1. Introduction
Let M be an n-dimensional complex manifold. Suppose h is a Hermitian metric on M with
$g=\textrm{Re } h$
the background Riemannian metric. There are two canonical connections associated to h and g, the Hermitian (or Chern) connection D and the Riemannian (or Levi-Civita) connection
$\nabla$
. As is well known, the Chern connection D coincides with the Levi-Civita
$\nabla$
if and only if h is Kähler [Reference Moroianu2]. Hence, curvatures associated with these two canonical connections are tightly related on Kähler manifolds.
Suppose that
$z=(z^{1},\cdots,z^{n})$
is a local holomorphic coordinate system on M. Denote by
$h=h_{\alpha\bar{\beta}}(z)dz^{\alpha}\otimes d\bar{z}^{\beta}$
a Hermitian metric on M, where
$h_{\alpha\bar{\beta}}(z)$
are smooth functions and
$H=(h_{\alpha\bar{\beta}}(z))$
is an
$n\times n$
positive definite Hermitian matrix. Let D denote the Chern connection of the Hermitian manifold (M, h). Its connection 1-forms are given by
\begin{equation*}\theta_{\beta}^{\alpha}={\varGamma}_{\beta\gamma}^{\alpha}dz^{\gamma},\end{equation*}
where
\begin{equation}\varGamma_{\beta\gamma}^{\alpha}=h^{\bar{\delta}\alpha}\displaystyle\frac{\partial h_{\beta\bar{\delta}}}{\partial z^{\gamma}}.\end{equation}
Since
$\partial\theta_{\beta}^{\alpha}-\theta_{\beta}^{\gamma}\wedge \theta_{\gamma}^{\alpha}=0$
, the curvature form of the Chern connection D is
\begin{equation*}\bar{\partial}\theta^\alpha_\beta =\Theta^{\alpha}_{\beta \mu\bar\nu}dz^{\mu}\wedge d\bar{z}^{\nu},\end{equation*}
where
\begin{equation}\Theta^{\alpha}_{\beta \mu\bar\nu}=-\displaystyle\frac{\partial {\varGamma}_{\beta\mu}^{\alpha}}{\partial \bar{z}^{\nu}}=-h^{\bar{\delta}{\alpha}}\displaystyle\frac{\partial^2 h_{{\beta}\bar{\delta}}}{\partial z^{\mu}\partial \bar{z}^{\nu}}+h^{\bar{\delta}{\alpha}}\displaystyle\frac{\partial h_{{\beta}\bar{\epsilon}}}{\partial z^{\mu}}h^{\bar{\epsilon}\gamma}\displaystyle\frac{\partial h_{\gamma{\bar\delta}}}{\partial \bar{z}^ {\nu}}.\end{equation}
The holomorphic sectional curvature tensor is defined by
\begin{equation}\Theta_{\alpha\bar{\beta}\mu\bar{\nu}}=h_{\gamma\bar{\beta}}\Theta^{\gamma}_{\alpha \mu\bar\nu}=-\displaystyle\frac{\partial^2 h_{\alpha\bar{\beta}}}{\partial z^{\mu}\partial \bar{z}^{\nu}}+\displaystyle\frac{\partial h_{\alpha\bar{\lambda}}}{\partial z^{\mu}}h^{\bar{\lambda}\kappa}\displaystyle\frac{\partial h_{\kappa\bar{\beta}}}{\partial \bar{z}^{\nu}}.\end{equation}
The holomorphic bisectional curvature in two directions
$v=v^{\alpha}\frac{\partial}{\partial z^{\alpha}}, \, w=w^{\alpha}\frac{\partial}{\partial z^{\alpha}} \in T^{1,0}_zM$
and the holomorphic sectional curvature in the direction
$v=v^{\alpha}\frac{\partial}{\partial z^{\alpha}}\in T^{1,0}_zM$
are, respectively, defined by
\begin{equation}\textrm{HBSC}{(z;\;v,w)} = \displaystyle\frac{\Theta_{\alpha\bar{\beta}\mu\bar{\nu}}v^{\alpha}\bar{v}^{\beta}w^{\mu}\bar{w}^{\nu}}{\, h_{\alpha\bar{\beta}}v^{\alpha}\bar{v}^{\beta}\cdot h_{\mu\bar{\nu}}w^{\mu}\bar{w}^{\nu}},\end{equation}
\begin{equation}\textrm{HSC}{(z;\;v)} = \displaystyle\frac{\Theta_{\alpha\bar{\beta}\mu\bar{\nu}}v^{\alpha}\bar{v}^{\beta}v^{\mu}\bar{v}^{\nu}}{\left(h_{\alpha\bar{\beta}}v^{\alpha}\bar{v}^{\beta}\right)^2}.\end{equation}
An n-dimensional complex manifold M is also a 2n-dimensional real manifold. Set
\begin{equation*}x=(\textrm{Re } (z),\textrm{Im } (z))\in {\Bbb R}^{2n}{,}\end{equation*}
and
\begin{equation} A(x) = \textrm{Re } (H(z)),\quad B(x) =\textrm{Im } (H(z)),\end{equation}
then
\begin{equation}G(x)=\left(g_{ij}(x)\right)_{2n\times 2n} = \left(\!\begin{array}{c@{\quad}c}A(x)&B(x)\\[5pt] -B(x)&A(x)\end{array}\! \right)\end{equation}
is a positive definite real symmetric matrix and
\begin{equation}G(x)^{-1} = \left(g^{ij}(x)\right)_{2n\times 2n}=\left(\!\begin{array}{c@{\quad}c}A_1(x)&B_1(x)\\[5pt]-B_1(x)&A_1(x)\end{array}\! \right),\end{equation}
where
\begin{equation} A_1(x) = \textrm{Re } (H^{-1}(z)),\quad B_1(x) =\textrm{Im } (H^{-1}(z)){\color{red}.}\end{equation}
Therefore,
$g=\textrm{Re }{h}=g_{ij}(x){d}x^i\otimes {d}x^j$
is a Riemannian metric on M, where
\begin{equation}{d}x^{\alpha}=\displaystyle\frac{1}{2}\left(dz^{\alpha}+d\bar{z}^{\alpha}\right), \quad {d}x^{\alpha+n}=-\displaystyle\frac{{i}}{2}\left(dz^{\alpha}-d\bar{z}^{\alpha}\right).\end{equation}
Let us denote by
$\nabla$
the Levi-Civita connection, then its connection 1-forms are given by
\begin{equation*}\omega_{i}^{k}=\gamma^{k}_{ij}dx^j,\end{equation*}
where
\begin{equation}\gamma_{ij}^{k}=\displaystyle\frac{1}{2}g^{kl}\left(\displaystyle\frac{\partial g_{il}}{\partial x^{j}}+\displaystyle\frac{\partial g_{jl}}{\partial x^{i}}-\displaystyle\frac{\partial g_{ij}}{\partial x^{l}}\right).\end{equation}
The curvature form of the Levi-Civita connection
$\nabla$
is
\begin{equation*}d\omega_k^l-\omega_k^h\wedge\omega_h^l=\frac{1}{2}R^{l}_{kij}{d}x^{i}\wedge {d}x^{j},\end{equation*}
where
\begin{equation}R^{l}_{kij}=\displaystyle\frac{\partial {\gamma}_{kj}^{l}}{\partial x^{i}}-\displaystyle\frac{\partial {\gamma}_{ki}^{l}}{\partial x^{j}}+\gamma^{h}_{kj}\gamma^l_{hi}-\gamma^h_{ki}\gamma^l_{hj}.\end{equation}
In local coordinates, we denote by
$\mathcal{R}=R^i_{jkl}dx^j\otimes\frac{\partial }{\partial x^i}\otimes dx^k \otimes dx^j$
. The sectional curvature tensor is defined by
\begin{equation*}R_{ijkl} =g_{ih} R^{h}_{jkl}.\end{equation*}
In local coordinates,
\begin{equation} R_{ijkl}=\displaystyle\frac{1}{2}\left(\displaystyle\frac{\partial^{2} g_{il}}{{\partial x^j}{\partial x^k}}+ \displaystyle\frac{\partial^{2} g_{jk}}{{\partial x^i}{\partial x^l}}- \displaystyle\frac{\partial^{2} g_{ik}}{{\partial x^j}{\partial x^l}}- \displaystyle\frac{\partial^{2} g_{jl}}{{\partial x^i}{\partial x^k}}\right) + g^{st}\left([jk,s][il,t]-[jl,s][ik,t]\right),\end{equation}
where
\begin{equation}[jk,s]=\displaystyle\frac{1}{2}\left(\displaystyle\frac{\partial g_{js}}{\partial x^k}+ \displaystyle\frac{\partial g_{ks}}{\partial x^j}-\displaystyle\frac{\partial g_{jk}}{\partial x_s}\right).\end{equation}
The sectional curvature of the 2-plane
$\Pi(u,y)$
spanned by two linearly independent tangent vectors
$u=u^i\frac{\partial }{\partial x^i},\,y=y^i\frac{\partial }{\partial x^i}\in T_xM$
of the Riemannian metric g is defined by
\begin{equation}\begin{split}K(x;\;u,y)=\displaystyle\frac{g(\mathcal{R}(u,y)y,u)}{g(u,y)g(y,y)-g(u,y)^2}=\displaystyle\frac{R_{ijkl}u^iy^ju^ky^l}{(g_{ij} g_{kl}-g_{il} g_{jk})u^iu^jy^ky^l}.\end{split}\end{equation}
For a Hermitian manifold (M, h), a 2-plane
$\Pi(u,Ju)$
is called a holomorphic plane section in [Reference Zheng4], where
$u\in TM$
and J is the complex structure on M.
Assume h is a Kähler metric, then the holomorphic sectional curvature is just half of the sectional curvature in a holomorphic plane section [Reference Zheng4], i.e.,
\begin{equation}\textrm{HSC}(z;\; v)=\displaystyle\frac{1}{2}K(x;\;u,Ju),\quad \forall v\in T^{1,0}_zM, \quad u=v+\bar{v}\in T_xM.\end{equation}
A nature question arises, when a Hermitian metric h satisfies (1.16), must it be Kähler? In this article, we give a positive answer.
Main Theorem Let (M, h) be a Hermitian manifold. If the holomorphic sectional curvature is just half of the sectional curvature in a holomorphic plane section, i.e., (1.16) holds, then h is a Kähler metric.
2. Proof of main theorem
In this section, we consider a Hermitian manifold such that the holomorphic sectional curvature is half of the sectional curvature in a holomorphic plane section.
For simplicity, we introduce some notations. Denote by
\begin{equation*}\frac{\partial}{\partial x}=\left(\frac{\partial}{\partial x^1},\cdots,\frac{\partial}{\partial x^{2n}}\right),\quad\frac{\partial}{\partial z}=\left(\frac{\partial}{\partial z^1},\cdots,\frac{\partial}{\partial z^{n}}\right),\quad\frac{\partial}{\partial \bar{z}}=\left(\frac{\partial}{\partial \bar{z}^1},\cdots,\frac{\partial}{\partial \bar{z}^{n}}\right),\end{equation*}
\begin{equation*}d{x}=\left(d x^1,\cdots,d x^{2n}\right), \quad dz=\left(d z^1,\cdots,d z^{n}\right),\quad d\bar{z}=\left(d \bar{z}^1,\cdots,d \bar{z}^{n}\right).\end{equation*}
Then
\begin{equation}\frac{\partial }{\partial x} = \left(\frac{\partial }{\partial z},\frac{\partial }{\partial \bar{z}}\right)F,\quad dx=\left(dz,d\bar{z}\right)\left(F^{{t}}\right)^{-1},\\\end{equation}
where
\begin{equation*}F = \left(\begin{array}{c@{\quad}c}I&{i}I\\I& -{i}I\end{array} \right),\quad\left(F^{{t}}\right)^{-1}= \displaystyle\frac{1}{2}\left(\begin{array}{c@{\quad}c}I&-{i}I\\I& {i}I\end{array} \right),\end{equation*}
I is the
$n\times n$
unit matrix, and
$F^{{t}}$
means transposition of F. We will denote by
\begin{equation*}J = \left(\begin{array}{c@{\quad}c}0&-I\\I& 0\end{array} \right)\end{equation*}
the
$2n\times 2n $
matrix associated with the complex structure J when no confusion can rise. Then
\begin{equation}FF^{{t}} = 2\left(\begin{array}{c@{\quad}c}0&I\\I&0\end{array} \right), \quad F\bar{F}^{{t}} = 2\left(\begin{array}{c@{\quad}c}I&0\\0&I\end{array} \right),\end{equation}
\begin{equation} FJF^{{t}} = -2{i}J,\quad FJF^{-1} ={i} \left(\begin{array}{c@{\quad}c}I&0\\0&-I\end{array} \right).\end{equation}
For any
$ v=v^{\alpha}\frac{\partial}{\partial z^{\alpha}} \in T_z^{1,0}M$
, we set
$u=u^i\frac{\partial}{\partial x^i}, \, \tilde{u}=\tilde{u}^i\frac{\partial}{\partial x^i} \in T_xM$
such that
\begin{equation*}u^{\alpha}=\tilde{u}^{\alpha+n}=\displaystyle\frac{1}{2}\left(v^{\alpha}+\bar{v}^{\alpha}\right), \quad u^{\alpha+n}=-\tilde{u}^{\alpha}=-\displaystyle\frac{{i}}{2}\left(v^{\alpha}-\bar{v}^{\alpha}\right).\end{equation*}
Then
\begin{equation}u= v +\bar{v}, \quad \tilde{u} = -{i} (v -\bar{v})=-Ju.\end{equation}
Set
$\unicode{x1D567}=(v^1, \ldots, v^n)$
,
$\unicode{x1D566}=(u^1,\ldots,u^{2n})$
,
$\tilde{\unicode{x1D566}}=(\tilde{u}^{1},\ldots,\tilde{u}^{2n})$
, we have
\begin{equation}\unicode{x1D566}= (\unicode{x1D567},\bar{\unicode{x1D567}})(F^{{t}})^{-1}, \quad \tilde{\unicode{x1D566}} = \unicode{x1D566} J.\end{equation}
It is easy to check that
$JG = GJ, \, JGJ^{{t}}=G$
, and
\begin{equation}FGF^{{t}} = 2\left(\begin{array}{c@{\quad}c}0&\bar{H}\\H&0\end{array} \right), \quad(F^{{t}})^{-1}GF^{-1} = \frac{1}{2}\left(\begin{array}{c@{\quad}c}0&H\\\bar{H}&0\end{array} \right),\end{equation}
\begin{equation}FG^{-1}F^{{t}} = 2\left(\begin{array}{c@{\quad}c}0&\bar{H}^{-1}\\H^{-1}&0\end{array} \right), \quad(F^{{t}})^{-1}G^{-1}F^{-1}= \frac{1}{2}\left(\begin{array}{c@{\quad}c}0&H^{-1}\\\bar{H}^{-1}&0\end{array} \right),\end{equation}
\begin{align}\unicode{x1D566} G = \left(\unicode{x1D567}H, \bar{\unicode{x1D567}}\bar{H}\right)\left(F^{{t}}\right)^{-1},\quad\tilde{\unicode{x1D566}}G =-{i} \left(\unicode{x1D567}H, -\bar{\unicode{x1D567}}\bar{H}\right)\left(F^{{t}}\right)^{-1}. \end{align}
Given
$w=w^\alpha\frac{\partial}{\partial z^\alpha}\in T^{1,0}_zM$
, we set
$y=y^i\frac{\partial}{\partial x^i}, \, \tilde{y}=\tilde{y}^i\frac{\partial}{\partial x^i} \in T_xM$
such that
\begin{equation*}y^{\alpha}=\tilde{y}^{\alpha+n}=\displaystyle\frac{1}{2}\left(w^{\alpha}+\bar{w}^{\alpha}\right), \quad y^{\alpha+n}=-\tilde{y}^{\alpha}=-\displaystyle\frac{{i}}{2}\left(w^{\alpha}-\bar{w}^{\alpha}\right),\end{equation*}
and
$\unicode{x1D568}=(w^1, \ldots, w^n)$
,
$\unicode{x1D56A}=(y^1,\ldots,y^{2n})$
,
$\tilde{\unicode{x1D56A}}=(\tilde{y}^{1},\ldots,\tilde{y}^{2n})$
. Then
\begin{align}\unicode{x1D566} G \, \unicode{x1D56A}^{{t}} =& \tilde{\unicode{x1D566}} G \tilde{\unicode{x1D56A}}^{{t}} =\displaystyle\frac{1}{2}\left(\unicode{x1D567} H \bar{\unicode{x1D568}}^{{t}} +\unicode{x1D568} H \bar{\unicode{x1D567}}^{{t}}\right), \end{align}
\begin{align}\unicode{x1D566} G \, \tilde{\unicode{x1D56A}}^{{t}} =&-\tilde{\unicode{x1D566}} G \unicode{x1D56A}^{{t}}=\displaystyle\frac{{i}}{2}\left(\unicode{x1D567}H\bar{\unicode{x1D568}}^{{t}}- \unicode{x1D568} H\bar{\unicode{x1D567}}^{{t}}\right).\end{align}
Especially,
$g(u,\tilde{u})=\unicode{x1D566} G \, \tilde{\unicode{x1D566}}^{{t}}=-\tilde{\unicode{x1D566}} G \unicode{x1D566}^{{t}}=0$
.
Lemma 2.1. Given a Hermitian manifold (M, h), then
\begin{align}\displaystyle\frac{1}{2}R_{ijkl} u^{i}\tilde{u}^{j}y^{k}\tilde{y}^{l} &=-\displaystyle\frac{1}{2} \displaystyle\frac{\partial h_{\alpha\bar\beta}}{\partial z^\mu\partial\bar{z}^\nu}\big(v^{\alpha}\bar{w}^{\beta}w^{\mu}\bar{v}^{\nu}+w^{\alpha}\bar{v}^{\beta}v^{\mu}\bar{w}^{\nu}\big)\nonumber \\[5pt] &\quad +\displaystyle\frac{1}{4}\left(v^{\alpha} w^{\mu}+w^{\alpha}v^{\mu}\right)\displaystyle\frac{\partial h_{\alpha\bar{\lambda}}}{\partial z^\mu}h^{\bar{\lambda}\gamma}\displaystyle\frac{\partial h_{\gamma\bar{\beta}}}{\partial \bar{z}^\nu}\big(\bar{w}^{\nu}\bar{v}^{\beta}+\bar{v}^{\nu}\bar{w}^{\beta}\big)\nonumber \\[5pt] &\quad -\displaystyle\frac{1}{4}v^{\alpha}\bar{w}^{\nu}\left(\displaystyle\frac{\partial h_{\alpha\bar{\lambda}}}{\partial \bar{z}^{\nu}}-\displaystyle\frac{\partial h_{\alpha \bar{\nu}}}{\partial \bar{z}^{\lambda}}\right)h^{\bar{\lambda}\gamma}\left(\displaystyle\frac{\partial h_{\gamma\bar{\beta}}}{\partial {z}^{\mu}}-\displaystyle\frac{\partial h_{\mu \bar{\beta}}}{\partial {z}^{\gamma}}\right){w}^{\mu}\bar{v}^{\beta}\\[5pt] &\quad -\displaystyle\frac{1}{4}w^{\alpha}\bar{v}^{\nu}\left(\displaystyle\frac{\partial h_{\alpha\bar{\lambda}}}{\partial \bar{z}^{\nu}}-\displaystyle\frac{\partial h_{\alpha \bar{\nu}}}{\partial \bar{z}^{\lambda}}\right)h^{\bar{\lambda}\gamma}\left(\displaystyle\frac{\partial h_{\gamma\bar{\beta}}}{\partial {z}^{\mu}}-\displaystyle\frac{\partial h_{\mu \bar{\beta}}}{\partial {z}^{\gamma}}\right){v}^{\mu}\bar{w}^{\beta}.\nonumber \end{align}
Proof. For simplicity, we denote by
\begin{equation*}L_{(u,y)}=u^i y^j\frac{\partial^2}{\partial x^i \partial x^j},\quad L_{(v,w)}=v^\alpha w^\beta\frac{\partial^2}{\partial z^\alpha \partial z^\beta},\end{equation*}
\begin{equation*}L_{(v,\bar{w})}=L_{(\bar{w},v)}=v^\alpha \bar{w}^\beta\frac{\partial^2}{\partial z^\alpha \partial \bar{z}^\beta},\quad L_{(\bar{v},\bar{w})}=\bar{v}^\alpha \bar{w}^\beta\frac{\partial^2}{\partial \bar{z}^\alpha \partial \bar{z}^\beta},\end{equation*}
and
\begin{equation*}v(H)=\left(v^\gamma\frac{\partial h_{\alpha\bar{\beta}}}{\partial z^\gamma}\right)_{n\times n},\quad u(G)=\left(u^k\frac{\partial g_{ij}}{\partial x^k}\right)_{2n\times 2n},\end{equation*}
\begin{equation*}L_{(u,y)}(G)=\left(u^i y^j\frac{\partial^2 g_{kl}}{\partial x^i \partial x^j}\right)_{2n\times 2n},\quad L_{(v,\bar{w})}(H)=\left(v^\alpha \bar{w}^\beta\frac{\partial^2 h_{\mu\bar{\nu}}}{\partial z^\alpha \partial \bar{z}^\beta}\right)_{n\times n}.\end{equation*}
In addition, we set
\begin{align*} a=\unicode{x1D567}H\bar{\unicode{x1D568}}^{{t}}-\unicode{x1D568} H\bar{\unicode{x1D567}}^{{t}},\quad b=\unicode{x1D567}H\bar{\unicode{x1D568}}^{{t}}+\unicode{x1D568} H\bar{\unicode{x1D567}}^{{t}},\end{align*}
\begin{align*} \mathscr{A}& =\unicode{x1D567}(w(H)) +\unicode{x1D568} (v(H))+\unicode{x1D567}(\bar{w}(H)) -\unicode{x1D568} (\bar{v}(H)) -\displaystyle\frac{\partial a}{\partial \bar{z}},\\ \mathscr{B} & =\unicode{x1D568} (v(H)) +\unicode{x1D567} (w(H))+\unicode{x1D568} (\bar{v}(H)) -\unicode{x1D567}(\bar{w}(H)) +\displaystyle\frac{\partial a}{\partial \bar{z}},\\\mathscr{C}&=\unicode{x1D567}(w(H)) +\unicode{x1D568} (v(H))-\unicode{x1D567}(\bar{w}(H)) -\unicode{x1D568} (\bar{v}(H)) +\displaystyle\frac{\partial b}{\partial \bar{z}},\\\mathscr{D}&=\unicode{x1D568} (v(H)) +\unicode{x1D567} (w(H))+\unicode{x1D568} (\bar{v}(H)) +\unicode{x1D567}(\bar{w}(H)) -\displaystyle\frac{\partial b}{\partial \bar{z}}. \end{align*}
A direct computation shows
\begin{align} L_{(u,y)} + L_{(\tilde{u},\tilde{y})}=&2\left(L_{(v,\bar{w})} + L_{(\bar{v},w)}\right), \end{align}
\begin{align}L_{(\tilde{u},y)} -L_{(u,\tilde{y})} =& 2{i} \left(L_{(\bar{v},w)} -L_{(v,\bar{w})}\right).\end{align}
Hence,
\begin{align*}\displaystyle\frac{1}{4}&\left(\displaystyle\frac{\partial^{2} g_{il}}{{\partial x^j}{\partial x^k}}+ \displaystyle\frac{\partial^{2} g_{jk}}{{\partial x^i}{\partial x^l}}- \displaystyle\frac{\partial^{2} g_{ik}}{{\partial x^j}{\partial x^l}}- \displaystyle\frac{\partial^{2} g_{jl}}{{\partial x^i}{\partial x^k}}\right)\, u^{i}\, \tilde{u}^{j}\, y^{k}\, \tilde{y}^{l}\\=&\displaystyle\frac{1}{4} \unicode{x1D566} \left(L_{(\tilde{u},y)}(G) -L_{(u,\tilde{y})}(G)\right)\tilde{\unicode{x1D56A}}^{{t}}-\displaystyle\frac{1}{4} \unicode{x1D566} \left(L_{(\tilde{u},\tilde{y})}(G) + L_{(u,y)}(G)\right)\unicode{x1D56A}^{{t}}\\=&-\displaystyle\frac{1}{2} \left[\unicode{x1D567}\left(L_{(\bar{v},w)}(H)\right)\bar{\unicode{x1D568}}^{{t}} +\unicode{x1D568}\left( L_{(v,\bar{w})}(H)\right)\bar{\unicode{x1D567}}^{{t}} \right]\\=&-\displaystyle\frac{1}{2} \left[\frac{\partial^2 h_{\alpha\bar{\beta}}}{\partial z^\mu \partial \bar{z}^\nu}w^\alpha\bar{v}^\beta v^\mu \bar{w}^\nu+\frac{\partial^2 h_{\alpha\bar{\beta}}}{\partial z^\mu \partial \bar{z}^\nu}v^\alpha\bar{w}^\beta w^\mu \bar{v}^\nu \right].\end{align*}
By a direct computation, we have
\begin{align*}\displaystyle\frac{\partial }{\partial x}\left(\tilde{\unicode{x1D566}} G \unicode{x1D56A}^{{t}}\right)=-\displaystyle\frac{i}{2}\left(\displaystyle\frac{\partial a}{\partial z},\displaystyle\frac{\partial a}{\partial \bar{z}}\right)F=-{i}\left(\displaystyle\frac{\partial a}{\partial \bar{z}},\displaystyle\frac{\partial a}{\partial{z}}\right)\left(F^{{t}}\right)^{-1},\\y\left(\tilde{\unicode{x1D566}}\, G\right) + \tilde{u}\left(\unicode{x1D56A} G\right)-\displaystyle\frac{\partial }{\partial x}\left(\tilde{\unicode{x1D566}} G \unicode{x1D56A}^{{t}}\right)=-{i}\left(\mathscr{A},-\,\bar{\!\mathscr{A\,}}\right)\left(F^{{t}}\right)^{-1}, \\u\left(\tilde{\unicode{x1D56A}} G\right) + \tilde{y}\left(\unicode{x1D566} G\right)-\displaystyle\frac{\partial }{\partial x}\left(\tilde{\unicode{x1D56A}} G \unicode{x1D566}^{{t}}\right)=-{i}\left(\mathscr{B},-\bar{\mathscr{B}}\right)\left(F^{{t}}\right)^{-1}. \end{align*}
Hence,
\begin{align*} \displaystyle\frac{1}{2} g^{st} [jk,s][il,t]\, u^{i}\, \tilde{u}^{j}\, y^{k} \, \tilde{y}^{l} &=\displaystyle\frac{1}{8} \left(\displaystyle\frac{\partial g_{js}}{\partial x^k} + \displaystyle\frac{\partial g_{ks}}{\partial x^j}- \displaystyle\frac{\partial g_{jk}}{\partial x^s} \right) \, \tilde{u}^{j}\, y^{k} \, g^{st}\left(\displaystyle\frac{\partial g_{it}}{\partial x^l}+ \displaystyle\frac{\partial g_{lt}}{\partial x^i} - \displaystyle\frac{\partial g_{il}}{\partial x^t}\right)\, u^{i}\, \tilde{y}^{l}\nonumber \\&=\displaystyle\frac{1}{8}\left(y\left(\tilde{\unicode{x1D566}} G\right) + \tilde{u}\left(\unicode{x1D56A} G\right)- \displaystyle\frac{\partial }{\partial x}\left(\tilde{u}G \unicode{x1D56A}^{{t}}\right)\right)G^{-1}\left(u\left(\tilde{\unicode{x1D56A}} G\right) + \tilde{y}\left(\unicode{x1D566} G\right)- \displaystyle\frac{\partial }{\partial x}\left(\tilde{\unicode{x1D56A}} G \unicode{x1D566}^{{t}}\right)\right)^{{t}}\nonumber \\&=-\displaystyle\frac{1}{8}\left(\mathscr{A},-\bar{\mathscr{A\,}}\right)\left(F^{{t}}\right)^{-1}G^{-1}F^{-1}\left(\mathscr{B},-\bar{\mathscr{B}}\right)^{{t}}\nonumber \\& =-\displaystyle\frac{1}{16}\left(\mathscr{A},-\bar{\mathscr{A}}\right)\left( \begin{array}{c@{\quad}c} 0 & H^{-1}\nonumber \\ \bar{H}^{-1} & 0 \\ \end{array}\right)\left(\mathscr{B},-\bar{\mathscr{B}}\right)^{{t}} \\&=\displaystyle\frac{1}{16}\left(\mathscr{A} H^{-1}\bar{\mathscr{B}}^{{t}}+\mathscr{B} H^{-1}\bar{\mathscr{A}}^{{t}}\right). \end{align*}
Similarly,
\begin{align*}\displaystyle\frac{\partial }{\partial x}\left(\tilde{\unicode{x1D566}} G \tilde{\unicode{x1D56A}}^{{t}}\right)=\displaystyle\frac{\partial }{\partial x}\left(\unicode{x1D566} G \unicode{x1D56A}^{{t}}\right)=\left(\displaystyle\frac{\partial b}{\partial \bar{z}},\displaystyle\frac{\partial b}{\partial{z}}\right)\left(F^{{t}}\right)^{-1},\\-\left(\tilde{u}\left(\tilde{\unicode{x1D56A}} G\right)+ \tilde{y}\left(\tilde{\unicode{x1D566}} G\right)- \displaystyle\frac{\partial }{\partial x}\left(\tilde{\unicode{x1D566}} G\tilde{\unicode{x1D56A}}^{{t}}\right)\right)= \left(\mathscr{C}, \bar{\mathscr{C}}\,\right)\left(F^{{t}}\right)^{-1},\\u\left(\unicode{x1D56A} G\right) + y\left(\unicode{x1D566} G\right)- \displaystyle\frac{\partial }{\partial x}\left(\unicode{x1D566} G\unicode{x1D56A}^{{t}}\right)= \left(\mathscr{D}, \bar{\mathscr{D}}\right)\left(F^{{t}}\right)^{-1},\end{align*}
and
\begin{align*} -\displaystyle\frac{1}{2} g^{st} [jl,s][ik,t] u^{i}\, \tilde{u}^{j}\, y^{k}\,\tilde{y}^{l}&= -\displaystyle\frac{1}{8}\left(\displaystyle\frac{\partial g_{js}}{\partial x^l}+ \displaystyle\frac{\partial g_{ls}}{\partial x^j}- \displaystyle\frac{\partial g_{jl}}{\partial x^s}\right)\tilde{u}^{j}\,\tilde{y}^{l}g^{st}\left(\displaystyle\frac{\partial g_{it}}{\partial x^k}+ \displaystyle\frac{\partial g_{kt}}{\partial x^i}- \displaystyle\frac{\partial g_{ik}}{\partial x^t}\right)u^{i}y^{k}\nonumber \\&= -\displaystyle\frac{1}{8}\Bigg(\tilde{u}\left(\tilde{\unicode{x1D56A}}G\right)+ \tilde{y}\left(\tilde{\unicode{x1D566}}G\right) \nonumber \\ & \quad -\displaystyle\frac{\partial }{\partial x}\left(\tilde{\unicode{x1D566}}G\tilde{\unicode{x1D56A}}^{{t}}\right)\Bigg)\;G^{-1} \left(u\left(\unicode{x1D56A} G\right) + y\left(\unicode{x1D566} G\right)- \displaystyle\frac{\partial }{\partial x}\left(\unicode{x1D566} G \unicode{x1D56A}^{{t}}\right)\right)^{{t}}\nonumber \\&=\displaystyle\frac{1}{8}\left(\mathscr{C}, \bar{\mathscr{C}}\,\right)\left(F^{{t}}\right)^{-1}G^{-1}\left(F\right)^{-1}\left(\mathscr{D}, \bar{\mathscr{D}}\right)^{{t}}\nonumber \\&=\displaystyle\frac{1}{16}\left(\mathscr{C},-\bar{\mathscr{C}}\,\right)\left( \begin{array}{c@{\quad}c} 0 & H^{-1} \\ \bar{H}^{-1} & 0 \\ \end{array}\right)\left(\mathscr{D},-\bar{\mathscr{D}}\right)^{{t}}\\&=\displaystyle\frac{1}{16}\left(\mathscr{C} H^{-1}\bar{\mathscr{D}}^{{t}}+\mathscr{D}{H}^{-1}\bar{\mathscr{C}}^{{t}}\right).\end{align*}
Therefore,
\begin{align*} & \displaystyle\frac{1}{2} g^{st}\left( [jk,s][il,t] - [jl,s][ik,t]\right) u^{i}\, \tilde{u}^{j}\, y^{k}\,\tilde{y}^{l}\\=&\displaystyle\frac{1}{4}\left[\unicode{x1D567}(w(H)) +\unicode{x1D568} (v(H)) \right]H^{-1}\overline{\left[\unicode{x1D567}(w(H)) +\unicode{x1D568} (v(H)) \right]}^{{t}}\\&-\displaystyle\frac{1}{4}\left[\unicode{x1D567}(\bar{w}(H)) -\displaystyle\frac{\partial \left(\unicode{x1D567}H\bar{\unicode{x1D568}}^{{t}}\right)}{\partial \bar{z}}\right]H^{-1}\overline{\left[\unicode{x1D567}(\bar{w}(H)) -\displaystyle\frac{\partial \left(\unicode{x1D567}H\bar{\unicode{x1D568}}^{{t}}\right)}{\partial \bar{z}}\right]}^{{t}}\\&-\displaystyle\frac{1}{4}\left[\unicode{x1D568} (\bar{v}(H)) -\displaystyle\frac{\partial \left(\unicode{x1D568} H\bar{\unicode{x1D567}}^{{t}}\right)}{\partial \bar{z}}\right]H^{-1}\overline{\left[\unicode{x1D568} (\bar{v}(H)) -\displaystyle\frac{\partial \left(\unicode{x1D568} H\bar{\unicode{x1D567}}^{{t}}\right)}{\partial \bar{z}}\right]}^{{t}}\\=&\displaystyle\frac{1}{4}\left(v^{\alpha} w^{\mu}+w^{\alpha}v^{\mu}\right)\displaystyle\frac{\partial h_{\alpha\bar{\lambda}}}{\partial z^\mu}h^{\bar{\lambda}\gamma}\displaystyle\frac{\partial h_{\gamma\bar{\beta}}}{\partial \bar{z}^\nu}\big(\bar{w}^{\nu}\bar{v}^{\beta}+\bar{v}^{\nu}\bar{w}^{\beta}\big)\\&-\displaystyle\frac{1}{4}v^{\alpha}\bar{w}^{\nu}\left(\displaystyle\frac{\partial h_{\alpha\bar{\lambda}}}{\partial \bar{z}^{\nu}}-\displaystyle\frac{\partial h_{\alpha \bar{\nu}}}{\partial \bar{z}^{\lambda}}\right)h^{\bar{\lambda}\gamma}\left(\displaystyle\frac{\partial h_{\gamma\bar{\beta}}}{\partial {z}^{\mu}}-\displaystyle\frac{\partial h_{\mu \bar{\beta}}}{\partial {z}^{\gamma}}\right){w}^{\mu}\bar{v}^{\beta}\\&-\displaystyle\frac{1}{4}w^{\alpha}\bar{v}^{\nu}\left(\displaystyle\frac{\partial h_{\alpha\bar{\lambda}}}{\partial \bar{z}^{\nu}}-\displaystyle\frac{\partial h_{\alpha \bar{\nu}}}{\partial \bar{z}^{\lambda}}\right)h^{\bar{\lambda}\gamma}\left(\displaystyle\frac{\partial h_{\gamma\bar{\beta}}}{\partial {z}^{\mu}}-\displaystyle\frac{\partial h_{\mu \bar{\beta}}}{\partial {z}^{\gamma}}\right){v}^{\mu}\bar{w}^{\beta}.\end{align*}
This completes the proof.
Now we prove Main Theorem.
Theorem 2.2. (Main Theorem). Let (M, h) be a Hermitian manifold. If the holomorphic sectional curvature is just half of the sectional curvature in a holomorphic plane section, i.e., (1.16) holds, then h is a Kähler metric.
Proof. Take
$y=u$
in (2.11), then
\begin{align} \displaystyle\frac{1}{2} R_{ijkl} u^{i}\tilde{u}^{j}u^{k}\tilde{u}^{l}-\Theta_{\alpha\bar{\beta}\mu\bar{\nu}}v^{\alpha}\bar{v}^{\beta}v^{\mu}\bar{v}^{\nu} = -\displaystyle\frac{1}{2}v^{\alpha}\bar{v}^{\nu}\left(\displaystyle\frac{\partial h_{\alpha\bar{\lambda}}}{\partial \bar{z}^{\nu}}-\displaystyle\frac{\partial h_{\alpha \bar{\nu}}}{\partial \bar{z}^{\lambda}}\right)h^{\bar{\lambda}\gamma}\left(\displaystyle\frac{\partial h_{\gamma\bar{\beta}}}{\partial {z}^{\mu}}-\displaystyle\frac{\partial h_{\mu \bar{\beta}}}{\partial {z}^{\gamma}}\right){v}^{\mu}\bar{v}^{\beta}. \end{align}
We can see
\begin{equation*}\displaystyle\frac{1}{2} R_{ijkl} u^{i}\tilde{u}^{j}u^{k}\tilde{u}^{l}=\Theta_{\alpha\bar{\beta}\mu\bar{\nu}}v^{\alpha}\bar{v}^{\beta}v^{\mu}\bar{v}^{\nu},\end{equation*}
if and only if
\begin{equation}v^{\alpha}\bar{v}^{\nu}\left(\displaystyle\frac{\partial h_{\alpha\bar{\lambda}}}{\partial \bar{z}^{\nu}}-\displaystyle\frac{\partial h_{\alpha \bar{\nu}}}{\partial \bar{z}^{\lambda}}\right)=0\end{equation}
holds for any
$v\in T^{1,0}_zM$
. Take
$\unicode{x1D567}=e_1=(1,0,\ldots,0)$
, then
$\displaystyle\frac{\partial h_{1\bar{\lambda}}}{\partial \bar{z}^{1}}=\displaystyle\frac{\partial h_{1\bar{1}}}{\partial \bar{z}^{\lambda}}$
,
$1\leq \lambda \leq n$
. Take
$\unicode{x1D567}=e_2=(0,1,\ldots,0)$
, then
$\displaystyle\frac{\partial h_{2\bar{\lambda}}}{\partial \bar{z}^{2}}=\displaystyle\frac{\partial h_{2\bar{2}}}{\partial \bar{z}^{\lambda}}$
,
$1\leq \lambda \leq n$
.
$\cdots$
Take
$\unicode{x1D567}=e_n=(0,0,\ldots,1)$
, then
$\displaystyle\frac{\partial h_{n\bar{\lambda}}}{\partial \bar{z}^{n}}=\displaystyle\frac{\partial h_{n\bar{n}}}{\partial \bar{z}^{\lambda}}$
,
$\hbox{$1\leq\!\lambda\,\leq\,n$}$
. Take
$\unicode{x1D567}=e_{\alpha}+e_{\beta}$
, where
$1\leq \alpha<\beta\leq n$
or
$1\leq \beta<\alpha\leq n$
, then
$\displaystyle\frac{\partial h_{\alpha\bar{\lambda}}}{\partial \bar{z}^{\beta}}=\displaystyle\frac{\partial h_{\alpha\bar{\beta}}}{\partial \bar{z}^{\lambda}}$
,
$1\leq \lambda \leq n$
. Hence, if
\begin{equation*}\textrm{HSC}(z;\; v)=\displaystyle\frac{1}{2}K(x;\;u,Ju),\end{equation*}
then
\begin{equation*} \displaystyle\frac{\partial h_{\alpha\bar{\lambda}}}{\partial \bar{z}^{\beta}}=\displaystyle\frac{\partial h_{\alpha\bar{\beta}}}{\partial \bar{z}^{\lambda}}, \quad 1\leq \alpha,\beta, \lambda \leq n,\end{equation*}
i.e., h is a Kähler metric.
As is well known, if h is a Kähler metric, then [Reference Lu and Xu1,Reference Siu3]
\begin{equation}2R_{ijkl}u^iy^ju^ky^l=\Theta_{\alpha\bar{\beta}\mu\bar{\nu}}\big(v^{\alpha}\bar{w}^{\beta}-w^{\alpha}\bar{v}^{\beta}\big)\big(w^{\mu}\bar{v}^{\nu}-v^{\mu}\bar{w}^{\nu}\big).\end{equation}
Now we have the following result.
Proposition 2.3. Let (M,h) be a Hermitian manifold such that (2.16) holds for any two directions
$u, \, y \in T_xM$
. Then h is a Kähler metric.
Proof. Take
$y=\tilde{u}$
, it follows from (2.16) that
$\displaystyle\frac{1}{2} R_{ijkl} u^{i}\tilde{u}^{j}u^{k}\tilde{u}^{l}-\Theta_{\alpha\bar{\beta}\mu\bar{\nu}}v^{\alpha}\bar{v}^{\beta}v^{\mu}\bar{v}^{\nu}=0$
holds for any
$u\in T_xM$
. By the proof of Main Theorem, we complete the proof.
Acknowledgements
The calculations of this paper benefit from the matrix theory taught by Prof. Yichao Xu. The authors express their heartfelt thanks to Prof. Yichao Xu. The authors are very grateful to the referee for providing many helpful suggestions.
