d}{{\rm d}s}\exp (sX)\mathsf {H} \big |_{s=0}$, where $\exp : \mathfrak {g} \to \mathsf {G}$ denotes the Lie exponential map of $\mathsf {G}$. We also identify any $\mathsf {G}$-invariant tensor field on $\mathsf {G}/\mathsf {H}$ with the corresponding $\operatorname {Ad}(\mathsf {H})$-invariant tensor on $\mathfrak {m}$ in the usual way.

Firstly, we define the $1$-parameter family $(g_r) \subset \mathrm {Sym}^{2}(\mathfrak {m}^{*})^{\operatorname {Ad}(\mathsf {H})}$ by

\[ g_r := (\phi_r)^{*}( {\mathbb g}|_{\mathcal{S}_r}). \]

We also set

\[ T_r := \frac{(\operatorname{d\!} \phi_r)_{e\mathsf{H}}^{-1}\Big(\mathbb{J}_{\gamma(r)}\dot{\gamma}(r)\Big)}{\big|(\operatorname{d\!} \phi_r)_{e\mathsf{H}}^{-1}\Big(\mathbb{J}_{\gamma(r)}\dot{\gamma}(r)\Big)\big|_Q} \in \mathfrak{m}, \quad \theta_r := Q(T_r,\cdot)|_{\mathfrak{m}} \in \mathfrak{m}^{*} . \]

Then, for any $r \in I$, the complex structure $\mathbb {J}$ induces a linear complex structure $J_r$ on $\mathfrak {p}_r := \ker (\theta _r) \subset \mathfrak {m}$ by setting

\[ J_r : \mathfrak{p}_r \to \mathfrak{p}_r, \quad J_r := (\operatorname{d\!} \phi_r)_{e\mathsf{H}}^{-1} \circ \mathbb{J}_{\gamma(r)} \circ (\operatorname{d\!} \phi_r)_{e\mathsf{H}}|_{\mathfrak{p}_r}. \]

The integrability of $\mathbb {J}$ implies that, for any $X,\,Y \in \mathfrak {p}_r$,

(2.8)\begin{equation} \begin{aligned} & [J_rX,Y]_{\mathfrak{m}}+[X,J_rY]_{\mathfrak{m}} \in \mathfrak{p}_r, \\ & J_r\Big([J_rX,Y]_{\mathfrak{m}}+[X,J_rY]_{\mathfrak{m}}\Big)=[J_rX,J_rY]_{\mathfrak{m}}-[X,Y]_{\mathfrak{m}}. \end{aligned} \end{equation}

Moreover, since the subgroup $\mathsf {H} \subset \mathsf {G}$ leaves any point of the geodesic $\gamma$ fixed and $\mathsf {G}$ acts by holomorphic isometries, it follows that

(2.9)\begin{equation} [\mathfrak{h},T_r]=0 \quad \text{ for any } r \in I \end{equation}

and so $(\theta _r,\,J_r)$ is a $1$-parameter family of $\mathsf {G}$-invariant CR structures on $\mathsf {G}/\mathsf {H}$, see e.g. [Reference Alekseevsky and Spiro3].

We recall now the following definition [Reference Podestà and Spiro48]: $\mathsf {G}/\mathsf {H}$ is said to be ordinary if $\mathsf {G}$ is semisimple, the normalizer $\mathsf {K}:= \mathsf {N}_{\mathsf {G}}(\mathsf {H}^{\operatorname {o}})$ of the connected component $\mathsf {H}^{\operatorname {o}}$ of $\mathsf {H}$ is the centralizer of a torus and $\dim \mathsf {K}=1+\dim \mathsf {H}$. This implies that:

• • $T \equiv T_r$ and $\mathfrak {p} \equiv \mathfrak {p}_r$ do not depend on $r \in I$;

• • the Lie algebra $\mathfrak {k} := \operatorname {Lie}(\mathsf {K})$ splits $Q$-orthogonally as $\mathfrak {k} = \mathfrak {h} \oplus \mathbb {R} T$.

Moreover, the complex structure $\mathbb {J}$ is said to be projectable if each $J_r$ is $\operatorname {Ad}(\mathsf {K})$-invariant. In this case, $(J_r)_{r \in I}$ is mapped onto a $1$-parameter family of $\mathsf {G}$-invariant complex structures on the flag manifold $\mathsf {G} / \mathsf {K}$. Since the set of invariant complex structures on a flag manifold is discrete [Reference Alekseevsky and Perelomov1, Reference Nishiyama42], it follows that $J \equiv J_r$ is constant.

In § 3, a distinguished kind of cohomogeneity one standard Hermitian manifolds will be investigated.

3.1 Chern connection of standard cohomogeneity one Hermitian manifolds

Let $(M,\,\mathbb {J},\, {\mathbb g})$ be a standard cohomogeneity one Hermitian manifolds acted effectively by holomorphic isometries by a compact, connected real Lie group $\mathsf {G}$.

Hereafter, we adopt the same notation introduced in § 2. Note that the complement $\mathfrak {m}$ of $\mathfrak {h}$ in $\mathfrak {g}$ admits a $Q$-orthogonal, $\operatorname {Ad}(\mathsf {H})$-invariant decomposition $\mathfrak {m}= \mathfrak {a} + \mathfrak {p}$, where $\mathfrak {a} := \mathbb {R} T$ is the trivial submodule. Since, by hypothesis, $\mathfrak {p}$ does not contain any trivial $\operatorname {Ad}(\mathsf {H})$-submodule, the metrics $g_r$ induced by $ {\mathbb g}$ on $\mathfrak {m}$ split uniquely as

\[ g_r = F(r)^{2} Q|_{\mathfrak{a} {\otimes} \mathfrak{a}} + g_r|_{\mathfrak{p} {\otimes} \mathfrak{p}} \]

by means of the Schur lemma, where $F: I \to \mathbb {R}$ is a smooth, positive function, possibly satisfying some appropriate boundary condition.

From now on, given $V \in \mathfrak {g}$, we denote by $V^{*} \in \Gamma (TM)$ the fundamental vector field on $M$ associated with $V$, that is $V^{*}_p := \frac {\rm d}{{\rm d}s}\exp (sV)\cdot p \big |_{s=0}$, and we also set

(3.1)\begin{equation} N:= F\frac{\partial}{\partial r}. \end{equation}

Note that, by construction,

(3.2)\begin{equation} (\mathbb{J} N)_{\gamma(r)} = T^{*}_{\gamma(r)} \end{equation}

and, for any $V,\,W \in \mathfrak {g}$, it holds

(3.3)\begin{equation} [V^{*},W^{*}]=-[V,W]^{*}, \quad [N,V^{*}]=0. \end{equation}

We observe also that, for any $X \in \mathfrak {p}$ and for any $Y_1,\,Y_2 \in \mathfrak {m}$, we have

(3.4)\begin{equation} {\mathbb g}(Y_1^{*},Y_2^{*})_{\gamma(r)} = g_r(Y_1,Y_2), \quad (\mathbb{J} X^{*})_{\gamma(r)} = (JX)^{*}_{\gamma(r)}. \end{equation}

By hypothesis, the fundamental vector fields $V^{*}$ are holomorphic, i.e.

(3.5)\begin{equation} [V^{*},\mathbb{J} A] = \mathbb{J} [V^{*},A] \quad \text{for any }A \in \Gamma(TM). \end{equation}

By using equations (2.2) and (2.3), one can directly obtain the explicit formulas for the Levi-Civita connection and the $3$-form $\operatorname {d\!}\omega$. Here and in the following statements, we consider the context and assume the notation as above.

Proposition 3.1 Let $X,\, Y,\, Z \in \mathfrak {p}$. Then, the non-vanishing components of the Levi-Civita connection are given by

\begin{align*} 2 {\mathbb g}\Big(D_{Y^{*}}X^{*},Z^{*}\Big)_{\gamma(r)} & = g_r([X,Y],Z)+g_r([Z,X],Y)+g_r([Z,Y],X),\\ 2 {\mathbb g}\Big(D_{Y^{*}}X^{*},T^{*}\Big)_{\gamma(r)} & = g_r([X,Y],T)+g_r([T,X],Y)+g_r([T,Y],X) ,\\ 2 {\mathbb g}\Big(D_{Y^{*}}X^{*},N\Big)_{\gamma(r)} & =-F(r)\frac{\partial}{\partial r}(g_r(X,Y)),\\ 2 {\mathbb g}\Big(D_{T^{*}}X^{*},Z^{*}\Big)_{\gamma(r)} & =-g_r([X,Z],T)-g_r([T,X],Z)-g_r([T,Z],X),\\ 2 {\mathbb g}\Big(D_NX^{*},Z^{*}\Big)_{\gamma(r)} & = F(r)\frac{\partial}{\partial r}(g_r(X,Z)),\\ 2 {\mathbb g}\Big(D_{Y^{*}}T^{*},Z^{*}\Big)_{\gamma(r)} & =-g_r([Y,Z],T) +g_r([T,Y],Z)-g_r([T,Z],Y),\\ 2 {\mathbb g}\Big(D_{T^{*}}T^{*},N\Big)_{\gamma(r)} & =-2F'(r)F(r)^{2},\\ 2 {\mathbb g}\Big(D_NT^{*},T^{*}\Big)_{\gamma(r)} & = 2F'(r)F(r)^{2},\\ 2 {\mathbb g}\Big(D_{Y^{*}}N,Z^{*}\Big)_{\gamma(r)} & = F(r)\frac{\partial}{\partial r}(g_r(Y,Z)) ,\\ 2 {\mathbb g}\Big(D_{T^{*}}N,T^{*}\Big)_{\gamma(r)} & = 2F'(r)F(r)^{2} ,\\ 2 {\mathbb g}\Big(D_NN,N\Big)_{\gamma(r)} & = 2F'(r)F(r)^{2}. \end{align*}

Proposition 3.2 Let $X,\,Y,\,Z \in \mathfrak {p}$. Then, the $3$-form $\operatorname {d\!}\omega$ is given by

\begin{align*} \operatorname{d\!}\omega(X^{*},Y^{*},Z^{*})_{\gamma(r)} & = g_r([X,Y],JZ) +g_r([Y,Z],JX) +g_r([Z,X],JY), \\ \operatorname{d\!}\omega(X^{*},Y^{*},T^{*})_{\gamma(r)} & = g_r([T,X],JY) +g_r([T,JY],X),\\ \operatorname{d\!}\omega(X^{*},Y^{*},N)_{\gamma(r)} & = F(r)\frac{\partial}{\partial r}\Big(g_r(JX,Y)\Big) +g_r([X,Y],T),\\ \operatorname{d\!}\omega(X^{*},T^{*},N)_{\gamma(r)} & = 0. \end{align*}

By combining propositions 3.1 and 3.2, and by equation (2.1), we obtain explicit formulas for the Chern connection along the geodesic $\gamma (r)$. More precisely

Proposition 3.3 Let $X,\,Y,\,Z \in \mathfrak {p}$. Then the non-vanishing components of the Chern connection are

\begin{align*} 2 {\mathbb g}\Big(\nabla_{Y^{*}}X^{*},Z^{*}\Big)_{\gamma(r)} & = g_r([X,Y],Z) +g_r([X,JY],JZ)\\ & \quad +g_r([Z,Y],X) -g_r([Z,JY],JX), \\ 2 {\mathbb g}\Big(\nabla_{Y^{*}}X^{*},T^{*}\Big)_{\gamma(r)} & = g_r([X,Y],T), \\ 2 {\mathbb g}\Big(\nabla_{Y^{*}}X^{*},N\Big)_{\gamma(r)} & = g_r([X,JY],T), \\ 2 {\mathbb g}\Big(\nabla_{T^{*}}X^{*},Z^{*}\Big)_{\gamma(r)} & = F(r)\frac{\partial}{\partial r}\Big(g_r(JX,Z)\Big) -g_r([T,X],Z) -g_r([T,Z],X), \\ 2 {\mathbb g}\Big(\nabla_NX^{*},Z^{*}\Big)_{\gamma(r)} & = F(r)\frac{\partial}{\partial r}(g_r(X,Z)) +g_r([T,JX],Z) +g_r([T,Z],JX),\\ 2 {\mathbb g}\Big(\nabla_{Y^{*}}T^{*},Z^{*}\Big)_{\gamma(r)} & = 2g_r([T,Y],Z) -g_r([Y,Z],T), \\ 2 {\mathbb g}\Big(\nabla_{T^{*}}T^{*},N\Big)_{\gamma(r)} & =-2F'(r)F(r)^{2}, \\ 2 {\mathbb g}\Big(\nabla_NT^{*},T^{*}\Big)_{\gamma(r)} & =+2F'(r)F(r)^{2}, \\ 2 {\mathbb g}\Big(\nabla_{Y^{*}}N,Z^{*}\Big)_{\gamma(r)} & = g_r([JY,Z],T),\\ 2 {\mathbb g}\Big(\nabla_{T^{*}}N,T^{*}\Big)_{\gamma(r)} & = 2F'(r)F(r)^{2}, \\ 2 {\mathbb g}\Big(\nabla_NN,N\Big)_{\gamma(r)} & = 2F'(r)F(r)^{2}. \end{align*}

As a direct consequence of proposition 3.3 and equation (2.6), we get

We can also characterize the Kähler metrics as follows. By proposition 3.2 it follows that

\[ \operatorname{d\!}\omega(X^{*},Y^{*},T^{*})_{\gamma(r)}=0 \quad \text{ for any } X,Y \in \mathfrak{p} \]

if and only if the restriction $g_r|_{\mathfrak {p}{\otimes }\mathfrak {p}}$ is $\operatorname {Ad}(\mathsf {K})$-invariant for any $r \in I$. This is equivalent to say that the metrics $(g_r)$ on $\mathsf {G}/\mathsf {H}$ are of submersion-type with respect to the homogeneous circle bundle $\mathsf {K}/\mathsf {H} \to \mathsf {G}/\mathsf {H} \to \mathsf {G}/\mathsf {K}$, namely, they induce metrics on the base such that the projection is a Riemannian submersion. Moreover, if $g_r|_{\mathfrak {p}{\otimes }\mathfrak {p}}$ is $\operatorname {Ad}(\mathsf {K})$-invariant, then the condition

\[ \operatorname{d\!}\omega(X^{*},Y^{*},Z^{*})_{\gamma(r)}=0 \quad \text{ for any } X,Y,Z \in \mathfrak{p} \]

holds true if and only if the induced metrics on the flag manifold $\mathsf {G}/\mathsf {K}$ are Kähler. Let us fix now a $Q$-orthogonal, $\operatorname {Ad}(\mathsf {K})$-invariant, irreducible decomposition

(3.6)\begin{equation} \mathfrak{p}= \mathfrak{p}_1+{\cdots}+\mathfrak{p}_{\ell}. \end{equation}

Since flag manifolds are equal rank homogeneous spaces, namely ${\rm rank}(\mathsf {K})={\rm rank}(\mathsf {G})$, it follows that their isotropy representations are always monotypic, namely $\mathfrak {p}_i \not \simeq \mathfrak {p}_j$ for any $1 \leq i < j \leq \ell$. Hence, the decomposition (3.6) is unique (up to order) and, by the Schur lemma, the metrics $g_r$ splits uniquely as

\[ g_r = F(r)^{2} Q|_{\mathfrak{a} {\otimes} \mathfrak{a}} +h_1(r)^{2} Q|_{\mathfrak{p}_1 {\otimes} \mathfrak{p}_1} + {\cdots} + h_{\ell}(r)^{2} Q|_{\mathfrak{p}_{\ell} {\otimes} \mathfrak{p}_{\ell}}. \]

Then, the condition

\[ \operatorname{d\!}\omega(X^{*},Y^{*},N)_{\gamma(r)}=0 \quad \text{ for any } X, Y \in \mathfrak{p} \]

holds true if and only if

(3.7)\begin{equation} \operatorname{ad}(T)|_{\mathfrak{p}_i} =-2\frac{h_i(r)h'_i(r)}{F(r)} J \quad \text{for any }1 \leq i \leq \ell. \end{equation}

3.2 Construction of Bérard-Bergery manifolds

Let us assume that $P = P(\mathsf {G},\,\mathsf {K}):= \mathsf {G}/\mathsf {K}$ is a simply-connected, irreducible compact Hermitian symmetric space, i.e. $\mathsf {G}$ is a connected, compact, simple Lie group and $\mathsf {K} \subset \mathsf {G}$ is a maximal, connected, compact subgroup with centre isomorphic to the circle group [Reference Helgason30, theorem 6.1]. Set $\dim _{\mathbb {R}}P=2(m-1)$, with $m\geq 2$, and let $p \in \mathbb {N}$ be the unique positive integer such that $p^{-1}c_1(P)$ is an indivisible (positive) class in the cohomology group $H^{2}(P;\mathbb {Z})$, see [Reference Borel and Hirzebruch13, chapter 5, § 16]. In the following, we list the possibilities for $P$, following [Reference Besse10, § 7.102 and § 9.124].

\[ \begin{array}{c|c|c|c} P & m & p & \text{conditions} \\ \hline \mathsf{SU}(k_1+k_2)/\mathsf{S}(\mathsf{U}(k_1){\times}\mathsf{U}(k_2)) & k_1k_2+1 & k_1+k_2 & k_2\geq k_1 \geq 1 \\ \mathsf{SO}(2k)/\mathsf{U}(k) & \dfrac{k(k-1)}{2}+1 & 2(k-1) & k\geq 5 \\ \mathsf{Sp}(k)/\mathsf{U}(k) & \dfrac{k(k+1)}{2}+1 & k+1 & k\geq 2 \\ \mathsf{SO}(k+2)/(\mathsf{SO}(2){\times}\mathsf{SO}(k)) & k+1 & k & k\geq 5 \\ \mathsf{E}_6/\mathsf{SO}(2)\mathsf{Spin}(10) & 17 & 12 & - \\ \mathsf{E}_7/\mathsf{SO}(2)\mathsf{E}_6 & 28 & 18 & - \\ \end{array} \]

Denote by $\mathsf {H} := [\mathsf {K},\,\mathsf {K}]$ the commutator of $\mathsf {K}$. By hypothesis, there exists an integer $s\geq 1$ such that $\mathsf {H} \cap \mathsf {Z}(\mathsf {K}) \simeq \mathbb {Z}_s$. Fix an isomorphism $\imath : \mathsf {U}(1) \to \mathsf {Z}(\mathsf {K})$ and consider the generator $\alpha := \imath ({\rm e}^{\sqrt {-1}({2\pi }/{s})})$ of $\mathsf {H} \cap \mathsf {Z}(\mathsf {K})$. Then, consider the right action of $\mathbb {Z}_s$ on $\mathsf {H} \times \mathsf {U}(1)$ given by $(h,\,z)\cdot j := (h\alpha ^{j},\,{\rm e}^{-\sqrt {-1}({2j\pi }/{s})}z)$ and take the quotient $\mathsf {H}{\cdot }\mathsf {U}(1) := \mathsf {H} \times _{\mathbb {Z}_s}\!\! \mathsf {U}(1)$. We can consider identified $\mathsf {K} = \mathsf {H}{\cdot }\mathsf {U}(1)$ by means of a Lie group isomorphism (see e.g. [Reference Bredon16, chapter 0, theorem 6.9]) and, for any positive integer $n$, we take the representation

\[ \rho_n : \mathsf{K} \to \mathsf{U}(1) \,, \quad \rho_n([h,z]) := z^{-sn}. \]

Then, consider the associated bundle

\[ \S_n = \S_n(\mathsf{G},\mathsf{K}) := \mathsf{G} \times_{\rho_n} \mathsf{U}(1) \]

with the projection $\pi _n: \S _n \to P$ given by $\pi _n([a,\,z]):= a \mathsf {K}$. Note that $\mathsf {G}$ acts transitively on the left on $\S _n$ by $\tilde {a} \cdot [a,\,z] := [\tilde {a}a,\,z]$ and the stabilizer of $[e,\,1]$ is the subgroup $\ker (\rho _n) = \mathsf {H} {\times } \mathbb {Z}_n \subset \mathsf {K}$, where

\[ \mathsf{H} {\times} \mathbb{Z}_n \simeq \mathsf{H} \times_{\mathbb{Z}_s}\!\! \mathbb{Z}_{sn} \subset \mathsf{H}{\cdot}\mathsf{U}(1) = \mathsf{K} \quad \text{ by } \quad (h,j) \mapsto \big[h,{\rm e}^{\sqrt{-1}({2j\pi}/n)}\big]. \]

We are ready to construct the family $M_{(i,n)}(\mathsf {G},\,\mathsf {K})$, with $1 \leq i \leq 4$ and $n \in \mathbb {N}$, of standard cohomogeneity one Hermitian manifolds in the following way.

1. (i) We set $M_{(1,n)}(\mathsf {G},\,\mathsf {K}) := \S _n \times \mathbb {R}$ and we let $\mathsf {G}$ act on $M_{(1,n)}$ via $a \cdot (x,\, r) := (a\cdot x,\,r)$. The orbit space is $\Omega =\mathbb {R}$.

2. (ii) We let $M_{(2,n)}(\mathsf {G},\,\mathsf {K}) := \S _n \times _{\mathsf {U}(1)} \mathbb {C}$ be the homogeneous complex line bundle over $P$ associated with $\pi _n: \S _n \to P$ by means of the standard action of $\mathsf {U}(1)$ on $\mathbb {C}$. Then, $M_{(2,n)}(\mathsf {G},\,\mathsf {K})$ is equivariantly diffeomorphic to the quotient of $\S _n \times [0,\,+\infty )$ by the fibration $\S _n \times \{0\} \to P$, on which $\mathsf {G}$ acts by left multiplication on the first factor.

3. (iii) We set $M_{(3,n)}(\mathsf {G},\,\mathsf {K}) := \S _n \times S^{1}$ and we let $\mathsf {G}$ act on $M_{(3,n)}(\mathsf {G},\,\mathsf {K})$ via $a \cdot (x,\, z) := (a\cdot x,\,z)$. The orbit space is $\Omega =S^{1}$.

4. (iv) We let $M_{(4,n)}(\mathsf {G},\,\mathsf {K}) := \S _n \times _{\mathsf {U}(1)} \mathbb {C} \mathbb {P}^{1}$ be the homogeneous $\mathbb {C} \mathbb {P}^{1}$-bundle over $P$ associated with $\pi _n: \S _n \to P$ by means of the standard action of $\mathsf {U}(1)$ on $\mathbb {C} \mathbb {P}^{1}$. Then, $M_{(4,n)}(\mathsf {G},\,\mathsf {K})$ is equivariantly diffeomorphic to the quotient of $\S _n \times [0,\,\pi ]$ under the identification of the two boundaries by means of the fibrations $\S _n \times \{0\} \to P$ and $\S _n \times \{\pi \} \to P$, on which $\mathsf {G}$ acts by left multiplication on the first factor.

We observe that the manifolds in families $M_{(3,n)}(\mathsf {G},\,\mathsf {K})$ and $M_{(4,n)}(\mathsf {G},\,\mathsf {K})$ are compact, and the manifolds in families $M_{(2,n)}(\mathsf {G},\,\mathsf {K})$ and $M_{(4,n)}(\mathsf {G},\,\mathsf {K})$ are simply connected. Moreover, manifolds in family $M_{(4,n)}(\mathsf {G},\,\mathsf {K})$ are almost-homogeneous in the sense of [Reference Huckleberry and Snow31], i.e. the complexified Lie group $\mathsf {G}^{\mathbb {C}}$ acts on them by biholomorphisms with one open orbit, see [Reference Podestà and Spiro49].

Let $M_{(i,n)}(\mathsf {G},\,\mathsf {K})$ be as above. We denote by $B$ be the Cartan–Killing form of $\mathfrak {g} := \operatorname {Lie}(\mathsf {G})$ and we set $\mathfrak {k} := \operatorname {Lie}(\mathsf {K})$, $\mathfrak {h} := \operatorname {Lie}(\mathsf {H})$, $\mathfrak {a} := \mathfrak {z}(\mathfrak {k}) = \operatorname {Lie}(\mathsf {Z}(\mathsf {K}))$. Then, the positive definite $\operatorname {Ad}(\mathsf {G})$-invariant scalar product $Q := \frac 1{4m}(-B)$ on $\mathfrak {g}$ determines a $\operatorname {Ad}(\mathsf {H})$-invariant, $Q$-orthogonal decomposition

\[ \mathfrak{g}=\underbrace{\mathfrak{h} + \mathfrak{a}}_{\mathfrak{k}} + \overbrace{\phantom{kiii} \mathfrak{p}}^{\mathfrak{m}}, \quad \text{with } [\mathfrak{k},\mathfrak{k}]=\mathfrak{h}, \ [\mathfrak{h},\mathfrak{a}] = \{0\}, \ [\mathfrak{k},\mathfrak{p}]\subset \mathfrak{p}, \ [\mathfrak{p},\mathfrak{p}] \subset \mathfrak{k}. \]

We fix a vector $T \in \mathfrak {a}$ with $Q(T,\,T)=1$ and we pick the only $\lambda >0$ such that

(3.8)\begin{equation} J := \lambda^{-1}\operatorname{ad}(T)|_{\mathfrak{p}} \end{equation}

is a linear complex structure on $\mathfrak {p}$ [Reference Kobayashi and Nomizu32, chapter XI, theorem 9.6]. Note that a direct computation implies

\[-4\,m = B(T,T) = \operatorname{Tr}(\operatorname{ad}(T) \circ \operatorname{ad}(T)) =-\lambda^{2} \operatorname{Tr}(\operatorname{Id}_{2(m-1)}) \]

and so

(3.9)\begin{equation} \lambda^{2}=\frac{2m}{m-1}. \end{equation}

Moreover, we stress that:

• • the restriction $Q_{\mathfrak {p}} := Q|_{\mathfrak {p}{\otimes }\mathfrak {p}}$ induces a $\mathsf {G}$-invariant Kähler–Einstein metric on the base space $P$ satisfying the equation $\operatorname {Ric}(Q_{\mathfrak {p}}) =2\,m Q_{\mathfrak {p}}$

• • the scalar product $\frac {2m(m-1)n^{2}}{p^{2}}Q_{\mathfrak {a}}$, with $Q_{\mathfrak {a}}:= Q |_{\mathfrak {a}{\otimes }\mathfrak {a}}$, corresponds to the standard metric of radius $1$ on the fibres of $\pi _n: \S _n \to P$

Then, being $P$ irreducible, any $\mathsf {G}$-invariant Hermitian metric $ {\mathbb g}$ on $(M_{(i,n)}(\mathsf {G},\,\mathsf {K}),\,\mathbb {J})$ which is of submersion-type with respect to $M_{(i,n)}(\mathsf {G},\,\mathsf {K}) \to P$ is completely determined by two positive, smooth functions $f,\,h: I \to \mathbb {R}$, satisfying some appropriate smoothness conditions, by means of the splitting

(3.10)\begin{equation} g_r = \frac{2m(m-1)n^{2}}{p^{2}}f(r)^{2}Q_{\mathfrak{a}} +h(r)^{2}Q_{\mathfrak{p}}, \end{equation}

where $(g_r)$ is again the 1-parameter family of $\mathsf {G}$-invariant metrics induced by $ {\mathbb g}$ on the principal orbits. Case by case, the smoothness conditions are the following (see e.g. [Reference Verdiani and Ziller54, p. 7] and [Reference Bérard-Bergery9, p. 39]).

1. (i) For $i=1$, $I=\mathbb {R}$ and there are no boundary conditions.

2. (ii) For $i=2$, $I=(0,\,+\infty )$ and the conditions are: $f$ is the restriction of a smooth odd function on $\mathbb {R}$ with $f'(0) = 1$ and $h$ is the restriction of a smooth even function on $\mathbb {R}$.

3. (iii) For $i=3$, $f,\,h$ need to be $S^{1}$-periodic.

4. (iv) For $i=4$, $I=(0,\,\pi )$ and the conditions are: $f$ is the restriction of a smooth odd function on $\mathbb {R}$ satisfying $f(\pi +r)=-f(\pi -r)$ with $f'(0) = 1 = -f'(\pi )$ and $h$ is the restriction of a smooth even function on $\mathbb {R}$ satisfying $h(\pi +r)=h(\pi -r)$.

Conversely, any pair of smooth functions $(f,\,h)$ satisfying the appropriate smoothness condition uniquely defines a smooth $\mathsf {G}$-invariant Hermitian metric $ {\mathbb g}= {\mathbb g}(f,\,h)$, which is of submersion-type.

By equations (3.7), (3.9) and (3.10), it follows that the metric $ {\mathbb g}(f,\,h)$ is Kähler if and only if the functions $f,\,h$ verify

(3.11)\begin{equation} h(r)h'(r)+\frac{mn}{p}f(r) = 0 \quad \text{ for any }r \in I. \end{equation}

From now on, we will adopt the following

Let us point out that the above construction can be performed in a more general setting, i.e. by requiring that the base space $P=\mathsf {G}/\mathsf {K}$ is a Kähler C-space, see [Reference Bérard-Bergery9]. However, in this work, we will just focus in the case of $P$ being symmetric and irreducible.

Example 3.8 Consider the Hermitian symmetric space $P=\mathbb {C} \mathbb {P}^{m-1}$, corresponding to $\mathsf {G}=\mathsf {SU}(m)$ and $\mathsf {K}=\mathsf {S}(\mathsf {U}(1){\times }\mathsf {U}(m-1))$. Here, the $\operatorname {Ad}(\mathsf {G})$-invariant scalar product $Q(A_1,\,A_2):= -\frac {1}{2}\operatorname {Tr}(A_1A_2)$ on the Lie algebra $\mathfrak {g}=\mathfrak {su}(m)$, defined following the above normalization, induces on $P$ the Fubini–Study metric with sectional curvature satisfying $1\leq \sec \leq 4$. In this case, the principal orbits are equivariantly diffeomorphic to the lens space $\S _n = \mathbb {Z}_n \backslash S^{2m-1}$, where $\mathbb {Z}_n$ acts on $S^{2m-1} \subset \mathbb {C}^{m}$ via $k \cdot z := {\rm e}^{-i({2k\pi }/n)}z$, and

\begin{align*} & M_{(1,n)}(\mathsf{G},\mathsf{K}) = \mathbb{Z}_n \backslash S^{2m-1} \times \mathbb{R} ,\qquad M_{(2,n)}(\mathsf{G},\mathsf{K}) = \mathcal{O}_{\mathbb{C} \mathbb{P}^{m-1}}(-n), \\ & M_{(3,n)}(\mathsf{G},\mathsf{K}) = \mathbb{Z}_n \backslash S^{2m-1} \times S^{1} , \qquad M_{(4,n)}(\mathsf{G},\mathsf{K}) = \mathbb{P}\Big(\mathcal{O}_{\mathbb{C} \mathbb{P}^{m-1}} \oplus \mathcal{O}_{\mathbb{C} \mathbb{P}^{m-1}}(-n)\Big). \end{align*}

Here, we denoted by $\mathcal {O}_{\mathbb {C} \mathbb {P}^{m-1}}$ the trivial line bundle over $\mathbb {C} \mathbb {P}^{m-1}$, by $\mathcal {O}_{\mathbb {C} \mathbb {P}^{m-1}}(-1)$ the tautological line bundle and by $\mathcal {O}_{\mathbb {C} \mathbb {P}^{m-1}}(-n) := \mathcal {O}_{\mathbb {C} \mathbb {P}^{m-1}}(-1)^{\otimes n}$. In particular:

• • if $(i,\,n)=(3,\,1)$, then we get a diagonal Hopf manifold;

• • if $(i,\,n)=(4,\,1)$, then we get the connected sum $\mathbb {C}\mathbb {P}^{m} \# \overline {\mathbb {C}\mathbb {P}}{}^{m}$;

• • if $m=2$ and $i=4$, then we get all the Hirzebruch surfaces.

Remark 3.11 Let $ {\mathbb g}= {\mathbb g}(f,\,h)$ a cohomogeneity one, submersion-type metric on a manifold $(M_{(i,n)}(\mathsf {G},\,\mathsf {K}),\,\mathbb {J})$ and $\varphi : I \to \mathbb {R}$ a positive, admissible function. Then, the metric $\varphi ^{2} {\mathbb g}$ is still a cohomogeneity one, submersion-type metric on $(M_{(i,n)}(\mathsf {G},\,\mathsf {K}),\,\mathbb {J})$ and $\varphi ^{2} {\mathbb g} = {\mathbb g}(\hat {f}_{\varphi },\,\hat {h}_{\varphi })$, with

\[ \hat{f}_{\varphi}(r) := \xi'_{\varphi}(\xi^{-1}_{\varphi}(r))f(\xi^{-1}_{\varphi}(r)) \,, \quad \hat{h}_{\varphi}(r) := \xi'_{\varphi}(\xi^{-1}_{\varphi}(r))h(\xi^{-1}_{\varphi}(r)) \,, \]

where $\xi _{\varphi }(r) := \int _0^{r}\varphi (t)\operatorname {d\!} t$. However we stress that, even if $ {\mathbb g}$ is complete, in general the conformal metric $\varphi ^{2} {\mathbb g}$ is not.

3.3 Curvature and torsion computations for Bérard-Bergery manifolds

We begin this section by listing the Levi-Civita connection and the Riemannian Ricci tensor of the manifolds $(M_{(i,n)}(\mathsf {G},\,\mathsf {K}),\,\mathbb {J},\, {\mathbb g})$. By straightforward computations, from proposition 3.1 and equations (3.10), (3.8) we get

Proposition 3.12 Let $X,\,Y,\,Z \in \mathfrak {p}$. Then

\begin{align*} D_{Y^{*}}X^{*}\big|_{\gamma(r)} & = \frac{\lambda}2\left(Q(JX,Y)T^{*}_{\gamma(r)} -\frac{p}{mn}\frac{h(r)h'(r)}{f(r)}Q(X,Y)N_{\gamma(r)}\right), \\ D_{T^{*}}X^{*}\big|_{\gamma(r)} & =-\frac 2{\lambda} \left(\frac{mn}p\right)^{2} \frac{f(r)^{2}}{h(r)^{2}}(JX)^{*}_{\gamma(r)}, \\ D_{N}X^{*}\big|_{\gamma(r)} & = \frac{2mn}{\lambda p} f(r)\frac{h'(r)}{h(r)}X^{*}_{\gamma(r)}, \\ D_{Y^{*}}T^{*}\big|_{\gamma(r)} & = \left(\lambda-\frac 2{\lambda} \left(\frac{mn}p\right)^{2}\frac{f(r)^{2}}{h(r)^{2}}\right)(JY)^{*}_{\gamma(r)}, \\ D_{T^{*}}T^{*}\big|_{\gamma(r)} & =-\frac{2mn}{\lambda p} f'(r)N_{\gamma(r)}, \\ D_{N}T^{*}\big|_{\gamma(r)} & = \frac{2mn}{\lambda p} f'(r)T^{*}_{\gamma(r)}, \\ D_{Y^{*}}N\big|_{\gamma(r)} & = \frac{2mn}{\lambda p} f(r)\frac{h'(r)}{h(r)}Y^{*}_{\gamma(r)}, \\ D_{T^{*}}N\big|_{\gamma(r)} & = \frac{2mn}{\lambda p} f'(r)T^{*}_{\gamma(r)}, \\ D_{N}N\big|_{\gamma(r)} & = \frac{2mn}{\lambda p} f'(r)N_{\gamma(r)}. \end{align*}

Moreover, from [Reference Grove and Ziller27, proposition 1.14], we directly obtain

Proposition 3.13 Let $X \in \mathfrak {p}$ with $Q(X,\,X)=1$. Then the Riemannian Ricci tensor is given by

\begin{align*} \operatorname{Ric}( {\mathbb g})(N,N)_{\gamma(r)} & = \frac{2m(m-1)n^{2}}{p^{2}}f(r)^{2}\left(-\frac{f''(r)}{f(r)} -2(m-1)\frac{h''(r)}{h(r)}\right), \\ \operatorname{Ric}( {\mathbb g})(T^{*},T^{*})_{\gamma(r)} & = \frac{2m(m-1)n^{2}}{p^{2}}f(r)^{2}\left(-\frac{f''(r)}{f(r)} -2(m-1)\frac{f'(r)}{f(r)}\frac{h'(r)}{h(r)}\right.\\ & \quad +\left.2(m-1)\left(\frac{mn}p\right)^{2}\frac{f(r)^{2}}{h(r)^{4}}\right), \\ \operatorname{Ric}( {\mathbb g})(X^{*},X^{*})_{\gamma(r)} & = h(r)^{2}\left(-\frac{h''(r)}{h(r)} -\frac{f'(r)}{f(r)}\frac{h'(r)}{h(r)}-(2m-3)\frac{h'(r)^{2}}{h(r)^{2}} \right.\\ & \quad -\left.2\left(\frac{mn}p\right)^{2}\frac{f(r)^{2}}{h(r)^{4}} +\frac{2m}{h(r)^{2}}\right) \end{align*}

and

\[ \operatorname{Ric}( {\mathbb g})(N,T^{*})_{\gamma(r)} = \operatorname{Ric}( {\mathbb g})(N,X^{*})_{\gamma(r)} = \operatorname{Ric}( {\mathbb g})(T^{*},X^{*})_{\gamma(r)} = 0 \,. \]

Furthermore, the Riemannian scalar curvature is

\begin{align*} \operatorname{scal}( {\mathbb g})(r) & = -2\frac{f''(r)}{f(r)} -4(m-1)\frac{h''(r)}{h(r)} -4(m-1)\frac{f'(r)}{f(r)}\frac{h'(r)}{h(r)}\\ & \quad -2(m-1)(2m-3)\frac{h'(r)^{2}}{h(r)^{2}} +\frac{4m(m-1)}{h(r)^{2}} -2(m-1)\left(\frac{mn}p\right)^{2}\frac{f(r)^{2}}{h(r)^{4}}. \end{align*}

We compute now the Chern connection and the Chern–Ricci tensors of the manifolds $(M_{(i,n)}(\mathsf {G},\,\mathsf {K}),\,\mathbb {J},\, {\mathbb g})$. From proposition 3.3 and equations (3.10), (3.8) we get

Proposition 3.14 Let $X,\,Y,\,Z \in \mathfrak {p}$. Then

\begin{align*} \nabla_{Y^{*}}X^{*}\big|_{\gamma(r)} & = \frac\lambda 2\Bigg(Q(JX,Y)T^{*}_{\gamma(r)} +Q(X,Y)N_{\gamma(r)}\Bigg), \\ \nabla_{T^{*}}X^{*}\big|_{\gamma(r)} & =\frac{2mn}{\lambda p}f(r)\frac{h'(r)}{h(r)}(JX)^{*}_{\gamma(r)}, \\ \nabla_{N}X^{*}\big|_{\gamma(r)} & =\frac{2mn}{\lambda p}f(r)\frac{h'(r)}{h(r)}X^{*}_{\gamma(r)}, \\ \nabla_{Y^{*}}T^{*}\big|_{\gamma(r)} & = \left(\lambda-\frac 2{\lambda} \left(\frac{mn}p\right)^{2}\frac{f(r)^{2}}{h(r)^{2}}\right)(JY)^{*}_{\gamma(r)}, \\ \nabla_{T^{*}}T^{*}\big|_{\gamma(r)} & =-\frac{2mn}{\lambda p}f'(r)N_{\gamma(r)}, \\ \nabla_{N}T^{*}\big|_{\gamma(r)} & = \frac{2mn}{\lambda p}f'(r)T^{*}_{\gamma(r)}, \\ \nabla_{Y^{*}}N\big|_{\gamma(r)} & =-\frac 2{\lambda} \left(\frac{mn}p\right)^{2} \frac{f(r)^{2}}{h(r)^{2}}Y^{*}_{\gamma(r)}, \\ \nabla_{T^{*}}N\big|_{\gamma(r)} & = \frac{2mn}{\lambda p}f'(r)T^{*}_{\gamma(r)}, \\ \nabla_{N}N\big|_{\gamma(r)} & = \frac{2mn}{\lambda p}f'(r)N_{\gamma(r)}. \end{align*}

We are ready to state the following proposition, whose proof will be given in Appendix A.

Given proposition 3.15, we are now able to study the second-Chern–Einstein equations and the constant Chern-scalar curvature equation for this special class of Hermitian cohomogeneity one manifolds. This will be done in § 5.

Finally, we recall that the torsion $\tau$ of the Chern connection is given by

(3.16)\begin{equation} -2 {\mathbb g}(\tau(A,B),C) = \operatorname{d\!}\omega(\mathbb{J} A,B,C)+\operatorname{d\!}\omega(A,\mathbb{J} B,C)\end{equation}

and that its trace $\vartheta (A) := \operatorname {Tr}(\tau (A,\,\cdot ))$ is called Lee form. We recall that it satisfies $\operatorname {d\!}\omega ^{m-1}=\vartheta \wedge \omega ^{m-1}$, see [Reference Gauduchon23, p 500].

From proposition 3.2 and formulas (3.8), (3.10) we obtain

Corollary 3.16 Let $X,\,Y,\,Z \in \mathfrak {p}$. Then

\[ \operatorname{d\!}\omega(X^{*},Y^{*},N)_{\gamma(r)} = \frac{4mn}{\lambda p}f(r)\Bigg(h(r)h'(r)+\frac{mn}pf(r)\Bigg)\rho(X,Y), \]

where $\rho (X,\,Y) := Q_{\mathfrak {p}}(JX,\,Y)$ is the $\mathsf {G}$-invariant Kähler–Einstein form on $P$, and

\[ \operatorname{d\!}\omega(X^{*},Y^{*},Z^{*})_{\gamma(r)} = \operatorname{d\!}\omega(X^{*},Y^{*},T^{*})_{\gamma(r)} = \operatorname{d\!}\omega(X^{*},T^{*},N)_{\gamma(r)} = 0. \]

and, by consequence

Proposition 3.17 Let $X,\,Y \in \mathfrak {p}$. Then it holds $\tau (N,\,T^{*})_{\gamma (r)} = \tau (X^{*},\,Y^{*})_{\gamma (r)} = 0$ and

\begin{align*} \tau(N,X^{*})_{\gamma(r)} & = \frac{2mn}{\lambda p} \frac{f(r)}{h(r)^{2}}\left(h(r)h'(r) +\frac{mn}p f(r)\right)X^{*}_{\gamma(r)},\\ \tau(T^{*},X^{*})_{\gamma(r)} & = \frac{2mn}{\lambda p} \frac{f(r)}{h(r)^{2}}\left(h(r)h'(r) +\frac{mn}p f(r)\right)(JX)^{*}_{\gamma(r)}. \end{align*}

Moreover, the Lee form $\vartheta$ satisfies

(3.17)\begin{equation} \vartheta(N)_{\gamma(r)} = \frac{4m(m-1)n}{\lambda p}\frac{f(r)}{h(r)^{2}}\left(h(r)h'(r) +\frac{mn}p f(r)\right) \end{equation}

and $\vartheta (T^{*})_{\gamma (r)} = \vartheta (X^{*})_{\gamma (r)} = 0$.

As a direct consequence of proposition 3.17, whose proof will be given in Appendix A, we get the following

Corollary 3.18 For any $A,\,B \in \Gamma (TM),$ it holds that

(3.18)\begin{equation} 2(m-1)\tau(A,B) = \Big(\vartheta(A)B-\vartheta(B)A\Big) -\Big(\vartheta(\mathbb{J} A)\mathbb{J} B-\vartheta(\mathbb{J} B)\mathbb{J} A\Big). \end{equation}

4.1 Proof of theorem A

We begin by pointing out the following

Proof. By [Reference Dragomir and Ornea19, corollary 1.1], we know that $ {\mathbb g}$ is locally conformally Kähler if and only if the complex structure $\mathbb {J}$ is parallel with respect to the Weyl connection associated with $( {\mathbb g},\,\frac {1}{m-1}\vartheta )$, equivalently, the following is satisfied:

\[ D_{A}\mathbb{J} B - \mathbb{J} D_A B = \frac{1}{2(m-1)} \Big(\vartheta(\mathbb{J} B)A - \vartheta(B)\mathbb{J} A + {\mathbb g}(A,B)\mathbb{J}\vartheta^\# + {\mathbb g}(\mathbb{J} A,B)\vartheta^\# \Big) \]

for any $A,\, B \in \Gamma (TM)$, where $\vartheta ^\# \in \Gamma (TM)$ is defined by the relation $ {\mathbb g}(\vartheta ^\#,\,\cdot ) = \vartheta$. By using equations (2.1) and (3.16), a straightforward computation shows that the above equation is equivalent to equation (3.18). Indeed, for any $C\in \Gamma (TM)$,

\begin{align*} & {\mathbb g}(D_{A}\mathbb{J} B - \mathbb{J} D_A B, C) \\ & \quad= {\mathbb g}(\nabla_{A}\mathbb{J} B,C) + {\mathbb g}(\nabla_{A}B,\mathbb{J} C) + \frac 12\operatorname{d\!}\omega(\mathbb{J} A, \mathbb{J} B, C) + \frac 12\operatorname{d\!}\omega(\mathbb{J} A, B, \mathbb{J} C) \\ & \quad=- {\mathbb g}(\tau(B,C),\mathbb{J} A) \\ & \quad= \frac 1{2(m-1)} {\mathbb g}\Big(\mathbb{J} A,(\vartheta(\mathbb{J} B)\mathbb{J} C-\vartheta(\mathbb{J} C)\mathbb{J} B)-(\vartheta(B)C-\vartheta(C)B)\Big) \\ & \quad= \frac 1{2(m-1)} {\mathbb g}\Big(\vartheta(\mathbb{J} B) {\mathbb g}(A,C) -\vartheta(B) {\mathbb g}(\mathbb{J} A,C) + {\mathbb g}(A,B) {\mathbb g}(\mathbb{J}\vartheta^\#,C) \\ & \qquad + {\mathbb g}(\mathbb{J} A,B) {\mathbb g}(\vartheta^\#,C)\Big) \\ & \quad= \frac 1{2(m-1)} {\mathbb g}\Big(\vartheta(\mathbb{J} B)A - \vartheta(B)\mathbb{J} A + {\mathbb g}(A,B)\mathbb{J}\vartheta^\# + {\mathbb g}(\mathbb{J} A,B)\vartheta^\#,C\Big) , \end{align*}

which shows the above mentioned equivalence.

Let now $ {\mathbb g}$ be a cohomogeneity one, submersion-type Hermitian metric on a complex manifold $(M_{(i,n)}(\mathsf {G},\,\mathsf {K}),\,\mathbb {J})$. By proposition 4.1, it follows that $\operatorname {d\!}\omega =\frac {1}{m-1}\vartheta \wedge \omega$ which in turn implies that $\mathcal {L}_A\vartheta =0$ for any holomorphic Killing vector field $A \in \Gamma (TM)$. Hence, by equations (3.1), (3.10) and proposition 3.12, the non-vanishing components of the Levi-Civita covariant derivative of $\vartheta$ are

(4.1)\begin{equation} \begin{aligned} (D_N\vartheta)(N)_{\gamma(r)} & = \frac{2mn}{\lambda p}\left(f(r) \frac{\partial}{\partial r}(\vartheta(N)_{\gamma(r)})-f'(r)\vartheta(N)_{\gamma(r)}\right), \\ (D_{T^{*}}\vartheta)(T^{*})_{\gamma(r)} & = \frac{2mn}{\lambda p}f'(r)\vartheta(N)_{\gamma(r)}, \\ (D_{Y^{*}}\vartheta)(X^{*})_{\gamma(r)} & = \frac{\lambda p}{2mn} \frac{h(r)h'(r)}{f(r)}Q(X,Y)\vartheta(N)_{\gamma(r)}, \end{aligned} \end{equation}

where $\lambda$ is given by equation (3.9). By corollary 4.2, the complex manifolds $(M_{(1,n)}(\mathsf {G},\,\mathsf {K}),\,\mathbb {J})$, $(M_{(2,n)}(\mathsf {G},\,\mathsf {K}),\,\mathbb {J})$ and $(M_{(4,n)}(\mathsf {G},\,\mathsf {K}),\,\mathbb {J})$ cannot admit cohomogeneity one, Hermitian metric of submersion-type that are Vaisman. Moreover, from equation (4.1), we get

Finally, we note that the manifolds $(M_{(1,n)} (\mathsf {G},\,\mathsf {K}),\,\mathbb {J})$ (respectively $(M_{(3,n)} (\mathsf {G},\,\mathsf {K}),\,\mathbb {J})$) are acted transitively by the larger group $\mathsf {G}\times \mathbb {R}$ (respectively $\mathsf {G}\times \mathsf {U}(1)$), and any metric $ {\mathbb g}= {\mathbb g}(f,\,h)$ is invariant under this action if and only if the functions $f$ and $h$ are constant. This completes the proof of theorem A.

4.2 Proof of theorem B

We begin by characterizing the Gauduchon condition as follows.

Proposition 4.6 A cohomogeneity one Hermitian metric $ {\mathbb g}$ of submersion-type on the complex manifolds $(M_{(i,n)}(\mathsf {G},\,\mathsf {K}),\,\mathbb {J})$ is Gauduchon if and only if it satisfies

(4.2)\begin{equation} \left(h(r)h'(r)+\frac{mn}{p} f(r)\right)f(r)h(r)^{2(m-2)}=k \quad \text{ for some }k \in \mathbb{R}. \end{equation}

Moreover, if $(M_{(i,n)}(\mathsf {G},\,\mathsf {K}),\,\mathbb {J})$ has singular orbits, then $ {\mathbb g}$ is Gauduchon if and only if it is Kähler.

Proof. Let $(\tilde {e}_{\alpha })$ be a $Q_{\mathfrak {p}}$-orthonormal basis for $\mathfrak {p}$. Then, a straightforward computation based on equation (4.1) yields

\begin{align*} \operatorname{d\!}{}^{*}\vartheta(r) & =-\left(\frac{\lambda p}{2mn}\right)^{2}f(r)^{-2}(D_N\vartheta)(N)_{\gamma(r)} \\ & \quad -\left(\frac{\lambda p}{2mn}\right)^{2}f(r)^{-2}(D_{T^{*}}\vartheta)(T^{*})_{\gamma(r)} -h(r)^{-2}\sum_{\alpha=1}^{2(m-1)}(D_{\tilde{e}_{\alpha}^{*}}\vartheta)(\tilde{e}_{\alpha}^{*})_{\gamma(r)} \\ & =-\frac{\lambda p}{2mn}f(r)^{-1} \frac{\partial}{\partial r}(\vartheta(N)_{\gamma(r)}) -(m-1)\frac{\lambda p}{mn}\frac{h'(r)}{f(r)h(r)}\vartheta(N)_{\gamma(r)}\\ & = 2(m-1)\frac 1{h(r)^{2}}\left(\left(h(r)h'(r)+\frac{mn}p f(r)\right)\left(\frac{f'(r)}{f(r)}+2(m-2)\frac{h'(r)}{h(r)}\right) \right.\\ & \qquad +\left.\left(h(r)h''(r)+h'(r)^{2}+\frac{mn}p f'(r)\right)\right) \end{align*}

and so $ {\mathbb g}$ is Gauduchon if and only if

\begin{align*} & \Bigg(h(r)h'(r)+\frac{mn}{p} f(r)\Bigg)\left(\frac{f'(r)}{f(r)}+2(m-2)\frac{h'(r)}{h(r)}\right) \\ & \quad +\left(h(r)h''(r)+h'(r)^{2}+\frac{mn}{p} f'(r)\right)=0. \end{align*}

Since $f(r),\, h(r)$ are positive for any $r \in I$ and

\begin{align*} & \frac 1{f(r)h(r)^{2(m-2)}}\frac{\operatorname{d\!}}{\operatorname{d\!} r}\left(\left(h(r)h'(r)+\frac{mn}{p} f(r)\right)f(r)h(r)^{2(m-2)}\right) \\ & \quad=\left(h(r)h'(r)+\frac{mn}{p} f(r)\right)\left(\frac{f'(r)}{f(r)}+2(m-2)\frac{h'(r)}{h(r)}\right) \\ & \qquad +\left(h(r)h''(r)+h'(r)^{2}+\frac{mn}{p} f'(r)\right), \end{align*}

it follows that $ {\mathbb g}$ is Gauduchon if and only if equation (4.2) is satisfied.

Let us assume now that $(M_{(i,n)}(\mathsf {G},\,\mathsf {K}),\,\mathbb {J})$ has a singular orbit, that is $i=2$ or $i=4$. Then, the smoothness conditions at $r=0$ imply that $k=0$ in equation (4.2). Therefore, in this case, $ {\mathbb g}$ is Gauduchon if and only if it is Kähler.

Since the balanced condition is equivalent to $\vartheta =0$, from proposition 3.17 and equation (3.11) we immediately get

Concerning the pluriclosed condition, a tedious but straightforward computation (see Appendix A) shows that

(4.3)\begin{equation} \operatorname{d\!}\operatorname{d\!}^{\,c}\omega(X^{*},Y^{*},Z^{*},W^{*})_{\gamma(r)} = 4\frac{mn}{p}f(r)\Bigg(h(r)h'(r)+ \frac{mn}{p}f(r)\Bigg)(\rho{\wedge}\rho)(X,Y,Z,W), \end{equation}

where $\rho (X,\,Y) = Q_{\mathfrak {p}}(JX,\,Y)$ is again the $\mathsf {G}$-invariant Kähler–Einstein form on $P$. Hence, together with equation (3.11), this proves the following

which completes the proof of theorem B.

5.1 The second-Chern–Einstein equations

Let $(M_{(i,n)}(\mathsf {G},\,\mathsf {K}),\,\mathbb {J},\, {\mathbb g})$ be a Bérard-Bergery standard cohomogeneity one Hermitian manifold and fix a unit speed geodesic $\gamma :\overline {I} \to M$ which intersects orthogonally any $\mathsf {G}$-orbit. Then, by means of proposition 3.15, the second-Chern–Einstein equation

\[ \operatorname{Ric}^{{\rm Ch}[2]}( {\mathbb g})=\frac{\lambda}{2m} {\mathbb g} \]

becomes

(5.1)\begin{equation} \begin{cases} -\dfrac{f''(r)}{f(r)} +\dfrac{2m(m-1)n}{p}\dfrac{f'(r)}{h(r)^{2}} +\dfrac{2m^{2}(m-1)n^{2}}{p^{2}}\dfrac{f(r)^{2}}{h(r)^{4}} = \dfrac{\lambda(r)}{2m} \\ - \dfrac{h''(r)}{h(r)} +\dfrac{h'(r)^{2}}{h(r)^{2}} -\dfrac{f'(r)}{f(r)}\dfrac{h'(r)}{h(r)} +\dfrac{2m(m-1)n}{p}f(r)\dfrac{h'(r)}{h(r)^{3}} \\ \quad -2\left(\dfrac{mn}{p}\right)^{2}\dfrac{f(r)^{2}}{h(r)^{4}} +\dfrac{2m}{h(r)^{2}} = \dfrac{\lambda(r)}{2m} \end{cases}, \end{equation}

where $\lambda : I \to \mathbb {R}$ is an admissible function (see remark 3.10). Note that, in this case, it holds that $\lambda =\operatorname {scal}^{\rm Ch}( {\mathbb g})$.

Our first result in this section concerns the local existence and uniqueness of second-Chern–Einstein metrics, with prescribed Chern-scalar curvature, in a neighbourhood of a singular orbit. More precisely

Proof. Fix a positive number $a>0$ and an admissible function $\lambda : I \to \mathbb {R}$. Let us write

(5.2)\begin{equation} x(r) := \frac{f(r)}{r}, \quad y(r):= x'(r), \ z(r) := h(r) , \ w(r) := h'(r). \end{equation}

Then, a straightforward computation shows that equations (5.1) become

(5.3)\begin{equation} \left\{ \begin{array}{@{}l} v'(r) = \dfrac 1r\,A\cdot v(r)+ N(r,v(r)) \\ v(0) = v_{\operatorname{o}} \end{array}\right., \end{equation}

with

\begin{align*} & v(r) = (x(r), y(r), z(r), w(r))^{t}, \quad A := \left(\begin{array}{@{}cccc@{}} 0 & 0 & 0 & 0 \\ 0 & -2 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 \end{array}\right), \\ & N(r,v) := \left(\begin{array}{@{}c@{}} y \\ \left(\dfrac{2m(m-1)n}{p}x^{2} -\dfrac{\lambda}{2m}x\right) +r\left(\dfrac{2m(m-1)n}{p}xy\right)\\ +\, r^{2}\left(\dfrac{2m^{2}(m-1)n^{2}}{p^{2}}\dfrac{x^{3}}{z^{4}}\right) \\ w \\ \left(\dfrac{w^{2}}{z} -\dfrac{yw}{x} +\dfrac{2m}{z} -\dfrac{\lambda}{2m}z\right) +r\left(\dfrac{2m(m-1)n}{p}\dfrac{xw}{z^{2}}\right)\\ +\, r^{2}\left(-\dfrac{2m^{2}(m-1)n^{2}}{p^{2}}\dfrac{x^{2}}{z^{3}}\right) \end{array}\right). \end{align*}

Moreover, the smoothness conditions for the functions $f$ and $h$, together with the equation $h(0)=a$, imply that

(5.4)\begin{equation} v_{\operatorname{o}} = (1, 0, a, 0)^{t}. \end{equation}

We stress now that the following conditions are satisfied:

• • the function $N=N(r,\,v)$ is smooth in a neighbourhood of $(0,\,v_{\operatorname {o}})$,

• • $A \cdot v_{\operatorname {o}} = 0$,

• • $\det (A -k \operatorname {Id}_4) \neq 0$ for any integer $k\geq 1$.

Then, by the Malgrange theorem [Reference Malgrange39, theorem 7.1], see also [Reference Böhm14, theorem 2.2], there exists a unique solution $v(r)$, defined on an interval $(-\varepsilon,\,\varepsilon )$, to equation (5.3) with initial condition (5.4), which depends continuously on the data $a$, $\lambda$.

By equation (5.2), we obtain a pair $(f,\,h)$ of smooth functions $f,\,h : (-\varepsilon,\,\varepsilon ) \to \mathbb {R}$ which satisfy equations (5.1) such that

(5.5)\begin{equation} f(0)=0, \quad f'(0)=1, \ f''(0)=0, \ h(0)=a, \ h'(0)=0. \end{equation}

Since the pair $(\hat {f},\,\hat {h})$ of functions defined by

\[ \hat{f},\hat{h} : (-\varepsilon,\varepsilon) \to \mathbb{R}, \quad \hat{f}(r) := -f(-r), \ \hat{h}(r) := h(-r) \]

satisfy equations (5.1) with the initial conditions (5.5), by uniqueness we conclude that $f$ is odd and $h$ is even. Therefore, these functions give rise to a smooth Hermitian metric $ {\mathbb g}= {\mathbb g}(f,\,h)$ on the open set $\mathcal {U}^{\rm reg}_{\varepsilon } \subset M_{(i,n)}(\mathsf {G},\,\mathsf {K})^{\rm reg}$, which admits a unique smooth extension over the singular orbit $\mathsf {G} \cdot \gamma (0)$.

Concerning complete solutions to equations (5.1), we point out the following

Remark 5.2 Fundamental examples of complete, non-Kähler, second-Chern– Einstein metrics can be easily found on the manifolds $(M_{(1,n)}(\mathsf {G},\,\mathsf {K}),\,\mathbb {J})$ and $(M_{(3,n)}(\mathsf {G},\,\mathsf {K}),\,\mathbb {J})$. Indeed, the constant functions

(5.6)\begin{equation} f(r) := \frac{p}{mn}, \quad h(r) := 1 \end{equation}

verify the smoothness conditions in case $i=1,\,3$ and so they give rise to homogeneous, smooth metrics which are non-Kähler by equation (3.11) and second-Chern–Einstein by equation (5.1) with constant Chern-scalar curvature $\lambda =4m(m-1)$. These examples include the standard metric on the linear Hopf manifold (see example 3.8).

We also stress that, on manifolds $(M_{(1,n)}(\mathsf {G},\,\mathsf {K}),\,\mathbb {J})$ and $(M_{(3,n)}(\mathsf {G},\,\mathsf {K}),\,\mathbb {J})$, all the metrics (not necessarily of cohomogeneity one) in the conformal class of $ {\mathbb g}= {\mathbb g}(f,\,h)$, with $f,\,h$ given by formula (5.6), are second-Chern–Einstein (see remark 2.4). In particular, by remark 3.11, for any admissible positive function $\phi : I \to \mathbb {R}$, the pair

(5.7)\begin{equation} f_{\phi}(r) := \frac{p}{mn}\phi(r) , \quad h_{\phi}(r) := \phi(r) \end{equation}

solve equations (5.1) with Chern-scalar curvature $\lambda (r) = 2m\big(-\frac {\phi ''(r)}{\phi (r)} +2(m-1)\frac {\phi '(r)+1}{\phi (r)}\big)$. Therefore, in the following, we will focus on Bérard-Bergery manifolds $(M_{(i,n)}(\mathsf {G},\,\mathsf {K}),\,\mathbb {J})$ with singular orbits, namely, the cases $i=2$ and $i=4$.

5.2 Complete second-Chern–Einstein metrics in case of singular orbits

In this section, we will construct complete second-Chern–Einstein metrics on the manifolds $(M_{(2,n)}(\mathsf {G},\,\mathsf {K}),\,\mathbb {J})$ by using the same technique as [Reference Bérard-Bergery9, § 11].

Let us start by noticing that, for the manifold $\mathcal {O}_{\mathbb {C}\mathbb {P}^{m-1}}(-1)$, the functions

(5.8)\begin{equation} f(r) := r, \quad h_k(r) := \sqrt{r^{2}+k^{2}}, \ r \in [0,+\infty), \ k>0 \end{equation}

solve the second-Chern–Einstein equations (5.1) and define Hermitian metrics $ {\mathbb g}_k= {\mathbb g}(f,\,h_k)$ which extend smoothly over the singular orbit. All the metrics $ {\mathbb g}_k$ are non-Kähler and their Chern-scalar curvatures are given by

\[ \operatorname{scal}^{\rm Ch}( {\mathbb g}_k)(r) = 4m(m-1)\frac{2r^{2}+k^{2}}{(r^{2}+k^{2})^{2}}. \]

Note that all these metrics are homothetic, indeed the satisfy $ {\mathbb g}_k = k^{2} {\mathbb g}_1$ (see remark 3.11). More in general, we have

Proof. Fix a manifold $(M_{(2,n)}(\mathsf {G},\,\mathsf {K}),\,\mathbb {J})$ and set

(5.9)\begin{equation} f_{\phi}(r) := \frac{p}{2mn}\phi'(r), \quad h_{\phi}(r) := \sqrt{\phi(r)} \end{equation}

for some smooth, positive, increasing function $\phi : [0,\,+\infty ) \to \mathbb {R}$. Note that, in this case

\[ h_{\phi}(r)h_{\phi}'(r)+\frac{mn}{p}f_{\phi}(r) = \phi'(r) \]

and so this metric is necessarily non-Kähler by equation (3.11). Setting the initial condition $h(0)=1$, second-Chern–Einstein equations (5.1) become

(5.10)\begin{equation} \left\{ \begin{array}{@{}l} \phi(r)\dfrac{\phi'''(r)}{\phi'(r)} -m\phi''(r)+2m=0 \\ \phi(0)=1, \quad \phi'(0)=0, \ \phi''(0)=\dfrac{2mn}{p} \end{array}\right.. \end{equation}

Cauchy problem (5.10) admits a unique smooth solution on some interval $[0,\,\varepsilon )$, which extends to an even smooth function on $(-\varepsilon,\,\varepsilon )$. Let us prove that this solution can be extended to the whole $[0,\,+\infty )$.

Assume that the solution to equation (5.10) is of the form

(5.11)\begin{equation} \phi'(r)=\sqrt{u(\phi(r))}. \end{equation}

Then, from equations (5.10) and (5.11), we get the following Cauchy problem for $u(t)$:

(5.12)\begin{equation} \left\{ \begin{array}{@{}l} tu''(t) -mu'(t) +4m=0 \\ u(1)=0, \quad u'(1)=\dfrac{4mn}{p} \end{array}\right.. \end{equation}

The unique solution to equation (5.12) is the function $u: [1,\,+\infty ) \to \mathbb {R}$ defined by

(5.13)\begin{equation} u(t) := -\frac{4m(n+p)}{p(m+1)}+4t+\frac{4(mn-p)}{p(m+1)}t^{m+1}, \end{equation}

which is smooth, positive and increasing. Hence, the function

\[ \varphi: [1,+\infty) \to \mathbb{R}, \quad \varphi(r) := \int_1^{r}\frac{\operatorname{d\!} t}{\sqrt{u(t)}} \]

is smooth, positive, increasing and, by construction, its inverse $\phi := \varphi ^{-1}$ solves the Cauchy problem (5.10). Therefore, by means of equation (5.9), the proof is completed.

Remark 5.4 The Chern-scalar curvature of the metric $ {\mathbb g}_{\phi }= {\mathbb g}(f_{\phi },\,h_{\phi })$ constructed from equation (5.9) by solving Cauchy problem (5.10) is given by

\[ \operatorname{scal}^{\rm Ch}( {\mathbb g}_{\phi})(r) = 2m\left(-\frac{\phi'''(r)}{\phi'(r)} +(m-1)\frac{\phi''(r)}{\phi(r)} +(m-1)\left(\frac{\phi'(r)}{\phi(r)}\right)^{2}\right) \,. \]

Note that, if $mn-p=0$, which correspond to the manifolds $\mathcal {O}_{\mathbb {C}\mathbb {P}^{m-1}}(-1)$, function (5.13) is $u(t)=4(t-1)$. Hence, we recover the family of examples introduced in formula (5.8).

Finally we observe that, concerning the compact simply-connected manifolds $(M_{(4,n)}(\mathsf {G},\,\mathsf {K}),\,\mathbb {J})$, we have the following

Remark 5.6 The projective space $\mathbb {C}\mathbb {P}^{m}$ is a standard cohomogeneity one manifold with respect to the action of $\mathsf {G}= \mathsf {SU}(m)$ given by $a \cdot [z^{0} : z] := [z^{0} : a \cdot z] \,.$ Even if it is not a Berard–Bérgéry manifold according to definition 3.7, all the formulas in § 3 and § 4 still apply to this specific case. In particular, all the $\mathsf {G}$-invariant metrics on $\mathbb {C}\mathbb {P}^{m}$ are of the form (3.10) with $n=1$ and $p=m$, where $f,\,h: [0,\,\frac {\pi }2] \to \mathbb {R}$ are smooth, positive function satisfying:

• • $f$ is the restriction of a smooth odd function on $\mathbb {R}$ satisfying

  \[ f\left(r+\frac{\pi}2\right) =-f\left(r-\frac{\pi}2\right) \quad \text{ and } \quad f'(0)=1=-f'\left(\frac{\pi}2\right); \]

• • $h$ is the restriction of a smooth even function on $\mathbb {R}$ satisfying

  \[ h\left(r+\frac{\pi}2\right) =-h\left(r-\frac{\pi}2\right) \quad \text{ and } \quad h'\left(\frac{\pi}2\right)=-1 \,. \]

Note that the Fubini–Study metric $ {\mathbb g}_{\rm FS}$ with sectional curvature $1 \leq \sec \leq 4$ corresponds to the functions

\[ f(r):= \frac 12\sin(2r), \quad h(r):= \cos(r). \]

As argued in the proof of proposition 5.5, all the $\mathsf {G}$-invariant second-Chern–Einstein metrics on $\mathbb {C}\mathbb {P}^{m}$ are necessarily conformal to the Fubini–Study metric $ {\mathbb g}_{\rm FS}$. For example, the functions

\[ f(r):= \frac 12\sin(2r), \quad h_k(r):= \cos(r)\sqrt{\sin(r)^{2}+k^{2}\cos(r)^{2}}, \ k>0 \]

define non-Kähler, second-Chern–Einstein metrics $ {\mathbb g}_k$ on $\mathbb {C}\mathbb {P}^{m}$ of the form

\[ {\mathbb g}_k = \varphi_k^{2}\, {\mathbb g}_{\rm FS}, \quad \text{ with } \quad \varphi_k(r) := \frac{k}{\cos(r)^{2}+k^{2}\sin(r)^{2}}. \]

5.3 Constant Chern-scalar curvature metrics in case of singular orbits

In this section, we construct complete constant Chern-scalar curvature metrics $ {\mathbb g}= {\mathbb g}(f,\,h)$ on the complex manifolds $(M_{(2,n)}(\mathsf {G},\,\mathsf {K}),\,\mathbb {J})$ and $(M_{(4,n)}(\mathsf {G},\,\mathsf {K}),\,\mathbb {J})$ by using again the technique exploited in [Reference Bérard-Bergery9, § 11].

Fix a complex manifold $(M_{(i,n)}(\mathsf {G},\,\mathsf {K}),\,\mathbb {J})$, with $i\in \{2,\,4\}$, and set

(5.14)\begin{equation} f_{\phi}(r) := \frac{p}{2mn}\phi(r)\phi'(r), \quad h_{\phi}(r) := \phi(r) \end{equation}

for some smooth, increasing, positive, function $\phi : I \to \mathbb {R}$. Note that, in this case,

\[ h_{\phi}(r)h_{\phi}'(r)+\frac{mn}{p}f_{\phi}(r) = \frac 12\phi(r)\phi'(r) \]

and so this metric is necessarily non-Kähler by equation (3.11). Let $c \in \mathbb {R}$ to be fixed later. Then, the constant Chern-scalar curvature equation

\[ \operatorname{scal}^{\rm Ch}( {\mathbb g}_{\phi})=c \]

for the metric $ {\mathbb g}_{\phi } := {\mathbb g}(f_{\phi },\,h_{\phi })$ becomes

(5.15)\begin{equation} \phi(r)^{2}\frac{\phi'''(r)}{\phi'(r)} +(m+2)\phi(r)\phi''(r) -m(m-1)\phi'(r)^{2} +\frac{c}2\phi(r)^{2}-2m(m-1)=0 \,. \end{equation}

We look for a solution of the form

(5.16)\begin{equation} \phi'(r)=\sqrt{u(\phi(r))} \end{equation}

for some smooth real function $u=u(t)$. Then, we get the following ODE

(5.17)\begin{equation} t^{2}u''(t) +(m+2)tu'(t) -2m(m-1)u(t) +ct^{2}-4m(m-1)=0, \end{equation}

which can be explicitly integrated. Indeed, the following cases occur.

• • If $m=2$, then the solutions to equation (5.17) are

  (5.18)\begin{equation} u_{a,b,c}(t) = at^{-4} -2 +bt-\frac{c}6t^{2}, \quad \text{with } a,b \in \mathbb{R} . \end{equation}
  In this case, the base space of the fibration $M_{(i,n)}(\mathsf {G},\,\mathsf {K}) \to P$ is necessarily $P=\mathbb {C} \mathbb {P}^{1}$ and so $p=2$.

• • If $m=3$, then the solutions to equation (5.17) are

  (5.19)\begin{equation} u_{a,b,c}(t) = at^{-6} -2 +bt^{2}-\frac{c}8\log(t)t^{2},\quad \text{with } a,b \in \mathbb{R}. \end{equation}
  In this case, the only possibilities for the base space of the fibration $M_{(i,n)}(\mathsf {G},\,\mathsf {K}) \to P$ are
  \[ P=\mathbb{C} \mathbb{P}^{2} = \mathsf{SU}(3)/\mathsf{S}(\mathsf{U}(1){\times}\mathsf{U}(2)), \quad P={\rm Gr}(2,\mathbb{R}^{5}) = \mathsf{Sp}(2)/\mathsf{U}(2) \]
  and so $p=3$.

• • If $m>3$, then the solutions to equation (5.17) are given by

  (5.20)\begin{equation} u_{a,b,c}(t) = at^{-2m} -2 +\frac{c}{2(m+1)(m-3)}t^{2} +bt^{m-1}, \quad \text{ with } a,b \in \mathbb{R}. \end{equation}

Then, by means of equations (5.14) and (5.16), we are able to construct constant Chern-scalar curvature metrics on the manifolds $(M_{(2,n)}(\mathsf {G},\,\mathsf {K}),\,\mathbb {J})$ and $(M_{(4,n)}(\mathsf {G},\,\mathsf {K}),\,\mathbb {J})$.

Proof. Setting $h(0)=1$, the smoothness conditions for $f,\,h$ imply that

(5.21)\begin{equation} \phi(0)=1, \quad \phi'(0)=0, \ \quad \phi''(0)=\frac{2mn}{p}. \end{equation}

Let us stress that, if there exists a smooth solution $\phi : [0,\,+\infty ) \to \mathbb {R}$ to equation (5.15) satisfying the boundary conditions (5.21), then it can be extended to a smooth even function on $\mathbb {R}$.

Note that, by means of equation (5.16), conditions (5.21) imply that the solution $u_{a,b,c}$ to the ODE (5.17) verifies

(5.22)\begin{equation} u_{a,b,c}(1)=0, \quad u_{a,b,c}'(1)=\frac{4mn}{p}. \end{equation}

Therefore, we obtain two values $a(c) ,\,b(c)$, depending on $c$, by imposing conditions (5.22):

• • if $m=2$, then by formula (5.18) we get

  \[ a(c):= \frac 25-\frac{c}{30}-\frac 45n , \quad b(c):= \frac 85+\frac{c}5+\frac 45n; \]

• • if $m=3$, then by formula (5.19) we get

  \[ a(c):= \frac 12-\frac{c}{64}-\frac{n}2, \quad b(c):= \frac 32+\frac{c}{64}+\frac{n}2; \]

• • if $m>3$, then by formula (5.20) we get

  \begin{align*} a(c) & := -\frac{(m+1)(4m(2n-p)+4p)+cp}{2p(m+1)(3m-1)}, \\ b(c) & := \frac{4m(m-3)(n+p)-cp}{p(3m-1)(m-3)}. \end{align*}

In all of these three cases it can be directly checked that, for any $c\leq 0$, the function $u_c := u_{a(c),b(c),c}$ is positive and increasing for $t \in (1,\,+\infty )$. Indeed, by means of conditions (5.22), there exists $\varepsilon _{\operatorname {o}}>0$ such that $u_c(t)>0$ and $u'_c(t)>0$ for any $t \in (1,\,1+\varepsilon _{\operatorname {o}})$. Assume by contradiction that there exists $t_{\operatorname {o}}>1$ such that $u'_c(t)>0$ for any $t \in [1,\,t_{\operatorname {o}})$ and $u'_c(t_{\operatorname {o}})=0$. Then, $t_{\operatorname {o}}$ is a local maximum point or a stationary point of inflection for $u(t)$, but equation (5.17) implies that

(5.23)\begin{equation} u_c''(t_{\operatorname{o}})= 2m(m-1)\frac{u_c(t_{\operatorname{o}})}{t_{\operatorname{o}}^{2}} -c +\frac{4m(m-1)}{t_{\operatorname{o}}^{2}} >0, \end{equation}

which is not possible. Hence

\[ \varphi_c: [1,+\infty) \to \mathbb{R}, \quad \varphi_c(r) := \int_1^{r}\frac{\operatorname{d\!} t}{\sqrt{u_c(t)}} \]

is smooth, increasing and, by construction, its inverse $\phi _c := \varphi _c^{-1}$ solves equation (5.15) with the initial conditions (5.21).

Constant Chern-scalar curvature metrics on Hirzebruch surfaces have been constructed by Koca and Lejmi by using the Bérard-Bergery ansatz in [Reference Koca and Lejmi33, theorem 1]. Since in complex dimension $m=2$ the complex manifolds $(M_{(4,n)}(\mathsf {G},\,\mathsf {K}),\,\mathbb {J})$ reduces to the Hirzebruch surfaces (see example 3.8), the next theorem extends their result to $m>2$.

Proof. Setting $h(0)=1$ and $h(\pi )=k>1$, the smoothness conditions for $f,\,h$ imply that

(5.24)\begin{equation} \begin{aligned} & \phi(0)=1, \quad \phi'(0)=0, \quad \phi''(0)=\frac{2mn}{p}, \\ & \phi(\pi)=k, \quad \phi'(\pi)=0, \quad \phi''(\pi)=-\frac{2mn}{kp}. \end{aligned} \end{equation}

Let us stress that, if there exists a smooth solution $\phi : [0,\,\pi ] \to \mathbb {R}$ to equation (5.15) satisfying the boundary conditions (5.24), then it can be extended to a smooth even function on $\mathbb {R}$ satisfying $\phi (\pi +r)=\phi (\pi -r)$. Since the case $m=2$ has already been addressed in [Reference Koca and Lejmi33], we limit ourselves to prove the statement for $m\geq 3$.

Assume $m=3$. Then, by means of equation (5.16), conditions (5.24) imply that the solution $u_{a,b,c}$ given in formula (5.19) verifies

\[ u_{a,b,c}(1)=0, \quad u_{a,b,c}(k)=0 , \quad u_{a,b,c}'(1)=4n, \quad u_{a,b,c}'(k)=-\frac{4n}{k}. \]

By imposing the first three conditions

\[ u_{a,b,c}(1)=0 , \quad u_{a,b,c}(k)=0 , \quad u_{a,b,c}'(1)=4n , \]

we obtain three values $a(k),\,b(k),\,c(k)$ depending on $k$, that are

\begin{align*} a(k) & := -\frac{2k^{6}(6(n-1)\log(k)k^{2}+3(k^{2}-1))}{(8\log(k)-1)k^{8}+1}, \\ b(k) & := \frac{32((n+3)k^{8}-4k^{6}-n+1)}{(8\log(k)-1)k^{8}+1}, \\ c(k) & := \frac{32((n+3)k^{8}-4k^{6}-n+1)}{(8\log(k)-1)k^{8}+1}. \end{align*}

Set $u_k := u_{a(k),b(k),c(k)}$ and observe that

\[ u_k'(k)+\frac{4n}{k}=\frac{\alpha(k)}{k(8\log(k)-1)k^{8}+1)}, \]

with

\begin{align*} \alpha(k) & := -4(n+3)k^{10} +32(n+1)k^{8}\log(k) -4(n-3)k^{8} \\ & \quad+ 32(n-1)k^{2}\log(k) +4(n+3)k^{2} +4(n-3). \end{align*}

Note that

\begin{align*} & \alpha(1)=\alpha'(1)=\alpha''(1)=0, \quad \alpha'''(1)=512n>0, \\ & \lim_{k \to +\infty} \alpha(k) =-\infty \end{align*}

and so there exists $\tilde {k}>1$ such that $\alpha (k)>0$ for any $1< k<\tilde {k}$ and $\alpha (\tilde {k})=0$. Then, we set $u := u_{\tilde {k}}$, so that the function

\[ \varphi: [1,\tilde{k}] \to [0,\pi], \quad \varphi(r) := \int_1^{r}\frac{\operatorname{d\!} t}{\sqrt{u(t)}} \]

is smooth, increasing and its inverse $\phi := \varphi ^{-1}$ solves the ODE (5.15) with the boundary conditions (5.24).

Assume $m>3$. Then, by means of equation (5.16), conditions (5.24) imply that the solution $u_{a,b,c}$ given in formula (5.19) verifies

\[ u_{a,b,c}(1)=0, \quad u_{a,b,c}(k)=0, \quad u_{a,b,c}'(1)=\frac{4mn}{p}, \quad u_{a,b,c}'(k)=-\frac{4mn}{kp}. \]

By imposing the first three conditions

\[ u_{a,b,c}(1)=0 , \quad u_{a,b,c}(k)=0, \quad u_{a,b,c}'(1)=\frac{4mn}{p}, \]

we obtain three values $a(k),\,b(k),\,c(k)$ depending on $k$, that are

\begin{align*} a(k) & := \frac{2\Big(2(mn-p)k^{m-1}+(mp-2mn-p)k^{2}-p(m-3)\Big)}{p\Big(2(m+1)k^{m-1}-(3m-1)k^{2}+(m-3)k^{-2m}\Big)}, \\ b(k) & := \frac{4\Big(-m(n+p)k^{2}+(mn-p)k^{-2m}+(m+1)p\Big)}{p\Big(2(m+1)k^{m-1}-(3m-1)k^{2}+(m-3)k^{-2m}\Big)}, \\ c(k) & := \frac{4\tilde{c}(k)}{p(2(m+1)k^{m-1}-(3m-1)k^{2}+(m-3)k^{-2m})} \,, \end{align*}

with

\begin{align*} \tilde{c}(k) & := 2m(m+1)(m-3)(n+p)k^{m-1}-(3m-1)(m+1)(m-3)p \\ & \phantom{:= \;} +(m^{3}(p-2n)+m^{2}(4n-3p) +m(6mn-p)+3p)k^{-2m}. \end{align*}

Set $u_k := u_{a(k),b(k),c(k)}$ and observe that

\[ u_k'(k)+\frac{4mn}{kp}=\frac{\alpha(k)}{pk^{2}\beta(k)}, \]

where $\alpha (k),\, \beta (k)$ are the polynomials in $k$ defined by

\begin{align*} \alpha(k) & := -4m(m-3)(n+p)k^{3m+2} +4(m+1)((m-1)p+2mn)k^{3m}\\ & \quad -4(3m-1)(mn+p)k^{2m+3} +4(3m-1)(mn-p)k^{m}\\ & \quad +4(m+1)((m-1)p-2mn)k^{3} -4m(m-3)(p-n)k, \\ \beta(k) & := 2(m+1)k^{3m-1} -(3m-1)k^{2(m+1)} +(m-3). \end{align*}

Note that $\beta (1)=0$ and

\[ \beta'(k)=2(m+1)(3m-1)(k^{m-3}-1)k^{2m+1} >0 \quad \text{for any }k>1, \]

hence $\beta (k)>0$ for any $k>1$. Moreover

\begin{align*} & \alpha(1)=\alpha'(1)=\alpha''(1)=0, \\ & \alpha'''(1)=8mn(3m-1)(m-3)(m-1)(m+1)>0, \\ & \lim_{k \to +\infty} \alpha(k) =-\infty \end{align*}

and so there exists $\tilde {k}>1$ such that $\alpha (k)>0$ for any $1< k<\tilde {k}$ and $\alpha (\tilde {k})=0$. Then, we set $u := u_{\tilde {k}}$, so that the function

\[ \varphi: [1,\tilde{k}] \to [0,\pi] , \quad \varphi(r) := \int_1^{r}\frac{\operatorname{d\!} t}{\sqrt{u(t)}} \]

is smooth, increasing and its inverse $\phi := \varphi ^{-1}$ solves the ODE (5.15) with conditions (5.24).

Finally, by means of an argument similar to the one used in the proof of theorem 5.7 (see equation (5.23)), it holds necessarily that $c>0$ in all the cases. This concludes the proof.