Riemannian Geometry of Noncommutative Surfaces

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Riemannian Geometry of Noncommutative Surfaces M.Chaichian ?and A.Tureanu ?Department of Physics,University of Helsinki and Helsinki Institute of Physics,P .O.Box 64,00014Helsinki,Finland R.B.Zhang ?School of Mathematics and Statistics,University of Sydney,Sydney,NSW 2006,Australia Xiao Zhang §Institute of Mathematics,Academy of Mathematics and Systems Science,Chinese Academy of Sciences,Beijing 100190,China A Riemannian geometry of noncommutative n -dimensional surfaces is developed as a ?rst step towards the construction of a consistent noncommutative gravitational theory.His-torically,as well,Riemannian geometry was recognized to be the underlying structure of Einstein’s theory of general relativity and led to further developments of the latter.The notions of metric and connections on such noncommutative surfaces are introduced and it is shown that the connections are metric-compatible,giving rise to the corresponding Rie-mann curvature.The latter also satis?es the noncommutative analogue of the ?rst and sec-ond Bianchi identities.As examples,noncommutative analogues of the sphere,torus and hyperboloid are studied in detail.The problem of covariance under appropriately de?ned general coordinate transformations is also discussed and commented on as compared with other treatments.I.INTRODUCTION In recent years there has been much progress in developing theories of noncommutative ge-ometry and exploring their applications in physics.Many viewpoints were adopted and different mathematical approaches were followed by different researchers.Connes’theory [1](see also Ref.[2])formulated within the framework of C ?-algebras is the most successful,which incorporates

cyclic cohomology and K-theory,and gives rise to noncommutative versions of index theorems.Theories generalizing aspects of algebraic geometry were also developed (see,e.g.,Ref.[3]for a review and references).A notion of noncommutative schemes was formulated,which seems to provide a useful framework for developing noncommutative algebraic geometry.

A major advance in theoretical physics in recent years was the deformation quantization of Poisson manifolds by Kontsevich (see Ref.[4]for the ?nal form of this work).This sparked intensive activities investigating applications of noncommutative geometries to quantum theory.Seiberg and Witten [5]showed that the anti-symmetric tensor ?eld arising from massless states

of

strings can be described by the noncommutativity of a spacetime,

[x μ,x ν]?=i θμν,θμνconstant matrix ,

(I.1)

where the multiplication of the algebra of functions is governed by the Moyal product

(f ?g )(x )=f (x )exp i 2θμν?μ??ν (I.3)

was used in Ref.

[16,17]to twist the universal enveloping algebra of the Poincar′e algebra,obtain-ing a noncommutative multiplication for the algebra of functions on the Poincar′e group closely related to the Moyal product (I.2).It is then natural to try to extend the procedure to other symme-tries of noncommutative ?eld theory and investigate whether the concept of twist provides a new symmetry principle for noncommutative spacetime.

The same Abelian twist element (I.3)was used in Refs.[9,10]for deforming the algebra of diffeomorphisms when attempting to obtain general coordinate transformations on the noncom-mutative space-time.It is interesting that Refs.[9,10]proposed gravitational theories which are different from the low energy limit of strings [18].However,based on physical arguments one would expect the Moyal product to be frame-dependent and transform under the general coordi-nate transformation.If the twist element is chosen as (I.3),the Moyal ?-product is ?xed once for all.This is likely to lead to problems similar to those observed in Ref.[19]when one attempted to deform the internal gauge transformations with the same twist element (I.3).Nevertheless twisting is expected to be a productive approach to the formulation of a noncommutative gravita-tional theory when implemented consistently.A “covariant twist”was proposed for internal gauge

transformations in Ref.[20],but it turned out that the corresponding star-product would not be associative.

Works on the noncommutative geometrical approach to gravity may be broadly divided into two types.One type attempts to develop noncommutative versions of Riemannian geometry ax-iomatically(that is,formally),while the other adapts general relativity to the noncommutative setting in an intuitive way.A problem is the lack of any safeguard against mathematical inconsis-tency in the latter type of works,and the same problem persists in the?rst type of works as well, since it is not clear whether nontrivial examples exist which satisfy all the axioms of the formally de?ned theories.

The aim of the present paper is to develop a theory of noncommutative Riemannian geom-etry by extracting an axiomatic framework from highly nontrivial and transparently consistent examples.Our approach is mathematically different from that of Refs.[9,10]and also quite far removed from the quantum group theoretical noncommutative Riemannian geometry[15](see also references in Ref.[15]and subsequent publications by the same author).

Recall that2-dimensional surfaces embedded in the Euclidean3-space provide the simplest yet nontrivial examples of Riemannian geometry.The Euclidean metric of the3-space induces a natural metric for a surface through the embedding;the Levi-Civita connection and the curvature of the tangent bundle of the surface can thus be described explicitly(for the theory of surfaces,see, e.g.,the textbook Ref.[21]).As a matter of fact,Riemannian geometry originated from Gauss’work on surfaces embedded in3-dimensional Euclidean space.

More generally,Whitney’s theorem enables the embedding of any smooth Riemannian mani-fold as a high dimensional surface in a?at Euclidean space of high enough dimension(see,e.g., Theorem9and Theorem11.1.1in Ref.[22]).The embedding also allows transparent construction and interpretation of all structures related to the Riemannian metric of M as in the2-dimensional case.

This paper develops noncommutative deformations of Riemannian geometry in the light of Whitney’s theorem.The?rst step is to deform the algebra of functions on a domain of the Eu-clidean space.We begin Section II by introducing the Moyal algebra A,which is a noncom-mutative deformation[23]of the algebra of smooth functions on a region of R2.The rest of Section II develops a noncommutative Riemannian geometry for noncommutative analogues of 2-dimensional surfaces embedded in3-space.Working over the Moyal algebra A,we show that much of the classical differential geometry for surfaces generalizes naturally to this noncommu-tative setting.In Section III,three illuminating examples are constructed,which are respectively noncommutative analogues of the sphere,torus and hyperboloid.Their noncommutative geome-tries are studied in detail.

We emphasize that the embeddings play a crucial role in our current understanding of the ge-ometry of the2-dimensional noncommutative surfaces.The metric of a noncommutative surface is constructed in terms of the embedding;the necessity of a left connection and also a right con-nection then naturally arises;even the de?nition of the curvature tensor is forced upon us by the context.Indeed,the extra information obtained by considering embeddings provides the guiding principles,which are lacking up to now,for building a theory of noncommutative Riemannian geometry.

Once the noncommutative Riemannian geometry of the2-dimensional surfaces is sorted out, its generalization to the noncommutative geometries corresponding to n-dimensional surfaces em-bedded in spaces of higher dimensions is straightforward.This is discussed in Section IV.

Recall that the basic principle of general relativity is general covariance.We study in Sec-tion V general coordinate transformations for noncommutative surfaces,which are brought about

by gauge transformations on the underlying noncommutative associative algebra A(over which noncommutative geometry is constructed).A new feature here is that the general coordinate trans-formations affect the multiplication of the underlying associative algebra A as well,turning it into another algebra nontrivially isomorphic to A.We make comparison with classical Rieman-nian geometry,showing that the gauge transformations should be considered as noncommutative analogues of diffeomorphisms.

The theory of surfaces developed over the deformation of the algebra of smooth functions on some region in R n now suggests a general theory of noncommutative Riemannian geometry of n-dimensional surfaces over arbitrary unital associative algebras with derivations.We present an outline of this general theory in Section VI.

We conclude this section with a remark on the presentation of the paper.As indicated above, we start from the simplest nontrivial examples of noncommutative Riemannian geometries and gradually extend the results to build up a theory of generality.This“experimental approach”is not the optimal format for presenting mathematics,as all special cases repeat the same pattern. However,it has the distinctive advantage that the general theory obtained in this way stands on ?rm ground.

II.NONCOMMUTATIVE SURFACES AND THEIR EMBEDDINGS

A.Noncommutative surfaces and their embeddings

The?rst step in constructing noncommutative deformations of Riemannian geometry is the deformation of algebras of functions.Let us?x a region U in R2,and write the coordinate of a point t in U as(t1,t2).Letˉh be a real indeterminate,and denote by R[[ˉh]]the ring of formal power series inˉh.Let A be the set of the formal power series inˉh with coef?cients being real smooth functions on https://www.sodocs.net/doc/714047251.html,ly,every element of A is of the form∑i≥0f iˉh i where f i are smooth functions on U.Then A is an R[[ˉh]]-module in the obvious way.

Given any two smooth functions f and g on U,we denote by f g the usual point-wise product of the two functions.We also de?ne their star-product(or more precisely,Moyal product)by

f?g=lim

t′→t

exp ˉh ??t′2???t′1 f(t)g(t′),(II.1)

where the exponential exp[ˉh(?

?t′2???t′1)]is to be understood as a power series in the differential operator?

?t′2???t′1.More explicitly,let

μp:A/ˉh A?A/ˉh A?→A/ˉh A,p=0,1,2,...,(II.2) be R-linear maps de?ned by

μp(f,g)=lim

t′→t 1

?t1

?

?t2

?

It has been known since the early days of quantum mechanics that the Moyal product is associative (see,e.g.,Ref.[4]for a reference),thus we arrive at an associative algebra over R[[ˉh]],which is a deformation[23]of the algebra of smooth functions on U.We shall usually denote this associative algebra by A,but when it is necessary to make explicit the multiplication of the algebra,we shall write it as(A,?).

Remark II.1.For the sake of being explicit,we restrict ourselves to consider the Moyal product (de?ned by(II.1))only in this section.As we shall see in Sections VI and V,the theory of noncommutative surfaces to be developed in this paper extends to more general star-products over algebras of smooth functions.

Write?i for?

which is equivalent to

g i j[q]=?

q

∑

n=1

q?n

∑

m=0

g ik[0]μn(g kl[m],g l j[q?n?m]).

Since the right-hand side involves only g l j[r]with r

In the same way,we can also show that there also exists a unique left inverse of g.It follows from the associativity of multiplication of matrices over any associative algebra that the left and right inverses of g are equal.

De?nition II.3.Given a noncommutative surface X,let

E i=?i X,i=1,2,

and call the left A-module T X and right A-module?T X de?ned by

T X={a?E1+b?E2|a,b∈A},?T X={E1?a+E2?b|a,b∈A}

the left and right tangent bundles of the noncommutative surface respectively.

Then T X?R[[ˉh]]?T X is a two-sided A-module.

Proposition II.4.The metric induces a homomorphism of two-sided A-modules

g:T X?R[[ˉh]]?T X?→A,

de?ned for any Z=z i?E i∈T X and W=E i?w i∈?T X by

Z?W→g(Z,W)=z i?g i j?w j.

It is easy to see that the map is indeed a homomorphism of two-sided A-modules,and it clearly coincides with the restriction of the dot-product to T X?R[[ˉh]]?T X.

Since the metric g is invertible,we can de?ne

E i=g i j?E j,?E i=E j?g ji,(II.6) which belong to T X and?T X respectively.Then

g(E i,E j)=δi j=g(E j,?E i),g(E i,?E j)=g i j.

Now any Y∈A3can be written as Y=y i?E i+Y⊥,with y i=Y??E i,and Y⊥=Y?y i?E i.We shall call y i?E i the left tangential component and Y⊥the left normal component of Y.Let

(T X)⊥={N∈A3|N?E i=0,?i},

which is clearly a left A-submodule of A3.In a similar way,we may also decompose Y into Y=E i??y i+?Y⊥with the right tangential component of Y given by?y i=E i?Y and the right normal component by?Y⊥=Y?E i??y i.Let

(?T X)⊥={N∈A3|E i?N=0,?i},

which is a right A-submodule of A3.Therefore,we have the following decompositions

A3=T X⊕(T X)⊥,as left A-module,

(II.7)

A3=?T X⊕(?T X)⊥,as right A-module.

It follows that the tangent bundles are?nitely generated projective modules over A.Follow-ing the general philosophy of noncommutative geometry[1],we may regard?nitely generated projective modules over A as vector bundles on the noncommutative surface.This justi?es the terminology of left and right tangent bundles for T X and?T X.

In fact T X and?T X are free left and right A-modules respectively,as E1and E2form A-bases for them.Consider T X for example.If there exists a relation a i?E i=0,where a i∈A,we have a i?E i?E j=a i?g i j=0,?j.The invertibility of the metric then leads to a i=0,?i.Since E1and E2generate T X,they indeed form an A-basis of T X.

One can introduce connections to the tangent bundles by following the standard procedure in the theory of surfaces[21].

De?nition II.5.De?ne operators

?i:T X?→T X,i=1,2,

by requiring that?i Z be equal to the left tangential component of?i Z for all Z∈T X.Similarly de?ne

??

?T X?→?T X,i=1,2,

i:

by requiring that??i?Z be equal to the right tangential component of?i?Z for all?Z∈?T X.Call the set consisting of the operators?i(respectively??i)a connection on T X(respectively?T X).

The following result justi?es the terminology.

Lemma II.6.For all Z∈T X,W∈?T X and f∈A,

?i(f?Z)=?i f?Z+f??i Z,??i(W?f)=W??i f+??i W?f.(II.8) Proof.By the Leibniz rule(II.3)for?i,

?i(f?Z)=(?i f)?Z+f?(?i Z),?i(W?f)=W?(?i f)+W?(?i f).

The lemma immediately follows from the tangential components of these relations under the de-compositions(II.7).

In order to describe the connections more explicitly,we note that there existΓk i j and?Γk i j in A such that

?i E j=Γk i j?E k,??i E j=E k??Γk i j.(II.9)

Because the metric is invertible,the elementsΓk i j and?Γk i j are uniquely de?ned by equation(II.9). We have

Γk i j=?i E j??E k?Γk i j=E k??i E j.(II.10)

It is evident thatΓk i j and?Γk i j are symmetric in the indices i and j.The following closely related objects will also be useful later:

Γi jk=?i E j?E k,?Γi jk=E k??i E j.

In contrast to the commutative case,Γk i j and?Γk i j do not coincide in general.We have

Γk i j=cΓi jl?g lk+?i jl?g lk,?Γk i j=g kl?cΓi jl?g kl??i jl,

where

cΓi jl=

1

2 ?i E j?E l?E l??i E j .

We shall call?i jl the noncommutative torsion of the noncommutative surface.Therefore the left and right connections involve two parts.The part cΓi jl depends on the metric only,while the noncommutative torsion embodies extra information.For a noncommutative surface embedded in A3,the noncommutative torsion depends explicitly on the embedding.In the classical limit with

ˉh=0,?k

i j vanishes and bothΓk i j and?Γk i j reduce to the standard Levi-Civita connection.

We have the following result.

Proposition II.7.The connections are metric compatible in the following sense

?i g(Z,?Z)=g(?i Z,?Z)+g(Z,??i?Z),?Z∈T X,?Z∈?T X.(II.11) This is equivalent to the fact that

?i g jk?Γi jk??Γik j=0.(II.12) Proof.Since g is a map of two-sided A-modules,it suf?ces to prove(II.11)by verifying the special case with Z=E j and?Z=E k.We have

?i g(E j,E k)=?i(E j?E k)

=?i E j?E k+E j??i E k

=g(?i E j,E k)+g(E j,??i E k),

where the second equality is equivalent(II.12).This proves both statements of the proposition. Remark II.8.In contrast to the commutative case,equation(II.12)by itself is not suf?cient to uniquely determine the connectionsΓi jk and?Γi jk;the noncommutative torsion needs to be speci-?ed independently.This is similar to the situation in supergeometry,where torsion is determined by other considerations.

At this point we should relate to the literature.The metric introduced here resembles similar notions in Refs.[14,24,25,26];also our left and right connections and their metric compatibility have much similarity with De?nitions2and3in Ref.[24].However,there are crucial differences. Our left(respectively right)tangent bundle is a left(respectively right)A-module only,while in Refs.[24,25]there is only one“tangent bundle”T which is a bimodule over some algebra(or Hopf algebra)B.The metrics de?ned in Refs.[14,24,25,26]are maps from T?B T to B. Remark II.9.A noteworthy feature of the metric in Ref.[14]is that a particular moving frame can be chosen to make all the components of the metric central(see equation(3.22)in Ref.[14]).In the context of the Moyal algebra,this amounts to that the metric is a constant matrix.

B.Curvatures and second fundamental form

Let[?i,?j]:=?i?j??j?i and[??i,??j]:=??i??j???j??i.Straightforward calculations show that for all f∈A,

[?i,?j](f?Z)=f?[?i,?j]Z,Z∈T X,

[??i,??j](W?f)=[??i,??j]W?f,W∈?T X.

Clearly the right-hand side of the?rst equation belongs to T X,while that of the second equation belongs to?T X.We re-state these important facts as a proposition.

Proposition II.10.The following maps

[?i,?j]:T X?→T X,[??i,??j]:?T X?→?T X

are left and right A-module homomorphisms respectively.

Since T X(respectively?T X)is generated by E1and E2as a left(respectively right)A-module, by Proposition II.10,we can always write

[?i,?j]E k=R l ki j?E l,[??i,??j]E k=E l??R l ki j(II.13) for some R l ki j,?R l ki j∈A.

De?nition II.11.We refer to R l ki j and?R l ki j respectively as the Riemann curvatures of the left and right tangent bundles of the noncommutative surface X.

The Riemann curvatures are uniquely determine by the relations(II.13).In fact,we have

R l ki j=g([?i,?j]E k,?E l),?R l ki j=g(E l,[??i,??j]E k).(II.14) Simple calculations yield the following result.

Lemma II.12.

R l ki j=??jΓl ik?Γp ik?Γl jp+?iΓl jk+Γp jk?Γl ip,

?R l

=??j?Γl ik??Γl jp??Γp ik+?i?Γl jk+?Γl ip??Γp jk

ki j

Proposition II.13.Let R lki j=R p ki j?g pl and?R lki j=?g kp??R p li j.The Riemann curvatures of the left and right tangent bundles coincide in the sense that R kli j=?R kli j.

Proof.By Proposition II.7,we have R lki j=(?i?j??j?i)E k?E l,which can be re-written as

R lki j=?i(?j E k?E l)??j E k???i E l

??j(?i E k?E l)+?i E k???j E l

=?i(?j E k?E l+E k???j E l)?E k???i??j E l

??j(?i E k?E l+E k???i E l)+E k???j??i E l.

Again by Proposition II.7,the?rst term on the far right-hand side can be written as?i?j g kl,and the third term can be written as??i?j g kl.Thus they cancel out,and we arrive at

R lki j=?E k?(??i??j???j??i)E l=?R lki j.

Because of the proposition,we only need to study the Riemannian curvature on one of the tangent bundles.Note that R kli j=?R kl ji,but there is no simple rule to relate R lki j to R kli j in contrast to the commutative case.

De?nition II.14.Let

R i j=R p ip j,R=g ji?R i j,(II.15) and call them the Ricci curvature and scalar curvature of the noncommutative surface respectively.

Then obviously

R i j=?g([?j,?l]E i,?E l),R=?g([?i,?k]E i,?E k).(II.16) In the theory of classical surfaces,the second fundamental form plays an important role.A similar notion exists for noncommutative surfaces.

De?nition II.15.We de?ne the left and right second fundamental forms of the noncommutative surface X by

h i j=?i E j?Γk i j?E k,?h i j=?i E j?E k??Γk i j.(II.17)

It follows from equation(II.9)that

h i j?E k=0,E k??h i j=0.(II.18) Remark II.16.Both the left and right second fundamental forms reduce to h0i j N in the commutative limit,where h0i j is the standard second fundamental form and N is the unit normal vector. The Riemann curvature R lki j=(?i?j??j?i)E k?E l can be expressed in terms of the second

fundamental forms.Note that

R lki j=?j E k??i E l??j E k???i E l??i E k??j E l+?i E k???j E l.

By De?nition II.15,

R lki j=?j E k??h il??i E k??h jl

=(?j E k+h jk)??h il?(?i E k+h ik)??h jl.

Equation(II.18)immediately leads to the following result.

Lemma II.17.The following generalized Gauss equation holds:

R lki j=h jk??h il?h ik??h jl.(II.19) Before closing this section,we mention that the Riemannian structure of a noncommutative surface is a deformation of the classical Riemannian structure of a surface by including quantum corrections.The embedding into A3is not subject to any constraints as the general theory stands. However,one may consider particular noncommutative surfaces with embeddings satisfying extra symmetry requirements similar to the way in which various star products on R3were obtained from the Moyal product on R4in Sections4and5in Ref.[27].

III.EXAMPLES

In this section,we consider in some detail three concrete examples of noncommutative surfaces: the noncommutative sphere,torus and hyperboloid.

A.Noncommutative sphere

Let U=(0,π)×(0,2π),and we writeθandφfor t1and t2respectively.Let X(θ,φ)= (X1(θ,φ),X2(θ,φ),X3(θ,φ))be given by

X(θ,φ)= sinθcosφcoshˉh,√coshˉh (III.1)

with the components being smooth functions in(θ,φ)∈U.It can be shown that X satis?es the following relation

X1?X1+X2?X2+X3?X3=1.(III.2) Thus we may regard the noncommutative surface de?ned by X as an analogue of the sphere S2.

and refer to it as a noncommutative sphere.We have

We shall denote it by S2ˉ

h

E1= cosθcosφcoshˉh,?√coshˉh ,

E2= ?sinθsinφcoshˉh,0 .

The components g i j=E i?E j of the metric g on S2ˉh can now be calculated,and we obtain

sinh2ˉh

g11=1,g22=sin2θ?

coshˉh sin2θ?cos2θ .

The components of this metric commute with one another as they depend onθonly.Thus it makes sense to consider the usual determinant G of g.We have

G=sin2θ+tanh2ˉh(cos22θ?cos2θ)

=sin2θ[1+tanh2ˉh(1?4cos2θ)].

The inverse metric is given by

sin2θ?tanh2ˉh cos2θ

g11=

,

sin2θ+tanh2ˉh(cos22θ?cos2θ)

tanhˉh cos2θ

g12=?g21=

Now we determine the connection and curvature tensor of the noncommutative sphere.The computations are quite lengthy,thus we only record the results here.For the Christoffel symbols,we have

Γ111=?Γ

111=0,Γ112=??Γ112=sin2θtanh ˉh ,Γ121=??Γ121=?sin2θtanh ˉh ,Γ122=?Γ122=

12sin2θ(1+tanh 2ˉh

),Γ221=?Γ

221=?13+4cos2θ)ˉh

3+O (ˉh 4),R 2112=?sin 2θ?

12(4+cos2θ?cos4θ)ˉh 2+O (ˉh 4),R 2212=?2sin 2θˉh ?(53cos2θ?4cos4θ)ˉh

3+O (ˉh 4).We can also compute asymptotic expansions of the Ricci curvature tensor

R 11=1+(6+4cos2θ)ˉh

2+O (ˉh 4),R 21=(2?cos2θ)ˉh

+13(16+29cos2θ+6cos4θ)ˉh

3+O (ˉh 4),R 22=sin 2θ+1

where a>1is a constant.Classically X is the torus.When we extend scalars from R to R[[ˉh]] and impose the star product on the algebra of smooth functions,X gives rise to a noncommutative

.We have

torus,which will be denoted by T2ˉ

h

E1=(cosθcosφ,cosθsinφ,?sinθ),

E2=(?(a+sinθ)sinφ,(a+sinθ)cosφ,0).

The components g i j=E i?E j of the metric g on T2ˉh take the form

g11=1+sinh2ˉh cos2θ,

g22=(a+coshˉh sinθ)2?sinh2ˉh cos2θ,

g12=?g21=?sinhˉh coshˉh cos2θ+a sinhˉh sinθ.

As they depend only onθ,the components of the metric commute with one another.The inverse metric is given by

(a+coshˉh sinθ)2?sinh2ˉh cos2θ

g11=

,

G

sinhˉh coshˉh cos2θ+a sinhˉh sinθ

g12=?g21=

sin2θcosh2ˉh,

2

1

Γ211=?sin2θsinhˉh coshˉh,Γ212=a cosθcoshˉh+

sin2θcoshˉh,Γ222=2a cosθsinhˉh+sin2θsinhˉh coshˉh.

2

We can?nd the asymptotic expansions of the curvature tensors with respect toˉh:

2sinθ(1+a sinθ)

R1112=

We can also compute asymptotic expansions of the Ricci curvature tensor

R11=

sinθ

2(a+sinθ)2

ˉh+O(ˉh3), R12=

sinθ(a+cos2θ+a sinθ)

a+sinθ

+O(ˉh2).

By settingˉh=0,we obtain from the various curvatures of T2ˉ

h the corresponding objects for the

usual torus T2.

C.Noncommutative hyperboloid

Another simple example is the noncommutative analogue of the hyperboloid described by X= (x,y,

2 ?1+cos2ˉh cosh2r ,

g12=?g21=?

1

2sinh2r cos2ˉh?1

sinh2r

,

g12=?g21=

tanˉh

Now we determine the curvature tensor of the noncommutative hyperboloid.For the connec-tion,we have

Γ111=cos2ˉh sinh2r,Γ112=?

1

2sin2ˉh sinh2r,Γ122=

1

2sin2ˉh sinh2r,Γ212=

1 2

cos2ˉh sinh2r,Γ222=

1 cosh2r

ˉh+O(ˉh2),

R2112=?

sinh2r

cosh2r

+O(ˉh2),

R2212=?

cosh2r+sinh22r

cosh2r

+O(ˉh2),

R21=

coth2r(2cosh2r?1)

cosh22r

ˉh+O(ˉh3), R22=

sinh2r

cosh22r

+O(ˉh2).

By settingˉh=0,we obtain from the various curvatures of H2ˉ

h the corresponding objects for the

usual hyperboloid H2.

IV.NONCOMMUTATIVE n-DIMENSIONAL SURFACES One can readily generalize the theory of Section II to higher dimensions,and we shall do this here.Noncommutative Bianchi identities will also be obtained.

Again for the sake of explicitness we restrict attention to the Moyal product on the smooth functions.However,as we shall see in Section V,it will be necessary to consider more general star-products in order to discuss“general coordinate transformations”of noncommutative surfaces.

A.Noncommutative n-dimensional surfaces

We take a region U in R n for a?xed n,and write the coordinate of t∈U as(t1,t2,...,t n).Let A denote the set of the smooth functions on U taking values in R[[ˉh]].Fix any constant skew symmetric n×n matrixθ.The Moyal product on A is de?ned by the following generalization of equation(II.1):

f?g=lim

exp ˉh∑i jθi j?i?′j f(t)g(t′)(IV.1)

t′→t

for any f,g∈A.Such a multiplication is known to be associative.Sinceθis a constant matrix, the Leibniz rule(II.3)remains valid in the present case:

?i(f?g)=?i f?g+f??i g.

For any?xed positive integer m,we can de?ne a dot-product

?:A m?R[[ˉh]]A m?→A m(IV.2) by generalizing(II.4)to A?B=a i?b i for all A=(a1,...,a m)and B=(b1,...,b m)in A m.As before,the dot-product is a map of two-sided A-modules.

Assume m>n.For X∈A m,we let E i=?i X,and de?ne g i j=E i?E j.Denote by g=(g i j)the n×n matrix with entries g i j.

De?nition IV.1.If g modˉh is invertible over U,we shall call X a noncommutative n-dimensional surface embedded in A m,and call g the metric of X.

The discussion on the metric in Section II carries over to the present situation;in particular,the invertibility of g modˉh implies that there exists a unique inverse(g i j).Now as in Section II,we de?ne the left tangent bundle T X(respectively right tangent bundle?T X)of the noncommutative surface as the left(respectively right)A-submodule of A m generated by the elements E i.The fact that the metric g belongs to GL n(A)enables us to show that the left and right tangent bundles are projective A-modules.

The connection?i on the left tangent bundle will be de?ned in the same way as in Section II, namely,by the composition of the derivative?i with the projection of A m onto the left tangent bundle.The connection??i on the right tangent bundle is de?ned similarly.Then?i and??i satisfy the analogous equation(V.6),and are compatible with the metric in the same sense as Proposition II.7.

One can show that

[?i,?j]:T X?→T X,[??i,??j]:?T X?→?T X

are left and right A-module homomorphisms respectively.This allows us to de?ne Riemann cur-vatures of the tangent bundles as in equation(II.13).Then the formulae given in Lemma II.12are still valid when the indices in the formulae are assumed to take values in{1,2,...,n}.Further-more,the left and right Riemann curvatures remain equal in the sense of Proposition II.13. Remark IV.2.One may de?ne a dot-product?:A m?R[[ˉh]]A m?→A m with a Minkowski signature by

a i?

b i

A?B=a0?b0?m?1∑

i=1

for any A=(a0,a1,...,a m?1)and B=(b0,b1,...,b m?1)in A m.This is still a map of two-sided A-modules.Then the afore developed theory can be adapted to this setting,leading to a theory of a noncommutative surface embedded in A m with a Minkowski signature.

For the sake of being concrete,we shall consider only noncommutative surfaces with Euclidean signature hereafter.

B.Bianchi identities

We examine properties of the Riemann curvature for arbitrary n and m.The main result in this subsection is the noncommutative analogues of Bianchi identities.

De?ne E i and?E l as in(II.6).Then

?p E l=??Γl pk?E k,??p?E l=??E k?Γl pk.(IV.3) These relations will be needed presently.Let

R l ki j;p=?p R l ki j?Γr pk?R l ri j?Γr pi?R l r jk?Γr p j?R l rki+R r ki j?Γl rp.(IV.4) Theorem IV.3.The Riemann curvature R i jkl satis?es the following relations

R l i jk+R l jki+R l ki j=0,R l ki j;p+R l k jp;i+R l kpi;j=0,(IV.5) which will be referred to as the?rst and second noncommutative Bianchi identities respectively. Proof.It follows from the relation?i E j=?j E i that

[?i,?j]E k+[?j,?k]E i+[?k,?i]E j=0.

This immediately leads to

g([?i,?j]E k,?E l)+g([?j,?k]E i,?E l)+g([?k,?i]E j,?E l)=0.

Using the de?nition of the Riemann curvature in this relation,we obtain the?rst Bianchi identity.

To prove the second Bianchi identity,note that

??p R l ki j+g(?p[?i,?j]E k,?E l)+g([?i,?j]E k,??p?E l)=0.

Cyclic permutations of the indices p,i,j lead to two further relations.Adding all the three relations together,we arrive at

??p R l ki j+g([?i,?j]?p E k,?E l)+g([?i,?j]E k,??p?E l)

??i R l k jp+g([?j,?p]?i E k,?E l)+g([?j,?p]E k,??i?E l)

??j R l kpi+g([?p,?i]?j E k,?E l)+g([?p,?i]E k,??j?E l)=0,(IV.6)

where we have used the following variant of the Jacobian identity

?p[?i,?j]+?i[?j,?p]+?j[?p,?i]=[?i,?j]?p+[?j,?p]?i+[?p,?i]?j.

By a tedious calculation one can show that

Q i jkp:=[?j,?k]?p E i+[?k,?i]?p E j

+[?p,?k]?i E j+[?k,?j]?i E p

+[?i,?k]?j E p+[?k,?p]?j E i

is identically zero.Now we add g(Q i jkp,?E l)to the left-hand side of(IV.6),obtaining an identity with?fteen terms on the left.Then the second Bianchi identity can be read off this equation by recalling(IV.3).

C.Einstein’s equation

Recall that in classical Riemannian geometry,the second Bianchi identity suggests the correct form of Einstein’s equation.Let us make some preliminary analysis of this point here.As we lack guiding principles for constructing an analog of Einstein’s equation,the material of this subsection is of a rather speculative nature.

In Section II,we introduced the Ricci curvature R i j and scalar curvature R.Their de?nitions can be generalized to higher dimensions in an obvious way.Let

R i j=g ik?R k j,(IV.7) then the scalar curvature is R=R i i.Let us also introduce the following object:

Θl p:=g([?p,?i]E i,?E l)=g ik?R l kpi.(IV.8) In the commutative case,Θl p coincides with R l p,but it is no longer true in the present setting. However,note that

Θl l=g ik?R l kli=g ik?R ki=R.(IV.9) By?rst contracting the indices j and l in the second Bianchi identity,then raising the index k to i by multiplying the resulting identity by g ik from the left and summing over i,we obtain the identity

0=?p R??i R i p+g [?i,?l]?p E i,?E l +g [?l,?p]?i E i,?E l

??lΘl p+g [?i,?l]E i,??p?E l +g [?p,?i]E i,??l?E l

+g [?p,?i]?l E i,?E l +g [?l,?p]E i,??i?E l .

Let us denote the sum of the last two terms on the right-hand side by?p.Then

?p=g ik?R r kpl?Γl ri??Γi lr?g rk?R l kpi.

In the commutative case,?p vanishes identically for all p.However in the noncommutative setting, there is no reason to expect this to happen.Let us now de?ne

R i p;i=?i R i p??Γi pr?R r i+?Γi ir?R r p,

Θl p;l=?lΘl p?Θr l?Γl rp+Θr p?Γl rl??p.(IV.10) Then the second Bianchi identity implies

R i p;i+Θi p;i??p R=0.(IV.11) The above discussions suggest that Einstein’s equation no longer takes its usual form in the noncommutative setting,but we have not been able to formulate a basic principle which enables us to derive a noncommutative analogue of Einstein’s equation.However,formulae(IV.11)and (IV.9)seem to suggest that the following is a reasonable proposal for a noncommutative Einstein equation in the vacuum:

R i j+Θi j?δi j R=0.(IV.12) We were informed by J.Madore that in other contexts of noncommutative general relativity,it also appeared to be necessary to include an object analogous toΘi j in the Einstein equation.

V.GENERAL COORDINATE TRANSFORMATIONS We investigate the effect of“general coordinate transformations”on noncommutative n-dimensional surfaces.This requires us to consider noncommutative surfaces de?ned over A en-dowed with star-products more general than the Moyal product.This should be compared with Refs.[9,10,12,13],where the only“general coordinate transformations”allowed were those keeping the Moyal product intact.

For the sake of being concrete,we assume that the noncommutative surface has Euclidean signature.

A.Gauge transformations

Denote by G(A)the set of R[[ˉh]]-linear mapsφ:A?→A satisfying the following conditions

φ(1)=1,φ=exp ∑iεi?i modˉh,(V.1) whereεi are smooth functions on U.Then clearly we have the following result.

Lemma V.1.The set G(A)forms a subgroup of the automorphism group of A as R[[ˉh]]-module.

For any givenφ∈G(A),de?ne an R[[ˉh]]-linear map

?φ:A?R[[ˉh]]A?→A,f?g→f?φg:=φ?1(φ(f)?φ(g)).(V.2) Lemma V.2. 1.The map?φis associative,thus there exists the associative algebra(A,?φ) over R[[ˉh]].Furthermore,φ:(A,?φ)?→(A,?)is an algebra isomorphism.

2.Let?φ=?φ?1,then for anyφ,ψ∈G(A)

ψ ψ?1(f)?φψ?1(g) =f?ψφg.(V.3) In this sense the de?nition of the new star-products respects the group structure of G(A). Proof.Because of the importance of this lemma for later discussions,we sketch a proof for it here, even though one can easily deduce a proof from Ref.[23].

For f,g,h∈A,we have

(f?φg)?φh=φ?1((φ(f)?φ(g))?φ(h))

=φ?1(φ(f)?(φ(g)?φ(h)))

=φ?1 φ(f)?φ(g?φh)

=f?φ(g?φh),

which proves the associativity of the new star-product.Asφis an R[[ˉh]]-module isomorphism by de?nition,we only need to show that it preserves multiplications in order to establish the isomorphism between the algebras.Nowφ(f?φg)=φ(f)?φ(g).This proves part(1).

Part(2)can be proven by unraveling the left-hand side of(V.3).

Adopting the terminology of Drinfeld from the context of quantum groups,we call an auto-morphismφ∈G(A)a gauge transformation,and call G(A)the gauge group.The star product?φwill be said to be gauge equivalent to the Moyal product(IV.1).However,note that our notion of gauge transformations is slightly more general than that in deformation theory[23],where the only type of gauge transformations allowed are of the special form

φ=id+ˉhφ1+ˉh2φ2+...,

withφi being R-linear maps on the space of smooth functions on U such thatφi(1)=0for all i. Such gauge transformations form a subgroup of G(A).

Remark V.3.The prime aim of the deformation theory[23]is to classify the gauge equivalence classes of deformations in this restricted sense but for arbitrary associative algebras.The seminal paper[4]of Kontsevich provided an explicit formula for a star-product from each gauge equiva-lence class of deformations of the algebra of functions on a Poisson manifold.

Remark V.4.General star-products gauge equivalent to the Moyal product were evaluated explic-itly up to the third order inˉh in Ref.[28].In Ref.[29],position-dependent star-products were also investigated and the ultra-violet divergences of a quantumφ4theory on4-dimensional spaces with such products were analyzed.

Given an elementφin the group G(A),we denote

u i:=φ?1(t i),i=1,2,...,n,

and refer to t→u as a general coordinate transformation of U.De?ne R[[ˉh]]-linear operators on A by

?φi=φ?1??i?φ.(V.4) Lemma V.5.The operators?φi have the following properties

?φi??φj??φj??φi=0,?φiφ?1(t j)=δj i,

and also satisfy the Leibniz rule

?φi(f?φg)=?φi(f)?φg+f?φ?φi(g),?f,g∈A.

Proof.The proof is easy but very illuminating.We have

?φi??φj??φj??φi=φ?1?(?i?j??j?i)?φ=0.

Also,?φiφ?1(t j)=φ?1(?i t j)=δj i,sinceφmaps a constant function to itself.

To prove the Leibniz rule,we note that

?φi(f?φg)=φ?1(?i(φ(f)?φ(g)))

=φ?1(?iφ(f)?φ(g))+φ?1(φ(f)??iφ(g))

=φ?1 φ(?φi f)?φ(g) +φ?1 φ(f)?φ(?φi g)

=?φi f?φg+f?φ?φi g.

This completes the proof of the lemma.

The Leibniz rule plays a crucial role in constructing noncommutative surfaces over(A,?φ).

Spheric geometry球面几何

Spheric geometry（球面几何） 是几何学的一门分科。 研究球面上图形的几何学。是古代从研究天体在天球上的“视运动”发展起来 的，其中专门研究球面上三角形的性质的称为“球面三角”。 球面几何学是在二维的球面表面上的几何学，也是非欧几何的一个例子。 在平面几何中，基本的观念是点和线。在球面上，点的观念和定义依旧不变， 但线不再是“直线”，而是两点之间最短的距离，称为最短线。在球面上，最短 线是大圆的弧，所以平面几何中的线在球面几何中被大圆所取代。同样的，在球 面几何中的角被定义在两个大圆之间。结果是球面三角学和平常的三角学有诸多 不同之处。例如：球面三角形的内角合大于180°。 对比于通过一个点至少有两条平行线，甚至无穷多条平行线的双曲面几何 学，通过特定的点没有平行线的球面几何学是椭圆几何学中最简单的模式。 球面几何学在航海学和天文学都有实际且重要的用途。 球面几何学的重要关键在塑造真实投影平面，通过辨认在球面上获得正相反 的对跖点(分列在边的两侧相对的点)。在当地，投影平面具有球面几何所有的特 性，但有不同的总体特性，特别是他是无定向的。 球面乃是空间中最完美匀称的曲面。两个半径相等的球面可以用一个平移把 它们叠合起来，而两个半径不相等的球面所相差者就是放大或缩小这种相似变 换，由此可见本质性的球面几何可以归纳到单位半径的球面来研讨。再者，在古 典天文学的研讨中，观察星星的方向可以用单位球面上的一个点来标记它，而两 个方向之间的角度（亦即方向差）则相应于单位球面上两点之间的球面距离 (spherical distance) 。 这也就是为什么古希腊天文学和几何学总是合为一体的，而且古希腊的几何 学家对于球面三角学(spherical trigonometry)的投入程度要远远超过他们对 于平面测量学的兴趣，因为「量天的学问」才是他们所致力去理解者；它的确比 丈量土地、计量财产等更引人入胜。 从现代的观点来看，球面几何乃是空间几何中蕴含在正交子群的部分，而向 量几何则是空间几何中蕴含在平移子群的部分，而且两者又密切相关、相辅相成， 例如向量运算都是正交协变的(orthogonal covariant)，所以向量代数又是研讨 球面几何的简明有力的利器。 七、球面幾何和球面三角學 項武義 ?單位球面的基本性質 ?球面三角學 球面乃是空間中最完美勻稱的曲面。兩個半徑相等的球面可以用一個平移把它們疊合起來，而兩徑不相等的球面所相差者就是放大或縮小這種相似變換，由此可見本質性的球面幾何可以歸納到

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二、组织架构图

三、各部门岗位职责： （一）董事会职责 根据《公司法》规定和公司章程，公司董事会是公司经营决策机构，也是股东会的常设权力机构。董事会向股东会负责。 1、负责召集股东会；执行股东会决议并向股东会报告工作； 2、决定公司的生产经营计划和投资方案； 3、决定公司内部管理机构的设置； 4、批准公司的基本管理制度； 5、听取执行董事及总经理的工作报告并作出决议； 6、制订公司年度财务预、决算方案和利润分配方案、弥补亏损 方案； 7、对公司增加或减少注册资本、分立、合并、终止和清算等重 大事项提出方案； 8、聘任或解聘公司总经理、副总经理、财务部门负责人，并决 定其奖惩。 （二）执行董事职责 １、主持召开股东大会、董事会议，并负责上述会议决议的贯彻落实。

Introduction to Geometry

Computer Science15-499C/15-881,Fall1997 Introduction to Geometry Instructor:Michael Erdmann(me+@https://www.sodocs.net/doc/714047251.html,) Guest Instructor:Yanxi Liu(yanxi+@https://www.sodocs.net/doc/714047251.html,) Location:Scaife324 Time:TR10:30–11:50 TA:German(german+@https://www.sodocs.net/doc/714047251.html,) Of?ce Hours:by appointment 1Course Outline This course will cover elementary differential and computational geometry.The purpose of the course is to prepare a student for advanced geometrical work in robotics and computer science.Increasingly,cutting-edge results in these areas require a working knowledge of differential geometry,algebraic geometry,algebraic topology, and computational geometry.Much of this work is inaccessible to a student just entering the?eld.In this course we will convey the basic tools,de?nitions,and results of differential geometry and the basic algorithms of computational geometry,so that a student can,either by self-study or through further courses,understand and implement the advanced results in computer science and robotics discovered in the past decade.We will touch upon some of these applications in the course.In particular,we will consider the robot motion planning problem as a core application. The topics are: Motion Planning: Con?guration Space,Visibility Graph,Non-Holonomic Motion Planning,Forces in Cspace. Frame Fields: Curves,Frenet Formulas,Covariant Derivatives,Differential Forms, Connection Forms,Structural Equations. Calculus on Surfaces: Surfaces,Patches,Tangent V ectors,Mappings,Differential Forms, Integration on Surfaces,Manifolds. Shape Operators: Surface Shape Operator,Normal Curvature,Gaussian Curvature. Point and Range Queries: One shot,repeated query,slab method,multidimensional binary tree.

大连海事大学船舶操纵复习提纲1到19条

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（3）对浅水，岸壁，船间效应缺乏应有的戒备； （4）不复诵车钟令和舵令； （5）未适应夜视而交接班 （6）狭水道，复杂水域航行时没有备车，备锚，增派了望人员； （7）在不应追越的水域，地段或情况下盲目追越； （8）未及时使用手操舵； （9）锚泊的水域或方法不当；或对本船或他船的走锚缺乏戒备 （10）了解地方特殊规定及避让习惯。 3.当时特殊情况可能要求的戒备上的疏忽 构成特殊情况的原因很多， 主要有：自然条件的突变；复杂的交通条件； 相遇船舶突然出现故障；出现《规则》条款没有提及的情况和格局等。 例如：（1）突遇浓雾，暴风雨等严重影响视距和船舶操纵性能的天气； （2）两艘以上的船舶相遇构成碰撞的局面； （3）夜间临近处突然发现不点灯的小船，或突然显示灯光的船舶； （4）他船突然采取具有危险性的背离《规则》的行动； （5）由于环境和条件的限制，使本船或他船无法按照《规则》的规定采取避碰行动。 三．“背离”的目的，条件与时机 1．目的:为避免紧迫危险。 2．条件: （1）“危险”确实存在，不是臆测或主观臆断的； （2）危险是紧迫； （3）“背离”是合理（且有效）的，不背离反而不利于避碰。 4.时机: 采取背离行动的时机显然只能在紧迫局面形成之后，“紧迫危险”尚未出现之前，不可过早或过晚。 NO.3 1.船舶: (1)显然，军舰专用船舶和从事海上勘探的各种钻井船等均属于船舶。 (2)潜水艇——当其在水面航行时,方为“船舶”。 (3)非排水船舶——航行时，基本上或完全不靠浮力支持船舶重量的船舶。 2. 机动船：这里为广义，但在第二章各条中，不包括： 失去控制的船舶，操限船和限于吃水的船舶，从事捕鱼的船舶。 3. 帆船Sailing vessel (指任何驶帆的船舶,如果装有推进器但不在使用.) 为单纯用帆行驶的船舶。机帆并用----为机动船。 4.从事捕鱼的船舶： （1）正在从事捕鱼,不论其是否对水移动； （2）作业时,所使用的渔具使其操纵性能受到限制。 5.水上飞机——水面航行时属“船舶”，水上超低空飞行时属“飞机”。

公司组织架构图和岗位说明书

广西建设有限公司 部门职能及岗位职责 （汇编） 2010年3月 目录 一、组织架构

-----------------------------------------------------------------------------------------2 1、公司行政组织架构---------------------------------------------------------------------------------2 2、工程项目部组织架构----------------------------------------------------------------------------3 二、总经理-----------------------------------------------------------------------------------------4 三、副总经理-----------------------------------------------------------------------------------------5 四、总工程师-----------------------------------------------------------------------------------------6 五、人力总监--------------------------------------------------------------------------------------7 六、财务总监--------------------------------------------------------------------------------------8 七、行政人事部--------------------------------------------------------------------------------------------9 1、人事主管---------------------------------------------------------------------------------------10 2、行政后勤主管------------------------------------------------------------------------------------11 3、文员-----------------------------------------------------------------------------------------12 八、财务部----------------------------------------------------------------------------------------13 1、部门经理-----------------------------------------------------------14 2、会计----------------------------------------------------------------------------------15 3、出纳--------------------------------------------------------------------------------------16 九、总工程师办公室-------------------------------------------------------------------------17 1、副总工程师--------------------------------------------------------18 2、预决算员---------------------------------------------------------19 3、质检员----------------------------------------------------------20 4、土建工程师--------------------------------------------------------21 十、采购部------------------------------------------------------------------------22 1、部门经理---------------------------------------------------------23 2、材料员-----------------------------------------------------24十一、设备部-------------------------------------------------------------------------------------------------- 25 1、部门经理-----------------------------------------------------------26 2、操作员----------------------------------------------------------------------------27十一、项目部----------------------------------------------------------------------------------------------- 28 3、部门经理-----------------------------------------------------------29 4、工长---------------------------------------------------------29 5、资料员------------------------------------------------------------30 6、试验员----------------------------------------------------------31 7、水电工 ---------------------------------------------------------------32 8、项目作业班长 ---------------------------------------------------------33 9、门卫-------------------------------------------------------34 一、建设公司行政组织机构：

XX科技公司组织结构及岗位说明书

XX科技公司组织结构及岗位说明书 北京XXX科技开发有限公司组织结构及岗位说明书

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物业管理员岗位说明书 (77) 行政主管岗位说明书 (79) 市场营销部 (82) 市场营销部组织结构： (82) 市场营销部部门职责： (82) 市场营销部岗位职责： (83) 市场营销部经理岗位说明书 (83) 产品规划岗位说明书 (86) 市场推广岗位说明书 (88) 客户经理岗位说明书 (90) 订单管理岗位说明书 (92) 营销管理岗位说明书 (94) 客户服务部 (98) 客户服务部组织结构： (98) 客户服务部部门职责： (98) 客户服务部岗位职责： (99) 客户服务部经理岗位说明书 (99) 客户服务岗位说明书 (102) 技术支持岗位说明书 (105) 备品备件管理岗位说明书 (107) 维修管理岗位说明书 (109) 维修工岗位说明书 (111) 财务部 (113) 财务部部门结构： (113) 财务部部门职责： (113) 财务部岗位职责： (114) 财务部经理岗位说明书 (114) 应付会计岗位说明书 (118) 应收会计岗位说明书 (120) 成本会计岗位说明书 (123) 总账会计岗位说明书 (125) 预算管理岗位说明书 (127) 出纳岗位说明书 (129) 质量部 (132) 质量部部门结构： (132) 质量部部门职责： (132) 质量部岗位职责： (133) 质量部经理岗位说明书 (133) 体系管理工程师岗位说明书 (137) 质量管理工程师岗位说明书 (140) 质检主管岗位说明书 (143) 进货检验岗位说明书 (146) 过程检验岗位说明书 (148)

有机化学第二章

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第三步：CH 2Cl 2+ Cl 2CHCl 3+HCl 第四步：CHCl 3+Cl 2 CCl 4+HCl 甲烷的四种氯代物均难溶于水，常温下，只有CH 3Cl 是气态，其余均为液态，CHCl 3俗称氯仿，CCl 4又叫四氯化碳，是重要的有机溶剂，密度比水大。 （2）乙烯 ①与卤素单质X 2加成 CH 2＝CH 2＋X 2→CH 2X —CH 2X ②与H 2加成 CH 2＝CH 2＋H 2 CH 3—CH 3 ③与卤化氢加成 CH 2＝CH 2＋HX →CH 3—CH 2X ④与水加成 CH 2＝CH 2＋H 2O CH 3CH 2OH ⑤氧化反应 常温下被氧化，如将乙烯通入酸性高锰酸钾溶液，溶液的紫色褪去。 ⑥易燃烧 CH 2＝CH 2＋3O 22CO 2+2H 2O 现象（火焰明亮，伴有黑烟） ⑦加聚反应 二烯烃的化学性质 二烯烃跟烯烃性质相似，由于含有双键，也能发生加成反应、氧化反应和加聚反应。这里我们主要介绍1，3-丁二烯与溴发生的两种加成反应。 当两个双键一起断裂，同时又生成一个新的双键，溴原子连接在1、4两个碳原子上，即1、4加成反应 (1)二烯烃的加成反应：（1，4一加成反应是主要的） 若两个双键中的一个比较活泼的键断裂，溴原子连接在1、2两个碳原子上，即1、2加成反应 催化剂 △ ?? →?催化剂??→ ? 点燃

STGeometry教程

ArcGIS GeoDatabase ST_Geometry简介 1使用ST_Geometry存储空间数据（oracle） 1.1简介 ArcSDE for Oracle提供了ST_Geometry类型来存储几何数据。ST_Geometry是一种遵循ISO和OGC规范的，可以通过SQL直接读取的空间信息存储类型。采用这种存储方式能够更好的利用oracle的资源，更好的兼容oracle的特征，比如复制和分区，并且能够更快的读取空间数据。使用ST_Geometry存储空间数据，可以把业务数据和空间数据存储到一张表中（使用SDENBLOB方式业务数据和空间数据是分开存储在B表和F表中的），因此可以很方便的在业务数据中增加空间数据（只需要在业务表中增加ST_Geometry列）。使用这种存储方式还能够简化多用户的读取，管理(只需要管理一张表)。 从ArcGIS 9.3开始，新的ArcSDE geodatabases for Oracle 会默认使用ST_Geometry 方式来存储空间数据。它实现了SQL3规范中的用户自定义类型（user-defined data types），允许用户使用ST_Geometry类型创建列来存储诸如界址点，街道，地块等空间数据。 使用ST_Geometry类型存储空间数据，具有以下优势： 1）通过SQL函数（ISO SQL/MM 标准）直接访问空间数据； 2）使用SQL语句存储、检索操纵空间数据，就像其他类型数据一样。 3）通过存储过程来进行复杂的空间数据检索和分析。 4）其他应用程序可以通过SQL语句来访问存储在geodatabase中的数据。从ArcGIS 9.3开始，新的ArcSDE geodatabases for Oracle 要求所以ST 函数调用的时候前面都要加上SDE schema名称。例如：要对查询出来的空间数据进行union操作，则SQL函数需要这样写："sde.ST_Union",在9，2版本之前，可以不加SDE schema名称。

公司组织架构图及岗位职责说明书

XX公司组织架构及部门岗位职责 说明书 一、总经理职责 总经理是公司经营管理的领导核心，是经营管理的最高决策人。 1.1职能 ①组织制订公司经营方针、经营目标、经营计划，分解到各部门并组织实施。 ②负责制订并落实公司各项规章制度、改革方案、改革措施。 ③提出公司组织机构设置方案。 ④提出公司经营理念，主导企业文化建设的基本方向，创造良好的工作环境、生活环境，培养员工归属感，提升企业的向心力、凝聚力、战斗力。 ⑤负责处理部门相互之间事务矛盾和问题。 ⑥负责公司投资项目选定。 ⑦负责审核公司经营费用支出。 ⑧决定公司各部门人员的聘用任免。对公司的经济效益负责，拥有经营指挥权和各种资源分配权 1.2权力 ①有权根据公司经营目标、经营方针、制订经营计划； ②有权实施公司改革方案、改革措施，制订公司制度。 ③有权提出公司机构设置建议。 ④有权聘用或解聘公司各部门经理、员工，并决定其薪酬待遇，有权对各部门员工进行工作调配。 ⑤有权审核公司经营费用支出与报销。 ⑥有权对公司员工作出奖惩决定 1.3 工作流程 ①依据公司各类信息，如财务报表、汇总的信息、各部门报告等，向各部门经理发出指令，提出工作安排。 ②接受指令人员，根据总经理要求，制订出相应制度、方案、政策、措施，做出决议报总经理。 ③总经理对提供的制度、方案、政策、决议等进行审阅，同意则签批给职能部门实施；不同意，则指令有关部门修订完善后，再审核、签批、下发、实施。 ④有关部门定期将各类制度、方案、政策、措施的执行情况，检查、落实后汇总上报总经理。 ⑤依据执行情况，总经理发出新指令。 二、总经办部门职能 1、项目工作的监督、管理：协助、监督公司重大项目工作的组织、实施、落实和绩效评估工作，部门工作重大问题的监控工作； 2、法律事务：依据公司工作开展的需要，全面负责公司内外法律事务，监管重大合同谈判，以确保公司的权益不受损害； 3、对外关系：代表公司参加有关会议，保持与政府部门、同行业机构等的联系，树立公司良好形象； 4、重大活动组织：协助、监管有关部门组织重大活动，企业文化建设，企业形象提升活动时，提供后勤保证；

使用ST_Geometry存储空间数据(oracle)

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