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2201v 1 [m a t h .D G ] 19 D e c 2001KILLING SPINOR EQUATIONS IN DIMENSION 7AND GEOMETRY OF
INTEGRABLE G 2-MANIFOLDS
THOMAS FRIEDRICH AND STEF AN IVANOV Abstract.We compute the scalar curvature of 7-dimensional G 2-manifolds admitting a connection with totally skew-symmetric torsion.We prove the formula for the general so-lution of the Killing spinor equation and express the Riemannian scalar curvature of the solution in terms of the dilation function and the NS 3-form ?eld.In dimension n =7the dilation function involved in the second fermionic string equation has an interpretation as a conformal change of the underlying integrable G 2-structure into a cocalibrated one of pure type W 3.Contents 1.Introduction 12.General properties of G 2-structures 33.Conformal transformations of G 2-structures 44.Connections with torsion,parallel spinors and Riemannian scalar curvature 55.Solutions to the Killing spinor equations in dimension 77References 81.Introduction Riemannian manifolds admitting parallel spinors with respect to a metric connection with totally skew-symmetric torsion became a subject of interest in theoretical and mathematical physics recently.One of the main reasons is that the number of preserved supersymmetries in string theory depends essentially on the number of parallel spinors.In 10-dimensional string theory,the Killing spinor equations with non-constant dilation Φand the 3-form ?eld strength H can be written in the following way [39],(see [25,24,16])(?)?Ψ=0,(d Φ?1
Received by the editors 1st February 2008.
Key words and phrases.G 2-structures,string equations.
Supported by the SFB 288”Di?erential geometry and quantum physics”of the DFG and the European Human Potential Program EDGE,Research Training Network HPRN-CT-2000-00101.S.Ivanov thanks ICTP for the support and excellent environment.
1
2THOMAS FRIEDRICH AND STEFAN IV ANOV
problems in string theory[32,37,21].One possible generalization of Calabi-Yau manifolds, hyper-K¨a hler manifolds,parallel G2-manifolds and parallel Spin(7)-manifolds are manifolds equipped with linear metric connections having skew-symmetric torsion and holonomy con-tained in SU(n),Sp(n),G2,Spin(7).One remarkable fact is that the existence(in small dimen-sions)of a parallel spinor with respect to a metric connection?with skew-symmetric torsion determines the connection in a unique way if its holonomy group is a subgroup of SU,Sp,G2, provided that some additional di?erential conditions on the structure are ful?lled[39,16],and always in dimension8for a subgroup of the group Spin(7)[23].The case of16-dimensional Riemannian manifolds with Spin(9)-structure was investigated in[13],homogeneous models are discussed in[2].The existence of?-parallel spinors in the dimensions4,5,6,7,8is studied in [39,10,24,16,17,23].In dimension7,the?rst consequence is that the manifold should be a G2-manifold with an integrable G2-structure[16],i.e.,the structure group could be reduced to the group G2and the corresponding3-formω3should obey d?ω3=θ∧?ω3for some special 1-formθ.In this paper we study solutions to the Killing spinor equations(?)in dimension 7and the geometry of integrable G2-manifolds.We?nd a formula for the Riemannian scalar curvature in terms of the fundamental3-form.Our?rst main result is the following Theorem1.1.Let(M,g,ω3)be an integrable G2-manifold with the fundamental3-formω3. The Riemannian scalar curvature Scal g is given in terms of the fundamental3-formω3by
Scal g=1
12
||T||2+3δθ,
(1.1)
whereθand T are the Lee form and the torsion of the unique G2-connection given by
(1.2)T=??dω3+1
3
?(?dω3∧ω3)=
1
12
·||T||2?6·△Φ,
where△Φ=δdΦis the Laplacian.The solution has constant dilation if and only if the G2-structure is cocalibrated of pure type W3.
Our proof relies on the existence theorem for a G2-connection with torsion,the Schr¨o dinger-Lichnerowicz formula for the connection with torsion(both established in[16])and the special properties of the Cli?ord action on the special parallel spinor.
KILLING SPINOR EQUATIONS IN DIMENSION7AND GEOMETRY OF INTEGRABLE G2-MANIFOLDS3
2.General properties of G2-structures
Let us consider R7endowed with an orientation and its standard inner product.Denote an oriented orthonormal basis by e1,...,e7.We shall use the same notation for the dual basis. We denote the monomial e i∧e j∧e k by e ijk.Consider the3-formω3on R7given by
(2.6)
ω3=e127+e135?e146?e236?e245+e347+e567.
The subgroup of SO(7)that?xesω3is the exceptional Lie group G2.It is a compact,simply-connected,simple Lie group of dimension14[34].The3-formω3corresponds to a real spinor and,therefore,G2is the isotropy group of a non-trivial real spinor.A G2-structure on a7-manifold M7is a reduction of the structure group of the tangent bundle to the exceptional group G2.This can be described geometrically by a nowhere vanishing di?erential3-formω3 on M7,which can be locally written as(2.6).The3-formω3is called the fundamental form of the G2-manifold M7(see[3])and it determines the metric completely.The action of G2on the tangent space gives an action of G2on k-forms and we obtain the following splitting[11,6]:Λ1(M7)=Λ17,Λ2(M7)=Λ27⊕Λ214,Λ3(M7)=Λ31⊕Λ37⊕Λ327,
where
Λ27={α∈Λ2(M7)|?(α∧ω3)=2α},Λ214={α∈Λ2(M7) ?(α∧ω3)=?α},Λ37={?(β∧ω3)|β∈Λ1(M7)},Λ327={γ∈Λ3(M7)|γ∧ω3=0,γ∧?ω3=0}. Following[8]we consider the1-formθde?ned by
(2.7)3θ=??(?dω3∧ω3)=?(δω3∧?ω3).
We shall call this1-form the Lee form associated with a given G2-structure.If the Lee form vanishes,then we shall call the G2-structure balanced.The classi?cation of the di?erent types of G2-structures was worked out by Fernandez-Gray[11],and Cabrera used the Lee form to characterize each of the16classes.An integrable G2-structure(or a structure of type W1⊕W3⊕W4)is characterized by the di?erential equation
d?ω3=θ∧?ω3,
and a cocalibrated G2-structure is de?ned by the condition
d?ω3=0.
A cocalibrated G2-structure of pure type W3is characterized by the two condtions d?ω3= 0,dω3∧ω3=0.Then the following proposition follows immediately.
Proposition2.1.If the Lee1-form is closed,then the G2-structure is locally conformal to a balanced G2-structure.
We shall call locally conformally parallel G2-manifolds that are not globally conformally parallel strict locally conformally parallel.
Example2.1.Any7-dimensional oriented spin Riemannian manifold admits a certain G2-structure,in general a non-parallel one(see for example[29]).The?rst known examples of complete parallel G2-manifold were constructed by Bryant and Salamon[7],the?rst compact examples by Joyce[26,27,28].There are many known examples of compact nearly parallel G2-manifolds:S7[11],SO(5)/SO(3)[7,35],the Alo?-Wallach spaces N(g,l)=SU(3)/U(1)g,l [9],any Einstein-Sasakian and any3-Sasakian space in dimension7[14,15].There are also some non-regular3-Sasakian manifolds(see[4,5]).Moreover,compact nearly parallel G2-manifolds with large symmetry group are classi?ed in[15].Compact integrable nilmanifolds are constructed and studied in[12].Any minimal hypersurface N in R8admits a cocalibrated G2-structure[11].Moreover,the structure is parallel,nearly parallel,cocalibrated of pure type if and only if the hypersurface N is totally geodesic,totally umbilic or minimal,respectively.
4THOMAS FRIEDRICH AND STEFAN IV ANOV
3.Conformal transformations of G2-structures
We study the conformal transformation of G2-structures(see[11]).
Proposition3.1.Letˉg=e2f·g,ˉω3=e3f·ω3be a conformal change of a G2-structure(g,ω3) and denote byˉθ,θthe corresponding Lee forms,respectively.Then
(3.8)ˉθ=θ+4d f.
Proof.We have the relations
volˉg=e7f·vol g,dˉω3=e3f·(3d f∧ω3+dω3).
We calculate
ˉ?dˉω3=e4f(?dω3+3?(d f∧ω3)),ˉ?dˉω3∧ˉω3=e7f(?dω3∧ω3?12?d f),
where we used the general identity?(ω3∧γ)∧ω3=4?γ,which is valid for any1-formγ. Consequently,we obtainˉθ=?1
3 ?(?dω3∧ω3)?12?2d f =θ+4d f. Proposition3.1allows us to?nd a distinguished G2-structure on a compact7-dimensional G2-manifold.
Theorem3.1.Let(M7,g,ω3)be a compact7-dimensional G2-manifold.Then there exists a unique(up to homothety)conformal G2-structure g0=e2f·g,ω30=e3f·ω3such that the corresponding Lee form is coclosed,δ0θ0=0.
Proof.We shall use the Gauduchon Theorem for the existence of a distinguished metric on a compact,hermitian or Weyl manifold[19,20].We shall use the expression of this theorem in terms of a Weyl structure(see[40],Appendix1).We consider the Weyl manifold(M7,g,θ,?W) with the Weyl1-formθ,where?W is a torsion-free linear connection on M7determined by the condition?W g=θ?g.Applying the Gauduchon Theorem we can?nd,in a unique way, a conformal metric g0such that the corresponding Weyl1-form is coclosed with respect to g0. The key point is that,by Proposition3.1,the Lee form transforms under conformal rescaling according to(3.8),which is exactly the transformation of the Weyl1-form under conformal rescaling of the metricˉg=e4f·g.Thus,there exists(up to homothety)a unique conformal G2-structure(g0,ω30)with coclosed Lee form. We shall call the G2-structure with coclosed Lee form the Gauduchon G2-structure. Corollary3.1.Let(M7,g,Φ)be a compact G2-manifold and(g,Φ)be the Gauduchon struc-ture.Then the following formula holds:
? dδω3∧?ω3 =||δω3||2.
In particular,if the structure is integrable,then
? dδω3∧?ω3 =24||θ||2.
https://www.doczj.com/doc/301603879.html,ing(2.7),we calculate that
0=3·δθ=?d(δω3∧?ω3)=? dδω3∧?ω3??d?ω3∧d?ω3 =? dδω3∧?ω3?||δω3||2·vol . If the structure is integrable,then||δω3||2=24||θ||2. Corollary3.2.On a compact G2-manifold with closed Lee form whose Gauduchon G2-structure is not balanced,the?rst Betti number satis?es b1(M)≥1.
For integrable G2-manifolds one can de?ne a suitable elliptic complex as well as cohomolgy groups?H i(M7)(see[12]).The?rst cohomolgy group is given by
?H1(M7)={α∈Λ1(M7):dα∧?ω3=0,d?α=0}.
KILLING SPINOR EQUATIONS IN DIMENSION7AND GEOMETRY OF INTEGRABLE G2-MANIFOLDS5 Corollary3.3.On a compact integrable manifold which is not globally conformally balanced, one has?b1≥1.
Proof.By the condition of the theorem the Gauduchon structure has a non-identically zero Lee form.Then0=δω3=?(dθ∧?ω3),since the structure is integrable.Adding the condition δθ=0,we obtain?b1≥1.
4.Connections with torsion,parallel spinors and Riemannian scalar curvature The Ricci tensor of an integrable G2-manifold was expressed in principle by the structure form ω3in the paper[16].Here we intend to?nd an explicit formula for the Riemannian scalar https://www.doczj.com/doc/301603879.html,ing the unique connection with skew-symmetric torsion preserving the given integrable G2-structure found in[16],we apply the Schr¨o dinger-Lichnerowicz formula for the Dirac operator of a metric connection with totally skew-symmetric torsion(see[16])in order to derive the formula for the scalar curvature.First,let us summarize the mentioned results from[16].
Theorem4.1.(see[16])Let(M7,g,ω3)be a G2-manifold.Then the following conditions are equivalent:
(1)The G2-structure is integrable,i.e.,d?ω3=θ∧?ω3;
(2)There exists a unique linear connection?preserving the G2-structure with totally skew-
symmetric torsion T given by
(4.9)T=??dω3+
1
7·(dω3,?ω3)?ω3,π47(dω3)=
3
6·λ·Ψ0?θ·Ψ0,λ=?
1
4
||T||2.
The4-formσT de?ned by the formula
σT=
1
6THOMAS FRIEDRICH AND STEFAN IV ANOV
Theorem 4.2.(see [16])Let Ψbe
a
parallel
spinor with respect to a metric connection ?
with totally
skew-symmetric torsion T on a Riemannian spin manifold M n .Then the following formulas hold
3·dT ·Ψ?2·σT ·Ψ+Scal ·Ψ=0,D(T ·Ψ)=dT ·Ψ+δT ·Ψ?2·σT ·Ψ.
Proof of Theorem 1.1.Let Ψ0be the ?-parallel spinor corresponding to the fundamental 3-form ω3.Then the Riemannian Dirac operator D g and the Levi-Civita connection ?g act on Ψ0by the rule
(4.12)?g X Ψ0=?14·T ·Ψ0=?74
·θ·Ψ0,where we used Theorem 4.1.We are going to apply the well known Schr¨o dinger-Lichnerowicz formula [31,38]
(D g )2=△g +1
8·D g λ·Ψ0 +364·λ2+94·δθ ·Ψ0?74·dθ·Ψ0+3
4
·n i =1 ?e i i e i T )·Ψ0?14·δT ·Ψ0?1
2·||T ||2
·Ψ0.Substituting (4.13)and (4.14)into the SL-formula,multiplying the obtained result by Ψ0and taking the real part,we arrive at (4.15) 4916·||θ||2+3322+18
·(σT ·Ψ0,Ψ0).On the other hand,using (4.10),we obtain
D(T ·Ψ0)=D(7
6·λ·Ψ0?θ·Ψ0)=7
6dλ
·Ψ0?d ?θ·Ψ0?δθ·Ψ0=dT ·Ψ0?2σT ·Ψ0+δT ·Ψ0.Multiplying the latter
equality by Ψ0and taking the real part,we obtain ?δθ·||Ψ0||2=(dT ·Ψ0,Ψ0)?(2σT ·Ψ0,Ψ0).Consequently,Theorem (4.2)and (4.11)imply (4.16) ?3·δθ?112·||dω3||2.
KILLING SPINOR EQUATIONS IN DIMENSION7AND GEOMETRY OF INTEGRABLE G2-MANIFOLDS7 Proof.In the case of a cocalibrated G2-structure of pure type,the torsion3-form T=??dω3. The claim follows from Theorem1.1. Using the results in[11]we derive immediately the following formula,which is essentially the reformulated Gauss equation.
Corollary4.4.Let M7be a hypersurface in R8the with second fundamental form S and mean curvature H.Then the Riemannian scalar curvature on M7is given by the formula
Scal g=49
12
·||S0||2,
(4.17)
where S0is the image of the traceless part of the second fundamental form via the isomorphism S20(R7)→Λ327.In particular,if M is a minimal hypersurface,then
Scal g=?
1
2||θ||2since the structure is locally conformally parallel.Then,Theo-
rem1.1leads to the formula
(4.19)Scal g=
15
6 M||θ||2d vol>0,
since the structure is strictly locally conformally parallel.The second assertion is a consequence of Corollary3.2.
5.Solutions to the Killing spinor equations in dimension7
We consider the Killing spinor equations(?)in dimension7.The existence of a?-parallel spinor is equivalent to the existence of a?-parallel integrable G2-structure and the3-form?eld strength H=T is given by(4.9).We now investigate the second Killing spinor equation(?). Proof of Theorem1.2.LetΨbe an arbitrary?-parallel such that(dΦ?T)·Ψ=0.The spinor ?eldΨde?nes a second G2-structureω30such thatΨ=Ψ0is the canonical spinor?eld.Since the connection preserves the spinor?eldΨ,it preserves the G2-structureω30,too.On the other hand,the connection preservingω30is unique.Consequently,the torsion T0coincides with the torsion form T and for the G2-structureω30we have
?Ψ0=0,(dΦ?T0)·Ψ0=0.
The Cli?ord action T0·Ψ0depends only on the(Λ31⊕Λ37)-part of https://www.doczj.com/doc/301603879.html,ing(4.9)and the algebraic formulas
?(γ∧ω30)·Ψ0=?γ(?ω30)·Ψ0=?4·γ·Ψ0,ω30·Ψ0=?7·Ψ0
we calculate
(5.20)T0·Ψ0=?θ·Ψ0?
1
8THOMAS FRIEDRICH AND STEFAN IV ANOV
Comparing with the second Killing spinor equation(?)we?nd2·dΦ=?β,(dω30,?ω30)=0 which completes the proof. As a corollary we obtain the result from[21],which states that any solution to both equations (?)has necessarily the NS three form H=T given by(1.4).A more precise analysis using Proposition3.1and Theorem1.1of the explicit solutions constructed in[21]shows that these solutions are conformally equivalent to a cocalibrated structure of pure type.In other words, the multiplication of the G2-structures(g±,ω3±)by(eΦ·g±,e(3/2)Φ·ω3±)is a new example of a cocalibrated G2-structure of pure type W3,and it is a solution to the Killing spinor equations with constant dilation.The same conclusions are valid for the solutions constructed in[1,37,32]. Theorem1.2allows us to construct a lot of compact solutions to the Killing spinor equations.If the dilation is a globally de?ned function,then any solution is globally conformally equivalent to a cocalibrated G2-structure of pure type.For example,any conformal transformation of a compact7-dimensional manifold with a Riemannian holonomy group G2constructed by Joyce [26,27]is a solution with non-constant dilation.Another source of solutions are conformal transformations of the cocalibrated G2-structures of pure type W3induced on any minimal hypersurface in R8.Summarizing,we obtain:
Corollary5.1.Any solution(M7,g,ω3)to the Killing spinor equations(?)in dimension7with non-constant globally de?ned dilation functionΦcomes from a solution with constant dilation by a conformal transformation(g=eΦ·g0,ω3=e(3/2)Φ·ω30),where(g0,ω30)is a cocalibrated G2-structure of pure type W3.
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Thomas Friedrich
Institut f¨u r Reine Mathematik
Humboldt-Universit¨a t zu Berlin
Sitz:WBC Adlershof
D-10099Berlin,Germany
friedric@mathematik.hu-berlin.de
Stefan Ivanov
F aculty of Mathematics and Informatics
University of Sofia“St.Kl.Ohridski”
blvd.James Bourchier5
1164Sofia,Bulgaria
ivanovsp@fmi.uni-sofia.bg
JOIN IN 学生用书1 Word List Starter Unit 1.Good afternoon 下午好 2.Good evening 晚上好 3.Good morning 早上好 4.Good night 晚安 5.Stand 站立 Unit 1 6.count [kaunt] （依次）点数 7.javascript:;eight [eit] 八 8.eleven [i'levn] 十一 9.four [f?:] 四 10.five [faiv] 五 11.flag [fl?g] 旗 12.guess [ges] 猜 13.jump [d??mp] 跳 14.nine [nain] 九 15.number ['n?mb?] 数字 16.one [w?n] 一 17.seven ['sevn] 七 18.six [siks] 六 19.ten [ten] 十 20.three [θri:] 三 21.twelve [twelv] 十二 22.two [tu:] 二 23.your [ju?] 你的 24.zero ['zi?r?u] 零、你们的 Unit 2 25.black [bl?k] 黑色26.blue [blu:] 蓝色 27.car [kɑ:] 小汽车 28.colour ['k?l?] 颜色 29.door [d?:] 门 30.favourite [feiv?rit]javascript:; 特别喜爱的 31.green [gri:n] 绿色 32.jeep [d?i:p] 吉普车 33.orange ['?:rind?] 橙黄色 34.pin k [pi?k] 粉红色 35.please [pli:z] 请 36.purple ['p?:pl] 紫色 37.red [red] 红色 38.white [wait] 白色 39.yellow ['jel?u] 黄色 Unit 3 40.blackboard ['bl?kb?:d] 黑板 41.book [buk] 书 42.chair [t???] 椅子 43.desk [desk] 桌子 44.pen [pen] 钢笔 45.pencil ['pensl] 铅笔 46.pencil case [keis] 笔盒 47.ruler ['ru:l?] 尺、直尺 48.schoolbag [sku:l] 书包 49.tree [tri:] 树 50.window ['wind?u] 窗、窗口 Unit 4 51.brown [braun] 棕色 52.cat [k?t] 猫
Join In剑桥小学英语(改编版)入门阶段Unit 1Hello,hello第1单元嗨,嗨 and mime. 1 听,模仿 Stand up Say 'hello' Slap hands Sit down 站起来说"嗨" 拍手坐下来 Good. Let's do up Say 'hello' Slap hands Sit down 好. 我们来再做一遍.站起来说"嗨"拍手坐下来 the pictures. 2 再听一遍给图画编号. up "hello" hands down 1 站起来 2 说"嗨" 3 拍手 4 坐下来说 3. A ,what's yourname 3 一首歌嗨,你叫什么名字 Hello. , what's yourname Hello. Hello. 嗨. 嗨. 嗨, 你叫什么名字嗨,嗨. Hello, what's yourname I'm Toby. I'm Toby. Hello,hello,hello.嗨, 你叫什么名字我叫托比. 我叫托比 . 嗨,嗨,嗨. I'm Toby. I'm Toby. Hello,hello, let's go! 我是托比. 我是托比. 嗨,嗨, 我们一起! Hello. , what's yourname I'm 'm Toby. 嗨.嗨.嗨, 你叫什么名字我叫托比.我叫托比. Hello,hello,hello. I'm 'm Toby. Hello,hello,let's go! 嗨,嗨,嗨.我是托比. 我是托比. 嗨,嗨,我们一起! 4 Listen and stick 4 听和指 What's your name I'm Bob. 你叫什么名字我叫鲍勃. What's your name I'm Rita. What's your name I'm Nick. 你叫什么名字我叫丽塔. 你叫什么名字我叫尼克. What's your name I'm Lisa. 你叫什么名字我叫利萨. 5. A story-Pit'sskateboard. 5 一个故事-彼德的滑板. Pa:Hello,Pit. Pa:好,彼德. Pi:Hello,:What's this Pi:嗨,帕特.Pa:这是什么 Pi:My new :Look!Pi:Goodbye,Pat! Pi:这是我的新滑板.Pi:看!Pi:再见,帕特! Pa:Bye-bye,Pit!Pi:Help!Help!pi:Bye-bye,skateboard! Pa:再见,彼德!Pi:救命!救命!Pi:再见,滑板! Unit 16. Let's learnand act 第1单元6 我们来边学边表演.
Join In剑桥小学英语(改编版)入门阶段 Unit 1Hello,hello第1单元嗨,嗨 1.Listen and mime. 1 听,模仿 Stand up Say 'hello' Slap hands Sit down 站起来说"嗨" 拍手坐下来 Good. Let's do itagain.Stand up Say 'hello' Slap hands Sit down 好. 我们来再做一遍.站起来说"嗨"拍手坐下来 2.listen again.Number the pictures. 2 再听一遍给图画编号. 1.Stand up 2.Say "hello" 3.Slap hands 4.Sit down 1 站起来 2 说"嗨" 3 拍手 4 坐下来说 3. A song.Hello,what's yourname? 3 一首歌嗨,你叫什么名字? Hello. Hello.Hello, what's yourname? Hello. Hello. 嗨. 嗨. 嗨, 你叫什么名字? 嗨,嗨. Hello, what's yourname? I'm Toby. I'm Toby. Hello,hello,hello. 嗨, 你叫什么名字? 我叫托比. 我叫托比 . 嗨,嗨,嗨. I'm Toby. I'm Toby. Hello,hello, let's go! 我是托比. 我是托比. 嗨,嗨, 我们一起! Hello. Hello.Hello, what's yourname? I'm Toby.I'm Toby. 嗨.嗨.嗨, 你叫什么名字? 我叫托比.我叫托比. Hello,hello,hello. I'm Toby.I'm Toby. Hello,hello,let's go! 嗨,嗨,嗨.我是托比. 我是托比. 嗨,嗨,我们一起! 4 Listen and stick 4 听和指 What's your name? I'm Bob. 你叫什么名字? 我叫鲍勃. What's your name ? I'm Rita. What's your name ? I'm Nick. 你叫什么名字? 我叫丽塔. 你叫什么名字? 我叫尼克. What's your name ? I'm Lisa. 你叫什么名字? 我叫利萨. 5. A story-Pit'sskateboard. 5 一个故事-彼德的滑板. Pa:Hello,Pit. Pa:好,彼德. Pi:Hello,Pat.Pa:What's this? Pi:嗨,帕特.Pa:这是什么? Pi:My new skateboard.Pi:Look!Pi:Goodbye,Pat! Pi:这是我的新滑板.Pi:看!Pi:再见,帕特! Pa:Bye-bye,Pit!Pi:Help!Help!pi:Bye-bye,skateboard! Pa:再见,彼德!Pi:救命!救命!Pi:再见,滑板! Unit 16. Let's learnand act 第1单元6 我们来边学边表演.
五年级下英语月考试卷-全能练考-Join in剑桥英语 姓名：日期：总分：100 一、根据意思，写出单词。（10′） Fanny:Jackie, ________[猜] my animal,my ________最喜欢的 animal.OK? Jackie:Ok!Mm…Has it got a nose? Fanny:Yes,it’s got a big long nose,…and two long ________[牙齿]. Jackie:Does it live in ________[美国]? Fanny:No,no.It lives ________[在] Africa and Asia. Jackie:Can it ________[飞]? Fanny:I’m sorry it can’t. Jackie:Is it big? Fanny:Yes,it’s very big and ________[重的]. Jackie:What ________[颜色] is it?Is it white or ________[黑色]? Fanny:It’s white. Jackie:Oh,I see.It’s ________[大象]. 二、找出不同类的单词。（10′） （）1 A.sofa B.table C.check （）2 A.noodle https://www.doczj.com/doc/301603879.html,k C.way （）3 A.fish B.wolf C.lion （）4 A.bike B.car C.write （）5 A.tree B.farm C.grass （）6 A.big B.small C.other ( ）7 A.eat B.listen C.brown
公司登记（备案）申请书 注：请仔细阅读本申请书《填写说明》，按要求填写。 □基本信息 名称 名称预先核准文号/ 注册号/统一 社会信用代码 住所 省（市/自治区）市（地区/盟/自治州）县（自治县/旗/自治旗/市/区）乡（民族乡/镇/街道）村（路/社区）号 生产经营地 省（市/自治区）市（地区/盟/自治州）县（自治县/旗/自治旗/市/区）乡（民族乡/镇/街道）村（路/社区）号 联系电话邮政编码 □设立 法定代表人 姓名 职务□董事长□执行董事□经理注册资本万元公司类型 设立方式 （股份公司填写） □发起设立□募集设立经营范围 经营期限□年□长期申请执照副本数量个
□变更 变更项目原登记内容申请变更登记内容 □备案 分公司 □增设□注销名称 注册号/统一 社会信用代码登记机关登记日期 清算组 成员 负责人联系电话 其他□董事□监事□经理□章程□章程修正案□财务负责人□联络员 □申请人声明 本公司依照《公司法》、《公司登记管理条例》相关规定申请登记、备案，提交材料真实有效。通过联络员登录企业信用信息公示系统向登记机关报送、向社会公示的企业信息为本企业提供、发布的信息，信息真实、有效。 法定代表人签字：公司盖章（清算组负责人）签字：年月日
附表1 法定代表人信息 姓名固定电话 移动电话电子邮箱 身份证件类型身份证件号码 （身份证件复印件粘贴处） 法定代表人签字：年月日
附表2 董事、监事、经理信息 姓名职务身份证件类型身份证件号码_______________ （身份证件复印件粘贴处） 姓名职务身份证件类型身份证件号码_______________ （身份证件复印件粘贴处） 姓名职务身份证件类型身份证件号码_______________ （身份证件复印件粘贴处）
《剑桥小学英语Join In ——Book 3 下学期》教材教法分析2012-03-12 18:50:43| 分类：JOIN IN 教案| 标签：|字号大中小订阅. 一、学情分析： 作为毕业班的学生，六年级的孩子在英语学习上具有非常显著的特点： 1、因为教材的编排体系和课时不足，某些知识学生已遗忘，知识回生的现象很突出。 2、有的学生因受学习习惯及学习能力的制约，有些知识掌握较差，学生学习个体的差异性，学习情况参差不齐、两级分化的情况明显，对英语感兴趣的孩子很喜欢英语，不喜欢英语的孩子很难学进去了。 3、六年级的孩子已经进入青春前期，他们跟三、四、五年级的孩子相比已经有了很大的不同。他们自尊心强，好胜心强，集体荣誉感也强，有自己的评判标准和思想，对知识的学习趋于理性化，更有自主学习的欲望和探索的要求。 六年级学生在英语学习上的两极分化已经给教师的教学带来很大的挑战，在教学中教师要注意引导学生调整学习方式： 1、注重培养学生自主学习的态度 如何抓住学习课堂上的学习注意力，吸引他们的视线，保持他们高涨的学习激情，注重过程的趣味化和学习内容的简易化。 2、给予学生独立思考的空间。 3、鼓励学生坚持课前预习、课后复习的好习惯。 六年级教材中的单词、句子量比较多，难点也比较多，学生课前回家预习，不懂的地方查英汉词典或者其它资料，上课可以达到事半功倍的效果，课后复习也可以很好的消化课堂上的内容。 4、注意培养学生合作学习的习惯。 5、重在培养学生探究的能力：学习内容以问题的方式呈现、留给学生更多的发展空间。 二、教材分析： （一）.教材特点： 1．以学生为主体，全面提高学生的素质。
有限合伙企业登记注册操作指南 风险控制部 20xx年x月xx日
目录 一、合伙企业的概念 (4) 二、有限合伙企业应具备的条件 (4) 三、有限合伙企业设立具备的条件 (4) 四、注册有限合伙企业程序 (5) 五、申请合伙企业登记注册应提交文件、证件 (6) （一）合伙企业设立登记应提交的文件、证件： (6) （二）合伙企业变更登记应提交的文件、证件： (7) （三）合伙企业注销登记应提交的文件、证件： (8) （四）合伙企业申请备案应提交的文件、证件： (9) （五）其他登记应提交的文件、证件： (9) 六、申请合伙企业分支机构登记注册应提交的文件、证件 (9) （一）合伙企业分支机构设立登记应提交的文件、证件 (10) （二）合伙企业分支机构变更登记应提交的文件、证件： (10) （三）合伙企业分支机构注销登记应提交的文件、证件： (11) （四）其他登记应提交的文件、证件： (12) 七、收费标准 (12) 八、办事流程图 (12) （一）有限合伙企业创办总体流程图（不含专业性前置审批） (12) （二）、工商局注册程序 (15)
（三）、工商局具体办理程序（引入网上预审核、电话预约方式） (16) 九、有限合伙企业与有限责任公司的区别 (16) （一）、设立依据 (16) （二）、出资人数 (16) （三）、出资方式 (17) （四）、注册资本 (17) （五）、组织机构 (18) （六）、出资流转 (18) （七）、对外投资 (19) （八）、税收缴纳 (20) （九）、利润分配 (20) （十）、债务承担 (21) 十、常见问题解答与指导 (21)
Quadratic Equations Let's glance at an example: The setup cost for the project to develop a new tank design is 50 thousand dollars. The development cost is a function of time. In other words, the cost increases with time. From previous contracts, the cost function is estimated to be t2 - 10 t where t is the time measured in working days. Suppose that a cost analyst has 650 thousand dollars allocated for the project, and he/she wants to know how long before he/she needs to allocate more resources to complete the project? Let's consider a solution
The total cost for the tank is equal to the sum of the initial cost and the development cost. t2 - 10t + 50 If $650 thousand has been given, we want to know how much time before more resources need to be allocated? t2 - 10t + 50 = 650 If we move 650 to the left of the equal sign, this is what we have t2 - 10t - 600 = 0 To solve the equation above, we just factor the terms as follows: (t + 20) (t - 30) The equation yields two answers: t = -20 and t = 30. Since negative time has no physical meaning, t = 30 is the only valid solution. Thus, it takes 30 working days before the cost analyst needs to allocate more resources to fund the project.
五 年 级 英 语 测 试 卷 学校 班级 姓名 听力部分（共20分） 一、Listen and colour . 听数字，涂颜色。（5分） 二、 Listen and tick . 听录音，在相应的格子里打“√”。 （6分） 三、Listen and number.听录音，标序号。（9分） pig fox lion cow snake duck
sheep 笔试部分（共80分） 一、Write the questions.将完整的句子写在下面的横线上。(10分) got it Has eyes on a farm it live sheep a it other animals eat it it Is 二、Look and choose.看看它们是谁，将字母填入括号内。(8分) A. B. C. D.
E. F. G. H. ( ) pig ( ) fox ( ) sheep ( ) cat ( ) snake ( ) lion ( ) mouse ( ) elephant 三、Look at the pictures and write the questions.看图片，根据答语写出相应的问题。(10分) No,it doesn’t. Yes,it is.
Yes,it does. Yes,it has. Yes,it does. 四、Choose the right answer.选择正确的答案。(18分) 1、it live on a farm？ 2. it fly？
3. it a cow？ 4. it eat chicken？ 5. you swim？ 6. you all right？ 五、Fill in the numbers.对话排序。(6分) Goodbye. Two apples , please. 45P , please. Thank you.
→客户提供：场所证明租赁协议身份证委托书三张一寸相片 →需准备材料：办理税务登记证时需要会计师资格证与财务人员劳动合同 →提交名称预审通知书→公司法定代表人签署的《公司设立登记申请书》→全体股东签署的《指定代表或者公共委托代理人的证明》（申请人填写股东姓名）→全体股东签署的公司章程（需得到工商局办事人员的认可）→股东身份证复印件→验资报告（需到计师事务所办理：需要材料有名称预审通知书复印件公司章程股东身份证复印件银行开具验资账户进账单原件银行开具询证函租赁合同及场所证明法人身份证原件公司开设临时存款账户的复印件）→任职文件（法人任职文件及股东董事会决议）→住所证明（房屋租赁合同）→工商局（办证大厅）提交所有材料→公司营业执照办理结束 →需带材料→公司营业执照正副本原件及复印件→法人身份证原件→代理人身份证→公章→办理人开具银行收据交款元工本费→填写申请书→组织机构代码证办
理结束 →需带材料→工商营业执照正副本复印件原件→组织机构正副本原件及复印件→公章→公司法定代表人签署的《公司设立登记申请书》→公司章程→股东注册资金情况表→验资报告书复印件→场所证明（租赁合同）→法人身份证复印件原件→会计师资格证（劳动合同）→税务登记证办理结束 →需带材料→工商营业执照正副本复印件原件→组织机构正副本原件及复印件→税务登记证原件及复印件→公章→法人身份证原件及复印件→代理人身份证原件及复印件→法人私章→公司验资账户→注以上复印件需四份→办理时间个工作日→办理结束 →需带材料→工商营业执照正副本复印件原件→组织机构正副本原件及复印件→公章→公司法定代表人签署的《公司设立登记申请书》→公司章程→股东注册资金情况表→验资报告书复印件→场所证明（租赁合同）→法人身份证复印件原件→会计师资格证（劳动合同）→会计制度→银行办理的开户许可证复印件→税务登记证备案办理结束
2011-2012 学年度上学期六年级复习题（Unit2-Unit3 ） 一、听力部分 1、听录音排序 ( ) () ()() () 2、听录音，找出与你所听到的单词属同一类的单词 （） 1. A. spaceman B. pond C . tiger （） 2. A.mascots B. potato C . jeans （） 3. A. door B. behind C . golden （） 4. A. sometimes B. shop C . prince （） 5. A. chair B. who C . sell 3、听录音，将下面人物与他的梦连线 Sarah Tim Juliet Jenny Peter 4、听短文，请在属于Mr. Brown的物品下面打√ ( ) ( ) ( ) ( ) ( ) ( ) ( ) 5、听问句选答句 （） 1. A. Yes, I am B. Yes, I have C . Yes, you do （） 2. A.Pink B. A friendship band C . Yes. （） 3. A. OK B. Bye-bye. C . Thanks, too. （） 4. A. Monday. B. Some juice. C . Kitty. （） 5. A. I ’ve got a shookbag. B. I ’m a student. C . It has got a round face. 6、听短文，选择正确答案 （） 1. Where is Xiaoqing from? She is from . A.Hebei B. Hubei C . Hunan （） 2. She goes to school at . A.7:00 B.7:30 C . 7:15 （） 3. How many classes in the afternoon? classes. A. four B. three C . two （） 4. Where is Xiaoqing at twelve o ’clock? She is . A. at home B. at school C .in the park （） 5. What does she do from seven to half past eight? She . A.watches TV B. reads the book C. does homework
Starter Unit Good to see you again知识总结 一. 短语 1. dance with me 和我一起跳舞 2. sing with me 和我一起唱歌 3. clap your hands 拍拍你的手 4. jump up high 高高跳起 5.shake your arms and your legs晃晃你的胳膊和腿 6. bend your knees 弯曲你的膝盖 7. touch your toes 触摸你的脚趾8. stand nose to nose鼻子贴鼻子站 二. 句子 1. ---Good morning. 早上好。 ---Good morning, Mr Li. 早上好，李老师。 2. ---Good afternoon. 下午好。 ---Good afternoon, Mr Brown. 下午好，布朗先生。 3. ---Good evening，Lisa. 晚上好,丽莎。 ---Good evening, Bob. 晚上好，鲍勃。 4. ---Good night. 晚安。 ----Good night. 晚安。 5. ---What’s your name? 你叫什么名字？ ---I’m Bob./ My name is Bob. 我叫鲍勃。 6. ---Open the window, please. 请打开窗户。 ---Yes ,Miss. 好的，老师。 7. ---What colour is it? 它是什么颜色？ 它是蓝红白混合的。 ---It’s blue, red and white. 皮特的桌子上是什么？ 8. ---What’s on Pit’s table? ---A schoolbag, an eraser and two books. 一个书包，一个橡皮和两本书。 9. ---What time is it? 几点钟？ 两点钟。 ---It’s two. 10.---What’s this? 这是什么？ ---My guitar. 我的吉他。
10. REACTION EQUATIONS Clicking the Reaction Equations button in the main menu shows the Reaction Equations Window, see Fig. 1. With this calculation option you can calculate the heat capacity, enthalpy, entropy and Gibbs energy values of a single species as well as of specified reactions between pure substances. See the reference state definitions, valid notations and abbreviations for the description of the chemical formulae in Chapter 28. Databases, chapter 2. 10.1 One Chemical Substance Fig. 1. Reaction Equations Window of HSC Chemistry.
Fig. 2.Thermodynamic data of A12O3 (alumina) displayed by the Reaction Equation option of HSC Chemistry. By entering a single chemical formula into the Formula box you will get similar tables of thermochemical data as presented in many thermochemical data books. HSC will, however, provide the results faster and exactly at those temperatures which you really want. Please follow these steps: 1.Write a chemical formula into the box, Fig. 1. For example: Fe, Na2SO4, Al2O3, SO4(-a), H(+a) or SO2(g). See the valid notation and syntax of chemical formulae in Chapter 21.2. 2.Select the lower limit, upper limit and temperature step. 3.Select the Temperature and Energy Units, by clicking the corresponding buttons. 4.Select the Format of the results. Normal (Absolute scale): H(species), S(species) and C(species) This format is used for example in the famous I. Barin, O. Knacke, and O. Kubaschewski data compilation1. Delta (Formation functions):
外研社剑桥小学英语Join_in四年级上册整体课时教案Starter Unit Let's begin 第一学时: The pupils learn to understand: How are you today? I’m fine/OK. Goodbye. The pupils learn to use: What’s your name? I’m (Alice). How are you? I’m fine. I’m OK. Activities and skills: Decoding meaning from teacher input. Using phrases for interaction in class. Greeting each other. I Introduce oneself. Asking someone’s mane. Teaching process: Step 1: Warm—up. Sing a song. Hello, what’s your name? Step 2: Presentation. 1. What’s your name? I’m…. (1) The teacher introduces herself, saying: Hello, /Good morning, I’m Miss Sun. (2) Asks a volunteer’s name:
What’s your name? The teacher prompt the pupil by whispering, I’m (Alice). (3) Asks all the pupils the same question and help those who need it by whispering I’m…. 2. How are you today? I’m fine. I’m OK. (1) The teacher explains the meaning of How are you today? (2) Tell the class to ask the question all together and introduce two answers, I’m (not very) fine/I’m Ok. (3) Asks all the pupils the same question and make sure that all the students can reply. Step3: Speak English in class. 1. Tells the pupils to open their books at page3, look at the four photographs, and listen to the tape. 2. Asks the pupils to dramatise the situations depicted in the four photographs. A volunteer will play the part of the parts of the other pupils, answering as a group. 第二学时: The pupils learn to use these new words: Sandwich, hamburger, hot dog, puller, cowboy, jeans, cinema, walkman, snack bar, Taxi, clown, superstar.
Join in三年级上册知识点 1.见面打招呼 Hello! = Hi! 你好 Good morning! 早上好！ Good afternoon! 下午好！ Good evening! 晚上好！ (注意：Good night的意思是“晚安”) 2.询问身体健康状况 How are you? 你好吗？ I’m fine. = I’m OK.我很好。 3.临走分别 Goodbye. = Bye bye. 再见。 See you tomorrow. 明天见。 4.Miss小姐Mr先生 5.询问名字 -What’s your name? 你叫什么名字？ -I’m Toby. = My name is Toby. = Toby.我叫托比。 6.Let’s+动词原形 如Let’s go! 我们走吧！ Let’s begin! 我们开始吧！ 7. Are you ready? 你准备好了吗？ I’m ready. 我准备好了。 8.十二个动词 look看listen听mime比划着表达speak讲read读write写draw画colour涂sing唱think想guess猜play玩9.询问物体 -What’s this? 这是什么？ -It’s my dog. / It’s a dog. / My dog. 这是我的狗。 10.my/your+名词 如my apple我的苹果 your apple你的苹果 11.数字0-10 zero零one一two二three三four四five五six六seven七eight八nine九ten十12.询问电话号码 -What’s your phone number? 你的电话号码是多少？ -It’s . / . 13.guess sth 猜东西 如Guess the number. 猜数字 14. What’s in the box? 盒子里有什么？ 15.询问数量 -How many? 多少？ -Nine. 九个。 16. Here is / are sth. 这是……
It Is Necessary To Clarify The Rights And Obligations Of The Parties, To Restrict Parties, And To Supervise Both Parties To Keep Their Promises And To Restrain The Act Of Reckless Repentance. 编订：XXXXXXXX 20XX年XX月XX日 代理公司注册登记协议书 简易版
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Chapter 1 First-Order ODEs C C h h a a p p t t e e r r 22 S S e e c c o o n n d d --O O r r d d e e r r L L i i n n e e a a r r O O D D E E s s ((二二階階線線性性常常微微分分方方程程式式)) Chapter 3 Higher-Order Linear ODEs Chapter 5 Series Solutions of ODEs Chapter 6 Laplace Transforms ? Ordinary differential equations may be divided into two large classes, linear (線性) and nonlinear (非線性) ODEs. Where nonlinear ODEs are difficult to solve, linear ODEs are much simpler because there are standard methods for solving many of these equations. 22..11 H H o o m m o o g g e e n n e e o o u u s s L L i i n n e e a a r r O O D D E E s s o o f f S S e e c c o o n n d d O O r r d d e e r r ((二二階階線線性性齊齊性性微微分分方方程程式式)) ? A second-order ODE is called linear (線性的) if it can be written as ()()()y p x y q x y r x '''++=. (1) ? 線性：方程式的每一項都不得出現()y x 和其導數(y ', y '',…)的乘積或自乘 In case ()0r x =, the equation is called homogeneous (齊性的). In case ()0r x ≠, the equation is called nonhomogeneous (非齊性的). The functions ()p x and ()q x are called the coefficients of the ODEs. ? Theorem 1 Superposition Principle for the Homogeneous Linear ODE (適用於線性 齊性常微分方程式的疊加原理) If both 1()y x and 2()y x are solutions of the homogeneous linear ODE ()()0y p x y q x y '''++=, (2) then a linear combination (線性組合) of 1y and 2y , say 1122()()c y x c y x +, is also a solution of the differential equation. Proof –