# Compact Hermitian surfaces with pointwise constant Gauduchon holomorphic
sectional curvature

Haojie Chen, Xiaolan Nie  Department of Mathematics, Zhejiang Normal
University, Jinhua Zhejiang, 321004, China<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

Motivated by a recent work of Chen-Zheng[8] on Strominger space forms, we
prove that a compact Hermitian surface with pointwise constant holomorphic
sectional curvature with respect to a Gauduchon connection $\nabla^{t}$ is
either Kähler, or an isosceles Hopf surface with an admissible metric and
$t=-1$ or $t=3$. In particular, a compact Hermitian surface with pointwise
constant Lichnerowicz holomorphic sectional curvature is Kähler. We further
generalize the result to the case for the two-parameter canonical connections
introduced by Zhao-Zheng[31], which extends a previous result by Apostolov-
Davidov-Muškarov[2].

††footnotetext: This research is partially supported by NSFC grants No.
11901530, No. 11801516, No. 12071161 and Zhejiang Provincial NSF grant No.
LY19A010017.

##  1\. Introduction

This note mainly concerns Hermitian surfaces with pointwise constant
holomorphic sectional curvature with respect to a Gauduchon connection
$\nabla^{t}$. Given a Hermitian manifold $(M,J,g)$, there is a one-parameter
family of canonical Hermitian connections $\nabla^{t}$ introduced by Gauduchon
[11] in 1997, which is the line joining the Chern connection $\nabla^{c}$, the
Strominger connection $\nabla^{s}$ (also known as Bismut connection) and the
Lichnerowicz connection $\nabla^{l}$. The last one is the restriction of the
Levi-Civita connection $\nabla^{LC}$ on the holomorphic tangent bundle
$T^{1,0}M$ (also known as the associated connection [10] or the induced Levi-
Civita connection [18]). For any $t\in\mathbb{R}$, the Gauduchon connection is
defined by

$\nabla^{t}=\frac{1+t}{2}\nabla^{c}+\frac{1-t}{2}\nabla^{s}=t\nabla^{c}+(1-t)\nabla^{l}.$

In particular, $\nabla^{1}=\nabla^{c}$, $\nabla^{-1}=\nabla^{s}$ and
$\nabla^{0}=\nabla^{l}$ (see also [31]). If $g$ is Kähler, then all
$\nabla^{t}$ coincide with the Levi-Civita connection. Otherwise, they are
mutually different. While each Gauduchon connection exhibits its own geometry,
the properties of the whole connection family may reflect the intrinsic
features of the Hermitian manifold.

There has been many recent studies on the curvatures of special connections on
Hermitian manifolds, e.g. for the Chern connection
([3],[4],[7],[16],[17],[20],[22]), the Levi-Civita connection ([2],[14],[21]),
the Strominger connection ([8],[25],[28],[29]) and the Lichnerowicz connection
([10],[13],[19]). There are also some recent work on the curvatures of general
Gauduchon connections (see [1],[23],[24],[27],[31] etc). Among various
curvature notions, the holomorphic sectional curvature is a natural substitute
of sectional curvature on Hermitian manifolds (see e.g. [15],[32] and the
references therein). A fundamental question is to understand Hermitian
manifolds with constant or pointwise constant holomorphic sectional curvature.
In 1985, Balas-Gauduchon [4] prove that a compact Hermitian surface with
constant nonpositive holomorphic sectional curvature with respect to
$\nabla^{c}$ must be Kähler. This result was beautifully extended by
Apostolov-Davidov-Muškarov [2] in 1996. They prove the following theorem, as
an application of their classification of compact self-dual Hermitian
surfaces.

######  Theorem 1.1 (Apostolov-Davidov-Muškarov).

Any compact Hermitian surface with point-wise constant holomorphic sectional
curvature with respect to the Levi-Civita connection or the Chern connection
must be Kähler.

Recently, an important new progress is made by S. Chen and F. Zheng in [8],
where they study the (weak) Srominger space forms, i.e., compact Hermitian
manifolds with (pointwise) constant Strominger holomorphic sectional
curvature. In particular, they give a complete classification in dimension two
and prove that

######  Theorem 1.2 (Chen-Zheng).

Any compact Hermitian surface $(M,g)$ with pointwise constant Strominger
holomorphic sectional curvature must be either Kähler or an isosceles Hopf
surface with an admissible metric.

Motivated by the above results, we study Hermitian surfaces with pointwise
constant holomorphic sectional curvature with respect to an arbitrary
Gauduchon connection. Using similar techniques as in [8], we prove the
following theorem.

######  Theorem 1.3.

Let $(M,g)$ be a compact Hermitian surface with pointwise constant holomorphic
sectional curvature with respect to a Gauduchon connection $\nabla^{t}$. Then
either (i) $g$ is Käher, or (ii) $(M,g)$ is an isosceles Hopf surface with an
admissible metric and in this case $t=-1$ or $t=3$.

Note that for a Kähler manifold of complex dimension $n\geq 2$, by the Schur’s
lemma, if the holomorphic sectional curvature is pointwise constant, it must
be constant. A complete Kähler manifold with constant holomorphic sectional
curvature is called a complex space form. It is known that a simply connected
complex space form is holomorphically isometric to
$\mathbb{CP}^{n},\mathbb{C}^{n}$ or $\mathbb{CH}^{n}$ ([6],[12]). Therefore,
in case (i) of Theorem 1.3, $(M,g)$ is isomorphic to one of the following:
$\mathbb{CP}^{2}$, complex torus, hyperelliptic surface and a compact quotient
of $\mathbb{CH}^{2}$. Admissible metrics on an isosceles Hopf surface are a
special class of Hermitian metrics which are conformal to the standard Hopf
metric (see section 5 for more details).

######  Remark 1.4.

The case for the Chern connection ($t=1$) is proved by Apostolov-Davidov-
Muškarov [2] and the case for the Strominger connection ($t=-1$ ) is proved by
Chen-Zheng [8].

As $\nabla^{0}=\nabla^{l}$, we have a direct corollary.

######  Corollary 1.5.

Any compact Hermitian surface with pointwise constant holomorphic sectional
curvature with respect to the Lichnerowicz connection is Kähler.

From the proof of Theorem 1.3, it follows that an admissible metric on
isosceles Hopf surfaces has the same pointwise constant holomorphic sectional
curvatures with respect to $\nabla^{s}$ and $\nabla^{3}$. However, the
curvature tensors of $\nabla^{s}$ and $\nabla^{3}$ are different. In
particular, the standard Hopf metric has zero curvature tensor for
$\nabla^{s}$ and nonzero curvature tensor for $\nabla^{3}$. This gives an
example of compact Hermitian surfaces with zero Gauduchon holomorphic
sectional curvature and nonvanishing curvature tensor (Corollary 5.4).

Next, we put the Levi-Civita connection into consideration. As is known, if
$g$ is not Kähler, the Levi-Civita connection $\nabla^{LC}$ is not a Hermitian
connection and does not coincide with any $\nabla^{t}$. As introduced by Zhao-
Zheng in [31], define the two-parameter canonical connections

$D^{t}_{s}=(1-s)\nabla^{t}+s\nabla^{LC}$

for any $(t,s)\in\mathbb{R}^{2}$. It is the connection plane spanned by the
Levi-Civita connection and the one-parameter Gauduchon connections
$\nabla^{t}$. Note that $D^{t}_{1}\equiv\nabla^{LC}$ and
$D^{t}_{0}=\nabla^{t}$. For a non-Kähler metric $g$ and $s\neq 0$, $D^{t}_{s}$
is not a Hermitian connection either. We obtain the following result which is
a generalization of Theorem 1.3.

######  Theorem 1.6.

Let $(M,g)$ be a compact Hermitian surface with pointwise constant holomorphic
sectional curvature with respect to a canonical $(t,s)$ connection
$D^{t}_{s}$. Then either (i) $g$ is Käher, or (ii) $(M,g)$ is an isosceles
Hopf surface with an admissible metric and in this case
$(1-t+ts)^{2}+s^{2}=4$.

######  Remark 1.7.

When $s=1$, $D_{1}^{t}$ is the Levi-Civita connection. In this case, the
result is proved in [2].

As a consequence of Theorem 1.6, we prove the following characterizations.

######  Corollary 1.8.

Let $(M,g)$ be a compact Hermitian surface. If one of the following is
satisfied,

  * (i)

$g$ has positive or negative pointwise constant holomorphic sectional
curvature with respect to $D^{t}_{s}$ for some $(t,s)\in\mathbb{R}^{2}$,

  * (ii)

$g$ has zero holomorphic sectional curvature with respect to $D^{t}_{s}$ for
$(t,s)$ satisfying $(1-t+ts)^{2}+s^{2}\neq 4$,

then $(M,g)$ is Kähler and a complex space form.

On the other side, if $g$ is non-Kähler and has zero holomorphic sectional
curvature with respect to some $D^{t}_{s}$, then it must be a scalar multiple
of the standard metric on an isosceles Hopf surface and
$(1-t+ts)^{2}+s^{2}=4$.

A key step to prove Theorem 1.3 and Theorem 1.6 is to show that a Hermitian
surface with pointwise constant holomorphic sectional curvature with respect
to $\nabla^{t}$ or $D^{t}_{s}$ must be a self-dual Riemannian 4-manifold. We
show that it also holds for general almost Hermitian surfaces (Proposition
3.3). A natural question is to classify compact almost Hermitian surfaces with
constant or pointwise constant Gauduchon holomorphic sectional curvature.

In section 5, we also study Hermitian metrics with pointwise constant
Gauduchon holomorphic sectional curvatures on higher dimensional isosceles
Hopf manifolds and prove the following (Proposition 5.3).

######  Proposition 1.9.

A Hermitian metric $\omega$ on an isosceles Hopf manifold which is conformal
to the standard Hopf metric has pointwise constant holomorphic sectional
curvature with respect to $D^{t}_{s}$ if and only if $(1-t+ts)^{2}+s^{2}=4$
and $\omega$ is an admissible metric.

The structure of the paper is as follows. In section 2, we briefly introduce
the canonical connections and holomorphic sectional curvature on an almost
Hermitian manifold. In section 3, we show that an almost Hermitian surface
with pointwise constant holomorphic sectional curvature must be self-dual. In
section 4, we compute the transformation of curvature components of the
canonical connections $\nabla^{t}$ and $D^{t}_{s}$ under conformal change of
Hermitian metrics on a complex manifold. In section 5, we study Hermitian
metrics with pointwise constant holomorphic sectional curvature on an
isosceles Hopf manifold. In section 6, we prove Theorem 1.6 and Corollary 1.8.
Throughout the paper, we assume $M$ to be connected.

Acknowlegments. We are deeply indebted to Professor Fangyang Zheng for many
stimulating and valuable discussions and particularly to the work [8]. We are
also very grateful to Professors Kefeng Liu and Jiaping Wang for helpful
communications and their encouragement.

##  2\. Canonical connections

In this section, we give some necessary notations. We refer the readers to[8]
[9] [26] for more details. Let $(M,J,g)$ be an almost Hermitian manifold.
Comparing to the Levi-Civita connection $\nabla^{LC}$ which preserves $g$ and
has vanishing torsion, the Chern connection is the unique connection
$\nabla^{c}$ which satisfies $\nabla^{c}g=0,\nabla^{c}J=0$ and has vanishing
$(1,1)$ part of the torsion. Fix any $q\in M$. Let
$\\{e_{1},e_{2},...,e_{n}\\}$ be a local unitary frame of $(1,0)$ vectors
around $q$ with $\\{\varphi^{1},\varphi^{2},...,\varphi^{n}\\}$ being the dual
frame. Assume that

$\displaystyle\nabla^{LC}e_{i}=(\theta_{1})^{j}_{i}e_{j}+(\overline{\theta_{2}})^{j}_{i}\bar{e}_{j},\
\ \ \nabla^{c}e_{i}=\theta^{j}_{i}e_{j},$ (2.1)

where $\theta_{1},\theta_{2},\theta$ are the matrices of connection one-forms
and the Einstein summation notation is used. Denote the torsion forms of the
Chern connection by

$\tau^{i}=T^{i}_{jk}\varphi^{j}\wedge\varphi^{k}+T^{i}_{\bar{j}\bar{k}}\overline{\varphi^{j}}\wedge\overline{\varphi^{k}},$

with $T^{i}_{jk}=-T^{i}_{kj},T^{i}_{\bar{j}\bar{k}}=-T^{i}_{\bar{k}\bar{j}}$.
It is known that $T^{i}_{\bar{j}\bar{k}}=\frac{1}{2}N^{i}_{\bar{j}\bar{k}}$,
where

$N^{i}_{\bar{j}\bar{k}}=-g([\bar{e}_{j},\bar{e}_{k}],\overline{e_{i}})$

are the components of the Nijenhuis tensor. For convenience, here
$T^{i}_{\bar{j}\bar{k}}$ and $N^{i}_{\bar{j}\bar{k}}$ may be different with
the notations in some literature up to a scalar. So if $J$ is integrable, then
$T^{i}_{\bar{j}\bar{k}}=0$. The following relations between $\nabla^{LC}$ and
$\nabla^{c}$ are known ( see e.g. [9][26]).

$\displaystyle\gamma^{j}_{i}$
$\displaystyle=(\theta_{1})^{j}_{i}-\theta^{j}_{i}=T_{ik}^{j}\varphi^{k}-\overline{T^{i}_{jk}}\overline{\varphi^{k}},$
(2.2) $\displaystyle(\theta_{2})^{j}_{i}$
$\displaystyle=\overline{T^{k}_{ij}}\varphi^{k}+(T^{j}_{\bar{i}\bar{k}}+T^{i}_{\bar{k}\bar{j}}+T^{k}_{\bar{i}\bar{j}})\overline{\varphi^{k}}.$
(2.3)

For a connection $\nabla$, denote the curvature tensor to be

$\displaystyle
R^{\nabla}(X,Y)Z=\nabla_{X}\nabla_{Y}Z-\nabla_{Y}\nabla_{X}Z-\nabla_{[X,Y]}Z,$
$\displaystyle R^{\nabla}(X,Y,Z,W)=g(R^{\nabla}(X,Y)Z,W)$

for any $X,Y,Z,W\in TM\otimes\mathbb{C}$. In particular, denote $R$ and
$R^{c}$ to be the curvature tensor of the Levi-Civita connection and the Chern
connection. Assume that $\Theta_{1},\Theta$ are the matrices of curvature
2-forms of $\nabla^{LC},\nabla^{c}$ with items
$(\Theta_{1})^{j}_{i}(X,Y)=R(X,Y,e_{i},\bar{e}_{j})$ and
$\Theta^{j}_{i}(X,Y)=R^{c}(X,Y,e_{i},\bar{e}_{j})$. From the structure
equations (see e.g. [26]), we have:

$\displaystyle(\Theta_{1})^{j}_{i}$
$\displaystyle=d(\theta_{1})^{j}_{i}-(\theta_{1})_{i}^{k}\wedge(\theta_{1})_{k}^{j}-(\overline{\theta_{2}})^{k}_{i}\wedge(\theta_{2})_{k}^{j},$
(2.4) $\displaystyle\Theta^{j}_{i}$
$\displaystyle=d\theta^{j}_{i}-\theta_{i}^{k}\wedge\theta_{k}^{j}.$ (2.5)

Then by (2.2),

$\displaystyle(\Theta_{1})^{j}_{i}-\Theta^{j}_{i}=d\gamma^{j}_{i}-\gamma^{k}_{i}\wedge\gamma_{k}^{j}-\gamma^{k}_{i}\wedge\theta_{k}^{j}-\theta^{k}_{i}\wedge\gamma_{k}^{j}-(\overline{\theta_{2}})^{k}_{i}\wedge(\theta_{2})_{k}^{j}.$
(2.6)

The Lichnerowicz connection $\nabla^{l}$ is the restriction of $\nabla^{LC}$
to $T^{1,0}M$. Therefore, we have

$\nabla^{l}e_{i}=(\theta_{1})^{j}_{i}e_{j}.$

######  Definition 2.1.

For any $t\in\mathbb{R}$, the Gauduchon connection $\nabla^{t}$ is defined to
be

$\nabla^{t}=t\nabla^{c}+(1-t)\nabla^{l}.$

Assume that $\nabla^{t}e_{i}=(\theta^{t})^{j}_{i}e_{j}$. Then

$\displaystyle(\theta^{t})_{i}^{j}=t\theta^{j}_{i}+(1-t)(\theta_{1})^{j}_{i}=\theta^{j}_{i}+(1-t)\gamma^{j}_{i}.$
(2.7)

Denote $\Theta^{t}$ to be the matrix of curvature 2-forms of $\nabla^{t}$.
Then

$\displaystyle(\Theta^{t})^{j}_{i}=d(\theta^{t})^{j}_{i}-(\theta^{t})^{k}_{i}\wedge(\theta^{t})_{k}^{j}.$
(2.8)

Therefore, by (2.5), (2.7) and (2.8)

$\displaystyle(\Theta^{t})^{j}_{i}-\Theta^{j}_{i}=(1-t)d\gamma^{j}_{i}-(1-t)\gamma^{k}_{i}\wedge\theta_{k}^{j}-(1-t)\theta^{k}_{i}\wedge\gamma_{k}^{j}-(1-t)^{2}\gamma^{k}_{i}\wedge\gamma_{k}^{j}.$
(2.9)

Then by (2.6) and (2.9)

$\displaystyle(\Theta^{t})^{j}_{i}-(\Theta_{1})^{j}_{i}=-td\gamma^{j}_{i}+t\gamma^{k}_{i}\wedge\theta_{k}^{j}+t\theta^{k}_{i}\wedge\gamma_{k}^{j}+(2t-t^{2})\gamma^{k}_{i}\wedge\gamma_{k}^{j}+(\overline{\theta_{2}})^{k}_{i}\wedge(\theta_{2})_{k}^{j}.$
(2.10)

Note that
$R^{t}_{k\bar{l}i\bar{j}}=(\Theta^{t})^{j}_{i}(e_{k},\bar{e}_{l}),R_{k\bar{l}i\bar{j}}=(\Theta_{1})^{j}_{i}(e_{k},\bar{e}_{l})$.
Then by (2.10) we get the following lemma which has been obtained in
Proposition 4.2 in [9].

######  Lemma 2.2.

$\displaystyle R^{t}_{k\bar{l}i\bar{j}}$
$\displaystyle=R_{k\bar{l}i\bar{j}}+t(T^{j}_{ik,\bar{l}}+\overline{T^{i}_{jl,\bar{k}}})+(t^{2}-2t)(T^{r}_{ik}\overline{T^{r}_{jl}}-T^{j}_{rk}\overline{T^{i}_{rl}})$
$\displaystyle-\overline{T^{k}_{rj}}T^{l}_{ir}+(T^{j}_{\bar{r}\bar{l}}+T^{r}_{\bar{l}\bar{j}}+T^{l}_{\bar{r}\bar{j}})(\overline{T^{r}_{\bar{i}\bar{k}}}+\overline{T^{i}_{\bar{k}\bar{r}}}+\overline{T^{k}_{\bar{i}\bar{r}}})$

If $J$ is integrable, then

$\displaystyle R^{t}_{k\bar{l}i\bar{j}}$
$\displaystyle=R_{k\bar{l}i\bar{j}}+t(T^{j}_{ik,\bar{l}}+\overline{T^{i}_{jl,\bar{k}}})+(t^{2}-2t)(T^{r}_{ik}\overline{T^{r}_{jl}}-T^{j}_{rk}\overline{T^{i}_{rl}})-\overline{T^{k}_{rj}}T^{l}_{ir}$

Here the subscripts in $T$ stands for covariant derivatives with respect to
$\nabla^{c}$.

Next, we introduce the two-parameter connections studied by Zhao-Zheng in
[31]. It is the connection plane spanned by the Levi-Civita connection and the
Gauduchon connections.

######  Definition 2.3.

For any $(t,s)\in\mathbb{R}^{2}$, define the $(t,s)$ canonical connection to
be

$D^{t}_{s}=(1-s)\nabla^{t}+s\nabla^{LC}.$

By (2.1), (2.2) and (2.7), writing $p=t-ts$, we have

$\displaystyle D^{t}_{s}e_{i}$
$\displaystyle=(\theta_{i}^{j}+(1-p)\gamma^{j}_{i})e_{j}+s(\overline{\theta_{2}})^{j}_{i}\bar{e}_{j}=(\theta^{p})_{i}^{j}e_{j}+s(\overline{\theta_{2}})^{j}_{i}\bar{e}_{j}.$
(2.11)

Denote $\Theta^{D}$ to be the matrix of curvature 2-forms of $D^{t}_{s}$. Then
we have

$\displaystyle(\Theta^{D})^{j}_{i}=d(\theta^{p})^{j}_{i}-(\theta^{p})_{i}^{k}\wedge(\theta^{p})_{k}^{j}-s^{2}\overline{(\theta_{2})}^{k}_{i}\wedge(\theta_{2})_{k}^{j}.$
(2.12)

So by (2.3), we have

$\displaystyle
R^{D}_{k\bar{l}i\bar{j}}=R^{p}_{k\bar{l}i\bar{j}}+s^{2}\\{\overline{T^{k}_{rj}}T^{l}_{ir}-(T^{j}_{\bar{r}\bar{l}}+T^{r}_{\bar{l}\bar{j}}+T^{l}_{\bar{r}\bar{j}})(\overline{T^{r}_{\bar{i}\bar{k}}}+\overline{T^{i}_{\bar{k}\bar{r}}}+\overline{T^{k}_{\bar{i}\bar{r}}})\\}.$
(2.13)

Combing with Lemma 2.2, we have

######  Lemma 2.4.

$\displaystyle R^{D}_{k\bar{l}i\bar{j}}$
$\displaystyle=R_{k\bar{l}i\bar{j}}+p(T^{j}_{ik,\bar{l}}+\overline{T^{i}_{jl,\bar{k}}})+(p^{2}-2p)(T^{r}_{ik}\overline{T^{r}_{jl}}-T^{j}_{rk}\overline{T^{i}_{rl}})$
$\displaystyle+(s^{2}-1)\\{\overline{T^{k}_{rj}}T^{l}_{ir}-(T^{j}_{\bar{r}\bar{l}}+T^{r}_{\bar{l}\bar{j}}+T^{l}_{\bar{r}\bar{j}})(\overline{T^{r}_{\bar{i}\bar{k}}}+\overline{T^{i}_{\bar{k}\bar{r}}}+\overline{T^{k}_{\bar{i}\bar{r}}})\\}$
(2.14)

If $J$ is integrable, then

$\displaystyle R^{D}_{k\bar{l}i\bar{j}}$
$\displaystyle=R_{k\bar{l}i\bar{j}}+p(T^{j}_{ik,\bar{l}}+\overline{T^{i}_{jl,\bar{k}}})+(p^{2}-2p)(T^{r}_{ik}\overline{T^{r}_{jl}}-T^{j}_{rk}\overline{T^{i}_{rl}})+(s^{2}-1)\overline{T^{k}_{rj}}T^{l}_{ir}$

Here the subscripts in $T$ stands for covariant derivatives with respect to
$\nabla^{c}$.

In the rest of the section, we discuss the holomorphic sectional curvature of
any metric connection. Given any metric connection $\nabla$, namely $\nabla
g=0$, the holomorphic sectional curvature of $\nabla$ on a $J$-invariant
tangent plane $\Sigma$ is denoted to be $H^{\nabla}(\Sigma)$ which is the
sectional curvature of $\nabla$ on $\Sigma$. If $\Sigma$ is spanned by
$\\{X,JX\\}$ for $X\in TM\setminus\\{0\\}$, then

$H^{\nabla}(\Sigma)=\dfrac{R^{\nabla}(X,JX,JX,X)}{|X|^{4}}.$

Let $\eta=\dfrac{1}{\sqrt{2}}(X-\sqrt{-1}JX)$. It holds that

$H^{\nabla}(\Sigma)=H^{\nabla}(\eta)=\dfrac{R^{\nabla}(\eta,\overline{\eta},\eta,\overline{\eta})}{|\eta|^{4}}.$

Assume that $R^{\nabla}_{i\bar{j}k\bar{l}}$ are the components of
$R^{\nabla}$. Define its symmetrization tensor components to be

$\displaystyle\widehat{R}^{\nabla}_{i\bar{j}k\bar{l}}=\dfrac{1}{4}(R^{\nabla}_{i\bar{j}k\bar{l}}+R^{\nabla}_{k\bar{j}i\bar{l}}+R^{\nabla}_{i\bar{l}k\bar{j}}+R^{\nabla}_{k\bar{l}i\bar{j}}).$
(2.15)

The following characterization of Hermitian metric of pointwise constant
holomorphic sectional curvature is given in [3] (see also [7]).

######  Lemma 2.5.

At any $q\in M$, $H^{\nabla}(\eta)=c$ for any $\eta\in
T_{q}^{1,0}M\setminus\\{0\\}$ if and only if
$\widehat{R}^{\nabla}_{i\bar{j}k\bar{l}}=\dfrac{c}{2}(\delta_{ij}\delta_{kl}+\delta_{il}\delta_{kj})$
with respect to any unitary frames near $q$.

##  3\. self-duality

In this section, we relate almost Hermitian surfaces with pointwise constant
holomorphic sectional curvature to Riemannian self-dual 4-manifolds. We refer
to [2],[5] and the reference therein for more details on self-dual
4-manifolds.

Given a Riemannian 4-manifold $(M,g)$, $g$ induces a metric on the bundle
$\Lambda^{2}TM$. The curvature of the Levi-Civita connection $\nabla^{LC}$
induces a self-adjoint endomorphism $\mathcal{R}$ of $\Lambda^{2}TM$ by
$g(\mathcal{R}(X\wedge Y),Z\wedge W)=-R(X,Y,Z,W)$, for $X,Y,Z,W\in TM$. The
Hodge $*$ operator on $\Lambda^{2}TM$ preserves $g$ and satisfies $*^{2}=id$.
The $+1,-1$ eigenbundles of $*$ are denoted by
$\Lambda^{2}_{+},\Lambda^{2}_{-}$. Define the Weyl tensor operator to be

$\mathcal{W}=\dfrac{1}{2}(\mathcal{R}+*\mathcal{R}*)-\dfrac{s_{g}}{12}id,$

where $s_{g}$ is the Riemannian scaler curvature of $g$. Define

$\mathcal{W}_{+}=\dfrac{1}{2}(\mathcal{W}+*\mathcal{W}),\ \ \
\mathcal{W}_{-}=\dfrac{1}{2}(\mathcal{W}-*\mathcal{W}).$

Then
$\mathcal{W}_{+}|_{\Lambda^{2}_{+}}=\mathcal{W}|_{\Lambda^{2}_{+}},\mathcal{W}_{+}|_{\Lambda^{2}_{-}}=0$
and
$\mathcal{W}_{-}|_{\Lambda^{2}_{-}}=\mathcal{W}|_{\Lambda^{2}_{-}},\mathcal{W}_{-}|_{\Lambda^{2}_{+}}=0$.
And for $\alpha_{+},\beta_{+}\in\Lambda^{2}_{+}$,
$\alpha_{-},\beta_{-}\in\Lambda^{2}_{-}$,

$\displaystyle g(\mathcal{W}_{+}(\alpha_{+}),\beta_{+})$
$\displaystyle=g(\mathcal{R}(\alpha_{+}),\beta_{+})-\dfrac{s_{g}}{12}g(\alpha_{+},\beta_{+}),$
$\displaystyle g(\mathcal{W}_{-}(\alpha_{-}),\beta_{-})$
$\displaystyle=g(\mathcal{R}(\alpha_{-}),\beta_{-})-\dfrac{s_{g}}{12}g(\alpha_{-},\beta_{-}).$
(3.1)

The operators $\mathcal{R},\mathcal{W},\mathcal{W}_{+},\mathcal{W}_{-}$ and
the inner product $g$ can be all extended complex linearly to
$\Lambda^{2}TM\otimes\mathbb{C}$.

######  Definition 3.1.

$(M,g)$ is called self-dual (anti-self-dual) if $\mathcal{W}_{-}=0$ (resp.
$\mathcal{W}_{+}=0$).

For an almost Hermitian surface $(M,J,g)$, fix $q\in M$ and assume
$e_{1},e_{2}$ to be a local unitary frame in $T^{1,0}M$. Then
$\\{e_{1}\wedge\bar{e}_{2},\dfrac{1}{\sqrt{2}}(e_{1}\wedge\bar{e}_{1}-e_{2}\wedge\bar{e}_{2}),\bar{e}_{1}\wedge
e_{2}\\}$ forms a unitary basis of $\Lambda^{2}_{-}\otimes\mathbb{C}$.
Applying (3.1) to this basis, the following characterization of self-duality
of almost Hermitian surfaces can be obtained, which has been obtained in Lemma
4.1 in [2] for the Hermitian case (see also Lemma 3 in [8]) and in the proof
of Theorem 1.3 in [16] for the almost Hermitian case.

######  Lemma 3.2.

Let $(M,J,g)$ be an almost Hermitian surface. Then $(M,g)$ is self dual if and
only if the following equations hold:

$\displaystyle R_{1\bar{2}1\bar{2}}=0,\ \ \
R_{1\bar{2}2\bar{2}}-R_{1\bar{2}1\bar{1}}=0,$ $\displaystyle
2R_{1\bar{2}2\bar{1}}+2R_{1\bar{1}2\bar{2}}-R_{1\bar{1}1\bar{1}}-R_{2\bar{2}2\bar{2}}=0.$

Then we prove

######  Proposition 3.3.

Let $(M,g)$ be an almost Hermitian surface and $(t,s)\in\mathbb{R}^{2}$.
Assume that the holomorphic sectional curvature with respect to $D^{t}_{s}$ is
pointwise constant. Then $M$ is self-dual.

###### Proof.

We follow the same steps as in [8]. As $D^{t}_{s}$ has pointwise constant
holomorphic sectional curvature $c$, by Lemma 2.5, for any unitary frame,

$\widehat{R}^{D}_{i\bar{j}k\bar{l}}=\dfrac{c}{2}(\delta_{ij}\delta_{kl}+\delta_{il}\delta_{kj}).$

In particular,

$\displaystyle\widehat{R}^{D}_{1\bar{2}1\bar{2}}=\widehat{R}^{D}_{1\bar{1}1\bar{2}}=\widehat{R}^{D}_{1\bar{2}2\bar{2}}=0,$
$\displaystyle\widehat{R}^{D}_{1\bar{1}1\bar{1}}=\widehat{R}^{D}_{2\bar{2}2\bar{2}}=c,\
\ \ \widehat{R}^{D}_{1\bar{1}2\bar{2}}=\dfrac{c}{2}.$ (3.2)

Next, from (2.14) we have the following:

$\displaystyle\widehat{R}^{D}_{k\bar{l}i\bar{j}}$
$\displaystyle=\widehat{R}_{k\bar{l}i\bar{j}}-\dfrac{1}{4}(p^{2}-2p-1+s^{2})(T^{j}_{rk}\overline{T^{i}_{rl}}+T^{j}_{ri}\overline{T^{k}_{rl}}+T^{l}_{rk}\overline{T^{i}_{rj}}+T^{l}_{ri}\overline{T^{k}_{rj}})$
$\displaystyle+(1-s^{2})(T^{j}_{\bar{r}\bar{l}}+T^{l}_{\bar{r}\bar{j}})(\overline{T^{i}_{\bar{k}\bar{r}}}+\overline{T^{k}_{\bar{i}\bar{r}}}).$

In particular, letting $b=p^{2}-2p-1+s^{2}$, we get

$\displaystyle\widehat{R}^{D}_{1\bar{2}1\bar{2}}$
$\displaystyle=\widehat{R}_{1\bar{2}1\bar{2}},$
$\displaystyle\widehat{R}^{D}_{1\bar{2}2\bar{2}}$
$\displaystyle=\widehat{R}_{1\bar{2}2\bar{2}}-\frac{1}{2}bT^{2}_{12}\overline{T^{1}_{12}}+2(1-s^{2})T^{2}_{\bar{1}\bar{2}}\overline{T^{1}_{\bar{2}\bar{1}}},$
$\displaystyle\widehat{R}^{D}_{1\bar{1}1\bar{2}}$
$\displaystyle=\widehat{R}_{1\bar{1}1\bar{2}}-\frac{1}{2}bT^{2}_{21}\overline{T^{1}_{21}}+2(1-s^{2})T^{2}_{\bar{2}\bar{1}}\overline{T^{1}_{\bar{1}\bar{2}}},$
$\displaystyle\widehat{R}^{D}_{1\bar{1}1\bar{1}}$
$\displaystyle=\widehat{R}_{1\bar{1}1\bar{1}}-b|T^{1}_{21}|^{2}-4(1-s^{2})|T^{1}_{\bar{2}\bar{1}}|^{2},$
$\displaystyle\widehat{R}^{D}_{2\bar{2}2\bar{2}}$
$\displaystyle=\widehat{R}_{2\bar{2}2\bar{2}}-b|T^{2}_{12}|^{2}-4(1-s^{2})|T^{2}_{\bar{1}\bar{2}}|^{2},$
$\displaystyle\widehat{R}^{D}_{1\bar{1}2\bar{2}}$
$\displaystyle=\widehat{R}_{1\bar{1}2\bar{2}}-\frac{1}{4}b(|T^{2}_{21}|^{2}+|T^{1}_{12}|^{2})-(1-s^{2})(|T^{2}_{\bar{1}\bar{2}}|^{2}+|T^{1}_{\bar{1}\bar{2}}|^{2}).$

So from (3.2), we get

$\displaystyle\widehat{R}_{1\bar{2}1\bar{2}}=0,\ \ \
\widehat{R}_{1\bar{1}1\bar{2}}-\widehat{R}_{1\bar{2}2\bar{2}}=0$
$\displaystyle
4\widehat{R}_{1\bar{1}2\bar{2}}-\widehat{R}_{1\bar{1}1\bar{1}}-\widehat{R}_{2\bar{2}2\bar{2}}=0$
(3.3)

For curvatures of the Levi-Civita connection, we have

$\displaystyle\widehat{R}_{1\bar{2}1\bar{2}}$
$\displaystyle={R}_{1\bar{2}1\bar{2}},\ \ \
\widehat{R}_{1\bar{1}1\bar{2}}=\dfrac{1}{2}({R}_{1\bar{1}1\bar{2}}+{R}_{1\bar{2}1\bar{1}})={R}_{1\bar{2}1\bar{1}},$
$\displaystyle\widehat{R}_{1\bar{2}2\bar{2}}$
$\displaystyle=\dfrac{1}{2}({R}_{1\bar{2}2\bar{2}}+{R}_{2\bar{2}1\bar{2}})={R}_{1\bar{2}2\bar{2}},$
$\displaystyle\widehat{R}_{1\bar{1}2\bar{2}}$
$\displaystyle=\dfrac{1}{4}({R}_{1\bar{1}2\bar{2}}+{R}_{2\bar{1}1\bar{2}}+{R}_{1\bar{2}2\bar{1}}+{R}_{2\bar{2}1\bar{1}})=\dfrac{1}{2}({R}_{1\bar{1}2\bar{2}}+{R}_{1\bar{2}2\bar{1}}),$
$\displaystyle\widehat{R}_{1\bar{1}1\bar{1}}$
$\displaystyle={R}_{1\bar{1}1\bar{1}},\ \ \
\widehat{R}_{2\bar{2}2\bar{2}}={R}_{2\bar{2}2\bar{2}}$

Therefore, putting them into (3.3), we get

$\displaystyle{R}_{1\bar{2}1\bar{2}}=0,\ \ \ \
{R}_{1\bar{2}1\bar{1}}-{R}_{1\bar{2}2\bar{2}}=0$ $\displaystyle
2({R}_{1\bar{1}2\bar{2}}+{R}_{1\bar{2}2\bar{1}})-{R}_{1\bar{1}1\bar{1}}-R_{2\bar{2}2\bar{2}}=0$

These are exactly the equations in Lemma 3.2. So we finish the proof. ∎

######  Remark 3.4.

When $(M,g)$ is almost Kähler, i.e. $d\omega=0$, the above result for the
Gauduchon connection $\nabla^{t}$ has been proved in [16].

##  4\. Conformal changes for curvatures of canonical connections

In this section, we compute the transformation of curvature components of the
canonical connections $\nabla^{t}$ and $D^{t}_{s}$ under conformal change of
Hermitian metrics on a complex manifold. This will be applied in the next
sections to prove the main results.

We first compute the case for $\nabla^{t}$ which is more illustrative. Let $g$
be a Hermitian metric on a complex manifold $(M,J)$. Fix any $q\in M$. As
before, let $\\{e_{1},e_{2},...,e_{n}\\}$ be a local unitary frame of $(1,0)$
vectors near $q$ with $\\{\varphi^{1},\varphi^{2},...,\varphi^{n}\\}$ being
the dual frame. For any $t\in\mathbb{R}$, denote $\nabla^{t}$ to be the
$t$-Gauduchon connection of $g$. Write
$\nabla^{t}e_{i}=(\theta^{t})^{j}_{i}e_{j}$. Then

$\displaystyle(\theta^{t})_{i}^{j}=\theta_{i}^{j}+(1-t)(T_{ik}^{j}\varphi^{k}-\overline{T_{jk}^{i}}\overline{\varphi}^{k}),$
(4.1)

where $\theta$ is the connection matrix of the Chern connection $\nabla^{c}$
of $g$ and $\\{T_{ik}^{j}\\}$ are the components of the torsion forms of the
$\nabla^{c}$ by $\tau^{i}=T^{i}_{jk}\varphi^{j}\wedge\varphi^{k}$ with
$T^{i}_{jk}=-T^{i}_{kj}$. Note that there are no components of
$T^{i}_{\bar{j}\bar{k}}$ since $J$ is integrable.

Let $\widetilde{g}=e^{2f}g$ be a metric conformal to $g$. Define
$\widetilde{e}_{i}=e^{-f}e_{i}$ and
$\widetilde{\varphi}^{i}=e^{f}\varphi^{i}$. Then $\\{\widetilde{e}_{i}\\}$
forms a unitary frame of $\widetilde{g}$ near $q$ with the unitary coframe
being $\\{\widetilde{\varphi}^{i}\\}$. Denote $\widetilde{\nabla}^{c}$ to be
the Chern connection of $\widetilde{g}$ with $\widetilde{\theta}$ being the
connection matrix and $\\{\widetilde{T}_{jk}^{i}\\}$ being the components of
the torsion forms $\widetilde{\tau}^{i}$ of $\widetilde{\nabla}^{c}$. Direct
calculations from the structure equations (see [8], [28]) give that

$\displaystyle(\widetilde{\theta})_{i}^{j}=\theta_{i}^{j}+(\partial
f-\bar{\partial}f)\delta_{ij},\ \ \ \
\tilde{\tau}^{i}=e^{f}(\tau^{i}+2\partial f\varphi^{i}).$ (4.2)

Then

$\displaystyle\widetilde{T}_{jk}^{i}=e^{-f}(T_{jk}^{i}+f_{j}\delta_{ik}-f_{k}\delta_{ij}),$
(4.3)

where $f_{j}=e_{j}f$. Denote $\widetilde{\nabla}^{t}$ to be the $t$-Gauduchon
connection of $\widetilde{g}$. Assume that
$\widetilde{\nabla}^{t}\widetilde{e}_{i}=(\widetilde{\theta}^{t})^{j}_{i}\widetilde{e}_{j}$.
Then by (4.1), it follows that

$\displaystyle(\widetilde{\theta}^{t})_{i}^{j}-(\theta^{t})_{i}^{j}=t(\partial
f-\bar{\partial}f)\delta_{ij}+(1-t)(f_{i}\varphi^{j}-f_{\bar{j}}\overline{\varphi^{i}}).$
(4.4)

Denote $\Theta^{t}$ and $\widetilde{\Theta}^{t}$ to be the curvature matrix of
$\nabla^{t}$ and $\widetilde{\nabla}^{t}$. As

$\displaystyle(\Theta^{t})_{i}^{j}=d(\theta^{t})_{i}^{j}-(\theta^{t})_{i}^{k}\wedge(\theta^{t})_{k}^{j},\
\ \ \
(\widetilde{\Theta}^{t})_{i}^{j}=d(\widetilde{\theta}^{t})_{i}^{j}-(\widetilde{\theta}^{t})_{i}^{k}\wedge(\widetilde{\theta}^{t})_{k}^{j},$

by (4.4), we have

$\displaystyle[(\widetilde{\Theta}^{t})_{i}^{j}]^{1,1}-[(\Theta^{t})_{i}^{j}]^{1,1}$
(4.5) $\displaystyle=$
$\displaystyle-2t\partial\bar{\partial}f\delta_{ij}+(1-t)(\bar{\partial}(f_{i}\varphi^{j})-\partial(f_{\bar{j}}\overline{\varphi^{i}}))-(1-t)^{2}(f_{k}f_{\bar{k}}\varphi^{j}\wedge\overline{\varphi^{i}}-f_{i}f_{\bar{j}}\varphi^{k}\wedge\overline{\varphi^{k}}).$

Here $[(\widetilde{\Theta}^{t})_{i}^{j}]^{1,1}$ means the $(1,1)$ part of the
curvature forms. Let

$\displaystyle R^{t}_{k\bar{l}i\bar{j}}$
$\displaystyle=(\Theta^{t})_{i}^{j}(e_{k},e_{\bar{l}})=R^{t}(e_{k},e_{\bar{l}},e_{i},e_{\bar{j}}),$
$\displaystyle\widetilde{R}^{t}_{k\bar{l}i\bar{j}}$
$\displaystyle=(\widetilde{\Theta}^{t})_{i}^{j}(\widetilde{e}_{k},\widetilde{e}_{\bar{l}})=\widetilde{R}^{t}(\widetilde{e}_{k},\widetilde{e}_{\bar{l}},\widetilde{e}_{i},\widetilde{e}_{\bar{j}}).$

By Lemma 1 in [8], we may choose a local unitary frame of (1,0) vectors near
$q$ such that $\theta^{t}|q=0$. By (4.1) and the structure equation
$d\varphi^{i}=-\theta^{i}_{j}\wedge\varphi^{j}+\tau^{i}$, we have at $q$

$\displaystyle\bar{\partial}\varphi^{i}=-(\theta^{i}_{j})^{0,1}\wedge\varphi^{j}=(1-t)\overline{T^{k}_{il}}\varphi^{k}\wedge\overline{\varphi^{l}}.$

Let
$f_{k\bar{l}}=\overline{e_{l}}e_{k}f-(\nabla^{t}_{\overline{e_{l}}}e_{k})f,\
f_{\bar{l}k}=e_{k}\overline{e_{l}}f-(\nabla^{t}_{e_{k}}\overline{e_{l}})f$,
i.e., $f_{k\bar{l}}$ and $f_{\bar{l}k}$ are the covariant derivatives of $f$
with respect to $\nabla^{t}$. Then

$\displaystyle
f_{\bar{j}k}-f_{k\bar{j}}=(1-t)(f_{\bar{r}}T_{rk}^{j}-f_{r}\overline{T_{rj}^{k}}).$
(4.6)

Then at $q$,

$\displaystyle\partial\bar{\partial}f(e_{k},\overline{e_{l}})=-\bar{\partial}\partial
f(e_{k},\overline{e_{l}})=f_{k\bar{l}}-(1-t)f_{r}\overline{T_{rl}^{k}}.$ (4.7)

Also,

$\displaystyle[\bar{\partial}(f_{i}\varphi^{j})-\partial(f_{\bar{j}}\overline{\varphi^{i}})](e_{k},\overline{e_{l}})$
$\displaystyle=$ $\displaystyle-
f_{i\bar{l}}\delta_{jk}-f_{k\bar{j}}\delta_{il}+(1-t)(f_{i}\overline{T_{jl}^{k}}-f_{\bar{r}}T_{rk}^{j}\delta_{il}+f_{r}\overline{T_{rj}^{k}}\delta_{il}+f_{\bar{j}}T_{ik}^{l}).$
(4.8)

It follows from (4.5) that

$\displaystyle
e^{2f}\widetilde{R}^{t}_{k\bar{l}i\bar{j}}-R^{t}_{k\bar{l}i\bar{j}}$
$\displaystyle=$
$\displaystyle-2tf_{k\bar{l}}\delta_{ij}+2t(1-t)f_{r}\overline{T_{rl}^{k}}\delta_{ij}-(1-t)(f_{i\bar{l}}\delta_{jk}+f_{k\bar{j}}\delta_{il})$
$\displaystyle+(1-t)^{2}(f_{i}\overline{T_{jl}^{k}}-f_{\bar{r}}T_{rk}^{j}\delta_{il}+f_{r}\overline{T_{rj}^{k}}\delta_{il}+f_{\bar{j}}T_{ik}^{l}-f_{r}f_{\bar{r}}\delta_{jk}\delta_{il}+f_{i}f_{\bar{j}}\delta_{kl})$
(4.9)

Note that the above equality is tensorial and then holds for any unitary
frames.
Denote $\widehat{R}^{t}_{i\bar{j}k\bar{l}}$ and
$\widehat{R}^{t}_{i\bar{j}k\bar{l}}$ to be the symmetrization of the curvature
$R^{t}$ and $\widetilde{R}^{t}$ as defined in (2.15). From (4.9) we get

$\displaystyle
e^{2f}\widehat{\widetilde{R}^{t}}_{k\bar{l}i\bar{j}}-\hat{R^{t}}_{k\bar{l}i\bar{j}}$
$\displaystyle=$
$\displaystyle-\dfrac{1}{2}(f_{i\bar{l}}\delta_{jk}+f_{k\bar{l}}\delta_{ij}+f_{i\bar{j}}\delta_{kl}+f_{k\bar{j}}\delta_{il})+\dfrac{(1-t)^{2}}{4}(f_{i}f_{\bar{j}}\delta_{kl}+f_{k}f_{\bar{j}}\delta_{il}+f_{i}f_{\bar{l}}\delta_{kj}+f_{k}f_{\bar{l}}\delta_{ij})$
$\displaystyle-\dfrac{(1-t)^{2}}{4}(f_{\bar{r}}T_{rk}^{j}\delta_{il}+f_{\bar{r}}T_{rk}^{l}\delta_{ij}+f_{\bar{r}}T_{ri}^{j}\delta_{kl}+f_{\bar{r}}T_{ri}^{l}\delta_{kj}+2f_{r}f_{\bar{r}}(\delta_{kl}\delta_{ij}+\delta_{jk}\delta_{il}))$
$\displaystyle+\dfrac{1-t^{2}}{4}(f_{r}\overline{T_{rl}^{k}}\delta_{ij}+f_{r}\overline{T_{rj}^{k}}\delta_{il}+f_{r}\overline{T_{rl}^{i}}\delta_{kj}+f_{r}\overline{T_{rj}^{i}}\delta_{kl}).$

Assume that $g$ is Kähler. Then
$R^{t}_{i\bar{j}k\bar{l}}=R_{i\bar{j}k\bar{l}}=\widehat{R}_{i\bar{j}k\bar{l}}$
for any $t\in\mathbb{R}$ and $T^{i}_{jk}=0$. So

$\displaystyle
e^{2f}\widehat{\widetilde{R}^{t}}_{k\bar{l}i\bar{j}}-R_{k\bar{l}i\bar{j}}=$
$\displaystyle-\dfrac{1}{2}(f_{i\bar{l}}\delta_{jk}+f_{k\bar{l}}\delta_{ij}+f_{i\bar{j}}\delta_{kl}+f_{k\bar{j}}\delta_{il})$
$\displaystyle+\dfrac{(1-t)^{2}}{4}(f_{i}f_{\bar{j}}\delta_{kl}+f_{k}f_{\bar{j}}\delta_{il}+f_{i}f_{\bar{l}}\delta_{kj}+f_{k}f_{\bar{l}}\delta_{ij}).$
$\displaystyle-\dfrac{(1-t)^{2}}{2}f_{r}f_{\bar{r}}(\delta_{kj}\delta_{il}+\delta_{ij}\delta_{kl}).$
(4.10)

Next, consider the two-parameter canonical connection
$D^{t}_{s}=(1-s)\nabla^{t}+s\nabla^{LC}$. Then by (2.11), we have

$D^{t}_{s}e_{i}=(\theta^{p})_{i}^{j}e_{j}+s(\overline{\theta_{2}})^{j}_{i}\bar{e}_{j},$

where $p=t-ts$ and $(\theta_{2})^{j}_{i}=\overline{T^{k}_{ij}}\varphi^{k}$ as
$J$ is integrable. Denote $\Theta^{D}$ to be the $(n\times n)$ curvature
matrix of 2-forms of $D^{t}_{s}$ given by
$(\Theta^{D})^{j}_{i}(X,Y)=R^{D}(X,Y,e_{i},\bar{e}_{j})$ for $X,Y\in TM$. Then
by the structure equation,

$\displaystyle(\Theta^{D})_{i}^{j}$
$\displaystyle=d(\theta^{p})_{i}^{j}-(\theta^{p})_{i}^{k}\wedge(\theta^{p})_{k}^{j}-s^{2}(\overline{\theta_{2}})_{i}^{k}\wedge(\theta_{2})_{k}^{j}$
$\displaystyle=d(\theta^{p})_{i}^{j}-(\theta^{p})_{i}^{k}\wedge(\theta^{p})_{k}^{j}-s^{2}T_{ik}^{l}\overline{T_{kj}^{r}}\overline{\varphi^{l}}\wedge\varphi^{r}.$
(4.11)

Denote $\widetilde{D}^{t}_{s}$ to be the $(r,s)$ canonical connection of
$\widetilde{g}$ with $\widetilde{\theta}^{D},\widetilde{\Theta}^{D}$ being the
connection and curvature matrix of $\widetilde{D}^{t}_{s}$. Similarly,

$\displaystyle(\widetilde{\Theta}^{D})_{i}^{j}$
$\displaystyle=d(\widetilde{\theta}^{p})_{i}^{j}-(\widetilde{\theta}^{p})_{i}^{k}\wedge(\widetilde{\theta}^{p})_{k}^{j}-s^{2}\widetilde{T}_{ik}^{l}\overline{\widetilde{T}_{kj}^{r}}\overline{\widetilde{\varphi}^{l}}\wedge\widetilde{\varphi}^{r}.$
(4.12)

By (4.4), (4.11) and (4.12), we get

$\displaystyle[(\widetilde{\Theta}^{D})^{j}_{i}]^{1,1}-[(\Theta^{D})_{i}^{j}]^{1,1}$
$\displaystyle=$
$\displaystyle-2p\partial\bar{\partial}f\delta_{ij}+(1-p)(\bar{\partial}(f_{i}\varphi^{j})-\partial(f_{j}\overline{\varphi}^{i}))$
$\displaystyle-((p-1)^{2}+s^{2})f_{r}f_{\bar{r}}\varphi^{j}\wedge\overline{\varphi}^{i}+((p-1)^{2}-s^{2})f_{i}f_{\bar{j}}\varphi^{r}\wedge\overline{\varphi}^{r}$
$\displaystyle-s^{2}(f_{\bar{j}}T^{l}_{ik}\varphi^{k}\wedge\overline{\varphi^{l}}-f_{i}\overline{T^{k}_{lj}}\varphi^{k}\wedge\overline{\varphi^{l}}+f_{\bar{r}}T^{l}_{ri}\varphi^{j}\wedge\overline{\varphi^{l}}+f_{r}\overline{T^{k}_{rj}}\varphi^{k}\wedge\overline{\varphi^{i}})$
$\displaystyle+s^{2}(f_{i}\varphi^{j}\wedge\bar{\partial}f+f_{\bar{j}}\partial
f\wedge\overline{\varphi}^{i})$

Choose a local unitary frame of (1,0) vectors near $q$ such that
$\theta^{p}|q=0$. By (4.1) and the structure equation, we have

$\displaystyle\bar{\partial}\varphi^{i}=-(\theta^{i}_{j})^{0,1}\wedge\varphi^{j}=-(1-p)\overline{T^{j}_{il}}\overline{\varphi^{l}}\wedge\varphi^{j}.$

Then at $q$, by (4.5) and (4.9), we have

$\displaystyle
e^{2f}\widetilde{R}^{D}_{k\bar{l}i\bar{j}}-R^{D}_{k\bar{l}i\bar{j}}$
$\displaystyle=$
$\displaystyle-2pf_{k\bar{l}}\delta_{ij}+2p(1-p)f_{r}\overline{T_{rl}^{k}}\delta_{ij}-(1-p)(f_{i\bar{l}}\delta_{jk}+f_{k\bar{j}}\delta_{il})$
$\displaystyle+(1-p)^{2}(f_{i}\overline{T_{jl}^{k}}-f_{\bar{r}}T_{rk}^{j}\delta_{il}+f_{r}\overline{T_{rj}^{k}}\delta_{il}+f_{\bar{j}}T_{ik}^{l}-f_{r}f_{\bar{r}}\delta_{jk}\delta_{il}+f_{i}f_{\bar{j}}\delta_{kl})$
$\displaystyle+s^{2}(f_{i}f_{\bar{l}}\delta_{jk}+f_{\bar{j}}f_{k}\delta_{il}-f_{i}f_{\bar{j}}\delta_{kl}-f_{r}f_{\bar{r}}\delta_{jk}\delta_{il})$
$\displaystyle+s^{2}(f_{i}\overline{T^{k}_{lj}}-f_{\bar{j}}T^{l}_{ik}-f_{\bar{r}}T^{l}_{ri}\delta_{jk}-f_{r}\overline{T^{k}_{rj}}\delta_{il}).$
(4.13)

Here, the subscripts under $f$ stand for covariant derivatives with respect to
$\nabla^{p}$. As before, the above equality holds under any unitary frames
since it is tensorial. Denote $\widehat{\widetilde{R}^{D}}_{k\bar{l}i\bar{j}}$
and $\widehat{R^{D}_{k\bar{l}i\bar{j}}}$ to be the components of
symmetrization of $\widetilde{R}^{D}$ and $R^{D}$. Then

$\displaystyle
e^{2f}\widehat{\widetilde{R}^{D}}_{k\bar{l}i\bar{j}}-\widehat{R^{D}_{k\bar{l}i\bar{j}}}$
$\displaystyle=$
$\displaystyle-\dfrac{1}{2}(f_{i\bar{l}}\delta_{jk}+f_{k\bar{l}}\delta_{ij}+f_{i\bar{j}}\delta_{kl}+f_{k\bar{j}}\delta_{il})+\dfrac{(p-1)^{2}+s^{2}}{4}(f_{i}f_{\bar{j}}\delta_{kl}+f_{k}f_{\bar{j}}\delta_{il}+f_{i}f_{\bar{l}}\delta_{kj}+f_{k}f_{\bar{l}}\delta_{ij})$
$\displaystyle-\dfrac{(p-1)^{2}+s^{2}}{4}(f_{\bar{r}}T_{rk}^{j}\delta_{il}+f_{\bar{r}}T_{rk}^{l}\delta_{ij}+f_{\bar{r}}T_{ri}^{j}\delta_{kl}+f_{\bar{r}}T_{ri}^{l}\delta_{kj}+2f_{r}f_{\bar{r}}(\delta_{kl}\delta_{ij}+\delta_{jk}\delta_{il}))$
$\displaystyle+\dfrac{1-p^{2}-s^{2}}{4}(f_{r}\overline{T_{rl}^{k}}\delta_{ij}+f_{r}\overline{T_{rj}^{k}}\delta_{il}+f_{r}\overline{T_{rl}^{i}}\delta_{kj}+f_{r}\overline{T_{rj}^{i}}\delta_{kl}).$

When $g$ is Kähler, we get

$\displaystyle
e^{2f}\widehat{\widetilde{R}^{D}}_{k\bar{l}i\bar{j}}-\widehat{R^{D}_{k\bar{l}i\bar{j}}}=$
$\displaystyle-\dfrac{1}{2}(f_{i\bar{l}}\delta_{jk}+f_{k\bar{l}}\delta_{ij}+f_{i\bar{j}}\delta_{kl}+f_{k\bar{j}}\delta_{il})$
$\displaystyle+\dfrac{(p-1)^{2}+s^{2}}{4}(f_{i}f_{\bar{j}}\delta_{kl}+f_{k}f_{\bar{j}}\delta_{il}+f_{i}f_{\bar{l}}\delta_{kj}+f_{k}f_{\bar{l}}\delta_{ij})$
$\displaystyle-\dfrac{(p-1)^{2}+s^{2}}{2}f_{r}f_{\bar{r}}(\delta_{kl}\delta_{ij}+\delta_{jk}\delta_{il}))$
(4.14)

##  5\. Isosceles Hopf manifolds

In this section, we apply the formulae in section 4 to study Hermitian metrics
which are conformal to the standard Hopf metric and have pointwise constant
holomorphic sectional curvature with respect to $\nabla^{t}$ or $D^{t}_{s}$ on
isosceles Hopf manifolds. Recall that an isosceles Hopf manifold $M_{\sigma}$
([8]) is the quotient of $\mathbb{C}^{n}\setminus\\{0\\}$ by the infinite
cyclic group $\mathbb{Z}$ generated by

$\sigma:(z_{1},\cdots,z_{n})\longrightarrow(a_{1}z_{1},\cdots,a_{n}z_{n})$

where $0<a=|a_{1}|=\cdots=|a_{n}|<1$. Denote
$\omega_{0}=\sqrt{-1}(dz_{1}\wedge d\bar{z}_{1}+\cdots+dz_{n}\wedge
d\bar{z}_{n})$ to be the Kähler form of the standard Euclidean metric $g_{0}$
on $\mathbb{C}^{n}$ and $\omega_{h}=\dfrac{\sqrt{-1}}{|z|^{2}}(dz_{1}\wedge
d\bar{z}_{1}+\cdots+dz_{n}\wedge d\bar{z}_{n})$, where
$|z|^{2}={}^{t}\bar{z}z=|z_{1}|^{2}+\cdots+|z_{n}|^{2}$ with
${}^{t}z=(z_{1},\cdots,z_{n})$ ($z$ is viewed as a column vector). As
$\omega_{h}$ is invariant under $\sigma$, it induces a Hermitian metric on
$M_{\sigma}$ which is called the standard Hopf metric.

In [8], Chen-Zheng study Hermitian metrics which are conformal to $\omega_{h}$
and have pointwise constant Strominger holomorphic sectional curvature and
prove the following result (Proposition 1 there).

######  Proposition 5.1 ([8]).

Let $M_{\sigma}$ be an isosceles Hopf manifold. A Hermitian metric
$\widetilde{\omega}$ on $M_{\sigma}$ conformal to $\omega_{h}$ has pointwise
constant Strominger holomorphic sectional curvature if and only if
$\widetilde{\omega}=\dfrac{c_{0}}{|z|^{2}+^{t}zAz+\overline{{}^{t}zAz}}\omega_{0}$
for some constant $c_{0}>0$ and symmetric $(n\times n)$ complex matrix $A$
satisfying $A\bar{A}<\dfrac{1}{4}I_{n}$ and $D_{\sigma}AD_{\sigma}=a^{2}A$,
where $D_{\sigma}=diag\\{a_{1},\cdots,a_{n}\\}$. $\widetilde{\omega}$ is then
called an admissible metric.

In the following, we will generalize Proposition 5.1 to the case for
$\nabla^{t}$ and $D^{t}_{s}$. Assume that $\widetilde{g}$ is a Hermitian
metric which is conformal to the standard Hopf metric. Write
$\widetilde{\omega}$ for the Kähler form of $\widetilde{g}$ such that
$\widetilde{\omega}=F\omega_{h}=e^{2f}\omega_{0}$, where
$e^{2f}=\dfrac{F}{|z|^{2}}$ and $F$ is a positive function on $M_{\sigma}$.
Take $\mathbb{C}^{n}\setminus\\{0\\}$ as a global parameter space of
$M_{\sigma}$ and view $F$ and $f$ as functions on
$\mathbb{C}^{n}\setminus\\{0\\}$. Then $\widetilde{g}$ is conformal to the
flat metric $g_{0}$ with the conformal factor $e^{2f}$. As
$\sigma^{*}|z|^{2}=a^{2}|z|^{2}$ and $\sigma^{*}F=F$, we get
$\sigma^{*}(e^{2f})=a^{-2}e^{2f}$. Now in the notation of section 4, as
$g=g_{0}$ is flat, $R^{t}=0$ and $T=0$. Let $\widetilde{R}^{t}$ and
$\widehat{\widetilde{R}^{t}}$ be the curvature and its symmetrization of the
$t$-Gauduchon connection of $\widetilde{g}$. Choose
$e_{i}=\frac{\partial}{\partial z_{i}}$, $1\leq i\leq n$. By (4.10), we have

$\displaystyle e^{2f}\widehat{\widetilde{R}^{t}}_{k\bar{l}i\bar{j}}=$
$\displaystyle-\dfrac{1}{2}(f_{i\bar{l}}\delta_{jk}+f_{k\bar{l}}\delta_{ij}+f_{i\bar{j}}\delta_{kl}+f_{k\bar{j}}\delta_{il})$
$\displaystyle+\dfrac{(1-t)^{2}}{4}(f_{i}f_{\bar{j}}\delta_{kl}+f_{k}f_{\bar{j}}\delta_{il}+f_{i}f_{\bar{l}}\delta_{kj}+f_{k}f_{\bar{l}}\delta_{ij}).$
$\displaystyle-\dfrac{(1-t)^{2}}{2}f_{r}f_{\bar{r}}(\delta_{kj}\delta_{il}+\delta_{ij}\delta_{kl}),$
(5.1)

where $f_{r},f_{i\bar{j}}$ are now ordinary partial derivatives. Assume that
$\widetilde{g}$ has pointwise constant holomorphic sectional curvature
$\widetilde{c}$ with respect to $\widetilde{\nabla}^{t}$, for some
$t\in\mathbb{R}$. By Lemma 2.5,

$\displaystyle\widehat{\widetilde{R}^{t}}_{k\bar{l}i\bar{j}}=\dfrac{\widetilde{c}}{2}(\delta_{ij}\delta_{kl}+\delta_{il}\delta_{kj}).$
(5.2)

Putting (5.2) into (5.1), we have

$\displaystyle(\dfrac{\widetilde{c}}{2}e^{2f}+\dfrac{(1-t)^{2}}{2}f_{r}f_{\bar{r}})(\delta_{ij}\delta_{kl}+\delta_{il}\delta_{kj})=-\dfrac{1}{2}(f_{i\bar{l}}\delta_{jk}+f_{k\bar{l}}\delta_{ij}+f_{i\bar{j}}\delta_{kl}+f_{k\bar{j}}\delta_{il})$
$\displaystyle+\dfrac{(1-t)^{2}}{4}(f_{i}f_{\bar{j}}\delta_{kl}+f_{k}f_{\bar{j}}\delta_{il}+f_{i}f_{\bar{l}}\delta_{kj}+f_{k}f_{\bar{l}}\delta_{ij}).$
(5.3)

We first argue that $t\neq 1$. Otherwise, if $t=1$, then (5.3) becomes

$\displaystyle-\widetilde{c}e^{2f}(\delta_{ij}\delta_{kl}+\delta_{il}\delta_{kj})=f_{i\bar{l}}\delta_{jk}+f_{k\bar{l}}\delta_{ij}+f_{i\bar{j}}\delta_{kl}+f_{k\bar{j}}\delta_{il}.$
(5.4)

When $n\geq 2$, by testing different indices in (5.4), we get For $i\neq j$,
$f_{i\bar{i}\bar{j}}=f_{i\bar{j}\bar{i}}=0$. So
$\frac{\partial}{\partial\bar{z}_{j}}(\widetilde{c}e^{2f})=0$. As
$\widetilde{c}e^{2f}$ is a real function, we get that $\widetilde{c}e^{2f}$ is
a constant. But this contradicts with $\sigma^{*}\widetilde{c}=\widetilde{c}$
and $\sigma^{*}e^{2f}=a^{-2}e^{2f}$. So $t\neq 1$ and
$\mu=-\dfrac{(1-t)^{2}}{2}\neq 0$. Let $\xi=e^{\mu f}$. Then

$\displaystyle\xi_{i}=\mu\xi f_{i},\ \ \ \
\xi_{i\bar{j}}=\mu\xi(f_{i\bar{j}}+\mu f_{i}f_{\bar{j}}).$ (5.5)

Putting (5.5) into (5.3), we get

$\displaystyle(-\mu\xi
e^{2f}\widetilde{c}+\dfrac{2}{\xi}\xi_{r}\xi_{\bar{r}})(\delta_{kj}\delta_{il}+\delta_{ij}\delta_{kl})=\xi_{i\bar{l}}\delta_{jk}+\xi_{k\bar{l}}\delta_{ij}+\xi_{i\bar{j}}\delta_{kl}+\xi_{k\bar{j}}\delta_{il}.$
(5.6)

Denote $\lambda=-\dfrac{1}{2}(\mu\xi
e^{2f}\widetilde{c}-\dfrac{2}{\xi}\xi_{r}\xi_{\bar{r}})$. By testing different
indices in (5.6), we get

$\displaystyle\xi_{i\bar{j}}=\lambda\delta_{\ij}.$ (5.7)

Then for $i\neq j$,
$\lambda_{\bar{j}}=\xi_{i\bar{i}\bar{j}}=\xi_{i\bar{j}\bar{i}}=0$. So
$\lambda$ is a constant. From (5.7), we get that
$\partial\bar{\partial}(\xi-\lambda|z|^{2})=0$. Then we may write

$\xi=\lambda|z|^{2}+\varphi+\overline{\varphi}$

for some holomorphic function $\varphi$ on $\mathbb{C}^{n}\setminus\\{0\\}$.
As $\sigma^{*}(e^{2f})=a^{-2}e^{2f}$ and $\xi=e^{\mu f}$, we have
$\sigma^{*}\xi=a^{-\mu}\xi$. It follows that

$\displaystyle\sigma^{*}(\varphi+\overline{\varphi})-a^{-\mu}(\varphi+\overline{\varphi})=\lambda(a^{-\mu}-a^{2})|z|^{2}.$

As $\varphi$ is holomorphic,
$\partial\bar{\partial}(\lambda(a^{-\mu}-a^{2})|z|^{2})=0$. Therefore

$\displaystyle\lambda(a^{-\mu}-a^{2})=0.$ (5.8)

We then divide the discussion into two cases. (i) If $t\neq-1$ and $t\neq 3$,
then $\mu=-\dfrac{(1-t)^{2}}{2}\neq-2$. By (5.8), we have $\lambda=0$ . Thus
$\varphi+\overline{\varphi}=\xi=e^{\mu f}>0$, i.e., $\varphi$ has positive
real part on $\mathbb{C}^{n}\setminus\\{0\\}$. When $n\geq 2$, by Hartogs
extension, $\varphi$ is holomorphic on $\mathbb{C}^{n}$. As
$\sigma^{*}\varphi=a^{-\mu}\varphi$. Expanding $\varphi$ in Taylor series, we
see that $\varphi$ is a homogeneous polynomial in $z$ with degree $-\mu$. This
is impossible since $\varphi$ has positive real part on
$\mathbb{C}^{n}\setminus\\{0\\}$. (ii) $t=-1$ or $t=3$, then $\mu=-2$. In this
case, $\xi=e^{-2f}$ and (5.1) becomes

$\displaystyle\widehat{\widetilde{R}^{t}}_{k\bar{l}i\bar{j}}=$
$\displaystyle\dfrac{1}{4}(\xi_{i\bar{l}}\delta_{jk}+\xi_{k\bar{l}}\delta_{ij}+\xi_{i\bar{j}}\delta_{kl}+\xi_{k\bar{j}}\delta_{il})-\dfrac{1}{2\xi}\xi_{r}\xi_{\bar{r}}(\delta_{kj}\delta_{il}+\delta_{ij}\delta_{kl}).$
(5.9)

If $\xi$ satisfies $\xi_{i\bar{j}}=\lambda\delta_{\ij}$ for some positive
constant $\lambda$ and $\sigma^{*}\xi=a^{2}\xi$, then by (5.9),
$\widetilde{g}=\dfrac{1}{\xi}g_{0}$ would define a Hermitian metric on
$M_{\sigma}$ and has pointwise constant holomorphic sectional curvature
$\lambda-\dfrac{1}{\xi}\xi_{r}\xi_{\bar{r}}$. This is exactly in the same
situation of the Strominger connection case (see Page 8 in [8]), which gives
that $\xi=\lambda(|z|^{2}+{}^{t}zAz+\overline{{}^{t}zAz})$. Here $A$ is a
symmetric $(n\times n)$ complex matrix satisfying $A\bar{A}<\dfrac{1}{4}I_{n}$
and $D_{\sigma}AD_{\sigma}=a^{2}A$ with
$D_{\sigma}=diag\\{a_{1},\cdots,a_{n}\\}$. In other words,
$\dfrac{1}{\xi}g_{0}$ is an admissible metric on $M_{\sigma}$. In summary, we
have proved the following result which generalizes Proposition 5.1.

######  Proposition 5.2.

Let $M_{\sigma}$ be an isosceles Hopf manifold. A Hermitian metric
$\widetilde{\omega}$ on $M_{\sigma}$ conformal to $\omega_{h}$ has pointwise
constant holomorphic sectional curvature with respect to
$\widetilde{\nabla}^{t}$ if and only if $t=-1$ or $t=3$ and
$\widetilde{\omega}$ is an admissible metric.

Note that from the above calculations, for an admissible metric
$\dfrac{1}{\xi_{A}}\omega_{0}$ with
$\xi_{A}=|z|^{2}+{}^{t}zAz+\overline{{}^{t}zAz}$, the holomorphic sectional
curvatures with respect to $\widetilde{\nabla}^{t}$ for $t=-1$ and $t=3$ are
both $-\dfrac{1}{\xi_{A}}(4\
{}^{t}zA\bar{A}\bar{z}+{}^{t}zAz+\overline{{}^{t}zAz})$. If $A\neq 0$, it is
impossible to be a constant (c.f. [8]).

Using the same techniques, we also derive the following more general result
for the two-parameter connection $\widetilde{D}^{t}_{s}$ whose proof is
omitted.

######  Proposition 5.3.

A Hermitian metric $\widetilde{\omega}$ on $M_{\sigma}$ which is conformal to
$\omega_{h}$ has pointwise constant holomorphic sectional curvature with
respect to $\widetilde{D}^{t}_{s}$ if and only if $(1-t+ts)^{2}+s^{2}=4$ and
$\widetilde{\omega}$ is an admissible metric. In this case, the holomorphic
sectional curvatures of $\widetilde{D}^{t}_{s}$ is $-\dfrac{1}{\xi_{A}}(4\
{}^{t}zA\bar{A}\bar{z}+{}^{t}zAz+\overline{{}^{t}zAz})$ up to a scalar.

When $n=2$, $M_{\sigma}$ is an isosceles Hopf surface. Let $A=0$ and
$\xi_{A}=|z|^{2}$. Then
$\widetilde{\omega}=\dfrac{1}{\xi_{A}}\omega_{0}=\omega_{h}$. So the standard
Hopf metric has zero holomorphic sectional curvature for
$\widetilde{\nabla}^{s}$ and $\widetilde{\nabla}^{3}$ (or generally
$\widetilde{D}^{t}_{s}$ with $(1-t+ts)^{2}+s^{2}=4$). Putting
$e^{-2f}=|z|^{2}$ into (4.9), direct calculation gives that the curvature for
$\widetilde{\nabla}^{s}$ is zero but nonzero for $\widetilde{\nabla}^{3}$. For
example, when $t=3$, we have
$\widetilde{R}^{t}_{k\bar{l}i\bar{j}}=3(\delta_{kl}\delta_{ij}-\delta_{il}\delta_{jk})+\dfrac{1}{|z|^{2}}(\bar{z}_{k}z_{j}\delta_{il}+\bar{z}_{i}z_{l}\delta_{jk}+\bar{z}_{i}z_{j}\delta_{kl}-3\bar{z}_{k}z_{l}\delta_{ij})$.
Then $\widetilde{R}^{t}_{1\bar{1}2\bar{2}}\neq 0$ for $t=3$. Therefore, we
have

######  Corollary 5.4.

There exist non-flat compact Hermitian surfaces with zero Gauduchon
holomorphic sectional curvature.

Recall that for a Kähler manifold, the holomorphic sectional curvature
determines the whole curvature tensor. In particular, a Kähler metric with
zero holomorphic sectional curvature must be flat. The above example shows
that it does not hold for compact Hermitian manifolds with respect to general
Gauduchon connections even in dimension two.

##  6\. Proof of the theorems

In this section, we prove Theorem 1.3, Theorem 1.6 and Corollary 1.8. As
Theorem 1.6 is a generalization of Theorem 1.3, it suffices to prove Theorem
1.6. Using the notations in section 4, assume that $(M,\widetilde{g})$ is a
compact Hermitian surface with pointwise constant holomorphic sectional
curvature with respect to $\widetilde{D}^{t}_{s}$ for some
$(t,s)\in\mathbb{R}^{2}$. By Proposition 3.3, $(M,\widetilde{g})$ is self-
dual. It follows from Theorem $1^{\prime}$ in [2] that $\widetilde{g}$ must be
conformal to one of the following metrics: (1) a Kähler meric $g$ with
constant holomorphic sectional curvature; (2) a non-flat Kähler meric $g$
which has local Kähler form $2\sqrt{-1}\dfrac{dz_{1}\wedge
d\bar{z}_{1}}{(1-|z_{1}|^{2})^{2}}+2\sqrt{-1}\dfrac{dz_{2}\wedge
d\bar{z}_{2}}{(1+|z_{2}|^{2})^{2}}$; (3) the standard Hopf metric $\omega_{h}$
on an isosceles Hopf surface (see also [8]).

With the computation formulae in section 4, we first prove that in case (1)
$\widetilde{g}$ must be a constant multiple of $g$. Assume
$\widetilde{g}=e^{2f}g$. As $\widetilde{g}$ has pointwise constant holomorphic
sectional curvature $\widetilde{c}$ with respect to $\widetilde{D}^{t}_{s}$
and $g$ has constant holomorphic sectional curvature $c$, by (4.14),

$\displaystyle(e^{2f}\dfrac{\widetilde{c}}{2}-\dfrac{c}{2})(\delta_{kl}\delta_{ij}+\delta_{jk}\delta_{il})=$
$\displaystyle-\dfrac{1}{2}(f_{i\bar{l}}\delta_{jk}+f_{k\bar{l}}\delta_{ij}+f_{i\bar{j}}\delta_{kl}+f_{k\bar{j}}\delta_{il})$
$\displaystyle+\dfrac{(p-1)^{2}+s^{2}}{4}(f_{i}f_{\bar{j}}\delta_{kl}+f_{k}f_{\bar{j}}\delta_{il}+f_{i}f_{\bar{l}}\delta_{kj}+f_{k}f_{\bar{l}}\delta_{ij})$
$\displaystyle-\dfrac{(p-1)^{2}+s^{2}}{2}f_{r}f_{\bar{r}}(\delta_{kl}\delta_{ij}+\delta_{jk}\delta_{il}).$
(6.1)

Let $\xi=e^{\mu f}$ with $\mu=-\dfrac{(p-1)^{2}+s^{2}}{2}$ and
$\lambda=-\dfrac{1}{2}(\mu\xi(e^{2f}\widetilde{c}-c)-\dfrac{2}{\xi}\xi_{r}\xi_{\bar{r}})$.
Then (6.1) becomes

$\displaystyle
2\lambda(\delta_{kl}\delta_{ij}+\delta_{jk}\delta_{il})=\xi_{i\bar{l}}\delta_{jk}+\xi_{k\bar{l}}\delta_{ij}+\xi_{i\bar{j}}\delta_{kl}+\xi_{k\bar{j}}\delta_{il}.$

By choosing different indices, we get

$\displaystyle\xi_{i\bar{j}}=\lambda\delta_{ij}.$ (6.2)

Since $g$ is Kähler ($R_{\bar{i}\bar{j}**}=0$),
$\lambda_{\bar{j}}=\xi_{i\bar{i}\bar{j}}=\xi_{i\bar{j}\bar{i}}=0$. So
$\lambda$ is a constant. Contracting (6.2) by $g$, we have
$\Delta\xi=2\lambda$ on $M$. As $M$ is compact, by the maximum principle,
$\lambda=0$ and $\xi$ is a constant. So $\widetilde{g}$ is a constant multiple
of $g$ and is Kähler.

Next, we show that case (2) is not possible, following [8]. Denote $g$ to be
the Kähler metric with local Kähler form $\omega=2\sqrt{-1}\dfrac{dz_{1}\wedge
d\bar{z}_{1}}{(1-|z_{1}|^{2})^{2}}+2\sqrt{-1}\dfrac{dz_{2}\wedge
d\bar{z}_{2}}{(1+|z_{2}|^{2})^{2}}$, where $(z_{1},z_{2})$ is a local
holomorphic coordinate. Then $g$ is the product metric of the Bergman metric
on $\mathbb{D}$ and the Fubini-Study metric on $\mathbb{P}^{1}$. Let
$\\{e_{1},e_{2}\\}$ be a unitary frame of $g$ which are parallel to
$\frac{\partial}{\partial z_{1}},\frac{\partial}{\partial z_{2}}$
respectively. Then $R_{1\bar{1}1\bar{1}}=-1,R_{2\bar{2}2\bar{2}}=1$ and all
other components $R_{i\bar{j}k\bar{l}}$ of $g$ are zero. Assume that
$\widetilde{g}=e^{2f}g$ for some global function $f$ and $\widetilde{g}$ has
pointwise constant holomorphic sectional curvature $\widetilde{c}$ with
respect to $\widetilde{D}^{t}_{s}$. Letting $(k\bar{l}i\bar{j})$ be
$(1\bar{1}1\bar{1}),(2\bar{2}2\bar{2})$ and $(1\bar{1}1\bar{2})$ in (4.14), we
get

$\displaystyle e^{2f}\widetilde{c}+1+((1-p)^{2}+s^{2})f_{r}f_{\bar{r}}$
$\displaystyle=-2f_{1\bar{1}}+((1-p)^{2}+s^{2})f_{1}f_{\bar{1}},$
$\displaystyle e^{2f}\widetilde{c}-1+((1-p)^{2}+s^{2})f_{r}f_{\bar{r}}$
$\displaystyle=-2f_{2\bar{2}}+((1-p)^{2}+s^{2})f_{2}f_{\bar{2}},$
$\displaystyle 0$
$\displaystyle=-2f_{1\bar{2}}+((1-p)^{2}+s^{2})f_{1}f_{\bar{2}}.$ (6.3)

If $(t,s)=(1,0)$, then $(1-p)^{2}+s^{2}=0$. The above equations become

$f_{1\bar{1}}=-\dfrac{1}{2}(e^{2f}\widetilde{c}+1),\hskip
14.22636ptf_{2\bar{2}}=-\dfrac{1}{2}(e^{2f}\widetilde{c}-1),\hskip
14.22636ptf_{1\bar{2}}=f_{2\bar{1}}=0.$

As $g$ is Kähler,
$f_{1\bar{1}\bar{2}}=f_{1\bar{2}\bar{1}}=0,f_{2\bar{2}\bar{1}}=f_{2\bar{1}\bar{2}}=0$.
So $\bar{e}_{1}(e^{2f}\widetilde{c})=\bar{e}_{2}(e^{2f}\widetilde{c})=0$.
Since $e^{2f}\widetilde{c}$ is a global real function, $e^{2f}\widetilde{c}$
must be a constant. Therefore, $\Delta f$ is a constant and by the maximum
principle, $f$ is a constant. Then
$0=e^{2f}\widetilde{c}+1=e^{2f}\widetilde{c}-1$, which is a contradiction. If
$(t,s)\neq(1,0)$, then $(1-p)^{2}+s^{2}\neq 0$. Let
$\mu=-\dfrac{(1-p)^{2}+s^{2}}{2}$, $\xi=e^{\mu f}$ and
$\lambda=-\dfrac{\mu\xi}{2}(e^{2f}\widetilde{c}-2\mu f_{r}f_{\bar{r}})$. Then
(6.3) becomes

$\displaystyle\xi_{1\bar{1}}=\lambda+\dfrac{\mu\xi}{2},\hskip
28.45274pt\xi_{2\bar{2}}=\lambda-\dfrac{\mu\xi}{2},\hskip
28.45274pt\xi_{1\bar{2}}=\xi_{2\bar{1}}=0.$

Let $\alpha=\lambda-\dfrac{\mu\xi}{2},\beta=\lambda+\dfrac{\mu\xi}{2}$, which
are two real global functions. Then $\alpha_{\bar{1}}=0,\beta_{\bar{2}}=0$. So
$\alpha$ depends only on $z_{2}$ and $\beta$ depends on $z_{1}$. Then
$\Delta\alpha=\alpha_{2\bar{2}}=(\beta-\mu\xi)_{2\bar{2}}=-\mu\alpha$. As
$\mu<0$, by the maximum principle, $\alpha=0$. So
$\Delta\xi=\xi_{1\bar{1}}+\xi_{2\bar{2}}=\mu\xi$. Then by the maximum
principle, the minimum of $\xi$ is nonpositive, which is a contradiction. In
conclusion, the case (2) is impossible.

Case (3) is just contained in the situation of Proposition 5.3. Then we deduce
that $(1-t+ts)^{2}+s^{2}=4$ and $\widetilde{g}$ is an admissible metric. The
proof of Theorem 1.6 is finished.

Last, we prove Corollary 1.8. Using the adjusted notation, assume that
$\widetilde{g}$ has pointwise constant holomorphic sectional curvature with
respect to $\widetilde{D}^{t}_{s}$ for some $(t,s)\in\mathbb{R}^{2}$ and is
non-Kähler. Then by Theorem 1.6, $\widetilde{g}$ must be an admissible metric
on an isosceles Hopf surface with $(t,s)$ satisfying $(1-t+ts)^{2}+s^{2}=4$.
In this case, by Proposition 5.3, the holomorphic sectional curvature is
$H=-\dfrac{1}{\xi_{A}}(4\
{}^{t}zA\bar{A}\bar{z}+{}^{t}zAz+\overline{{}^{t}zAz})$, where
$\xi_{A}=|z|^{2}+{}^{t}zAz+\overline{{}^{t}zAz}$ and $A$ is symmetric and
satisfies $A\bar{A}<\dfrac{1}{4}I_{n}$. Diagonalizing $A$ with a unitary
matrix $U$ (see page 8 in [8]), direct calculation shows that $H$ can not be a
positive function or a negative function. So Corollary 1.8 is proved.

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