I don't live in California and I haven't been down there very often, but I often waver between thinking the place is either right on the mark or off the wall on many issues. One issue in particular that's garnered a lot of news lately is the growing movement toward enshrining the right to repair stuff. The movement seems to have originated among farmers who want to be able to repair broken farming equipment themselves or have it fixed by local third-party repair shops instead of having to pay the exorbitant fees that manufacturers of such equipment charge for such services. A lobbying organization has been set up in the USA to help propel the movement forward, and up here in Canada a Member of Parliament of the Province of Ontario has submitted a bill proposing legislation that allows consumers to have the legal right to repair the devices they've purchased. The European Union is also considering strengthening legislation in this area, in particular to address the growing problem of tech waste or e-waste (although broken dishwashers seem to be more of a problem over there).
Apple has opposed such legislation in the past and has lobbied in various jurisdictions to kill any proposed legislation that would make it easy for users to repair their broken iPhones and other devices, though they have begun making some small steps towards this direction. But U.S. state initiatives to introduce right to repair legislation continue to gain ground, and in all the hurry and rush it might be worthwhile for us to pause for a moment and ask ourselves whether such initiatives will help us as consumers or end up harming us instead.
But let's say that the right to repair movement wins the current conflict and legislation to this effect gets enshrined by federal, state and provincial governments here in North America. Does this mean that we as consumers can celebrate that we've won?
Not necessarily. If you look into some of the right to repair legislation closely you'll discover things like rules requiring companies that manufacture devices like smartphones to keep making parts for phones they no longer manufacture. In some cases proposed legislation would require companies like Apple to continue making parts available for iPhone models up to seven years after the models themselves are no longer being manufactured. Changes like this would necessitate a complete retooling of the company's manufacturing processes, and it could cost they would incur doing this could be huge. And guess who will ultimately have to bear the brunt of these costs? Us consumers.
Then there are proposals in such legislation that would require manufacturers to allow third parties to make and sell replacement parts for the devices they manufacture. Now really, does that sound like a good idea to you? Take that most exasperating but necessary part of today's laptop or smartphone: the battery. While most of the parts used to make iPhones and other devices are manufactured in China by companies under license to Apple, there are also a ton of Apple-clone parts out there being manufactured and sold by unlicensed Chinese companies, many of which are shady. (I encourage you to read Paul Midler's book "Poorly Made in China: An Insider's Account of the China Production Game" if you have any doubts about this area. His earlier book "What's Wrong with China" is even more disturbing -- or hilarious, depending on how you see things.) With batteries this can be a real problem because Lithium ion batteries can be prone to exploding if they're not manufactured properly according to strict industrial standards. And I think the last thing you want to have happen to you is to have your iPhone blow up in your hand because you had a local repair shop replace the battery in your phone with one made by someone other than Apple. And requiring Apple itself to continue manufacturing batteries for iPhones years after the phones themselves are being manufactured or sold is also problematic because lithium ion batteries have a limited shelf life when they're not being used.
There's also another problem which has to do with the security and privacy of your devices. Many of us who own Macs and iPads have felt we needed longer Lightning cables than the ones that came with our tablets, and to save money we bought such cables on Amazon instead of from Apple. Well, recently a security researcher demonstrated you can actually hide a WiFi chip inside a Lightning cable and use it to gain control over a Mac you plug the cable into. Now I wonder how long it will be until cables like that show up on Amazon Marketplace sites? What I mean is that by passing legislation that requires manufacturers to release detailed technical documentation of their devices so third-party companies can manufacture compatible parts and repair shops can undertake repairs with these parts, legislators who appear to have our best interests at heart as consumers may actually be facilitating the spread of malware that can compromise the security of our tech gear and eavesdrop on our private communications.
My point in all this is to simply point out that legislation can often break as many things as it tries to fix. So maybe it's time we all took a breath and reconsidered the scope of what the right to repair movement is trying to accomplish through proposed legislation and live with the fact that life in this world will always be a mix of yin and yang.
Hi Mitch -- Glad to see you writing about Right to Repair, but in fairness to your readers -- you have mis-stated important specifics about the bills themselves. First -- R2R legislation does not require OEMS to continue to make parts over time. There is existing law in California with a time requirement, but that isn't R2R. Second, there is no requirement anywhere that any OEMS must make third part parts available. R2R only requires that the OEM sell their parts, if they exist, on fair and reasonable terms. If a consumer wants to buy a 3rd party battery -- thats already their choice.
As to releasing design information that might compromise safety and security -- thats not in any of our legislation either. Manufacturers have already put their patents in the public eye. Repair documentation never includes any trade secrets, and its obviously the case that the Genius employees at Apple aren't the source of design secrets. If the OEM techs doesn't get the secret sauce -- neither does anyone else.


text:
About Trinity Hall
Trinity Hall is the fifth oldest College in the University of Cambridge. It was founded in 1350 by Bishop Bateman, originally for the study of canon and civil law. The College consists of around 650 students, both undergraduates and postgraduates across a range of subjects, 60 Fellows, 130 staff and 8,000 alumni worldwide. The College is located in central Cambridge by the river, with a further site at Wychfield off Storey's Way and accommodation on Thompson's Lane.
In a city central location by the river, Trinity Hall is the fifth oldest College in the University of Cambridge and home to around 650 students.
The College consists of around 650 students, both undergraduates and graduates across a range of subject, 60 Fellows, 130 staff and 8,000 alumni worldwide.
Discover Trinity Hall's expansive and varied gardens across two sites, one in the heart of the city and Wychfield, a short distance west.
For students and visitors alike, our old and new library are the most popular areas to visit.
All genres of music are encouraged at Trinity Hall and there are countless opportunities to get involved.
The Chapel exists for every member of Trinity Hall, and is available for private prayer, meditation, or just quietness and reflection any hour.
Students enjoy a wide variety of sports at Trinity Hall. Whether you are an established athlete or interested in trying something new, there is sure to be something here for you.
Key sources of support, reporting options, and disciplinary procedures.
View and apply for our academic, postdoctoral and staff vacancies.

redpajama_set_name: RedPajamaCommonCrawl


This exciting Nine-day tour offers you the opportunity to see and experience the best that the United Arab Emirates have to offer. You will have the opportunity to explore the old and modern part of cosmopolitan Dubai and visit its major highlights. Later you will be in for a thrilling journey over rolling dunes deep into the heart of the desert and enjoy a BBQ dinner and Belly dancing performance. You will go on an exceptional adventure exploring the East coast of UAE with its quaint fishing villages to mountain canyons and beautiful beaches. On this tour, you will also spend your day at regions largest integrated theme park destination. Last but not the least, you will go on a full day excursion to Abu Dhabi, the capital city of UAE where you get to marvel at the majestic Grand Mosque, which is one of the largest mosques in the world.
Prices for this tour are a person, based on a twin/double room sharing basis.


LAW LIK-AH (Kevin Cheng) used to practice law for personal fame and glory but after setting up a Legal Advice Centre with his cousin GEORGE MIKE JR. (Lam Chi Sin) and his good friend TING KA-FU (Sam Lee) in Sham Shui Po, LIK-AH changes his mind and realises that legal representation is not only a privilege of the rich.
On the other hand, barrister WONG SZ-FU (Myolie Wu), who used to admire LIK-AH for his invincible performances, starts to disdain him for downgrading to a ghetto lawyer. However, after a few court cases with him, the two start to appreciate each other and fall in love.
Besides succeeding in her relationship, SZ-FU tries to become an elite practitioner in town. In order to gain advancement, she seizes every chance to get close to the renowned barrister TSEUNG PAK-KEI (Shek Sau). As PAK-KEI enjoys getting people into trouble, he makes sure SZ-FU is accidentally embroiled in a lawsuit.
In order to save SZ-FU, LIK-AH has no choice but to challenge the law.
I agree, the only people I like in this show is Kevin Cheng, Myolie Wu and Sharon Chan.
I don't like the cast one bit! I've never liked Kevin Cheng, and I'm only ok with Myolie, but her recent roles are kinda boring. But it's been a while since I last watched a legal drama so I decided to give this a shot. And it seems pretty good! Very promising. Can I just say I like the theme song?! It's super random and not even completely in Cantonese but it's very catchy.
I agree w u! Kevin's good at this bad boy swagger thing but didn't he do this is Only You too? But i'm def nt a fan of myolie. The thing I like about this show is that it's a legal drama, which (hopefully) means it'll b more structured and has more focus.
i really like this series too!!! kevin has the funniest facial expressions! hahaha and like you said, his interactions with myolie are hilarious hahha 🙂 are you going to write any posts on this?
Up to episode 6 now! I hope the series will keep it up!
The actors are of a good calibre and are really endearing and funny. I really like Kevin, Myolie & Joyce also never seen Sam Lee in a tv series. I watched the first episodes it has the makings of good series!
yeah I think Sam Lee is an ok actor. I wonder how's Sharon's character will turn out to be. Kevin is far too cute and hilarious in this series. He does actually make a really good lawyer. Joyce Tang's acting is good I guess, not much improve from her last series I've seen her in Beauty Knows No Pain. I don't know I guess she plays this good girl roles most of the time. Myolie is an ok lawyer I guess, but I'm interested in how will Kevin and Myolie will end up together?
It's all just like jumping from one pic/scene or snapshot or whatever to the next… not really a clip? If you know what I mean… lol.
I'm liking it so far though (: I like Kevin, Myolie, Joyce (but her hair in this series…), and Sharon so this is sorta win xD & I haven't watched any TVB series since… Relic of an Emissary. Hmm, debating if I should watch My Sister of Eternal Flower or not… only appeal is Raymond and Charmaine but HER HAIR! D: yeah, I have a thing with hair haha and her hair… sigh.
I sometimes like to watch Alex Lam as Mike George, I think he has improved ever since I watched him from My Sister of Eternal Flower and now this show.
what's with woman's hairs these days, first charmaine and now joyce..
This series looked so weird from the opening credits video, but I was pleasantly surprised by how entertaining and interesting it became 😀 I think it's quite refreshing with this mix of cast, and definitely nice to see Kevin play a different type of role. Quite likeable and amusing.
I also like the look of Myolie here, kinda cool cut, haha.. her personality can be rather irrational at times but I guess it balances out with Kevins.
I have to say I think the ending could have been better :/ I was glad PK was in jail but LA…. Noooo…anyways as they say there might be series two…. I'll definitely want to watch to see what happens.
I had that same reaction too. I thought that L.A going to jail was some sort of a joke so that Myolie can get back with him but it turns out to be not. 🙁 But I think it was sweet that Lik A did for Si fu. I didn't really like Myolie's character in the show one bit ever since episode 1.
And it's sad that Sharon died without pursuing her dream to open a French restaurant. I enjoyed watching Sharon and Sam Lee together 🙂 I guess they made the show a little bit better for me to continue watching this show.
I'm still confused with the last episode though. Is PK now good or evil? And is mike george jr and leung si lai together or not?
OMY Myolie is a lawer, I have to watch this series. I like her and kevin too.
I love justice series. I like this move so far, Myolie has cool hair.


Content questions are used to ask about the identify of a person, an object, a time, a location, or a quantity, or to seek an explanation or process.
In English, we can ask questions using "Who, What, When, How, Where, Why".
Today we are going to learn how to ask these type of questions in Chinese.
1. Who's at the door?
tā zài gēn shuí shuō huà ?
2. What is your ideal in life?
nǐ de lǐ xiǎng shì shén me ?
nǐ jǐ shí huí lái ?
2. When is your vacation?
nǐ men shén me shí hou fàng jià ?
1. How should I address her?
wǒ gāi zěn me chēng hu tā ?
2. How did you spend your holiday?
jià qī guò de zěn yàng ?
nǐ zhù zài nǎ lǐ ?
2. Where is he hiding?
tā cáng zài nǎ er ?
nǐ shì shén me dì fang de rén ?
1. Why were you late?
2. Why must he do it that way?
wèi shén me tā yào nà yàng zuò ?
We have learnt most of the characters for these content questions before. Try to practice how to write those characters.
哪儿 means where, and the pinyin is nǎr or nǎ	er if you split them.


text:
Home > News > Etah : 3 jihadi cattle thieves held for gang-rape of 10-year-old girl
Etah : 3 jihadi cattle thieves held for gang-rape of 10-year-old girl
Agra : Three days after a 10-year-old girl was abducted from her house and gang raped in a farm field, Etah district police has arrested three cattle thieves allegedly involved in the heinous crime.
The three accused were identified as Arif Khan of Nadrai, Kasganj district, Israr and Kalua Khan of Agra district. The trio is charged under IPC section 376 (D) (Gangrape), 363 (punishment for kidnapping) and POCSO Act.
According to police, the three accused abducted the survivor on August 25 in the wee hours while she was sleeping on a cot along with her younger sister in the open area of her house which is adjacent to the main road of the village. The incident happened under Awagadh police jurisdiction.
Etah superintendent of police Sunil Kumar Singh said, "Initially, the objective of three accused was to steal the two buffaloes from the survivor's house for which they recceed the place before. However, when they reached survivor's house, they couldn't find the cattle as it was tied in her uncle's barn. Infuriated, the accused abducted the teenage girl by gagging her and raped her in a nearby farm field."
"The matter came to light when the girl returned home bleeding profusely from the abdomen and alerted her parents who rushed her to the nearby community health centre after informing the police. She was referred to Aligarh for higher treatment. During probe, our team found the blood soaked blanket used in the rape and also the rope brought by the thieves to steal the buffaloes. Forensic team will now collect semen sample from the three accused to match with the survivor's vaginal swab sample," he added.
CO Jalesar Gurmeet Singh, who visited the survivor at Aligarh government hospital on Thursday, said, "The condition of the girl is now stable. She is able to narrate the sequence of events. We will conduct an identification parade of the three accused."
FanaticsTrending Topics
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BJP MLA Raja Singh put under house arrest, says MIM behind the attacks in Bhainsa


Excellently placed close to the white sandy beach in the bustling town of EL Arenal, the Blue Sea Tower (previously: Blue Sea Arenal Tower) offers contemporary accommodation that is convenient to a large selection of activities to keep all ages entertained. Make yourself at home in one of the guest rooms that all have come with a basic amenities, including a private bathroom and a balcony with pool or town views. A safe and telephone all come as standard and Wi-Fi is available at the reception if you wish to stay connected. The multi-lingual staff are also on hand around the clock to ensure even the most demanding of guests needs are met for a truly enjoyable stay. Guests of the hotel can start their day with a filling buffet breakfast each morning making for an ideal start to any day. Leisure facilities at the Blue Sea Tower include an attractive outdoor swimming pool, which is lined with a selection of loungers making it an ideal place to spend the day relaxing under the sun. There is also a pool found on the rooftop with stunning views of the entire Bahia de Palma and El Arenal. For those wishing to explore the area, the world famous Aqualand waterpark and several golf courses can be found nearby. A great selection of water sports can be availed of at the beach also.
The hotel has 142 guest rooms all offering private balconies, bathrooms with a shower/tub and bidet and city or sea views. Wi-Fi is available at a surcharge.
Price includes: Prices includes Flights & 07 nights AL in the Blue Sea Tower.
Price & availability correct as of 15th February 2016 at 12:55, based on 2 adults sharing, monthly prices from.
ABTA P4936. Agent for ATOL Protected Operators.


text:
\section{Introduction}
In 1985, M.Gromov (\cite{[G-85]}) introduced the beautiful theory of pseudoholomorphic curves in symplectic manifold, which has profoundly influenced the study of symplectic geometry and symplectic topology. One can refer to D.Macduff and D.Salamon's book \cite{[MS-12]} and references therein. Moreover,  Gromov's compactness theorem of pseudoholomorphic curves was also a interesting probelm from the analytic point of views.
T.H.Parker and J.G.Wolfson (\cite{[PW-93]}), R.G.Ye (\cite{[Y-94]}) independently gave the analytic proofs of Gromov's compactness theorem.
This field has been extended to the case of domain manifold with higher dimention, i.e., the pseudoholomorphic map between almost Hermitian manifolds.
Ch.Y.Wang (\cite{[W1-03]}) studied the regularity and blow-up analysis of pseudoholomorphic maps by using the techniques developed in the theory of harmonic maps. T.Rivi\'ere and G.Tian (\cite{[RT-04]}) studied the almost complex four-manifold into algebraic varieties  in connection with C.Taubes' works. For the triholmorphic maps, Ch,Y.Wang (\cite{[W2-03]}), C.Bellettini and G.Tian (\cite{[BT-19]}) obtian the quantization of energy, compactness results respectively. In this paper, we wish to focus on the geometry of pseudoholomorphic maps by using the canonical connection on almost Hermitian manifold instead of the Levi-Civita connection.

The pseudoholomorphic map is closed related to the theory of harmonic map. It is known that (\cite{[EL-78]}, \cite{[X-96]}) a holomorphic map between K\"ahlerian manifolds is harmonic. Conversely, how to judge a harmonic map to be a holomorphic one is an important topic in the theory of harmonic map.
Y.T.Siu (\cite{[S-80]}) proved the holomorphicity result of harmonic maps between compact K\"ahler manifold under the target manifold has strongly negative curvature. J.Jost and S.T.Yau (\cite{[JY-93]}) used the Hermitian harmonic maps to study maps from Hermitian manifold to Riemannian manifods, and they also obtianed rigidity results in Hermitian geometry. K.F.Liu and X.K. Yang (\cite{[LY-14]}) extended Siu's result to Hermitian manifolds, and they also studied several types of harmonic maps from Hermitian manifold. Y.X.Dong (\cite{[D-13]}) has used the monotonicity formulae to derive holomorphicity and Liouville type results for pluriharmonic maps and harmonic maps between K\"ahlerian manifolds. Y.L.Xin (\cite{[X-96]}) found a sufficient condition of partial energe density to judge the holomorphicity of a harmonic maps from Riemannian surfaces into complex projective space.
L.Ni (\cite{[N-19]}) use the $\partial\bar{\partial}$-Bochner formulae to study the general Schwarz lemma and their applications on K\"ahler manifolds.
K.Tang (\cite{[T-19]}) generalized works of L.Ni to

\noindent------------------------------------------

*The corresponding author.

\noindent  the case of Hermitian manifolds.
It is clear that the Bochner technique, i.e., the Bochner type formulae of different geometric quantity, palys an important role in studing holomorphicity and Liouville type results of harmonic maps.

The pseudoholomorphic isometry is a special class of pseudoholomorphic map that preserves the metric. If we view from theory of submanifold, the geometry of pseudoholomorphic isometry between almost Hermitain manifolds is very similar to the minimal one between Riemannian manifolds. For examples, one can choose adapted unitary frame field along a pseudoholomorphic isometry, and the trace of canonical second fundamental form is vanishing. In order to solve the Bernstein problem, J.Simons (\cite{[J-68]}) calculated the Laplacian of squre norm of the second fundamental form of closed minimal submanifolds in the unit sphere, and he obtianed the famous Simons integral inequality. K.Ogiue (\cite{[O-74]}) derived the complex version of the formula for holomorphic isometry between K\"ahler manifolds. H.W.Xu (\cite{[X-95]}) generalized Simons integral inequality to closed minimal submanifold in pinched Riemannian manifolds. We wish to develop these works to pseudoholomorphic isometry between almost Hermitian manifolds.

Notice that the canonical connection on almost Hermitian manifold  is the unique one preserves the metric, the almost complex structure and also has vanishing $(1,1)$-part of torsion. So, it's reasonable and natural to use the canonical connetion to study the properties of maps between almost Hermitian manifolds which are independent of Levi-Civita connection, such as  the Liouville type results, the pseudoholomorphicity and so on. Inspired by the works of V.Tosatti (\cite{[T-07]}), V.Tosatti, B.Wenkove and S.T.Yau (\cite{[TWY-08]}), X.Zhang (\cite{[Zh-12]}), by using the canonical connection, one can find the Bochner formulae in complex version is quite like the one of harmonic map. We utilize techniques and methods of harmonic maps in the nice book (\cite{[X-96]}) by Y.L.Xin to study the pseudoholomorphic maps  and  pluriharmonic maps. These are the main motivation of our paper.

The paper organized as follows. In section 2, we recall the almost Hermitian geometry with respect to the canonical connection. In section 3, we calculate the canonical Laplacian of partial energy density of pluriharmonic maps, and Laplacian of the half norm of canonical second fundamental form $S$ of pseudoholomorphic maps. In section 4 and 5, we study the Liouville type results of pseudoholomorphic  maps and pseudoholomorphicity of pluriharminic maps by using the Bochner type formulae obtained, respectively. In section 6, we derivie Simons integral inequality for pseudoholomorphic isometry, and we also give the bound of $S$ under the condition of parallel canonical second fundamental form.

\section{Geometries of almost Hermitian manifold}
The purpose of this section is to establish notation, review some well known facts (see \cite{[T-07]}, \cite{[TWY-08]}) concerning almost Hermitian geometry, and to establish the basic identities of smooth map between almost Hermitian manifolds related to the canonical connection. For the curvature properties, one can refer to the detailed papers \cite{[FZ-19]}, \cite{[Y-17]} and references therein.

\subsection{Geometry of canonical connection}

Let $(M, J, g)$ be a $2m$-dimensional almost Hermitian manifold, which means that $J$ is an almost complex structure on $M$ and Riemannian metric $g$ is $J$-invariant.
The tangent bundle of $M$ denoted by $TM$, and its complexification $TM\otimes\mathbb{C}$ denoted by $T^\mathbb{C}M$. By extending the almost complex structure
$\mathbb{C}$-linearly to $T^\mathbb{C}M$, we obtain the decomposition
\begin{equation}\label{2-1-1}
T^\mathbb{C}M=T'M\oplus T''M,
\end{equation}
where $T'M$ and $T''M$ are the eigenspaces of $J$ corresponding to eigenvalues $\sqrt{-1}$ and $-\sqrt{-1}$ respectively. Similarly, by extending $J$ to forms, one can decompose $k$-forms into a sum of $(p,q)$-forms with $p+q=k$. Extending the almost Hermitian metric $\mathbb{C}$-linearly to $T^\mathbb{C}M$, then it induces a Hermitian metric on $T'M$, still denoted by $g$. Locally, we can choose a unitary frame $\{e_1,\ldots,e_m\}$ with the dual coframe $\{\theta^1,\ldots,\theta^m\}$ so that
\begin{equation}\label{2-1-2}
g=\theta^i\otimes\overline{\theta^i},\hspace{1cm} dV_g=(\sqrt{-1})^m\theta^1\wedge\overline{\theta^1}\wedge\cdots\wedge\theta^m\wedge\overline{\theta^m}.
\end{equation}
The fundamental 2-form corresponding to the Hermitian metric $g$ is given by
\begin{equation}\label{2-1-3}
\omega=\sqrt{-1}\;\theta^i\wedge\overline{\theta^i}.
\end{equation}

Let $\nabla$ be an affine connection on $TM$, which is called \emph{almost-Hermitian} if it satifies
\begin{equation}\label{2-1-4}
\nabla J=0,\hspace{1cm} \nabla g=0.
\end{equation}
It is easy to see that such connections always exist on any almost Hermitian manifold. One can refer to \cite{[FZ-19]}, \cite{[G-97]} for details.  Locally, it follows from (\ref{2-1-4}) that the \emph{connection $1$-forms} $\{\theta^j_i\}$ are given by
\begin{equation}\label{2-1-5}
\nabla e_i=\theta_i^j e_j,\hspace{1cm}\theta^j_i+\overline{\theta^i_j}=0.
\end{equation}
The first and second structure equations of $\nabla$ are determined by
\begin{equation}\label{2-1-6}
d\theta^i=-\theta^i_j\wedge\theta^j+\Theta^i,
\end{equation}
\begin{equation}\label{2-1-7}
d\theta^i_j=-\theta^i_k\wedge\theta^k_j+\Omega^i_j,\hspace{0.5cm}\Omega^j_i+\overline{\Omega^i_j}=0,
\end{equation}
where the $2$-forms $\Theta^i$, $\Omega^i_j$ are called \emph{torsion} and \emph{curvature} of $\nabla$, respectively.
It is well known (see \cite{[G-97]}) that there exists a unique almost Hermitian connection $\nabla$ on $TM$ whose $(1,1)$-part satisfies
\begin{equation}\label{2-1-8}
(\Theta^i)^{(1,1)}=0.
\end{equation}
which is called \emph{canonical connection} on $(M,J,g)$. It is just the Chern connection when $J$ is integrable. We will always to use $\nabla$ to stand for the canonical connection on almost Hermitian manifold.

There are several important families of almost Hermitian manifold defined by using the fundamental 2-form, Levi-Civita connection and almost complex structure. We say that $(M,J,g)$ is \emph{almost-K\"ahlerian} if $d\omega=0$, that it is \emph{quasi-K\"ahlerian} if $(d\omega)^{(1,2)}=0$, and that it is \emph{semi-K\"ahlerian} if $d^*\omega=0$, where $d^*$ is the codifferential operator of $d$ with respect to the metric $g$. To character them, we set
\begin{equation}\label{2-1-11}
\Theta^i:=L^i_{jk}\,\theta^j\wedge\theta^k+N^i_{\bar{j}\,\bar{k}}\,\overline{\theta^j}\wedge\overline{\theta^k}
\end{equation}
with $L^i_{jk}=-L^i_{kj}$, $N^i_{\bar{j}\,\bar{k}}=-N^i_{\bar{k}\,\bar{j}}$. Then, in terms of $L^i_{jk}$ and $N^i_{\bar{j}\,\bar{k}}$, it is easy to check that $(M,J,g)$ is almost-K\"ahlerian if and only if
\begin{equation}\label{2-1-12}
L^i_{jk}=0\hspace{0.3cm} and \hspace{0.3cm} N^i_{\bar{j}\,\bar{k}}+N^j_{\bar{k}\,\bar{i}}+N^k_{\bar{i}\,\bar{j}}=0,
\end{equation}
it is quasi-K\"ahlerian if and only if
\begin{equation}\label{2-1-13}
L^i_{jk}=0,
\end{equation}
and it is semi-K\"ahlerian if and only if
\begin{equation}\label{2-1-14}
L_j:=\sum\limits_{i}L^i_{ij}=0.
\end{equation}
On the other hand, by using the Levi-Civita connection, we call $(M,J,g)$ is \emph{nearly-K\"ahlerian} if $(D_XJ)X=0$ for any tangent vector $X$ on $M$. This is equivalent to
\begin{equation}\label{2-2-60}
L^i_{jk}=0\hspace{0.3cm}and \hspace{0.3cm} N^i_{\bar{j}\,\bar{k}}=N^j_{\bar{k}\,\bar{i}}.
\end{equation}

We also set
\begin{equation}\label{2-1-15}
\Omega^i_j:=R^i_{jk\ell}\,\theta^k\wedge\theta^\ell+R^i_{jk\bar{\ell}}\,\theta^k\wedge\overline{\theta^\ell}+R^i_{j\bar{k}\,\bar{\ell}}\,\overline{\theta^k}\wedge\overline{\theta^\ell}
\end{equation}
with $R^i_{jk\ell}=-R^i_{j\ell k}$, $R^i_{j\bar{k}\,\bar{\ell}}=-R^i_{j\bar{\ell}\,\bar{k}}$. The skew-Hermitian of curvature forms $\Omega^i_j$ implies
\begin{equation}\label{2-1-16}
R^i_{jk\ell}=\overline{R^j_{i\bar{\ell}\,\bar{k}}},\hspace{1cm}R^i_{jk\bar{\ell}}=\overline{R^j_{i\ell\bar{k}}}.
\end{equation}
For a nonzero vectors $X\in T'M$, we call $HS(X):=R^j_{ik\bar{\ell}}\,X^i\overline{X^j}X^k\overline{X^\ell}/|X|^4$ the \emph{holomorphic sectional curvature} of canonical connection in direction $X$, and
for two nonzero vectors $X,Y\in T'M$ we call
$HB(X,Y):=R^j_{ik\bar{\ell}}\,X^i\overline{X^j}Y^k\overline{Y^\ell}/(|X|^2|Y|^2)$ the \emph{holomophic bisectional curvature} of canonical connection in the directions $X$, $Y$. We say the holomorphic bisectional curvature is bounded below by $A$ and bounded above by $B$, if $A\leq HB(X,Y)\leq B$ for all nonzero $(1,0)$-type vectors $X,Y$. The tensors $R'_{k\bar{\ell}}:=R^i_{ik\bar{\ell}}$, $R''_{k\bar{\ell}}:=R^\ell_{ki\bar{i}}$ are called the \emph{first Ricci curvature}, \emph{second Ricci curvature} of the canonical connection, which will be denoted by $Ric'_M$, $Ric''_M$, respectively. We say the first (resp. second) Ricci curvature is bounded below by $A$ if $R'_{k\bar{\ell}}X^k\overline{X^\ell}\geq A|X|^2$ (resp. $R''_{k\bar{\ell}}X^k\overline{X^\ell}\geq A|X|^2$) for all $(1,0)$-type vector $X$. Similar bounds of torsion and $(2,0)$-part of curvature can be defined. We will say $(M,J,g)$ has \emph{bouded geometry} if
its curvature, convariant derivatives of curvature are uniformly bounded.

\vspace{0.2cm}

Let $u$ be a smooth function on $M$, its differential can be written as
\begin{equation}\label{2-1-32}
du=u_i\,\theta^i+u_{\bar{i}}\,\overline{\theta^i},
\end{equation}
where $u_{\bar{i}}=\overline{u_i}$. Taking the exterior derivative of (\ref{2-1-32}) and using (\ref{2-1-6}), we obtain
\begin{equation}\label{2-1-33}
(u_{i,j}\,\theta^j+u_{i,\bar{j}}\,\overline{\theta^j})\wedge\theta^i+u_i\,\Theta^i+(u_{\bar{i},j}\,\theta^j+u_{\bar{i},\bar{j}}\,\overline{\theta^j})\wedge\overline{\theta^i}+u_{\bar{i}}\,\overline{\Theta^i}=0,
\end{equation}
where $u_{i,j}$, $u_{i,\bar{j}}$, $u_{\bar{i},j}$, $u_{\bar{i},\bar{j}}$ are defined by
\begin{equation}\label{2-1-34}
u_{i,j}\,\theta^j+u_{i,\bar{j}}\,\overline{\theta^j}:=du_i-u_j\,\theta^j_i,
\end{equation}
\begin{equation}\label{2-1-35}
u_{\bar{i},j}\,\theta^j+u_{\bar{i},\bar{j}}\,\overline{\theta^j}:=du_{\bar{i}}-u_{\bar{j}}\,\overline{\theta^j_i}.
\end{equation}
Comparing the coefficients of (\ref{2-1-33}), we obtain
\begin{equation}\label{2-1-36}
u_{i,j}-u_{j,i}=2u_kL^k_{ij}+2u_{\bar{k}}\overline{N^k_{\bar{i}\,\bar{j}}},
\end{equation}
\begin{equation}\label{2-1-37}
u_{i,\bar{j}}-u_{\bar{j},i}=0.
\end{equation}
The \emph{canonical Hessian} of a smooth function $u$ on $M$ is defined by $\nabla du$, which can be expressed as
\begin{equation}\label{2-1-44}
\nabla du=u_{i,j}\,\theta^j\otimes\theta^i+u_{i,\bar{j}}\,\overline{\theta^j}\otimes\theta^i+u_{\bar{i},j}\,\theta^j\otimes\overline{\theta^i}
+u_{\bar{i},\bar{j}}\,\overline{\theta^j}\otimes\overline{\theta^i}.
\end{equation}
By taking the trace of (\ref{2-1-44}), the \emph{canonical Laplacian} of $u$ is defined by
\begin{equation}\label{2-1-45}
\Box u:=\sum\limits_{i}(u_{i,\bar{i}}+u_{\bar{i},i})=2\sum\limits_{i}u_{i,\bar{i}}.
\end{equation}
We call a smooth function $u$ is \emph{strictly plurisubharmonic} if $\nabla du(X,\overline{X})=u_{i,\bar{j}}X^i\overline{X^j}>0$ for any nonzero vector $X\in T'M$.

Let $\Delta$ be the usual Laplacian related to Levi-Civita connection of $g$. The difference (see \cite{[T-07]}, \cite{[TWY-08]}) between $\Delta$ and $\Box$ are given by
\begin{equation}\label{2-2-62}
\Delta u=\Box u-2\langle\nabla u,X_L\rangle,
\end{equation}
for a function $u\in C^2(M)$,
where $\nabla u=u_{\bar{i}}\,e_i+u_i\,\overline{e_i}$, $X_L=\overline{L_i}\,e_i+L_i\,\overline{e_i}$ and $L_i$ defined in (\ref{2-1-14}).
Similarly, for a vector field $X=X^k\,e_k+\overline{X^k}\,\overline{e_k}$, by using the canonical connection and Levi-Civita connection, one can define the \emph{canonical divergence} and usual divergence of $X$, denoted by $\mbox{div}^c(X)$ and $\mbox{div}(X)$ respectively. It is easy to check that the difference of such two divergences is given by
\begin{equation}\label{2-2-62-1}
\mbox{div}(X)=\mbox{div}^c(X)-2\langle X,X_L\rangle.
\end{equation}
Thus, these two Laplacian and two divergences are equal respectively, when $(M,J,g)$ is semi-K\"ahlerian, or quasi-K\"ahlerian, or almost-K\"ahlerian, or nearly-K\"ahlerian.

\subsection{Smooth map between almost Hermitian manifolds}
Let $(M,J,g)$ and $(\widetilde{M},\tilde{J}, \tilde{g})$ be two almost Hermitiant manifolds of dimension $2m$ and $2n$, with the canonical connections $\nabla$, $\widetilde{\nabla}$, respectively.
Let $f$ be a smooth map from $M$ into $\widetilde{M}$.
Locally, choosing a unitary frame $\{e_i\}$ with the dual coframe $\{\theta^i\}$ on $M$, and choosing a unitary frame $\{\tilde{e}_\alpha\}$ with the dual coframe $\{\tilde{\theta}^\alpha\}$ on $\widetilde{M}$. Set
\begin{equation}\label{2-3-69}
f^*\tilde{\theta}^\alpha:=a^\alpha_i\,\theta^i+a^\alpha_{\bar{i}}\,\overline{\theta^i},\hspace{0.5cm}
f^*\overline{\tilde{\theta}^\alpha}:=a^{\bar{\alpha}}_i\,\theta^i+a^{\bar{\alpha}}_{\bar{i}}\,\overline{\theta^i},
\end{equation}
where $a^\alpha_i$, $a^\alpha_{\bar{i}}$, $a^{\bar{\alpha}}_i$, $a^{\bar{\alpha}}_{\bar{i}}$ satisfy
\begin{equation}\label{2-3-70}
\overline{a^\alpha_i}=a^{\bar{\alpha}}_{\bar{i}}, \hspace{1cm}\overline{a^\alpha_{\bar{i}}}=a^{\bar{\alpha}}_i.
\end{equation}

Taking the exterior derivative of the first identity in (\ref{2-3-69}), using the first structure equations of $\nabla$ and $\widetilde{\nabla}$, we obtain
\begin{equation}\label{2-3-72}
(a^\alpha_{i,j}\,\theta^j+a^\alpha_{i,\bar{j}}\,\overline{\theta}^j)\wedge\theta^i+(a^\alpha_{\bar{i},j}\,\theta^j+a^\alpha_{\bar{i},\bar{j}}\,\overline{\theta}^j)\wedge\overline{\theta^i}+a^\alpha_p\,\Theta^p+a^\alpha_{\bar{p}}\,\overline{\Theta^p}-\widetilde{\Theta}^\alpha=0,
\end{equation}
where $a^\alpha_{i,j}$, $a^\alpha_{i,\bar{j}}$, $a^\alpha_{\bar{i},j}$,  $a^\alpha_{\bar{i},\bar{j}}$ are defined by
\begin{equation}\label{2-3-73}
a^\alpha_{i,j}\,\theta^j+a^\alpha_{i,\bar{j}}\,\overline{\theta^j}:=da^\alpha_i-a^\alpha_j\,\theta^j_i+a^\beta_i\,\tilde{\theta}^\alpha_\beta,
\end{equation}
\begin{equation}\label{2-3-74}
a^\alpha_{\bar{i},j}\,\theta^j+a^\alpha_{\bar{i},\bar{j}}\,\overline{\theta^j}:=da^\alpha_{\bar{i}}-a^\alpha_{\bar{j}}\,\overline{\theta^j_i}+a^\beta_{\bar{i}}\,\tilde{\theta}^\alpha_\beta.
\end{equation}
Here we have omitted the pull-back $f^*$ acts on  $\widetilde{\Theta}^\alpha$, $\tilde{\theta}^\alpha_\beta$, and similar conventions will be used in the sequal. Comparing the coefficients of (\ref{2-3-72}), we obtain
\begin{equation}\label{2-3-75}
a^\alpha_{i,j}-a^\alpha_{j,i}=2a^\alpha_p\,L^p_{ij}+2a^\alpha_{\bar{p}}\,\overline{N^p_{\bar{i}\,\bar{j}}}-2a^\beta_ia^\gamma_j\tilde{L}^\alpha_{\beta\gamma}-2\overline{a^\beta_{\bar{i}}}\,\overline{a^\gamma_{\bar{j}}}\,\widetilde{N}^\alpha_{\bar{\beta}\bar{\gamma}},
\end{equation}
\begin{equation}\label{2-3-76}
a^\alpha_{i,\bar{j}}-a^\alpha_{\bar{j},i}=(a^\beta_{\bar{j}}a^\gamma_i-a^\beta_ia^\gamma_{\bar{j}})\tilde{L}^\alpha_{\beta\gamma}+(\overline{a^\beta_j}\,\overline{a^\gamma_{\bar{i}}}-\overline{a^\beta_{\bar{i}}}\,\overline{a^\gamma_j})\widetilde{N}^\alpha_{\bar{\beta}\bar{\gamma}},
\end{equation}
\begin{equation}\label{2-3-77}
a^\alpha_{\bar{i},\bar{j}}-a^\alpha_{\bar{j},\bar{i}}=2a^\alpha_p\,N^p_{\bar{i}\,\bar{j}}+2a^\alpha_{\bar{p}}\,\overline{L^p_{ij}}-2a^\beta_{\bar{i}}a^\gamma_{\bar{j}}\tilde{L}^\alpha_{\beta\gamma}-2\overline{a^\beta_{i}}\,\overline{a^\gamma_{j}}\,\widetilde{N}^\alpha_{\bar{\beta}\bar{\gamma}}.
\end{equation}

Taking the exterior differential of (\ref{2-3-73}), one have
\begin{equation}\label{2-3-78}
(a^\alpha_{i,jk}\,\theta^k+a^\alpha_{i,j\bar{k}}\,\overline{\theta^k})\wedge\theta^j+(a^\alpha_{i,\bar{j}k}\,\theta^k+a^\alpha_{i,\bar{j}\bar{k}}\,\overline{\theta^k})\wedge\overline{\theta^j}+a^\alpha_{i,p}\Theta^p+a^\alpha_{i,\bar{p}}\overline{\Theta^p}=-a^\alpha_p\,\Omega^p_i+a^\beta_i\widetilde{\Omega}^\alpha_\beta,
\end{equation}
which implies
\begin{eqnarray}\label{2-3-79}
a^\alpha_{i,jk}-a^\alpha_{i,kj}&=&2a^\alpha_p\,R^p_{ijk}-2a^\beta_ia^\gamma_ja^\delta_k\widetilde{R}^\alpha_{\beta\gamma\delta}-a^\beta_i(a^\gamma_ja^{\bar{\delta}}_k-a^\gamma_ka^{\bar{\delta}}_j)\widetilde{R}^\alpha_{\beta\gamma\bar{\delta}}\nonumber\\
&{}&-2a^\beta_ia^{\bar{\gamma}}_ja^{\bar{\delta}}_k\widetilde{R}^\alpha_{\beta\bar{\gamma}\bar{\delta}}+2a^\alpha_{i,p}L^p_{jk}+2a^\alpha_{i,\bar{p}}\,\overline{N^p_{\bar{j}\,\bar{k}}},
\end{eqnarray}
\begin{eqnarray}\label{2-3-80}
a^\alpha_{i,j\bar{k}}-a^\alpha_{i,\bar{k}j}=a^\alpha_pR^p_{ij\bar{k}}-2a^\beta_ia^\gamma_ja^\delta_{\bar{k}}\widetilde{R}^\alpha_{\beta\gamma\delta}
-a^\beta_i(a^\gamma_ja^{\bar{\delta}}_{\bar{k}}-a^\gamma_{\bar{k}}a^{\bar{\delta}}_{j})\widetilde{R}^\alpha_{\beta\gamma\bar{\delta}}-2a^\beta_ia^{\bar{\gamma}}_ja^{\bar{\delta}}_{\bar{k}}\widetilde{R}^\alpha_{\beta\bar{\gamma}\bar{\delta}},
\end{eqnarray}
\begin{eqnarray}\label{2-3-81}
a^\alpha_{i,\bar{j}\,\bar{k}}-a^\alpha_{i,\bar{k}\,\bar{j}}&=&2a^\alpha_p\,R^p_{i\bar{j}\,\bar{k}}-2a^\beta_ia^\gamma_{\bar{j}}a^\delta_{\bar{k}}\widetilde{R}^\alpha_{\beta\gamma\delta}-a^\beta_i(a^\gamma_{\bar{j}}a^{\bar{\delta}}_{\bar{k}}-a^\gamma_{\bar{k}}a^{\bar{\delta}}_{\bar{j}})\widetilde{R}^\alpha_{\beta\gamma\bar{\delta}}\nonumber\\
&{}&-2a^\beta_ia^{\bar{\gamma}}_{\bar{j}}a^{\bar{\delta}}_{\bar{k}}\widetilde{R}^\alpha_{\beta\bar{\gamma}\bar{\delta}}+2a^\alpha_{i,p}N^p_{\bar{j}\,\bar{k}}+2a^\alpha_{i,\bar{p}}\,\overline{L^p_{jk}},
\end{eqnarray}
where $a^\alpha_{i,jk}$, $a^\alpha_{i,j\bar{k}}$, $a^\alpha_{i,\bar{j}k}$, $a^\alpha_{i,\bar{j}\bar{k}}$ defined by
\begin{equation}\label{2-3-82}
a^\alpha_{i,jk}\,\theta^k+a^\alpha_{i,j\bar{k}}\,\overline{\theta^k}:=da^\alpha_{i,j}-a^\alpha_{p,j}\,\theta^p_i-a^\alpha_{i,p}\,\theta^p_j+a^\beta_{i,j}\tilde{\theta}_\beta^\alpha,
\end{equation}
\begin{equation}\label{2-3-83}
a^\alpha_{i,\bar{j}k}\,\theta^k+a^\alpha_{i,\bar{j}\,\bar{k}}\,\overline{\theta^k}:=da^\alpha_{i,\bar{j}}-a^\alpha_{p,\bar{j}}\,\theta^p_i-a^\alpha_{i,\bar{p}}\,\overline{\theta^p_j}+a^\beta_{i,\bar{j}}\tilde{\theta}_\beta^\alpha.
\end{equation}

Taking the exterior derivative of (\ref{2-3-74}), we have
\begin{equation}\label{2-3-83-1}
(a^\alpha_{\bar{i},jk}\theta^k+a^\alpha_{\bar{i},j\bar{k}}\overline{\theta^k})\wedge\theta^j
+(a^\alpha_{\bar{i},\bar{j}k}\theta^k+a^\alpha_{\bar{i},\bar{j}\,\bar{k}}\overline{\theta^k})\wedge\overline{\theta^j}
+a^\alpha_{\bar{i},p}\Theta^p+a^\alpha_{\bar{i},\bar{p}}\overline{\Theta^p}
=-a^\alpha_{\bar{p}}\overline{\Omega^p_i}+a^\beta_{\bar{i}}\widetilde{\Omega}^\alpha_\beta,
\end{equation}
which implies
\begin{eqnarray}\label{2-3-83-2}
a^\alpha_{\bar{i},jk}-a^\alpha_{\bar{i},kj}&=&2a^\alpha_{\bar{p}}\overline{R^p_{i\bar{j}\bar{k}}}-2a^\beta_{\bar{i}}a^\gamma_ja^\delta_k\widetilde{R}^\alpha_{\beta\gamma\delta}-a^\beta_{\bar{i}}(a^\gamma_ja^{\bar{\delta}}_k-a^\gamma_ka^{\bar{\delta}}_j)\widetilde{R}^\alpha_{\beta\gamma\bar{\delta}}\nonumber\\
&{}&-2a^\beta_{\bar{i}}a^{\bar{\gamma}}_ja^{\bar{\delta}}_k\widetilde{R}^\alpha_{\beta\bar{\gamma}\bar{\delta}}+2a^\alpha_{\bar{i}p}L^p_{jk}+2a^\alpha_{\bar{i}\bar{p}}\overline{N^p_{\bar{j}\bar{k}}},
\end{eqnarray}
\begin{eqnarray}\label{2-3-83-3}
a^\alpha_{\bar{i},j\bar{k}}-a^\alpha_{\bar{i},\bar{k}j}=-a^\alpha_{\bar{p}}\overline{R^p_{ik\bar{j}}}
-2a^\beta_{\bar{i}}a^\gamma_ja^\delta_{\bar{k}}\widetilde{R}^\alpha_{\beta\gamma\delta}
-a^\beta_{\bar{i}}(a^\gamma_ja^{\bar{\delta}}_{\bar{k}}-a^\gamma_{\bar{k}}a^{\bar{\delta}}_j)\widetilde{R}^\alpha_{\beta\gamma\bar{\delta}}
-2a^\beta_{\bar{i}}a^{\bar{\gamma}}_ja^{\bar{\delta}}_{\bar{k}}\widetilde{R}^\alpha_{\beta\bar{\gamma}\bar{\delta}},
\end{eqnarray}
\begin{eqnarray}\label{2-3-83-4}
a^\alpha_{\bar{i},\bar{j}\,\bar{k}}-a^\alpha_{\bar{i},\bar{k}\,\bar{j}}&=&2a^\alpha_{\bar{p}}\overline{R^p_{ijk}}-2a^\beta_{\bar{i}}a^\gamma_{\bar{j}}a^\delta_{\bar{k}}\widetilde{R}^\alpha_{\beta\gamma\delta}-a^\beta_{\bar{i}}(a^\gamma_{\bar{j}}a^{\bar{\delta}}_{\bar{k}}-a^\gamma_{\bar{k}}a^{\bar{\delta}}_{\bar{j}})\widetilde{R}^\alpha_{\beta\gamma\bar{\delta}}\nonumber\\
&{}&-2a^\beta_{\bar{i}}a^{\bar{\gamma}}_{\bar{j}}a^{\bar{\delta}}_{\bar{k}}\widetilde{R}^\alpha_{\beta\bar{\gamma}\bar{\delta}}+2a^\alpha_{\bar{i}p}N^p_{\bar{j}\bar{k}}+2a^\alpha_{\bar{i}\bar{p}}\overline{L^p_{jk}},
\end{eqnarray}
where $a^\alpha_{\bar{i},jk}$, $a^\alpha_{\bar{i},j\bar{k}}$, $a^\alpha_{\bar{i},\bar{j}k}$ and $a^\alpha_{\bar{i},\bar{j}\bar{k}}$ defined by
\begin{equation}\label{2-3-83-5}
a^\alpha_{\bar{i},jk}\theta^k+a^\alpha_{\bar{i},j\bar{k}}\,\overline{\theta^k}:=da^\alpha_{\bar{i},j}-a^\alpha_{\bar{k},j}\overline{\theta^k_i}
-a^\alpha_{\bar{i},k}\theta^k_j+a^\beta_{\bar{i},j}\tilde{\theta}_\beta^\alpha,
\end{equation}
\begin{equation}\label{2-3-83-6}
a^\alpha_{\bar{i},\bar{j}k}\theta^k+a^\alpha_{\bar{i},\bar{j}\,\bar{k}}\,\overline{\theta^k}:=da^\alpha_{\bar{i},\bar{j}}-a^\alpha_{\bar{k},\bar{j}}\overline{\theta^k_i}
-a^\alpha_{\bar{i},\bar{k}}\overline{\theta^k_j}+a^\beta_{\bar{i},\bar{j}}\tilde{\theta}_\beta^\alpha.
\end{equation}
Similarly, taking the exterior derivative of (\ref{2-3-82}), by comparing the (1,1)-part, one can get
\begin{eqnarray}\label{2-3-84}
a^\alpha_{i,jk\bar{\ell}}-a^\alpha_{i,j\bar{\ell}k}&=&a^\alpha_{p,j}R^p_{ik\bar{\ell}}+a^\alpha_{i,p}R^p_{jk\bar{\ell}}-2a^\beta_{i,j}a^\gamma_ka^\delta_{\bar{\ell}}\widetilde{R}^\alpha_{\beta\gamma\delta}-a^\beta_{i,j}(a^\gamma_ka^{\bar{\delta}}_{\bar{\ell}}-a^\gamma_{\bar{\ell}}a^{\bar{\delta}}_k)\widetilde{R}^\alpha_{\beta\gamma\bar{\delta}}\nonumber\\
&{}&-2a^\beta_{i,j}a^{\bar{\gamma}}_ka^{\bar{\delta}}_{\bar{\ell}}\widetilde{R}^\alpha_{\beta\bar{\gamma}\,\bar{\delta}},
\end{eqnarray}
where the covariant derivatives $a^\alpha_{i,jk\bar{\ell}}$, $a^\alpha_{i,j\bar{\ell}k}$ defined in the natural way as before.

In terms of unitary frame, from (\ref{2-3-69}), the differential of $f$ can be expressed by
\begin{equation}
df=a^\alpha_i\,\theta^i\otimes\tilde{e}_\alpha+a^\alpha_{\bar{i}}\,\overline{\theta^i}\otimes\tilde{e}_\alpha
+a^{\bar{\alpha}}_i\,\theta^i\otimes\overline{\tilde{e}_\alpha}+a^{\bar{\alpha}}_{\bar{i}}\,\overline{\theta^i}\otimes\overline{\tilde{e}_\alpha},
\end{equation}
which is a smooth section of bundle $T^*M\otimes f^{-1}T\widetilde{M}$ or $(T^{\mathbb{C}}M)^*\otimes f^{-1}T^{\mathbb{C}}\widetilde{M}$. Taking the covariant differntial of $df$ by using the connection induced by the canonical connection $\nabla$ and $\widetilde{\nabla}$, one can get the \emph{canonical second fundamental form} $\nabla df$ of $f$. More explicitly, we have
\begin{eqnarray}
\nabla df&=&a^\alpha_{i,j}\,\theta^j\otimes\theta^i\otimes\tilde{e}_\alpha+a^\alpha_{i,\bar{j}}\,\overline{\theta^j}\otimes\theta^i\otimes\tilde{e}_\alpha
+a^\alpha_{\bar{i},j}\,\theta^j\otimes\overline{\theta^i}\otimes\tilde{e}_\alpha+a^\alpha_{\bar{i},\bar{j}}\,\overline{\theta^j}\otimes\overline{\theta^i}\otimes\tilde{e}_\alpha\hspace{0.8cm}\nonumber\\
&+&a^{\bar{\alpha}}_{i,j}\,\theta^j\otimes\theta^i\otimes\overline{\tilde{e}_\alpha}\!+a^{\bar{\alpha}}_{i,\bar{j}}\,\overline{\theta^j}\otimes\theta^i\otimes\overline{\tilde{e}_\alpha}
\!+a^{\bar{\alpha}}_{\bar{i},j}\,\theta^j\otimes\overline{\theta^i}\otimes\overline{\tilde{e}_\alpha}\!+a^{\bar{\alpha}}_{\bar{i},\bar{j}}\,\overline{\theta^j}\otimes\overline{\theta^i}\otimes\overline{\tilde{e}_\alpha}.
\end{eqnarray}
Such a map is called \emph{canonical geodesic} if $\nabla df=0$; and it is called \emph{pluriharmonic} if $(\nabla df)^{(1,1)}=0$, which means that $a^\alpha_{i,\bar{j}}=a^\alpha_{\bar{i},j}=a^{\bar{\alpha}}_{i,\bar{j}}=a^{\bar{\alpha}}_{\bar{i},j}=0$.

The \emph{partial energy densities} $e'(f)$, $e''(f)$, \emph{energy density} $e(f)$ (see \cite{[EL-78]}, \cite{[X-96]}) of a smooth map $f$ between almost Hermitian manifolds defined by
\begin{equation}\label{2-3-86}
e'(f):=\sum\limits_{i,\alpha}|a^\alpha_i|^2,\hspace{0.5cm}
e''(f):=\sum\limits_{i,\alpha}|a^\alpha_{\bar{i}}|^2,\hspace{0.5cm}
e(f):=e'(f)+e''(f),
\end{equation}
and the corresponding \emph{partial energies} and \emph{energy} of $f$ defined by
\begin{equation}\label{2-3-87}
E'(f):=\int_{M}e'(f)\,dV_g,\hspace{0.5cm}
E''(f):=\int_{M}e''(f)\,dV_g,\hspace{0.5cm}
E(f):=\int_{M}e(f)\,dV_g,
\end{equation}
respectively.
A smooth map is called \emph{harmonic} if it is a critical point of the energy functional $E$. For more details about harmonic map one can refer to the books \cite{[EL-78]}, \cite{[X-96]} and references therein.

Let $f$ be a smooth map between $(M,J,g)$ and $(\widetilde{M},\tilde{J}, \tilde{g})$, it is called \emph{pseudoholomorphic} if its differential $df$ satisfies
\begin{equation}\label{2.49}
df J=\tilde{J}df.
\end{equation}
 In terms of differential forms, one can check that the indentity (\ref{2.49}) is equivalent to
\begin{equation}\label{2-3-71}
a^\alpha_{\bar{i}}=a^{\bar{\alpha}}_i=0,
\end{equation}
which means that $f^*$ preserves the forms of $(1,0)$-type. Thus, the definition (\ref{2-3-74}) and the identity (\ref{2-3-76}) imply
\begin{equation}\label{2.52}
a^\alpha_{\bar{i},j}=a^\alpha_{\bar{i},\bar{j}}=a^\alpha_{i,\bar{j}}=0.
\end{equation}
This tell us that a pseudoholomorphic map is naturally a pluriharmonic one.

\section{Bochner type formulae}

In this section we wish to calculate the canonical Laplacian of the partial energy densities of pluriharmonic maps, and the canonical Laplacian of half norm of canonical second fundamental form of pseudoholomorphic map.
\begin{Proposition}
Let $f$ be a pluriharmonic map between almost Hermitian manifolds $(M,J,g)$ and $(\widetilde{M},\tilde{J}, \tilde{g})$, then we have

\begin{eqnarray*}
(\,1\,) \;\frac{1}{2}\Box \,e'=|a^\alpha_{i,j}|^2+\overline{a^\alpha_i}a^\alpha_jR''_{i\bar{j}}-4\emph{Re}(\overline{a^\alpha_i}a^\beta_ia^\gamma_ja^\delta_{\bar{j}}\widetilde{R}^\alpha_{\beta\gamma\delta})-\overline{a^\alpha_i}a^\beta_i(a^\gamma_j\overline{a^\delta_j}-a^\gamma_{\bar{j}}\overline{a^\delta_{\bar{j}}})\widetilde{R}^\alpha_{\beta\gamma\bar{\delta}};
\end{eqnarray*}
\begin{eqnarray*}
(\,2\,) \;\frac{1}{2}\Box \,e''=|a^\alpha_{\bar{i},\bar{j}}|^2+a^\alpha_{\bar{i}}\overline{a^\alpha_{\bar{j}}}R''_{i\bar{j}}+4\emph{Re}(a^\alpha_{\bar{i}}\overline{a^\beta_{\bar{i}}}a^\gamma_{\bar{j}}a^\delta_{j}\widetilde{R}^\alpha_{\beta\gamma\delta})+a^\alpha_{\bar{i}}\overline{a^\beta_{\bar{i}}}(\overline{a^\gamma_j}a^\delta_j-\overline{a^\gamma_{\bar{j}}}a^\delta_{\bar{j}})\widetilde{R}^\alpha_{\beta\gamma\bar{\delta}}.
\end{eqnarray*}
\end{Proposition}

\emph{Proof}. Since $f$ is pluriharmonic, i.e., $a^\alpha_{i,\bar{j}}=a^\alpha_{\bar{i},j}=0$, we have
\begin{eqnarray}\label{3-1}
\frac{1}{2}\Box\,e'&=&(a^\alpha_i\,\overline{a^\alpha_i}),_{j\bar{j}}\nonumber\\
&=&(a^\alpha_{i,j}\,\overline{a^\alpha_i}+a^\alpha_i\,\overline{a^\alpha_{i,\bar{j}}}),_{\bar{j}}\nonumber\\
&=&a^\alpha_{i,j\bar{j}}\,\overline{a^\alpha_i}+a^\alpha_{i,j}\,\overline{a^\alpha_{i,j}}.
\end{eqnarray}
By using (\ref{2-3-80}) with $a^\alpha_{i,\bar{j}j}=0$, we obtain
\begin{eqnarray}\label{3-2}
a^\alpha_{i,j\bar{j}}&=&a^\alpha_pR^p_{ij\bar{j}}-2a^\beta_ia^\gamma_ja^\delta_{\bar{j}}\widetilde{R}^\alpha_{\beta\gamma\delta}-a^\beta_i(a^\gamma_j\overline{a^\delta_j}-a^\gamma_{\bar{j}}\overline{a^\delta_{\bar{j}}})\widetilde{R}^\alpha_{\beta\gamma\bar{\delta}}
-2a^\beta_i\,\overline{a^\gamma_{\bar{j}}}\,\overline{a^\delta_{j}}\widetilde{R}^\alpha_{\beta\bar{\gamma}\bar{\delta}}.
\end{eqnarray}
On the other hand, by (\ref{2-1-16}), we have
\begin{eqnarray}\label{3-3}
\overline{\overline{a^\alpha_i}\,a^\beta_i\,\overline{a^\gamma_{\bar{j}}}\,\overline{a^\delta_{j}}\widetilde{R}^\alpha_{\beta\bar{\gamma}\bar{\delta}}}
&=&a^\alpha_i\,\overline{a^\beta_i}a^\gamma_{\bar{j}}\,a^\delta_{j}\widetilde{R}^\beta_{\alpha\delta\gamma}=\overline{a^\alpha_i}a^\beta_ia^\gamma_ja^\delta_{\bar{j}}\widetilde{R}^\alpha_{\beta\gamma\delta}.
\end{eqnarray}
The first identity follows from (\ref{3-1}), (\ref{3-2}) and (\ref{3-3}). Similar caculations do for the second identity.

\hfill$q.e.d$

The following Bochner formula belongs to V.Tosatti, which appeared in his work on Schwarz Lemma of pseudoholomorphic map between almost Hermitian manifolds.
\begin{Corollary}\label{Corollary4.2}\emph{(\cite{[T-07]})}
Let $f$ be a pseudoholomorphic map between almost Hermitian manifolds $(M,J,g)$ and $(\widetilde{M},\tilde{J}, \tilde{g})$, we have
\begin{eqnarray*}
\frac{1}{2}\Box \,e=|a^\alpha_{i,j}|^2+\overline{a^\alpha_i}a^\alpha_jR''_{i\bar{j}}-\overline{a^\alpha_i}a^\beta_ia^\gamma_j\overline{a^\delta_j}\widetilde{R}^\alpha_{\beta\gamma\bar{\delta}}.
\end{eqnarray*}
\end{Corollary}

\emph{Proof}. Since a pseudoholomorphic map is pluriharmonic, it follows from that $a^\alpha_{\bar{i}}=0$ and $e=e'$.

\hfill$q.e.d$

For the square of the length second fundamental form of pseudoholomorphic maps, we have
\begin{Proposition}\label{Proposition4.4}
Let $f$ be a pseudoholomorphic map between almost Hermitian manifolds $(M,J,g)$ and $(\widetilde{M},\tilde{J}, \tilde{g})$, and let $S$ be the half norm of the second fundamental form of $f$. Then
\begin{eqnarray*}
\frac{1}{2}\Box S&=&|a^\alpha_{i,jk}|^2+|a^\alpha_{i,j\bar{k}}|^2+ \overline{a^\alpha_{i,j}}\big(a^\alpha_{p,j}R''_{i\bar{p}}+a^\alpha_{i,p}R''_{j\bar{p}}+2a^\alpha_{p,k}R^p_{ij\bar{k}}-a^\beta_{i,j}a^\gamma_k\overline{a^\delta_k}\widetilde{R}^\alpha_{\beta\gamma\bar{\delta}}\big)\\
&{}&+2\emph{Re}\big(\overline{a^\alpha_{i,j}}a^\alpha_pR^p_{ij\bar{k},k}\big)-2\emph{Re}\big(\overline{a^\alpha_{i,j}}(a^\beta_ia^\gamma_j\overline{a^\delta_k}\widetilde{R}^\alpha_{\beta\gamma\bar{\delta}})_{,k}\big).
\end{eqnarray*}
\end{Proposition}

\emph{Proof}. Under the unitary frame field, we have $S=|a^\alpha_{i,j}|^2$, so
\begin{eqnarray}\label{4-4}
\frac{1}{2}\Box S&=&(a^\alpha_{i,j}\overline{a^\alpha_{i,j}})_{,k\bar{k}}\nonumber\\
&=&(a^\alpha_{i,jk}\overline{a^\alpha_{i,j}}+a^\alpha_{i,j}\overline{a^\alpha_{i,j\bar{k}}})_{,\bar{k}}\nonumber\\
&=& |a^\alpha_{i,jk}|^2+|a^\alpha_{i,j\bar{k}}|^2+(a^\alpha_{i,jk})_{,\bar{k}}\,\overline{a^\alpha_{i,j}}+a^\alpha_{i,j}\,(\overline{a^\alpha_{i,j\bar{k}}})_{,\bar{k}}.
\end{eqnarray}
By using the pseudoholomorphicity, the Ricci identities (\ref{2-3-80}) and (\ref{2-3-84}) give
\begin{equation}\label{4-5}
a^\alpha_{i,j\bar{k}}=a^\alpha_pR^p_{ij\bar{k}}-a^\beta_ia^\gamma_j\overline{a^\delta_k}\widetilde{R}^\alpha_{\beta\gamma\bar{\delta}},
\end{equation}
\begin{equation}\label{4-6}
(a^\alpha_{i,jk})_{,\bar{k}}=(a^\alpha_{i,j\bar{k}})_{,k}+a^\alpha_{p,j}R^p_{ik\bar{k}}+a^\alpha_{i,p}R^p_{jk\bar{k}}-a^\beta_{i,j}a^\gamma_k\overline{a^\delta_k}\widetilde{R}^\alpha_{\beta\gamma\bar{\delta}}.
\end{equation}
Then the half of the canonical Laplacian of $S$ is follows from (\ref{4-4}), (\ref{4-5}), (\ref{4-6}) and the curvature properties (2.15).

\hfill$q.e.d$

Appling the Bochner formulae above, we can get a $C^2$-estimate of pseudoholomorphic map. That is

\begin{Theorem}
Let $f$ be a pseudoholomorphic from closed almost Hermitian manifold $(M,J,g)$ into almost Hermitian manifold $(\widetilde{M},\tilde{J}, \tilde{g})$ with bounded geometry. If there is a constant $\Lambda$ such that $e(f)\leq\Lambda$, then
$$S\leq C,$$
where $C$ is a constant only depend on $\Lambda$, the curvatures and their covariant derivatives of $g$, $\tilde{g}$.
\end{Theorem}

\emph{Proof}. It follows from Corollary \ref{Corollary4.2} and Proposition \ref{Proposition4.4}, there exist constants $C_1$, $C_2$ and $C_3$ only depend on $\Lambda$, the correponding curvatures and their covariant derivatives such that
\begin{equation*}
\frac{1}{2}\;\Box e(f)\geq S-C_1,\hspace{0.8cm}\frac{1}{2}\Box S\geq -C_2S-C_3,
\end{equation*}
which imply
\begin{equation*}
\frac{1}{2}\Box\big(S+(1+C_2)\,e(f)\big)\geq S-(C_1+C_1C_2+C_3).
\end{equation*}
By using the maximum principle to $S+(1+C_2)\,e(f)$ we can get the desired upper bound of $S$.

\hfill$q.e.d$

\section{Liouville type theorems}
We will use the properties of pseudoholomorphic maps and the Bochner formulae to get Liouville type results, which are similar to the theory of harmonic map. The main tools are the maximum principle and integral estimates.

\begin{Theorem}\label{Theorem5.1}
Let $f$ be a pseudoholomorphic map from closed almost Hermitian manifold $(M, J, g)$ into almost Hermitian manifold $(\widetilde{M},\tilde{J},\tilde{g})$, and $f(M)\subset V$. If $u$ is a strictly plurisubharmonic function on $V$, then $f$ is constant.
\end{Theorem}

\emph{Proof}. By using the pseudoholomorphicity, we have $a^\alpha_{\bar{i}}=0$ and $a^\alpha_{i,\bar{j}}=0$. So, the canonical Laplacian of $u\circ f$ is given by
\begin{eqnarray*}
\frac{1}{2}\Box (u\circ f)&=&(u\circ f)_{,k\bar{k}}\\
&=&(u_\alpha a^\alpha_k+u_{\bar{\alpha}}\, \overline{a^\alpha_{\bar{k}}})_{,\bar{k}}\\
&=&(u_\alpha a^\alpha_k)_{,\bar{k}}\\
&=&(u_{\alpha,\beta}\,a^\beta_{\bar{k}}+u_{\alpha,\bar{\beta}}\,a^{\bar{\beta}}_{\bar{k}})a^\alpha_k+u_\alpha a^\alpha_{k,\bar{k}}\\
&=&u_{\alpha,\bar{\beta}}\,a^\alpha_k \,\overline{a^\beta_k}.
\end{eqnarray*}
Notice that $u$ is strictly plurisubharmonic, so the maximum principle implies that $e(f)=0$, i.e., $f$ is a constant.

\hfill$q.e.d$

\begin{Theorem}
Let $f$ be a pseudoholomorphic map from closed almost Hermitian manifold $(M, J, g)$ into almost Hermitian manifold $(\widetilde{M},\tilde{J},\tilde{g})$
with $Ric''_M\geq 0$ and $HB_{\widetilde{M}}\leq 0$. Then $f$ is canonical totall geodesic. Furthermore, $f$ is constant if $Ric''_M$ is positive at a point in $M$.
\end{Theorem}

\emph{Proof}. Recall the Bochner formule obtained in Corollary \ref{Corollary4.2}, that is
\begin{equation}\label{5-1}
\frac{1}{2}\Box \,e=|a^\alpha_{i,j}|^2+\overline{a^\alpha_i}a^\alpha_jR''_{i\bar{j}}-\overline{a^\alpha_i}a^\beta_ia^\gamma_j\overline{a^\delta_j}\widetilde{R}^\alpha_{\beta\gamma\bar{\delta}}.
\end{equation}
The curvature conditons imply that $\Box e\geq 0$, so the maximum principle implies $e$ is a constant. At this time, all the terms in the right hand side of (\ref{5-1}) are nonnegative and their summation is equal to zero, so we obtain $|a^\alpha_{i,j}|^2=0$, i.e., $\nabla df=0$. If $Ric''_M$ is positive at a piont, the second term in the right hand side of (\ref{5-1}) implies $e=0$.

\hfill$q.e.d$

We can also obtain a similar result when the holomorphic bisectional curvature of target manifold might be positive.

\begin{Theorem}\label{Theorem5.3}
Let $(M, J, g)$, $(\widetilde{M},\tilde{J},\tilde{g})$ be two  almost Hermitian manifolds with $M$ is closed, and $Ric''_M\geq A$, $HB_{\widetilde{M}}\leq B$, where $A$, $B$ are positive constants. If the pseudoholomorphic map $f:M\longrightarrow\widetilde{M}$ satisfies $e(f)\leq \frac{A}{B}$, then $f$ is canonical totally geodesic and $e(f)=0$ or $e(f)=\frac{A}{B}$. Furthermore, $f$ is constant if $e(f)< \frac{A}{B}$.
\end{Theorem}

\emph{Proof}. By the curvture conditions, the Bochner formule (\ref{5-1}) implies
\begin{equation*}
\frac{1}{2}\Box\,e(f)\geq |a^\alpha_{i,j}|^2+\big(A-B\,e(f)\big)e(f).
\end{equation*}
So, if $e(f)\leq \frac{A}{B}$, we have $\Box\,e(f)\geq 0$, then the maximum principle implies $e(f)$ is a constant, $|a^\alpha_{i,j}|^2=0$ and $\big(A-B\,e(f)\big)e(f)=0$. Thus, we have $e(f)=0$ if  $e(f)< \frac{A}{B}$, which means that $f$ is constant.

\hfill$q.e.d$

For the domain manifold is complete and non-compact, we can obtain the following Liouville type result provied the energy is finite.
\begin{Theorem}
Let $(M, J, g)$ be a complete, non-compact almost Hermitian manifold with infinite volume and $Ric''_M\geq 0$, let $(\widetilde{M},\tilde{J},\tilde{g})$ be an almost Hermitian manifold with $HB_{\widetilde{M}}\leq 0$. If the pseudoholomorphic $f:M\longrightarrow\widetilde{M}$ has finite energy and
$$\int_{M}\langle \nabla e(f), X_L\rangle\,dV_g=0,$$
then $f$ is constant.
\end{Theorem}

\emph{Proof}. The Bochner formule (\ref{5-1}) and Cauchy-Schwarz inequality imply
\begin{equation}\label{5-2}
\Box\,e(f)\geq 2S, \hspace{1cm} |\nabla\,e(f)|^2\leq 2\,e(f)\,S,
\end{equation}
respectively. For any $\epsilon >0$, set $u_\epsilon:=\sqrt{e(f)+\epsilon}$, then we have
\begin{eqnarray*}
\frac{1}{2}\Box u_\epsilon&=&(u_\epsilon)_{,k\bar{k}}\\
&=& \frac{1}{4}u_\epsilon^{-1}\big(\Box\, e(f)-\frac{|\nabla e(f)|^2}{2(e(f)+\epsilon)}\big).
\end{eqnarray*}
Together with (\ref{5-2}), this implies
\begin{equation}\label{5-3}
\Box u_\epsilon\geq \frac{1}{2}u_\epsilon^{-1}S\big(1-\frac{e(f)}{2(e(f)+\epsilon)}\big)\geq0.
\end{equation}

Let $x_0\in M$ be a fixed point. We denote the geodesic balls centered at $x_0$ with radii $R$, $2R$ by $B_R$, $B_{2R}$, respectively. Choosing a cut-off function $\eta$ such that $0\leq \eta \leq1$, $\eta(x)=1$ for $x\in B_R$, $\eta(x)=0$ for $x\in M\setminus B_{2R}$. Moreover, we further assume that $|\nabla \eta|\leq \frac{C}{R}$, where $C$ is a positive constant. By using (2.24), the inequality (\ref{5-3}) gives
\begin{eqnarray}\label{5-4}
0&\leq& \int_{B_{2R}}\eta^2\, u_\epsilon\,\Box u_\epsilon\,dV_g\nonumber\\
&=&\int_{B_{2R}}\eta^2\, u_\epsilon\,\big(\Delta u_\epsilon+2\langle  \nabla u_\epsilon, X_L\rangle\big)\,dV_g\nonumber\\
&=&-2\int_{B_{2R}}\eta\,u_\epsilon\langle \nabla\eta,\nabla u_\epsilon\rangle\, dV_g-\int_{B_{2R}}\eta^2|\nabla u_\epsilon|^2+\int_{B_{2R}}\eta^2\langle \nabla e(f),X_L\rangle\,dV_g\nonumber\\
&\leq& 2\Big(\int_{B_{2R}\setminus B_R}\eta^2|\nabla u_\epsilon|^2\,dV_g\Big)^{1/2}\Big(\int_{B_{2R}\setminus B_R}u_\epsilon^2|\nabla \eta|^2\,dV_g\Big)^{1/2}
-\int_{B_{2R}\setminus B_R} \eta^2|\nabla u_\epsilon|^2\,dV_g\nonumber\\
&{}&-\int_{B_R} |\nabla u_\epsilon|^2\,dV_g+\int_{B_{2R}}\eta^2\langle \nabla e(f),X_L\rangle\,dV_g.
\end{eqnarray}
If we view (\ref{5-4}) as a quadratic inequality of the term
$\Big(\int_{B_{2R}\setminus B_R} \eta^2|\nabla u_\epsilon|^2\,dV_g\Big)^{1/2}$, we get
\begin{eqnarray}\label{5-5}
\int_{B_R} |\nabla u_\epsilon|^2\,dV_g-\int_{B_{2R}}\eta^2\langle \nabla e(f),X_L\rangle\,dV_g&\leq& \int_{B_{2R}\setminus B_R}u_\epsilon^2|\nabla \eta|^2\,dV_g\nonumber\\
&\leq& \frac{C^2}{R^2}\int_{B_{2R}} u^2_\epsilon\;dV_g.
\end{eqnarray}
Notice that $|\nabla u_\epsilon|^2=\frac{|\nabla e(f)|^2}{4(e(f)+\epsilon)}$, the inequality (\ref{5-5}) implies
\begin{eqnarray}\label{5-6}
\int_{\Sigma \cap B_R} \frac{|\nabla e(f)|^2}{4(e(f)+\epsilon)}\,dV_g-\int_{B_{2R}}\eta^2\langle \nabla e(f),X_L\rangle\,dV_g
&\leq& \frac{C^2}{R^2}\int_{B_{2R}} (e(f)+\epsilon)\;dV_g,
\end{eqnarray}
where $\Sigma:=\{x\in M\,|\,e(f)(x)\neq 0\}.$ Letting $\epsilon\rightarrow 0$, $R\rightarrow +\infty$ and using $f$ has finite energy, we obtain
\begin{eqnarray}\label{5-6}
\int_{\Sigma} \frac{|\nabla e(f)|^2}{4e(f)}\,dV_g
\leq 0.
\end{eqnarray}
This tells us that $e(f)$ is a constant. Hence, $e(f)=0$ from the facts that $f$ has finite energy and $(M,g)$ has infinite volume.

\hfill$q.e.d$

\emph{Remark.} The condition that $(M, g)$ has infinite volume can be replaced by a curvature condition. The integration condition is always stisfied when $(M,J,g)$ is semi-K\"ahlerian, quasi-K\"ahlerian, or almost-K\"ahlerian, or nearly-K\"ahlerian by the fact that $X_L=0$.

\section{Pseudoholomorphicity of pluriharmonic maps}
We will investigate the pseudoholomorphicity of pluriharmonic maps by using the Bochner formulae and maximum principle. Two cases with respect to the topology of domain manifold will be considered.

\begin{Theorem}
Let$(M, J, g)$ be a closed almost Hermitian manifold with $Ric''_M\geq A$, and let  $(\widetilde{M},\tilde{J},\tilde{g})$ be an almost Hermitian manifold with $HB_{\widetilde{M}}\leq B$ and the $(2,0)$-part of the curvature is bounded by $C$. If a pluriharmonic map $f:M\longrightarrow\widetilde{M}$ satisfies $(B+2C)\,e(f)< A$,
then $f$ is pseudoholomorphic.
\end{Theorem}

\emph{Proof}. Under the curvature conditions, the second Bochner formule in Proposition 3.1 gives
\begin{eqnarray}\label{6-1}
\frac{1}{2}\Box e''(f)&\geq& |a^\alpha_{\bar{i},\bar{j}}|^2+e''(f)\big(A-4C\,\sqrt{e''(f)}\sqrt{e'(f)}-B\,e(f)\big)\nonumber\\
&\geq&  |a^\alpha_{\bar{i},\bar{j}}|^2+e''(f)\Big(A-2C\,e(f)-B\,e(f)\Big).
\end{eqnarray}
By using the maximum principle, the differential inequality (\ref{6-1}) implies
$$e''(f)\big(A-2C\,e(f)-B\,e(f)\big)=0.$$
Thus, we have $e''(f)=0$ by the fact that $(B+2C)\,e(f)< A$.

\hfill$q.e.d$

To proceed, we need the following maximum principle (Proposition 4.1 in \cite{[T-07]}) due to V.Tosatti.

\begin{Lemma}\label{Lemma6.2}\emph{(\cite{[T-07]})}
Let $(M,J,g)$ be a complete almost Hermitian manifold with second Ricci curvature bounded below, with torsion and $(2,0)$-part of the curvature bounded. Let $u$ be a real function that is bounded from below. Then ginven any $\epsilon>0$ there exists a point $x_\epsilon\in M$ such that
\begin{equation*}
\liminf\limits_{\epsilon\rightarrow 0}u(x_\epsilon)=\inf\limits_{M}u,\hspace{0.6cm}
|\nabla u|(x_\epsilon)\leq \epsilon,\hspace{0.6cm}
\Box\, u(x_\epsilon)\geq -\epsilon.
\end{equation*}
\end{Lemma}

\begin{Theorem}\label{Theorem6.3}
Let $(M,J,g)$ be a complete almost Hermitian manifold with $Ric''_M\geq A$ and with torsion and $(2,0)$-part of the curvature bounded by $C$. Let  $(\widetilde{M},\tilde{J},\tilde{g})$ be an almost Hermitian manifold with $B_1\leq
HB_{\widetilde{M}}\leq B_2$. If a pluriharmonic map $f:M\longrightarrow \widetilde{M}$ satisfies $(B_2+2C)e''(f)\leq A+(B_1-2C)e'(f)-\delta$ for a positive constant $\delta>0$, then $f$ is pseudoholomorphic.
\end{Theorem}

\emph{Proof}. For any $\lambda>0$, set $u_\lambda:=\big(e''(f)+\lambda\big)^{-1/2}$, through direct calculation, we have
\begin{equation}\label{6-2}
\Box\,u_\lambda=3\,u_\lambda^{-1}\,|\nabla u_\lambda|^2-\frac{1}{2}u_\lambda^{3}\,\Box e''(f).
\end{equation}
On the other hand, by using the second Bochner formula in Proposition 4.1, we obtain
\begin{eqnarray}\label{6-3}
\frac{1}{2}\Box\,e''(f)&\geq& A\,e''(f)-4C\,\big(e'(f)\big)^{1/2}\big(e''(f)\big)^{3/2}+B_1\,e'(f)\,e''(f)-B_2\big(e''(f)\big)^2\nonumber\\
&=& e''(f)\Big[A-4C\,\big(e'(f)\big)^{1/2}\big(e''(f)\big)^{1/2}+B_1\,e'(f)-B_2\,e''(f)\Big]\nonumber\\
&\geq& e''(f)\big[A+(B_1-2C)\,e'(f)-(B_2+2C)\,e''(f)\big]\nonumber\\
&\geq& \delta\,e''(f).
\end{eqnarray}
Applying the Lemma 5.2 to $u_\lambda$, for any $\epsilon>0$, there exisits a point $x_\epsilon\in M$ such that
\begin{equation*}
u_\lambda(x_\epsilon)\leq\inf\limits_{M}u_\lambda+\epsilon,\hspace{0.6cm}
|\nabla u_\lambda|(x_\epsilon)\leq \epsilon,\hspace{0.6cm}
\Box\, u_\lambda(x_\epsilon)\geq -\epsilon.
\end{equation*}
So, at point $x_\epsilon$, the identity (\ref{6-2}) and the inequality (\ref{6-3}) imply
\begin{eqnarray}\label{6-4}
\frac{\delta\,e''(f)}{(e''(f)+\lambda)^2}=\delta\,e''(f)\,u_\lambda^{4}&\leq& 3\epsilon^2+\epsilon\,u_\lambda\nonumber\\
&\leq&3\epsilon^2+\epsilon\big(\inf\limits_{M}u_\lambda+\epsilon\big).
\end{eqnarray}
Letting $\epsilon\rightarrow 0$, then $u_\lambda(x_\epsilon)\rightarrow \inf\limits_{M}u_\lambda$ and hence $e''(f)(x_\epsilon)\rightarrow\sup\limits_M e''(f)$. Thus, the inequality (\ref{6-4}) implies that $\sup\limits_M e''(f)\leq 0$, i.e., $f$ is pseudoholomorphic.

\hfill$q.e.d$

\section{Pseudoholomorphic isometric immersion}
The purpose of this section is to derive Simons integral inequality, to give bounds of the norm of parallel canonical second fundamental form, of the   pseudoholomorphic isometry between the almost Hermitian manifolds. The idear inspired from H.W.Xu's work (\cite{[X-95]}) on closed minimal submanifold in pinched Riemannian manifolds. We will adopt the following range of indices:
$$1\leq i,j,k,\ell\ldots\leq m;\hspace{0.3cm}m+1\leq \lambda,\mu,\nu,\sigma,\ldots\leq n; \hspace{0.3cm}1\leq \alpha,\beta,\gamma,\delta\ldots\leq n.$$

We first recall the structure equations of almost Hermitian manifold $(\widetilde{M},\tilde{J},\tilde{g})$. Locally, by choosing a unitary frame field $\{\tilde{e}_\alpha\}$ with dual $\{\tilde{\theta}^\alpha\}$, the first and second structure equations are given by
\begin{equation}\label{7-1}
d\tilde{\theta}^\alpha=-\tilde{\theta}^\alpha_\beta\wedge\tilde{\theta}^\beta+\widetilde{\Theta}^\alpha,\hspace{0.3cm}\tilde{\theta}^\alpha_\beta+\overline{\tilde{\theta}^\beta_\alpha}=0,
\end{equation}
\begin{equation}\label{7-2}
d\tilde{\theta}^\alpha_{\beta}=-\tilde{\theta}^\alpha_\gamma\wedge\tilde{\theta}^\gamma_\beta+\widetilde{\Omega}^\alpha_\beta,\hspace{0.3cm}\widetilde{\Omega}^\alpha_\beta+\overline{\widetilde{\Omega}^\beta_\alpha}=0,
\end{equation}
where $\tilde{\theta}^\alpha_\beta$, $\widetilde{\Theta}^\alpha$, $\widetilde{\Omega}^\alpha_\beta$ are connection 1-forms, torsion and curvature forms respectively.

Let $f$ be a pseudoholomorphic isometry from $(M,J,g)$ into $(\widetilde{M},\tilde{J},\tilde{g})$ with $\mbox{dim}M=2m$ and  $\mbox{dim}\widetilde{M}=2n$. We choose an adapted unitary frame field $\{\tilde{e}_\alpha\}$ along $f$, which means that $\{\tilde{e}_i\}$ are tangent to $f(M)$ and $\{\tilde{e}_\lambda\}$ are normal to $f(M)$. So, we have
\begin{equation}\label{7-3}
f^*\tilde{\theta}^i=\theta^i,\hspace{1cm}f^*\tilde{\theta}^\lambda=0,
\end{equation}
i.e., $a^\alpha_i=\delta_{\alpha\,i}$. Then the pull-back of (\ref{7-1}) with $\alpha=i$ gives
\begin{equation}\label{7-4}
\tilde{\theta}^i_j=\theta^i_j,\hspace{1cm}\widetilde{\Theta}^i=\Theta^i,
\end{equation}
where we have omitted the pull-back $f^*$ on the forms $\tilde{\theta}^i_j$, $\widetilde{\Theta}^i$. Substituting $\tilde{\theta}^i_j=\theta^i_j$ and $a^\alpha_i=\delta_{\alpha i}$ into (\ref{2-3-73}) with $\alpha=k$, $\lambda$ respectively, we have
\begin{equation}\label{7-5}
a^k_{i,j}=0,\hspace{1cm}\tilde{\theta}^\lambda_i=a^\lambda_{i,j}\,\theta^j.
\end{equation}
Taking $(\alpha,\beta)=(i,j)$ in (\ref{7-2}), together with the first identities in(\ref{7-4}), we obtain the Gauss equation
\begin{equation*}
\Omega^i_j+(-\overline{\tilde{\theta}^\lambda_i})\wedge\tilde{\theta}^\lambda_j=\widetilde{\Omega}^i_j,
\end{equation*}
which implies
\begin{equation}\label{7-6}
R^i_{jk\bar{\ell}}+a^\lambda_{j,k}\overline{a^\lambda_{i,\ell}}=\widetilde{R}^i_{jk\bar{\ell}},
\end{equation}
and
\begin{equation}\label{7-7}
R^i_{jk\ell}=\widetilde{R}^i_{jk\ell},\hspace{0.6cm} R^i_{j\bar{k}\;\bar{\ell}}=\widetilde{R}^i_{j\bar{k}\;\bar{\ell}}.
\end{equation}
Similarly, taking $(\alpha,\beta)=(\lambda,\mu)$ in (\ref{7-2}), we get the Ricci equation
\begin{equation}\label{7-8}
\Omega^\lambda_\mu+\tilde{\theta}^\lambda_i\wedge(-\overline{\tilde{\theta}^\mu_i})=\widetilde{\Omega}^\lambda_\mu,
\end{equation}
where $\Omega^\lambda_\mu$ is the normal curvature forms defined by $\Omega^\lambda_\mu:=d\tilde{\theta}^\lambda_\mu+\tilde{\theta}^\lambda_\nu\wedge\tilde{\theta}^\nu_\mu$. If we set
\begin{equation*}
\Omega^\lambda_\mu:=R^\lambda_{\mu jk}\,\theta^j\wedge\theta^k +R^\lambda_{\mu j\bar{k}}\,\theta^j\wedge\overline{\theta^k} +R^\lambda_{\mu \bar{j}\,\bar{k}}\,\overline{\theta^j}\wedge\overline{\theta^k},
\end{equation*}
from (\ref{7-8}), we have
\begin{equation}\label{7-9}
R^\lambda_{\mu j\bar{k}}=a^\lambda_{i,j}\overline{a^\mu_{i,k}}+\widetilde{R}^\lambda_{\mu j\bar{k}},
\end{equation}
\begin{equation}\label{7-10}
R^\lambda_{\mu j k}=\widetilde{R}^\lambda_{\mu j k},\hspace{0.6cm}R^\lambda_{\mu \bar{j}\, \bar{k}}=\widetilde{R}^\lambda_{\mu \bar{j}\, \bar{k}}.
\end{equation}
We call $R^\perp:=\sum\limits_{i,\lambda}R^\lambda_{\lambda i i}$ the \emph{normal scalar curvature} of $M$ in $\widetilde{M}$.
The Codazzi equation is follows from (\ref{2-3-79})-(\ref{2-3-81}) or by taking $(\alpha, \beta)=(\lambda,i)$ in (\ref{7-2}), that is
\begin{equation}\label{7-11}
a^\lambda_{i,j\bar{k}}=-\widetilde{R}^\lambda_{ij\bar{k}}, \hspace{0.5cm}
a^\lambda_{i,jk}-a^\lambda_{i,kj}=2a^\lambda_{i,p}L^p_{jk}-2\widetilde{R}^\lambda_{ijk},\hspace{0.5cm}
a^\lambda_{i,p}N^p_{\bar{j}\,\bar{k}}=\widetilde{R}^\lambda_{i\bar{j}\,\bar{k}}.
\end{equation}

There are some basic properties of pseudoholomorphic isometry.

\begin{Proposition}
Let $(M, J, g)$, $(\widetilde{M},\tilde{J},\tilde{g})$ be two almost Hermitian manifolds with $HB_{\widetilde{M}}\leq B$. If $f:M \rightarrow \widetilde{M}$ is a pseudoholomorphic isometry, then we have

\emph{(1)} $Ric'_M-mBg$, $Ric''_M-mBg$ is negative semi-definite.

\emph{(2)} $R\leq m^2 B$.

\emph{(3)} ${HS}_M\leq B$.
\end{Proposition}

\emph{Proof}. For any $X=X^i\,e_i\in T'M$, by using the Gauss equation (\ref{7-6}), we have
\begin{eqnarray*}
R'_{j\bar{k}}X^j\overline{X^k}-mB|X|^2&\leq&\sum\limits_{i}R^i_{ij\bar{k}}X^j\overline{X^k}-\sum\limits_{i}\widetilde{R}^i_{ij\bar{k}}X^j\overline{X^k}\\
&=&-\sum\limits_{i,\lambda}a^\lambda_{i,j}\overline{a^\lambda_{i,k}}X^j\overline{X^k}\leq 0.
\end{eqnarray*}
Similar proof for $Ric''_M-mBg$. The second and third inequalities are also follwow from Gauss equation (\ref{7-6}) and  $HB_{\widetilde{M}}\leq B$.

\hfill$q.e.d$

\emph{Remark}. The Proposition 6.1 provides various curvature criterion to detect that whether an almost Hermitian manifold can be pseudoholomorphically immersed into another one.

\begin{Proposition}
Let $(M, J, g)$, $(\widetilde{M},\tilde{J},\tilde{g})$ be two almost Hermitian manifolds with $HB_{\widetilde{M}}\leq B$. If $f:M \rightarrow \widetilde{M}$ is a pseudoholomorphic isometry, then $f$ is canonical totally geodesic if $f$ satisfies one of the following conditions\emph{:}

\emph{(1)} $Ric'_M=mBg$, or $Ric''_M=mBg$.

\emph{(2)} $R= m^2 B$.

\emph{(3)} ${HS}_M=B$.
\end{Proposition}

\emph{Proof}. By the Gauss equation (\ref{7-6}), any one of the conditions implies that the canonical second fundamental form $a^\alpha_{i,j}=0$, i.e.,
$\nabla df=0$.

\hfill$q.e.d$

For the scalar curvature and normal scalar curvature, we have

\begin{Proposition}
Let $(M, J, g)$, $(\widetilde{M},\tilde{J},\tilde{g})$ be two almost Hermitian manifolds with $B_1\leq HB_{\widetilde{M}}\leq B_2$. For a pseudoholomorphic isometry $f:M \rightarrow \widetilde{M}$, we have

$$mnB_1\leq R+R^\perp\leq mnB_2.$$
\end{Proposition}

\emph{Proof}. By the Gauss equation (\ref{7-6}) and Ricci equation (\ref{7-9}), we have
$$R+R^\perp=\sum\limits_{i,j}\widetilde{R}^i_{ij\bar{j}}+\sum\limits_{i,\lambda}\widetilde{R}^\lambda_{\lambda i\bar{i}}.$$
So, the inequalities follow from the bounds of holomorphic bisectional curvature of $\widetilde{M}$.

\hfill$q.e.d$

We will give an upper bound of the curvature tensor in terms of the bound of holomorphic bisectional curvature.

\begin{Lemma}\label{Lemma6.4}
Let $(M, J, g)$ be an almost Hermitian manifold with $a \leq HB_M\leq b$. Under a unitary frame field, the components of $(1,1)$-part of the curvature denoted by $R^i_{jk\bar{\ell}}$, then for fixed indices $i,j,k,\ell$, we have

\emph{(1)} $|R^j_{ik\bar{\ell}}X^i\overline{X^j}|^2\leq 2 (b-a)^2|X|^4$ for $k\neq\ell$ and any $X\in T'M$.

\emph{(2)} $|R^i_{jk\bar{\ell}}X^k\overline{X^\ell}|^2\leq 2 (b-a)^2|X|^4$ for $i\neq j$ and any $X\in T'M$.

\emph{(3)} $|R^j_{ik\bar{\ell}}|^2 \leq 4(1+\sqrt{2})^2(b-a)^2$ for $i\neq j$ and $k\neq\ell$.
\end{Lemma}

\emph{Proof}. For any $X=X^i\,e_i$, $Y=Y^j\,e_j$, $Z=Z^k\,e_k\in T'M$, we have
\begin{eqnarray}\label{7-12}
R^j_{ik\bar{\ell}}\,X^i\overline{X^j}Y^k\overline{Z^\ell}+R^j_{ik\bar{\ell}}\,X^i\overline{X^j}Z^k\overline{Y^\ell}&=&R^j_{ik\bar{\ell}}\,X^i\overline{X^j}(Y^k+Z^k)(\overline{Y^\ell}+\overline{Z^\ell})\hspace{2cm}\nonumber\\
&{}&-R^j_{ik\bar{\ell}}\,X^i\overline{X^j}Y^k\overline{Y^\ell}-R^j_{ik\bar{\ell}}\,X^i\overline{X^j}Z^k\overline{Z^\ell}.
\end{eqnarray}
Taking $X=X^i\,e_i$, $Y=e_k$, $Z=e_\ell$ with $k\neq \ell$ in (\ref{7-12}) and together $\overline{R^i_{jk\bar{\ell}}}=R^j_{i\ell\,\bar{k}}$, we obtain
\begin{eqnarray}\label{7-13}
2\,\mbox{Re}\big(R^j_{ik\bar{\ell}}X^i\overline{X^j}\big)&=&R^j_{ik\bar{\ell}}X^i\overline{X^j}+R^j_{i\ell\,\bar{k}}X^i\overline{X^j}\hspace{7cm}\nonumber\\
&=& R^j_{ip\,\bar{q}}X^i\overline{X^j}(\delta_{pk}+\delta_{p\ell})(\delta_{qk}+\delta_{q\ell})-R^j_{ik\,\bar{k}}X^i\overline{X^j}-R^j_{i\ell\,\bar{\ell}}X^i\overline{X^j},
\end{eqnarray}
which implies
\begin{equation}\label{7-14}
-(b-a)|X|^2\leq \mbox{Re}(R^j_{ik\bar{\ell}}X^i\overline{X^j})\leq (b-a)|X|^2.
\end{equation}
By taking $X=X^ie_i$, $Y=-\sqrt{-1}e_k$, $Z=e_\ell$ with $k\neq \ell$ in (\ref{7-12}), we can obtain
\begin{equation}\label{7-15}
-(b-a)|X|^2\leq \mbox{Im}(R^j_{ik\bar{\ell}}X^i\overline{X^j})\leq (b-a)|X|^2.
\end{equation}
So, inequlities (\ref{7-14}) and (\ref{7-15}) give the desired upper bound of $|R^j_{ik\bar{\ell}}X^i\overline{X^j}|^2$, and similar proof for $|R^i_{jk\bar{\ell}}X^k\overline{X^\ell}|^2$.

On the other hand, for any $X=X^i\,e_i$, $Y=Y^j\,e_j$, $Z=Z^k\,e_k$, $W=W^\ell\,e_\ell\in T'M$, we have
\begin{eqnarray}\label{7-16}
R^j_{ik\bar{\ell}}\,X^i\overline{Y^j}Z^k\overline{W^\ell}+R^j_{ik\bar{\ell}}\,Y^i\overline{X^j}Z^k\overline{W^\ell}&=&R^j_{ik\bar{\ell}}(X^i+Y^i)(\overline{X^j}+\overline{Y^j})Z^k\overline{W^\ell}\hspace{2cm}\nonumber\\
&{}&-R^j_{ik\bar{\ell}}\,X^i\overline{X^j}Z^k\overline{W^\ell}-R^j_{ik\bar{\ell}}\,Y^i\overline{Y^j}Z^k\overline{W^\ell}.
\end{eqnarray}
Taking $X=e_i$, $Y=e_j$, $Z=e_k$, $W=e_\ell$ with  $i\neq j$ and $k\neq \ell$ in (\ref{7-16}), we obtain
\begin{eqnarray}\label{7-17}
R^j_{ik\bar{\ell}}+R^i_{jk\bar{\ell}}=R^q_{pk\bar{\ell}}\,(\delta_{pi}+\delta_{pj})(\delta_{qi}+\delta_{qj})-R^i_{ik\bar{\ell}}-R^j_{jk\bar{\ell}},
\end{eqnarray}
and by taking $X=\sqrt{-1}e_i$, $Y=e_j$, $Z=e_k$, $W=\sqrt{-1}e_\ell$ with  $i\neq j$ and $k\neq \ell$ in (\ref{7-16}), we obtain
\begin{eqnarray}\label{7-18}
R^j_{ik\bar{\ell}}-R^i_{jk\bar{\ell}}=-\sqrt{-1}R^q_{pk\bar{\ell}}\,(\sqrt{-1}\delta_{pi}+\delta_{pj})(-\sqrt{-1}\delta_{qi}+\delta_{qj})+\sqrt{-1}R^i_{ik\bar{\ell}}+\sqrt{-1}R^j_{jk\bar{\ell}}.\hspace{0.3cm}
\end{eqnarray}
Thus, from (\ref{7-17}) and (\ref{7-18}), we get
\begin{eqnarray}\label{7-19}
2R^j_{ik\bar{\ell}}&=&R^q_{pk\bar{\ell}}\,(\delta_{pi}+\delta_{pj})(\delta_{qi}+\delta_{qj})-\sqrt{-1}R^q_{pk\bar{\ell}}\,(\sqrt{-1}\delta_{pi}+\delta_{pj})(-\sqrt{-1}\delta_{qi}+\delta_{qj})\nonumber\\
&{}&-(1-\sqrt{-1})R^i_{ik\bar{\ell}}-(1-\sqrt{-1})R^j_{jk\bar{\ell}}.
\end{eqnarray}
So, by using the estimates (1) and (2) have been obtained, we get
\begin{eqnarray}
2|R^j_{ik\bar{\ell}}| &\leq& |R^q_{pk\bar{\ell}}\,(\delta_{pi}+\delta_{pj})(\delta_{qi}+\delta_{qj})|+|R^q_{pk\bar{\ell}}\,(\sqrt{-1}\delta_{pi}+\delta_{pj})(-\sqrt{-1}\delta_{qi}+\delta_{qj})|\nonumber\\
&{}& +\sqrt{2}\,|R^i_{ik\bar{\ell}}|+\sqrt{2}\,|R^j_{jk\bar{\ell}}|\nonumber\\
&\leq& \sqrt{2}(b-a)|e_i+e_j|^2+ \sqrt{2}(b-a)|\sqrt{-1}e_i+e_j|^2 +4(b-a)\nonumber\\
&=&4(1+\sqrt{2})(b-a),
\end{eqnarray}
which gives the desired estimate of $|R^j_{ik\ell}|^2$ with $i\neq j$ and $k\neq\ell$.

\hfill$q.e.d$

We wish to give a lower bound of $\frac{1}{2}\Box S$ by using the bounds of holomorphic bisectional curvature of target manifold $(\widetilde{M},\tilde{J},\tilde{g})$. In the sequal, we assume that the holomorphic bisectional curvature pointwisely satisfies $a(x)\leq HB_{\widetilde{M}}\leq b(x)$ for $x\in \widetilde{M}$. According to the expression of $\frac{1}{2}\Box S$, for convenience, we define three terms as follows:
\begin{equation*}
\mbox{(I)}:= 2\mbox{Re}\big(\overline{a^\alpha_{i,j}}a^\alpha_pR^p_{ij\bar{k},k}\big),
\end{equation*}
\begin{equation*}
\mbox{(II)}:= |a^\alpha_{i,j\bar{k}}|^2-2\mbox{Re}\big(\overline{a^\alpha_{i,j}}(a^\beta_ia^\gamma_j\overline{a^\delta_k}\widetilde{R}^\alpha_{\beta\gamma\bar{\delta}})_{,k}\big),
\end{equation*}
\begin{equation*}
\mbox{(III)}:= \overline{a^\alpha_{i,j}}\big(a^\alpha_{p,j}R''_{i\bar{p}}+a^\alpha_{i,p}R''_{j\bar{p}}+2a^\alpha_{p,k}R^p_{ij\bar{k}}-a^\beta_{i,j}a^\gamma_k\overline{a^\delta_k}\widetilde{R}^\alpha_{\beta\gamma\bar{\delta}}\big).
\end{equation*}

\begin{Lemma}\label{Lemma6.5}
For the terms \emph{(I)} and \emph{(II)}, we have

\emph{(1)} $\emph{(I)}=0$.

\emph{(2)} $\emph{(II)}\geq -\mbox{\emph{div}}^c(X)- P(m,n)(b-a)^2-m^2\,\emph{max}\{a^2,b^2\}$, where $X$ defined in \emph{(\ref{6.22})} and $P(m,n)$ is a polynomial with respect to $m,n$ defined in \emph{(\ref{6.25})}.
\end{Lemma}

\emph{Proof}. By choosing an adapted unitary frame field, we have $a^k_{i,j}=0$ and $a^\lambda_i=0$, which imply $\mbox{(I)}=0$. Notice that
\begin{eqnarray}\label{6.21}
2\mbox{Re}\big(\overline{a^\alpha_{i,j}}(a^\beta_ia^\gamma_j\overline{a^\delta_k}\widetilde{R}^\alpha_{\beta\gamma\bar{\delta}})_{,k}\big)
&=& \overline{a^\alpha_{i,j}}(a^\beta_ia^\gamma_j\overline{a^\delta_k}\widetilde{R}^\alpha_{\beta\gamma\bar{\delta}})_{,k}+a^\alpha_{i,j}(\overline{a^\beta_ia^\gamma_j\overline{a^\delta_k}\widetilde{R}^\alpha_{\beta\gamma\bar{\delta}}})_{,\bar{k}}\nonumber\\
&=&X^k_{,k}
+\overline{X^k_{,k}}-\overline{a^\alpha_{i,j\bar{k}}}\widetilde{R}^\alpha_{ij\bar{k}}-a^\alpha_{i,j\bar{k}}\overline{\widetilde{R}^\alpha_{ij\bar{k}}}\nonumber\\
&=&\mbox{div}^c(X)-\overline{a^\alpha_{i,j\bar{k}}}\widetilde{R}^\alpha_{ij\bar{k}}-a^\alpha_{i,j\bar{k}}\overline{\widetilde{R}^\alpha_{ij\bar{k}}}
\end{eqnarray}
where
\begin{equation}\label{6.22}
X=X^k\,e_k+\overline{X^k}\,\overline{e_k},\hspace{0.6cm}X^k=\overline{a^\alpha_{i,j}}a^\beta_ia^\gamma_j\overline{a^\delta_k}\widetilde{R}^\alpha_{\beta\gamma\bar{\delta}}.
\end{equation}
 Thus, by (\ref{6.21}) and Cauchy-Schwarz inequality, we have
\begin{eqnarray}\label{6.23}
\mbox{(II)}&=& |a^\alpha_{i,j\bar{k}}|^2-\mbox{div}^c(X)+\overline{a^\alpha_{i,j\bar{k}}}\widetilde{R}^\alpha_{ij\bar{k}}+a^\alpha_{i,j\bar{k}}\overline{\widetilde{R}^\alpha_{ij\bar{k}}}\nonumber\\
&\geq& |a^\alpha_{i,j\bar{k}}|^2-\mbox{div}^c(X)-|a^\alpha_{i,j\bar{k}}|^2-|\widetilde{R}^\alpha_{i,j\bar{k}}|^2\nonumber\\
&=&-\mbox{div}^c(X)-|\widetilde{R}^\alpha_{ij\bar{k}}|^2.
\end{eqnarray}
On the other hand, by using Lemma 6.4 repeatedly, we have
\begin{eqnarray}\label{6.24}
\sum\limits_{i,j,k,\alpha}|\widetilde{R}^\alpha_{ij\bar{k}}|^2&=&\sum\limits_{i,j\neq k}|\widetilde{R}^i_{ij\bar{k}}|^2+\sum\limits_{i\neq\ell,j}|\widetilde{R}^\ell_{ij\bar{j}}|^2+\sum\limits_{i,j}|\widetilde{R}^i_{ij\bar{j}}|^2+\sum\limits_{i\neq\ell,j\neq k}|\widetilde{R}^\ell_{ij\bar{k}}|^2\nonumber\\
&{}&+\sum\limits_{i,j,\lambda}|\widetilde{R}^\lambda_{ij\bar{j}}|^2+\sum\limits_{i,j\neq k,\lambda}|\widetilde{R}^\lambda_{ij\bar{k}}|^2\nonumber\\
&\leq& 2m^2(m-1)(b-a)^2+2m^2(m-1)(b-a)^2+m^2\mbox{max}\{a^2,b^2\}\nonumber\\
&{}&+4(1+\sqrt{2})^2m^2(m-1)^2(b-a)^2+2m^2(n-m)(b-a)^2\nonumber\\
&{}&+4(1+\sqrt{2})^2m^2(n-m)(m-1)(b-a)^2\nonumber\\
&:=&P(m,n)(b-a)^2+m^2\,\mbox{max}\{a^2,b^2\},
\end{eqnarray}
where
\begin{equation}\label{6.25}
P(m,n)=2m^2(m+n-2)+4(1+\sqrt{2})^2\,m^2(m-1)(n-1).
\end{equation}
\hfill$q.e.d$

\begin{Lemma}\label{Lemma6.6}
For the term \emph{(III)} , we have
\begin{eqnarray*}
\emph{(III)}\geq (m+2)\,b\,S-P(m)(b-a)S-4S^2,
\end{eqnarray*}
where $P(m)=2(m+2)+4(1+\sqrt{2})$.
\end{Lemma}

\emph{Proof}. Under an adapted unitary frame field, using the Gauss equation (\ref{7-6}), we have
\begin{eqnarray}\label{6.26}
\mbox{(III)}&=& \overline{a^\lambda_{i,j}}\big(a^\lambda_{p,j}\widetilde{R}^p_{ik\bar{k}}+a^\lambda_{i,p}\widetilde{R}^p_{jk\bar{k}}+2a^\lambda_{p,k}\widetilde{R}^p_{ij\bar{k}}-a^\mu_{i,j}\widetilde{R}^\lambda_{\mu k\bar{k}}\big)\nonumber\\
&{}& -\overline{a^\lambda_{i,j}}a^\lambda_{p,j}a^\mu_{i,k}\overline{a^\mu_{p,k}}
-\overline{a^\lambda_{i,j}}a^\lambda_{i,p}a^\mu_{j,k}\overline{a^\mu_{p,k}}
-2\overline{a^\lambda_{i,j}}a^\lambda_{p,k}a^\mu_{i,j}\overline{a^\mu_{p,k}}.
\end{eqnarray}
By using the bounds of holomorphic bisectional curvature, we have
\begin{eqnarray}\label{6.27}
\sum\limits_{i,j,k,p,\lambda}\overline{a^\lambda_{i,j}}a^\lambda_{p,j}\widetilde{R}^p_{ik\bar{k}}&=&\sum\limits_{j,k,\lambda}\sum\limits_{i,p}\overline{a^\lambda_{i,j}}a^\lambda_{p,j}\widetilde{R}^p_{ik\bar{k}}\nonumber\\
&\geq&a\sum\limits_{k}\sum\limits_{i,j,\lambda}|a^\lambda_{i,j}|^2=maS.
\end{eqnarray}
Similarly, we can get
\begin{equation}\label{6.28}
\sum\limits_{i,j,k,p,\lambda}\overline{a^\lambda_{i,j}}a^\lambda_{i,p}\widetilde{R}^p_{jk\bar{k}}\geq maS,\hspace{0.8cm}
\sum\limits_{i,j,k,\lambda,\mu}\overline{a^\lambda_{i,j}}a^\mu_{i,j}\widetilde{R}^\lambda_{\mu k\bar{k}}\leq mbS.
\end{equation}
On the other hand, by Lemma 6.4, the term
\begin{eqnarray}\label{6.29}
\sum\limits_{i,j,k,p,\lambda}\overline{a^\lambda_{i,j}}a^\lambda_{p,k}\widetilde{R}^p_{ij\bar{k}}&=&\sum\limits_{i,j,k,\lambda}\overline{a^\lambda_{i,j}}a^\lambda_{i,k}\widetilde{R}^i_{ij\bar{k}}+\sum\limits_{i,j,p,\lambda}\overline{a^\lambda_{i,j}}a^\lambda_{p,j}\widetilde{R}^p_{ij\bar{j}}\nonumber\\
&{}&-\sum\limits_{i,j,\lambda}\overline{a^\lambda_{i,j}}a^\lambda_{i,j}\widetilde{R}^i_{ij\bar{j}}+\sum\limits_{i\neq p,j\neq k,\lambda}\overline{a^\lambda_{i,j}}a^\lambda_{p,k}\widetilde{R}^p_{ij\bar{k}}\nonumber\\
&\geq& a \sum\limits_{i,\lambda}\sum\limits_{j}|a^\lambda_{i,j}|^2+a \sum\limits_{j,\lambda}\sum\limits_{i}|a^\lambda_{i,j}|^2\nonumber\\
&{}&-b\sum\limits_{i,j,\lambda}|a^\lambda_{i,j}|^2-2(1+\sqrt{2})(b-a)\sum\limits_{i\neq p,j\neq k,\lambda}|\overline{a^\lambda_{i,j}}|\,|a^\lambda_{p,k}|
\nonumber\\
&\geq& (2a-b)S-2(1+\sqrt{2})(b-a)S.
\end{eqnarray}

To estimate the remainning terms in (III), we set
\begin{equation*}
S'_{p\bar{i}}:=\sum\limits_{j,\lambda}a^\lambda_{p,j}\overline{a^\lambda_{i,j}},\hspace{0.6cm}
S''_{p\bar{j}}:=\sum\limits_{i,\lambda}a^\lambda_{i,p}\overline{a^\lambda_{i,j}}, \hspace{0.6cm}
S'''_{\lambda\,\bar{\mu}}:=\sum\limits_{i,j}a^\lambda_{i,j}\overline{a^\mu_{i,j}}.
\end{equation*}
It is clear that the matrices $S'=(S'_{p\bar{i}})$,  $S''=(S'_{p\bar{j}})$,  $S'''=(S'''_{\lambda\,\bar{\mu}})$ are Hermitian and semi-positive definite.
They are satisfy $S=\mbox{tr}(S')=\mbox{tr}(S'')=\mbox{tr}(S''')$. Locally, one can choose unitary frame to diagonalize $S'$, so we have
\begin{eqnarray}\label{6.30}
\sum\limits_{i,j,k,p,\lambda,\mu}\overline{a^\lambda_{i,j}}a^\lambda_{p,j}a^\mu_{i,k}\overline{a^\mu_{p,k}}
&=&\sum\limits_{i}S'_{i\bar{i}}S'_{i\bar{i}}\leq(\sum\limits_{i}S'_{i\bar{i}})^2=S^2.
\end{eqnarray}
Similarly, we also have
\begin{equation}\label{6.31}
\sum\limits_{i,j,k,p,\lambda,\mu}\overline{a^\lambda_{i,j}}a^\lambda_{p,k}a^\mu_{i,j}\overline{a^\mu_{p,k}}=\sum\limits_{\lambda}S'''_{\lambda\bar{\lambda}}S'''_{\lambda\bar{\lambda}}\leq(\sum\limits_{\lambda}S'''_{\lambda\bar{\lambda}})=S^2.
\end{equation}
By the trick of diagonalization, the term
\begin{eqnarray}\label{6.32}
\sum\limits_{i,j,k,p,\lambda,\mu}\overline{a^\lambda_{i,j}}a^\lambda_{i,p}a^\mu_{j,k}\overline{a^\mu_{p,k}}&=&\sum\limits_{j}S'_{j\bar{j}}S''_{j\bar{j}}\nonumber\\
&\leq& \big(\sum\limits_{j}(S'_{j\bar{j}})^2\big)^{1/2}\big(\sum\limits_{j}(S''_{j\bar{j}})^2\big)^{1/2}\leq S^2.
\end{eqnarray}
Thus, the lower bound of (III) follows from (\ref{6.26})-(\ref{6.32}).

\hfill$q.e.d$

Now we can prove the Simons integral inequality of pseudoholmorphic isometry.

\begin{Theorem}
Let $(M,J,g)$ be a closed semi-K\"ahler manifold, and let $(\widetilde{M},\tilde{J},\tilde{g})$ be an almost Hermitian manifold pointwisely satisfies $a(x)\leq HB_{\widetilde{M}}\leq b(x)$ for $x\in \widetilde{M}$. Let $f:M\rightarrow \widetilde{M}$ be a pseudoholomorphic isometric immersion, then
\begin{equation*}
\int_M\Big(\!-4S^2+\big[(m+2)b\!-\!P(m)(b-a)\big]S\!-\!P(m,n)(b-a)^2\!-m^2\,\emph{max}\{a^2,b^2\}\!\Big)dV_g\leq 0,
\end{equation*}
where $P(m)\!=\!2(m+2)+4(1+\sqrt{2})$, $P(m,n)\!=\!2m^2(m+n-2)+4(1+\sqrt{2})^2\,m^2(m-1)(n-1)$.
\end{Theorem}

\emph{Proof}. By Proposition 3.3, Lemma 6.5 and Lemma 6.6, we have
\begin{eqnarray}\label{6.33}
\frac{1}{2}\Box S&\geq& -\mbox{div}^c(X)-4S^2+\big[(m+2)b-P(m)(b-a)\big]S\nonumber\\
&{}&- P(m,n)(b-a)^2-m^2\,\mbox{max}\{a^2,b^2\}.
\end{eqnarray}
Notic that $(M,J,g)$ is closed and semi-K\"ahlerian, so $\int_M\,\Box S\,dV_g=0$ and $\int_M\,\mbox{div}^c(X)\,dV_g=0$, then the integral inequality follows from integrating on both sides of (\ref{6.33}).

\hfill$q.e.d$

\emph{Remark.} The semi-K\"ahlerian condition is to ensure that the terms $\Box S$ and $\mbox{div}^c(X)$ integrated on $M$ are zero. So, the same result holds when $(M,J,g)$ is quasi-K\"ahlerian, or almost-K\"ahlerian, or nearly-K\"ahlerian.

\begin{Theorem}
Let $f:M\rightarrow \widetilde{M}$ be a pseudoholomorphic isometric immersion from almost Hermitian manifold $(M,J,g)$ into $(\widetilde{M},\tilde{J},\tilde{g})$ with parallel canonical second fundamental form. Suppose $(\widetilde{M},\tilde{J},\tilde{g})$ pointwisely satisfies $a(x)\leq HB_{\widetilde{M}}\leq b(x)$ for $x\in \widetilde{M}$, where $a(x)$ and $b(x)$ are bounded functions on $\widetilde{M}$. Then $S=0$, or
$$\frac{1}{4}\big[(m+2)b_0-P(m)(b_0-a_0)\big]\leq S\leq \frac{m(n-m)}{m+n}\big[(m+2)a_0+P(m)(b_0-a_0)\big]$$ provided $P(m)(b_0-ma_0)< (m+2)b_0$, where  $P(m)=2(m+2)+4(1+\sqrt{2})$, $a_0:=\inf\limits_{x\in\widetilde{M}} a(x)$ and $b_{0}:=\sup\limits_{x\in \widetilde{M}} b(x)$.
\end{Theorem}

\emph{Proof}. Under the condition that $f$ has parallel canonical fundamental form, we have
\begin{equation}\label{6.34}
a^\alpha_{i,jk}=a^\alpha_{i,j\bar{k}}=0,
\end{equation}
and hence the Codazzi equation (\ref{7-11}) gives
\begin{equation}\label{6.35}
\widetilde{R}^\lambda_{ij\bar{k}}=0.
\end{equation}
The identity (\ref{4-4}) and (\ref{6.34}) imply that $S$ is a constant. For $X^k$ defined in (\ref{6.22}), the identity (\ref{6.35}) gives
\begin{equation}\label{6.36}
X^k=\overline{a^\lambda_{i,j}}\widetilde{R}^\lambda_{ij\bar{k}}=0.
\end{equation}
Thus, the term $\mbox{(II)}=0$ follows from (\ref{6.21}), (\ref{6.35}) and (\ref{6.36}). By using that $S$ is a constant, from Lemma 6.6, we have
\begin{equation}\label{6.37}
0\geq \big[(m+2)b_0-P(m)(b_0-a_0)-4S\big]S,
\end{equation}
where $a_0:=\inf\limits_{x\in\widetilde{M}} a(x)$ and $b_{0}:=\sup\limits_{x\in \widetilde{M}} b(x)$.

On the other hand, modify the proof of Lemma 6.6, we can get the following estimates:
\begin{equation}\label{6.38}
\sum\limits_{i,j,k,p,\lambda}\overline{a^\lambda_{i,j}}a^\lambda_{p,j}\widetilde{R}^p_{ik\bar{k}}\leq mb_0S,
\end{equation}
\begin{equation}\label{6.39}
\sum\limits_{i,j,k,p,\lambda}\overline{a^\lambda_{i,j}}a^\lambda_{i,p}\widetilde{R}^p_{jk\bar{k}}\leq mb_0S,
\end{equation}
\begin{equation}\label{6.40}
\sum\limits_{i,j,k,\lambda,\mu}\overline{a^\lambda_{i,j}}a^\mu_{i,j}\widetilde{R}^\lambda_{\mu k\bar{k}}\geq ma_0S,
\end{equation}
\begin{eqnarray}\label{6.41}
\sum\limits_{i,j,k,p,\lambda}\overline{a^\lambda_{i,j}}a^\lambda_{p,k}\widetilde{R}^p_{ij\bar{k}}
\leq (2b_0-a_0)S+2(1+\sqrt{2})(b_0-a_0)S,
\end{eqnarray}
\begin{eqnarray}\label{6.42}
\sum\limits_{i,j,k,p,\lambda,\mu}\overline{a^\lambda_{i,j}}a^\lambda_{p,j}a^\mu_{i,k}\overline{a^\mu_{p,k}}
\geq\frac{1}{m}S^2,
\end{eqnarray}
\begin{equation}\label{6.43}
\sum\limits_{i,j,k,p,\lambda,\mu}\overline{a^\lambda_{i,j}}a^\lambda_{p,k}a^\mu_{i,j}\overline{a^\mu_{p,k}}\geq \frac{1}{n-m}S^2,
\end{equation}
\begin{eqnarray}\label{6.44}
\sum\limits_{i,j,k,p,\lambda,\mu}\overline{a^\lambda_{i,j}}a^\lambda_{i,p}a^\mu_{j,k}\overline{a^\mu_{p,k}}\geq 0.
\end{eqnarray}
From (\ref{6.38})-(\ref{6.44}) and Proposition 3.3, together with $S$ is a constant, we have
\begin{equation}\label{6.45}
0\leq \big[(m+2)a_0+P(m)(b_0-a_0)-\frac{n+m}{m(n-m)}S\big]S.
\end{equation}
Then, the bounds of $S$ follows from identities (\ref{6.37}) and (\ref{6.45}).

\hfill$q.e.d$

\hspace{0.6cm}

\noindent \textbf{Acknowledgments}. The second author would like to express his gratitude to professor X. Zhang for share the reference \cite{[Zh-12]}, and to X.Ch.Zhou for suggestions and comments in pareparing this paper. This project is supported by the NSFC (Grant No.11871445) and the Fundamental Research Funds for the Central Universities.

redpajama_set_name: RedPajamaArXiv


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Five months after Islamic State (IS) was toppled in a fierce battle over the tiny sliver left of its so-called caliphate, mysterious fires keep breaking out.
Local farmers who were hoping for a good year after a decade of war and drought are now trying to save their crops from arsonists determined to destroy their livelihoods.
The region's Kurdish administration has blamed extremist groups including Islamic State for causing $50 million in damage to this year's wheat and barley harvests.
Agriculture Minister Salman Barudo said multiple fires were lit deliberately each day, with 50,000 hectares destroyed since May.
"After they were defeated on the front line, they wanted to fight us economically," Mr Barudo said.
Mr Barudo said IS had already claimed responsibility for some of the fires.
"It was clear. [Islamic State] announced it on social media. They said, 'we are burning the fields that belong to the infidels, the Kurds,'" he said.
Photo The arson attacks have cost local farmers at least $50 million.
ABC News: Tom Hancock
Some Kurds are also convinced the blazes are not solely the work of Islamic State.
Sleeper cells that support Syria's Assad regime and Turkey may have also launched copycat attacks.
It is a bitter blow to the Kurds, who lost 12,000 fighters in the six-year battle to defeat IS.
"They are trying to create a perception that the [Kurdish-led Government] is not able to protect the region here," Mr Barudo said.
Farmers die trying to save their livelihoods
So far, 14 people have died in the fires while they desperately trying to protect their farms.
Photo At least 14 people have died trying to battle the crop fires in Syria.
The ABC witnessed one blaze rip through two fields near the Kurdish city of Qamishli.
Over about 30 minutes, two large fields were completely engulfed by flames.
In hot and windy conditions, locals using sacks and shovels struggled to extinguish the fire, which flared up whenever the winds picked up or the blaze hit a patch of chaff or unharvested wheat.
"This is happening every day, every single day. It's too much for us," one young local man said.
In the village of Amouda, Abdul Aziz lost his 67-year-old father who was burned while trying to dig a firebreak around one crop fire.
"The fire burned his tractor, and he too was burned," Abdul Aziz said.
His father died of his injuries as he was rushed to a hospital.
In a neighbouring village, local woman Jazzaya said she was burned while carrying water through the fields to the local men who were fighting a fire.
She has deep burns on her legs and abdomen.
Jazzaya has already had surgery but has been told she needs more skin grafts, which she cannot afford.
"There are no hospitals here to do the surgery. They told me that I need to go to a private hospital," she said.
Photo This year was meant to be a profitable one for Syrian farmers after a wet winter.
The fires this season are unprecedented and some farmers who spoke to the ABC said the extent of the blazes could be fuelled by an exceptionally wet winter.
Big rains led to much more spring growth: there is simply a lot more grain that is available to burn.
Most of the crops have now been harvested but the fires are continuing, fuelled by several inches of stubble that remains on the ground.
Cooler weather should help them get the blazes under control, but with temperatures still in the 40s, that could be months away.
Posted Tue 9 Jul 2019, 12:03 AM AEST
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