Weyl structures on quaternionic manifolds. A state of the art
a r X i v :m a t h /0105041v 1 [m a t h .D G ] 5 M a y 2001WEYL STRUCTURES ON QUATERNIONIC MANIFOLDS.A STATE OF THE
ART.
LIVIU ORNEA This is a survey on quaternion Hermitian Weyl (locally conformally quaternion K¨a hler)and hyper-hermitian Weyl (locally conformally hyperk¨a hler)manifolds.These geometries appear by requesting the compatibility of some quaternion Hermitian or hyperhermitian structure with a Weyl structure.The motivation for such a study is two-fold:it comes from the constantly growing interest in Weyl (and Einstein-Weyl)geometry and,on the other hand,from the necessity of understanding the existing classes of quaternion Hermitian manifolds.Various geometries are involved in the following discussion.The ?rst sections give the minimal back-ground on Weyl geometry,quaternion Hermitian geometry and 3-Sasakian geometry.The reader is sup-posed familiar with Hermitian (K¨a hler and,if possible,locally conformally K¨a hler)and metric contact (mainly Sasakian)geometry.All manifolds and geometric objects on them are supposed di?erentiable of class C ∞.1.Weyl structures We present here the necessary background concerning Weyl structures on conformal manifolds.We refer to [16],[18],[21]or to the most recent survey [14]for more details and physical interpretation (motivation)for Weyl and Einstein-Weyl geometry.Let M be a n -dimensional,paracompact,smooth manifold,n ≥2.A CO(n )?O(n )×R +structure on M is equivalent with the giving of a conformal class c of Riemannian metrics.The pair (M,c )is a conformal manifold .For each metric g ∈c one can consider the Levi-Civita connection ?g ,but this will not be compatible with the conformal class.Instead,we shall work with CO(n )-connections.Precisely:De?nition 1.1.A Weyl connection D on a conformal manifold (M,c )is a torsion-free connection which preserves the conformal class c .We say that D de?nes a Weyl structure on (M,c )and (M,c,D )is a Weyl manifold.Preserving the conformal class means that for any g ∈c ,there exists a 1-form θg (called the Higgs ?eld)such that
Dg =θg ?g.This formula is conformally invariant in the following sense:
if h =e f g,f ∈C ∞(M ),then θh =θg ?d f.
(1.1)Conversely,if one starts with a ?xed Riemannian metric g on M and a ?xed 1-form θ(with T =θ?),the connection D =?g ?
11991Mathematics Subject Classi?cation.53C15,53C25,53C55,53C10.
Key words and phrases.Weyl structure,quaternion Hermitian manifold,locally conformally K¨a hler geometry,3-Sasakian geometry,Einstein manifold,homogeneous manifold,foliation,complex structure,Riemannian submersion,QKT structure.The author is a member of EDGE,Research Training Network HPRN-CT-2000-00101,supported by The European Human Potential Programme.
1
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is a Weyl connection,preserving the conformal class of g.Clearly,(g,θ)and(e f,θ?d f)de?ne the same Weyl structure.
On a Weyl manifold(M,c,D),Weyl introduced the distance curvature function,a2-form de?ned by Θ=dθg.By(1.1),the de?nition does not depend on g∈c.IfΘ=0,the cohomology class[θg]∈H1dR(M) is independent on g∈c.A Weyl structure withΘ=0is called closed.
All these geometric objects can be interpreted as sections in tensor bundles of the bundle of scalars of weight1,associated to the bundle of linear frames of M via the representation GL(n,R)?A→| det A|1/n. E.g.c is a section of S2T?M?L2,θis a connection form in L whose curvature form is exactly the distance curvature function etc.This also motivates the terminology.We refer to[18]for a systematic treatment of this viewpoint.
A fundamental result on Weyl structures is the following”co-closedeness lemma”:
Theorem1.1.[17]Let(M,c)be a compact,oriented,conformal manifold of dimension>2.For any Weyl structure D preserving c,there exists a unique(up to homothety)g0∈c such that the associated Higgs?eldθg
is g0-coclosed.
The metric g0provided by the theorem is called the Gauduchon metric of the Weyl structure.
In Weyl geometry,the good notion of Einstein manifold makes use of the Ricci tensor associated to the Weyl connection:
Ric D=
1
n
Scal D g·g?n?2
n dScal D g=
2n
n
dScal D g+2δg?gθ?δg dθ++2?g Tθ+(n?3)d θ 2g=0. Contracting here withθyields
Dθ=
1
WEYL STRUCTURES ON QUATERNIONIC MANIFOLDS 3
This,together with the relation between D and ?g prove the ?rst statement of the following extremely important result (the second statement will be proved in a more particular situation):
Theorem 1.2.[18]Let D be an Einstein-Weyl structure on a compact,oriented manifold (M,c )of dimension >2.Let g be the Gauduchon metric in c associated to D and θthe corresponding Higgs ?eld.If the Weyl structure is closed,but not exact,then
1)θis ?g parallel:?g θ=0(in particular,also g -harmonic).
2)Ric D =0.
Odd dimensional spheres and products of spheres S 1×S 2n +1admit Einstein-Weyl structures (note that S 1×S 2and S 1×S 3can bear no Einstein metric,cf.[22]).Further examples,with Ric D =0,will be the compact quaternion Hermitian Weyl and hyperhermitian Weyl manifolds.
2.Quaternionic Hermitian manifolds
This section is devoted to the introduction of quaternion Hermitian geometry.The standard references are [40],[4],[43],[3].
De?nition 2.1.Let (M,g )be a 4n -dimensional Riemannian manifold.Suppose End (T M )has a rank 3subbundle H with transition functions in SO(3),locally generated by orthogonal almost complex structures I α,α=1,2,3satisfying the quaternionic relations.Precisely:
I 2α=?Id,I αI β=εαβγI γ,g (I α·,I α·)=g (·,·)α,β,γ=1,2,3
(2.1)where εαβγis 1(resp.?1)when (αβγ)is an even (resp.odd)permutation of (123)(such a basis of H is called admissible.)The triple (M,g,H )is called a quaternionic Hermitian manifold whose quaternionic bundle is H .
Any (local)or global section of H is called compatible ,but in general,H has no global section.A striking example is H P n ,the quaternionic projective space.The three canonical almost complex structures of H n +1induced by multiplication with the imaginary quaternionic units descend to only local almost complex structures on H P n generating the bundle H .The metric is the one projected by the ?at one on H n +1,i.e.the Fubini-Study metric written in quaternionic coordinates.Note that H P 1is di?eomorphic with S 4,hence cannot bear any almost complex structure.Consequently,no greater dimensional H P n can have an almost complex structure neither,because this would be induced on any quaternionic projective line H P 1,contradiction.
This shows that the case when H is trivial is of a special importance and motivates
De?nition 2.2.A quaternionic Hermitian manifold with trivial quaternionic bundle is called a hyper-hermitian manifold .
In this terminology,an admissible basis of a quaternion Hermitian manifold is a local almost hyper-hermitian structure.
For a hyperhermitian manifold we shall always ?x a (global)basis of H satisfying the quaternionic relations,so we shall regard it as a manifold endowed with three Hermitian structures (g,I α)related by the identities (2.1)The simplest example is H n .But we shall encounter other many examples.
The analogy with Hermitian geometry suggests imposing conditions of K¨a hler type.Let ?g be the Levi-Civita connection of the metric g .
De?nition 2.3.A quaternionic Hermitian manifold (M,g,H )of dimension at least 8is quaternion
K¨a hler if ?g parallelizes H ,i.e.?g I α=a βα?I β(with sqew-symmetric matrix of one forms (a βα)).
A hyperhermitian manifold is called hyperk¨a hler if ?g I α=0for α=1,2,3.
Remark 2.1.This de?nition of quaternion K¨a hler manifold is redundant in dimension 4.As S.Marchi-afava proved (see [29])that any four-dimensional isometric submanifold of a quaternion K¨a hler manifold
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whose tangent bundle is invariant to each element of H is Einstein and self-dual,one takes this as a de?nition.We won’t be concerned with dimension4in this report.
Note that,unless in the complex case,here the parallelism of H does not imply the integrability of the single almost complex structures.
Example2.1.H n with its?at metric is hyperk¨a hler.By a result of A.Beauville,the K3surfaces also, see[4],Chapter14.The irreducible,symmetric quaternion K¨a hler were classi?ed by J.Wolf.Apart H P n,the compact ones are:the Grassmannian of oriented4-planes in R m,the Grassmannian of complex 2-planes of C m and?ve other exceptional spaces(see[4],loc.cit.)
From the holonomy viewpoint,equivalent de?nitions are obtained as follows:A Riemannian manifold is quaternion K¨a hler(resp.hyperk¨a hler)i?its holonomy is contained in Sp(n)·Sp(1)=Sp(n)×Sp(1)/Z2 (resp.Sp(n)).
On a quaternion quaternionic Hermitian manifold,the usual K¨a hler forms are only local:on any trivializing open set U,one has the2-formsωα(·,·)=g(I a·,·).But the4-formω= 3α=1ωα∧ωαis global (because the transition functions of H are in SO(3)),nondegenerate and,if the manifold is quaternion K¨a hler,parallel.Hence,it gives a nontrivial4-cohomology class,precisely[ω]=8π2p1(H)∈H4(M,R) ([27]).
To get a converse,let H be the algebraic ideal generated by H inΛ2T?M(by identifying,as usual, a local almost complex structure with the associated K¨a hler2-form).It is a di?erential ideal if for any admissible basis of H,one has dωα= 3β=1ηαβ∧ωβfor some local1-formsηαβ.Then we have: Theorem2.1.[43]A quaternion Hermitian manifold of dimension at least12with closed4-formωis quaternion K¨a hler.
A quaternion Hermitian manifold of dimension8is quaternion K¨a hler i?ωis closed and H is a di?erential ideal.
Swann’s proof uses representation theory.A more direct one can be found in[2].
Remark2.2.It is important to note that the condition of being a di?erential ideal is conformally invariant and,moreover,invariant to di?erent choices of admissible basis.
For an almost almost quaternionic Hermitian manifold(M,g,H),we de?ne its structure tensor by
1
T H=
trace{g?1?H X g}.
(8(n+1)
A direct(but lengthy)computation proves:
Lemma2.1.[2]Let(M,g,H)be a quaternion Hermitian manifold such that H is a di?erential ideal. For any admissible basis of H,the following formulae for T H and?H hold good:
1
T H X Y=
WEYL STRUCTURES ON QUATERNIONIC MANIFOLDS5 (?H Z g)(X,Y)=
1
n+2
Ric g(IγX,Y),dim M=4n.
From these one gets:
g(R g(X,I1X)Z,I1Z)+g(R g(X,I1X)I2Z,I3Z)+
+g(R g(I2X,I3X)Z,I1Z)+g(R g(I2X,I3X)I2Z,I3Z)=
=4
n+4
Ric g(Z,Z) X 2
for any X and Z,hence Ric g(X,X)=λg(X,X)and(M,g)is Einstein.
On the other hand,hyperk¨a hler manifolds have holonomy included in Sp(n)?SU(2n),hence they are Ricci-?at,in particular Einstein(see[4]).
Although apparently hyperk¨a hler manifolds form a subclass of quaternion K¨a hler ones,this is not quite true.Besides the holonomy argument,the following result motivates the dichotomy:
Theorem2.3.[5]A quaternion K¨a hler manifold is Ricci-?at i?its reduced holonomy group is contained in Sp(n).And if it is not Ricci-?at,then it is de Rham irreducible.
From these results it is clear that when discussing quaternion K¨a hler manifolds,one is mainly interested in the non-zero scalar curvature.
Ricci?at quaternion K¨a hler manifolds are called locally hyperk¨a hler.Similarly,P.Piccinni discussed in [37],[38]the class of locally quaternion K¨a hler manifolds,having the reduced holonomy group contained in Sp(n)·Sp(1)and proved:
Proposition2.1.[37]Any complete locally quaternion K¨a hler manifold with positive scalar curvature is compact,locally symmetric and admits a?nite covering by a quaternion K¨a hler Wolf symmetric space.
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As the local sections of H are generally non-integrable,one cannot use the methods of complex geometry directly on quaternion K¨a hler manifolds.However,one can construct an associated bundle whose total space is Hermitian.Let p:Z(M)→M be the unit sphere subbundle of H.Its?bre Z(M)m is the set of all almost complex structures on T m M.This is called the twistor bundle of https://www.docsj.com/doc/d216896406.html,ing the Levi-Civita connection?g,one splits the tangent bundle of Z(M)in horizontal and vertical parts.Then an almost complex structure J can be de?ned on Z(M)as follows:each z∈Z(M)represents a complex structure on T p(z)M;as the horizontal subspace in z is naturally identi?ed with T p(z)M,the action of J on horizontal vectors will be the tautological one,coinciding with the action of z.The vertical subspace in z is isomorphic with the tangent space of the?bre S2.Hence we let J act on vertical vectors as the canonical complex structure of S2.Happily,J is integrable.Moreover:
Theorem2.4.[40]Let(M,g,H)be a quaternion K¨a hler manifold with positive scalar curvature.Then (Z(M),J)admits a K¨a hler-Einstein metric with positive Ricci curvature with respect to which p becomes
a Riemannian submersion.
3.Local and global3-Sasakian manifolds
We now describe the odd dimensional analogue,within the frame of contact geometry,of hyperk¨a hler manifolds,as well as a local version of it.We send the interested reader to the excellent recent survey [9],where also a rather exhaustive list of references is given and to[33]for the local version.
De?nition3.1.A4n+3dimensional Riemannian manifold(N,h)such that the c?o ne metric dr2+r2h on R+×N is hyperk¨a hler is called a3-Sasakian manifold.
This is equivalent to the existence of three mutually orthogonal unit Killing vector?eldsξ1,ξ2,ξ3, each one de?ning a Sasakian structure(i.e.:?α:=?hξαsatis?es the di?erential equation?h?α= Id?ξ?α?h?ξα)and related by:
[ξ1,ξ2]=2ξ3,[ξ2,ξ3]=2ξ1,[ξ3,ξ1]=2ξ2.
3-Sasakian manifolds are necessarily Einstein([24])with positive scalar curvature and their Einstein constant is4n+2.
Starting with a3-Sasakian manifold N,one has to consider the foliation generated by the three structure vector?eldsξα.It is easy to compute the curvature of the leaves:it is precisely one.Hence,the leaves are spherical space forms.If the foliation is quasi-regular(it is enough to have compact leaves), then the quotient space is a quaternion K¨a hler orbifold M of positive sectional curvature(see[10]for a thorough discussion about the geometry and topology of orbifolds and their applications in contact geometry).As all the geometric constructions we are interested in can be carried out in the category of orbifolds,one considers now the twistor space Z(M).The triangle is closed by observing that,?xing one of the contact structures of N,one has an S1-bundle N→Z(M)whose Chern class is,up to torsion,the one of an induced Hopf bundle(this is a particular case of a Boothby-Wang?bration,cf.[6]).Moreover, all three orbifold?brations involved in this commutative triangle are Riemannian submersions.
Conversely,given a positive quaternion K¨a hler orbifold(M,g,H),one constructs its K¨a hler-Einstein twistor space(it will be an orbifold)and an SO(3)-principal bundle over M.The total space N will then be a3-Sasakian orbifold which,as above,?bers in S1over Z(M)closing the diagram.One of the deepest results in this theory was the determination of conditions under which N is indeed a manifold(cf.[13]).
A local version of3-Sasakian structure will be also useful in the sequel:
De?nition3.2.[33]A Riemannian manifold(N,h)is said to be a locally3-Sasakian manifold if a rank 3vector subbundle K?T N is given,locally spanned by an orthonormal tripleξ1,ξ2,ξ3of Killing vector ?elds satisfying:
(i)[ξα,ξβ]=2ξγfor(α,β,γ)=(1,2,3)and circular permutations.
(ii)Any two such triplesξ1,ξ2,ξ3andξ′1,ξ′2,ξ′3are related on the intersections U∩U′of their de?nition open sets by matrices of functions with values in SO(3).
WEYL STRUCTURES ON QUATERNIONIC MANIFOLDS 7
(iii)If ?α=?h ξα,(α=1,2,3),then (?h Y ?α)Z =ξ?α(Z )Y ?h (Y,Z )ξα,for any local vector ?elds
Y,Z .
Clearly,if K can be globally trivialized with Killing vector ?elds as above,(N,h )is 3-Sasakian.It is easily seen that locally 3-Sasakian manifolds share the local properties with the (global)3-Sasakian spaces:they are Einstein with positive scalar curvature;hence,by Myers’theorem we have
Proposition https://www.docsj.com/doc/d216896406.html,plete locally and globally 3-Sasakian manifolds are compact.
But a speci?c property of the local case is:
Proposition 3.2.[33]The bundle K of a locally 3-Sasakian manifold is ?at.
Proof.Let (ξ1,ξ2,ξ3),(ξ′1,ξ′2,ξ′3)be two local orthonormal triples of Killing ?elds trivializing K on U ,U ′.Then,on U ∩U ′=?we have ξ′λ=f σλξσ.We shall show that f σλare https://www.docsj.com/doc/d216896406.html,pute ?rst the bracket
2ξ′ν=[ξ′λ,ξ′μ]={f ρλξρ(f σμ)?f ρμξρ(f σλ)}ξσ+f ρλf σμ[ξρ,ξσ].
From (f μλ)∈SO(3)and [ξρ,ξσ]=2ξτ((ρ,σ,τ)=(1,2,3)and cyclic permutations),we can derive:
f ρλf σμ[ξρ,ξσ]=2{f ρλf σμ?f σλf ρμ}ξτ=2ξ′ν.
Hence
f ρλξρ(f σμ)?f ρμξρ(f σλ)=0.Thus,for any λ,μ,σ=1,2,3:ξ′λ(f μσ)?ξ′μ(f σλ)=0.It follows:
ξλ(f μσ)?ξμ(f λσ)=0.
(3.1)Now we use the Killing condition applied to ξ′λ=f μλξμ:
Y (f μλ)h (ξμ,Z )+Z (f μλ)h (ξm u,Y )=0,Y,Z ∈X (M )
which yields,on one hand Z (f μλ)=0for any Z ⊥span {ξ1,ξ2,ξ3}and,on the other hand ξρ(f σλ)+ξσ(f ρλ)=0.
(3.2)This and (3.1)imply ξσ(f ρλ)=0and the proof is complete.
The vector bundle K generates a 3-dimensional foliation that,for simplicity,we equally denote K .It can be shown that K is Riemannian.As in the global case,if the leaves of K are compact,the leaf space M =N/K is a compact orbifold.The metric h projects to a metric g on P making the natural projection πa Riemannian submersion with totally geodesic ?bers.The locally de?ned endomorphisms ?λcan be projected on M producing locally de?ned almost complex structures:J αX π(x )=π?(?α(?X
x )),where ?X is the horizontal lift of X w.r.t.the submersion.As ?α??β=??γ+ξα?ξ?β,P can be covered with open
sets endowed with local almost hyperhermitian structures {J α}.As the transition functions of K are in SO(3),so are the transition functions of the bundle F locally spanned by the ?α.Hence,two di?erent almost hyperhermitian structures are related on their common domain by transition functions in SO(3).This means that the bundle H they generate is https://www.docsj.com/doc/d216896406.html,ing the O’Neill formulae,it is now seen,as in the global case,that (M,g,H )is a quaternion K¨a hler orbifold.Summing up we can state:
Proposition 3.3.[33]Let (N,h,K )be a locally 3-Sasakian manifold such that K has compact leaves.Then the leaf space M =N/K is a quaternion K¨a hler orbifold with positive scalar curvature and the natural projection π:N →M is a Riemannian,totally geodesic submersion which ?bers are (generally inhomogeneous)3-dimensional spherical space forms.
Remark 3.1.P.Piccinni proved in [37]that some global 3-Sasakian manifolds also project over local quaternion K¨a hler manifolds with positive scalar curvature.
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A further study of the(supposed compact)leaves of K will show a very speci?c property of locally 3-Sasakian manifolds.To this end,we recall,following[41],some aspects of the classi?cation of3-dimensional spherical space forms S3/G,with G a?nite group of isometries of S3,hence a?nite subgroup of SO(4).The?nite subgroups of S3are known:they are cyclic groups of any order or binary dihedral, tetrahedral,octahedral,icosahedral and,of course,the identity.In all these cases,S3/G is a homogeneous 3-dimensional space form carrying an induced(global)3-Sasakian structure,see[11].The other?nite subgroups of SO(4),not contained in but acting freely on S3,are characterized by being conjugated in SO(4)to a subgroup ofΓ1=U(1)·Sp(1)orΓ2=Sp(1)·U(1).Observe that the right(resp.left) isomorphism between H and C2induces an isomorphism betweenΓ1(resp.Γ2)and U(2).Hence,any ?nite subgroupΓofΓ1orΓ2will preserve two structures of S3:the locally3-Sasakian structure induced by the hyperhermitian structure of C2and a global Sasakian structure induced by some complex Hermitian structure of C2belonging to the given hyperhermitian one.Moreover,alteringΓby conjugation in SO(4)does not a?ect the above preserved structures;only the global Sasakian structure will come from a hermitian structure of R4conjugate with the standard one.Altogether,we obtain:
Proposition3.4.[33]On any locally3-Sasakian manifold,the compact leaves of K are locally3-Sasakian 3-dimensional space-forms carrying a global almost Sasakian structure.
We end with another consequence of Proposition3.4:
Corollary3.1.[33]Let?K→?N be the pull-back of the bundle K→N to the universal Riemannian covering space of a locally3-Sasakian manifold.Then?K is globally trivialized by a global3-Sasakian structure on?N.
Proof.By Proposition3.2,the bundle?K→?N is trivial.However,this is not enough to deduce that the trivialization can be realized with Killing?elds generating a su(2)algebra.E.g.the inhomogeneous 3-dimensional spherical space forms are parallelizable but locally,not globally3-Sasakian.To overcome this di?culty,start with the induced locally3-Sasakian structure of?N.Let X1be the global Sasakian structure of?N provided by Proposition3.4and consider an open set?U on which?K is trivialized by a local3-Sasakian structure incuding X1.
The manifold?N is simply connected and Einstein,hence analytic(see[4],Theorem5.26).By a result of Nomizu(cf.[31])each local Killing vector?eld on?N can be extended uniquely to the whole?N. We thus extend the above three local Killing?elds.Clearly,the extension Y1of X1coincides with X1. The extension Y2of X2is thus orthogonal to Y1and belongs to?K in every point of?N.It follows from Proposition3.2that Y2is a global Sasakian structure.Now Y3=1
WEYL STRUCTURES ON QUATERNIONIC MANIFOLDS9 2’)(M,g,H)is hyperhermitian for any g∈c;
3’)DI=0for any section of H.
The above de?nition is clearly inspired by the complex case,where the theory of Hermitian-Weyl (locally conformal K¨a hlerian in other terminology)is widely studied(see[15]for a recent survey).Indeed, the following equivalent de?nition is available:
Proposition4.1.[34](M4n,c,H,D)is quaternion-Hermitian-Weyl
(resp.hyperhermitian Weyl)if and only if(M,g,H)is locally conformally quaternion K¨a hler(resp. locally conformally hyperk¨a hler)(i.e.g|
U i
=e f i g′i,where the g′i are quaternion K¨a hler(resp.hyperk¨a hler) over open neighbourhoods{U i}covering M)for each g∈c.
Proof.Let(M4n,c,H,D)be quaternion-Hermitian Weyl.Fix a metric g∈c and choose an open set U on which H is trivialized by an admissible basis I1,I2,I3.Then Dg=θg?g together with condition2) of the de?nition imply Dωα=θ?ωα+aβα?ωβ,hence
dωα=θ∧ωα+aβα∧ωβ.
(4.1)
This implies that H is a di?erential ideal and,on the other hand,the derivative of the fundamental four-form is dω=θg∧ω.Di?erentiating here we get0=d2ω=dθg∧ω.Asωis nondegenerate,this means dθg=0.Consequently,locally,on some open sets U i,θg=d f i for some di?erentiable functions de?ned on U i.It is now easy to see that for each g′i=e?f i g|
U i
,the associated4-form is closed,hence, taking into account Proposition2.1and Remark2.2,the local metrics g′i are quaternion K¨a hler.
Conversely,starting with the local quaternion K¨a hler metrics g′i=e?f i g|
U i ,de?ne(θg)|
U i
=d f i.It
can be seen that these local one forms glue together to a global,closed one-form and dω=θg∧ω.Then construct the Weyl connection associated to g andθg:
D=?g?1
q i)?1 i dq i?d
10LIVIU ORNEA
Before going over,let us note the following result:
Proposition4.2.[33]A quaternion Hermitian manifold(M,g,H)admits a unique quaternion Hermi-tian Weyl structure.
Proof.We have to prove that there exists a unique torsion free connection preserving both H and[g]. Indeed,if D1,D2are such,letθ1,θ2be the associated Lee forms.Then the fundamental4-formωsatis?es
(4.3)
dω=θ1∧ω=θ2∧ω.
Using the operator L:Λ1T?M→Λ5T?M,Lα=α∧ω,(4.3)yields L(θ1?θ2)=0.But L is injective, because it is related to its formal adjointΛbyΛL=(n?1)Id.Henceθ1=θ2.Finally,formula(4.2) proves that D1=D2.
Remark4.1.[37]For hyperhermitian Weyl manifolds,this uniqueness property is implied by the char-acterization of the Obata connection as the unique torsion-free hypercomplex connection.It must then coincide with our Weyl connection D.In general,the set of torsion-free quaternionic connections has an a?ne structure modelled on the space of1-forms.However,only one torsion-free connection can preserve a given conformal class of hyperhermitian metrics.This follows from the fact that the exterior multiplication with the fundamental four-form of the metric maps injectivelyΛ1(T?M)intoΛ5(T?M).
is in fact the Levi Civita connection of the local quaternion K¨a hler Note that the connection D|U
i
metric g′i.As quaternion-K¨a hler manifolds are Einstein,we obtain the following fundamental result: Proposition4.3.[34]Quaternion Hermitian Weyl manifolds are Einstein Weyl.
Hence,as dθ=0,i.e.the Weyl structure(M,c,D)is closed and not exact,because the D is the Levi-Civita connection of local metrics(the Weyl structure is only locally exact),the quoted Theorem 1.2of P.Gauduchon implies:
Proposition4.4.[34]On any compact quaternion-Weyl(hyperhermitian Weyl)manifold which is not globally conformal quaternion K¨a hler(hyperk¨a hler)there exists a representative g∈c(the Gauduchon metric)such that the associated Lee formθg be?g-parallel.
In the sequel,the parallel Lee form of the Gauduchon metric will always be supposed of unit length. Corollary4.3.Let g be the above metric with parallel Lee form on a compact hyperhermitian Weyl manifold and{Iα}an adapted hyperhermitian structure.Then(g,Iα)are Vaisman structures on M(cf.
[15]).
Proposition4.5.[32]On a compact quaternion Weyl manifold which is not globally conformal quater-nion K¨a hler,the local quaternion K¨a hler metrics g′i are Ricci-?at.
Proof.This result follows directly from Theorem1.2,2),but we prefer to give here a direct proof,adapted to our situation.
On each U i,the relation between the scalar curvatures Scal′i and Scal of g′i and g is(cf.[4],p.59):
Scal′i=e?f i Scal|U i?(4n?1)(2n?1)
2 .
As bothθand Scal are global objects on M,it follows thatθis exact,contradiction.But if Scal′i=0on
=(4n?1)(2n?1)
some U i,then Scal=Scal|U
i
WEYL STRUCTURES ON QUATERNIONIC MANIFOLDS 11
Remark 4.2.The above result says that quaternion-Hermitian Weyl manifolds are locally conformally locally hyperk¨a hler.In particular,the open subsets U i can always be taken simply connected and endowed with admissible basis made of by integrable,parallel almost complex structures.But this does not mean that M would be a locally conformal K¨a hler manifold,because a global K¨a hler structure might not exist.
Another characterization,using the di?erential ideal H is the following (recall that the di?erential ideal condition is conformally invariant,so one can speak about the di?erential ideal of a conformal manifold):Theorem 4.1.[1]A quaternionic conformal manifold (M,c,H )of dimension at least 12is quaternion-Hermitian Weyl if and only if H is a di?erential ideal.
The following result is essential in the author’s proof,also motivating the restriction on the dimension:Lemma 4.1.[1]Let (M,g,H )be an almost hyperhermitian manifold with dim M ≥12.Suppose 3
α=1φα∧ωα=0for some 2-forms φα.Then there exists the sqew-symmetric matrix of real func-tions f αρsuch that φα= ρ=αf αρωρ.
Proof.Let F αbe the 1?1tensor ?elds metrically equivalent with the 2-forms φα.The identity in the statement can be rewritten as:
3 ρ=1
{?φρ(X,Y )I ρZ +φρ(X,Z )I ρY +ωρ(Y,Z )F ρX ??φρ(Y,Z )I ρX +ωρ(Z,X )F ρY +ωρ(X,Y )F ρZ }=0.
(4.4)Let now X be unitary,?xed.In the orthogonal complement of H X ={X,I 1X,I 2X,I 3X }we choose a unitary Z and let Y =I αZ .With these choices,the above identity reads:
F α(X )=3 ρ=1{φρ(X,I ρZ )I ρZ ?φρ(X,Z )I ρI αZ }+
3 ρ=1φρ(I ρZ,Z )I ρX.Here we use the assumption n ≥3to obtain:F α(X )=
3 ρ=1φρ(I ρZ,Z )I ρX,hence φαhave the form φα= ρ=αf αρωρwhich,introduced in the equation (4.4),gives:
3 α=1f αα{?ωα(X,Y )I αZ +ωα(X,Z )I αY ?ωα(Y,Z )I αX }++
ρ=α(f αρ+f ρα){ωα(X,Y )I ρZ +ωα(X,Z )I ρY +ωα(Y,Z )I ρX }=0.
Again using n ≥3,we may choose Y and Z orthogonal to H X and get:?f ααωα(Y,Z )? ρ=α
(f αρ+f ρα)ωα(Y,Z )=0.
Now it remains to take Z =I αY to derive the sqew-symmetry of (f αρ).
Proof.(of Theorem 4.1).Fix g ∈c and an admissible basis for H .Starting from equations (2.2),(2.3)and dωα= 3β=1ηαβ∧ωβ,one can derive the following formula:
dωα=ηγ∧ωβ?ωβ∧ωγ+1
3dη∧ωα+(dηγ+ηα∧ηβ)∧ωβ?(dηβ+ηγ∧ηα)∧ωγ=0.
12LIVIU ORNEA
The previous Lemma applies and provides:
1
dσ∧ωα,
3
an equation similar to(4.1).The rest and the converse are obvious.
Remark4.3.It is still unknown if this result is true in dimension8too.
Remark4.4.For quaternion Hermitian manifolds,various adapted canonical connections were introduced by V.Oproiu,M.Obata and others.A uni?ed treatement can be found in some recent papers of D. Alekseevski,E.Bonan,S.Marchiafava(see e.g.[3]and the references therein).In particular,in[30], one?nds a characterization of hyperhermitian Weyl manifolds in terms of canonical connections and structure tensors of the subordinated quaternionic Hermitian structure.
We end this section with a characterizations of quaternion K¨a hler manifolds among(non compact)qua-ternion Hermitian Weyl manifolds by means of submanifolds(compare with[44]for the complex case): Proposition4.6.[32]A quaternion Hermitian Weyl manifold(M,g,H)of dimension at least8is quaternion K¨a hler if and only if through each point of it passes a totally geodesic submanifold of real dimension4h≥8which is quaternion K¨a hler with respect to the structure induced by(g,H).
Proof.On a given submanifold of M,locally one can induce the metric g and the quaternion K¨a hler one g′i.Correspondingly,there are two second fundamental forms b and b′i.As g and g′i are conformally related on U i,the relation between b and b′i is
1
b′i=b+
WEYL STRUCTURES ON QUATERNIONIC MANIFOLDS 13
where T (X,Y,Z )=g (T or ?(X,Y ),Z )and T or ?(X,Y )=?X Y ??Y X ?[X,Y ].The holonomy of such a connection is contained in Sp(n )·Sp(1).These structures appear naturally on the target space of (4,0)supersymmetric two-dimensional sigma model with Wess-Zumino term and seem to be of growing interest for physicists.Let us introduce the 1-forms:
t α(X )=?1
2n +1 αt α∧ωαand dt =0.
4.2.The canonical foliations.From now on (M,c,H,D )will be compact,non globally conformal quaternion K¨a hler.According to Proposition 4.4,we let g ∈c be the Gauduchon metric whose Lee form θ:=θg is parallel w.r.t.the Levi-Civita connection ?:=?g .Hence we look at the quaternion Hermitian manifold (M,g,H ).We also suppose θ=0meaning that M is not quaternion K¨a hler,see Corollary 4.1.We recall that,being parallel,we can suppose θnormalised,i.e.|θ|=1.We denote T :=θ?and let T α=I αT and θα=θ?I α.
The following proposition gathers the computational formulae we need:
Proposition 4.8.[32]Let (M,g,H )be a compact quaternion Hermitian Weyl manifold and {I 1,I 2,I 2}a local admissible basis of H with I αintegrable and parallel (as in remark 4.2).The following formulae hold good:
L T I α=0,
L T g =0,L T ωα=0,L T ω=0
(4.6)?I α=12{θ?θα?θα?θ?ωα}(4.10)
dθα=?ωα+θ∧θα(4.11)
L T αωα=0,L T αωβ=ωβ,L T αω=0
(4.12)
where L is the operator of Lie derivative.The proof is by direct computation and mimics the corresponding one for Vaisman manifolds,see [15].In particular,from (4.6)and (4.8),we obtain according to [36]:
Corollary 4.4.The vector ?elds T and T αare in?nitesimal automorphisms of the quaternion Hermitian structure.
There are two interesting foliations on any compact quaternion Hermitian Weyl manifold:
?the (4n ?1)-dimensional F ,spanned by the kernel of θand ?the 4-dimensional D ,locally generated by T,T 1,T 2,T 3.
Here are their properties:
Proposition 4.9.[33],On a compact quaternion Hermitian Weyl manifold,F is a Riemannian,totally geodesic foliation.Its leaves have an induced locally 3-Sasakian structure.
14LIVIU ORNEA
Proof.The?rst statement is a consequence of(4.6).As for the second one,the bundle K is locally generated by the(rescaled to be unitary)local vector?elds Tα.Indeed,they are Killing by the last equation of(4.8);the?rst condition of de?nition3.2is given by(4.9);the transition functions of K are in SO(3)because the transition functions of H are so;?nally,condition3)of the de?nition is implied by (4.10).
Corollary4.5.[32]On a compact hyperhermitian Weyl manifold,F is a Riemannian,totally geodesic foliation whose leaves have an induced(global)3-Sasakian structure.
Proposition4.10.[32]On a compact quaternion Hermitian Weyl manifold,the foliation D is Riemann-ian,totally geodesic.Its leaves are conformally?at4-manifolds(H?{0})/G,with G a discrete subgroup of GL(1,H)·Sp(1)inducing an integrable(in the sense of G-structures)quaternionic structure. Proof.Let X be a leaf of D and let the superscript′refer to restrictions of objects from M to X.A local orthonormal basis of tangent vectors for X is provided by{T′,T′1,T′2,T′3}.As X is totally geodesic,?′θ′=0and a direct computation of the curvature tensor of the Weyl connection R D on this basis proves R D=0on X.Hence X is conformally?at and the curvature tensor of the Levi-Civita connection is
R′(U,Y)Z=θ′(U)θ′(Z)Y?θ′(Y)θ′(Z)U?θ′(U)g′(Y,Z)T′+
(4.13)
+θ′(Y)g′(U,Z)T′+g′(Y,Z)U?g′(U,Z)Y.
It follows that the Ricci tensor Ric′=g′?θ′?θ′is g′-parallel and,on the other hand,the sectional curvature is non-negative and strictly positive on any plane of the form{T′α,T′β}.Now recall that the universal Riemannian covering spaces of conformally?at Riemannian manifolds with parallel Ricci tensor were classi?ed in[28].By the above discussion and the reducibility of X(due to?′T′=0),the only class ?tting from Lafontaine’s classi?cation is that with universal cover R4?{0}equipped with the conformally ?at metric written in quaternionic coordinate(h h.We still have to determine the allowed deck groups.
Happily,Riemannian manifolds with such universal cover were studied in[18]and,in arbitrary dimen-sion,in[47].Here it is proved that equation(4.13)forces the deck group of the covering to contain only conformal transformations of the form(in real coordinates)?x i=ρa i j x j whereρ>0and(a i j)∈SO(4). This leads to the following form of G:
(4.14)
G={ht k0;h∈G0,k∈Z}
where t0is a conformal transformation of maximal module0<ρ<1and G′is one of the?nite subgroups of U(2)listed in[25].Finally,as CO+(4)?GL(1,H)·Sp(1),X has an induced integrable quaternionic structure.
Corollary4.6.[32]On a compact hyperhermitian Weyl manifold,the foliation D is Riemannian,to-tally geodesic.Its leaves,if compact,are complex Hopf surfaces(non-primary,in general)admitting an integrable hypercomplex structure.
Proof.Only the second statement has to be proved.It is clear that the leaves inherit a hyperhermitian Weyl,non hyperk¨a hler(becauseθ=0)structure.The compact hyperhermitian surfaces are classi?ed in [8]and the only class having the stated property is that of Hopf surfaces.
As above,here integrable hypercomplex structure is intended in the sense of G-structures,i.e.of the existence of a local quaternionic coordinate such that the di?erential of the change of coordinate belongs to H?.For further use we recall the following:
Theorem4.2.(cf.[26])A complex Hopf surface S admits an integrable hypercomplex structure if and only if S=(H?{0})/Γwhere the discrete groupΓis conjugate in GL(2,C)to any of the following subgroups G?H??GL(2,C):
WEYL STRUCTURES ON QUATERNIONIC MANIFOLDS15 (i)G=Z m×Γc with Z m andΓc both cyclic generated by left multiplication by a m=e2πi/m,m≥1, and c∈C?.
(ii)G=L×Γc,where c∈R?and L is one of the following:D4m,the dihedral group,m≥2,generated
√
by the quaternion j andρm=eπi/m;T24,the tetrahedral group generated byζ2and1/
2(ζ3+ζ3j);I120,the icosahedral group generated √
by?3,j,1/
16LIVIU ORNEA
Corollary 4.7.[32]The class of compact hyperhermitian Weyl manifolds,not hyperk¨a hler and having a
quasi-regular (resp.regular)T coincide with the class of ?at principal S 1-bundles over compact 3-Sasakian
orbifolds (resp.manifolds).
4.3.2.The link with quaternion K¨a hler geometry.We now describe the leaf space of the foliation D ,when it exists.
Theorem 4.4.[32],[34]Let (M,g,H )be a compact quaternion Hermitian Weyl (resp.hyperhermitian Weyl)manifold,non quaternion K¨a hler (resp.non hyperk¨a hler)whose foliation D has compact leaves.Then the leaves space P =M/D is a compact quaternion K¨a hler orbifold with positive scalar curvature,the projection is a Riemannian,totally geodesic submersion and a ?bre bundle map with ?bres as described in Proposition 4.10(resp.4.6).
Proof.In the local case of quaternion Hermitian Weyl M ,we have to explain how to project the struc-ture of M over P .The key point is that locally,H has admissible basis formed by ?-parallel (hence integrable)complex structures.Then formulae (4.6),(4.8)show that H is projectable.The foliation being Riemannian,g is also projectable.The compatibility of the projected quaternion bundle with the projected metric is clear.To show that the projected structure is quaternion K¨a hler,let ωP be the 4-form of the projected structure.As the projection is a totally geodesic Riemannian submersion,ωP coincides with the restriction of ωto basic vector ?elds on M .Hence,it is enough to show that ?ω=0on basic vector ?elds.But ?ω= α?ωα∧ωα+ωα∧?ωαand the result follows from equation (4.7).The scalar curvature of (P,g )is easily computed using O’Neill formulae.
The global case of a hyperhermitian Weyl M now follows.
Remark 4.5.The above ?bration can never be trivial,according to Proposition 4.11.
Let now M be hyperhermitian Weyl,T be the foliation generated by the vector ?eld T and V the 2-dimensional foliation generated by T and JT ,where J is a ?xed compatible global complex structure belonging to H .Theorem 4.4,together with the structure of 3-Sasakian manifolds described in section 3,furnish the following structure theorem:
Theorem 4.5.[32],[33]Let (M,g,H )be a compact hyperhermitian Weyl manifold,non hyperk¨a hler,such that the foliations D ,V ,T and K have compact leaves.There exists the following commutative diagram of ?bre bundles and Riemannian submersions in the category of orbifolds:
P
Z
e e e e e e e e e e e
S 2N c d d d d d s S 1T 1C Here N is globally 3-Sasakian.The ?bres of M →P are Kato’s integrable hypercomplex Hopf surfaces (S 1×S 3)/G ,non necessarily primary and non necessarily all homeomorphic if M is hyperhermitian Weyl.The S 1-bundle P →Z is a Boothby-Wang ?bration.
WEYL STRUCTURES ON QUATERNIONIC MANIFOLDS 17
Note that all arrows appearing in the diagram are canonical,except for M →Z ,which depends on the choice of the compatible global complex structure on M .However,di?erent choices of this complex structure produce analytically equivalent complex manifolds Z .
Remark 4.6.The diagram 4.5holds also if dim (M )=8.In this case P is still Einstein by the above discussion.The integrability of the complex structure on its twistor space implies it is also self-dual (cf.
[4]).Then just recall that a 4-dimensional N is usually de?ned to be quaternionic K¨a hler if it is Einstein and self-dual.
Remark 4.7.For the hyperhermitian Weyl manifold M =S 1×S 4n ?1,diagram 4.5becomes the well-known:
H P n ?1
C P n ?1
e e e e e e e e e e e
S 2S 4n ?1c d d d d d s S 1T 1C which was the model for the general one.Also,examples of quaternion Hermitian Weyl manifolds will be obtained by considering appropriate quotients of the manifolds in the vertices of this diagram.
Remark 4.8.It is proved in [11]that in every dimension 4k ?5,k ≥3there are in?nitely many distinct homotopy types of complete inhomogeneous 3-Sasakian manifolds.Thus,by simply making the product with S 1,we obtain in?nitely many non-homotopically equivalent examples of compact hyperhermitian Weyl manifolds.
4.3.3.Some topological consequences of diagram 4.9.A ?rst consequence of the diagram 4.5concerns cohomology.Note ?rst that the property ?θ=0implies the vanishing of the Euler characteristic of M .Then,applying twice the Gysin sequence in the upper triangle one ?nds the relations between the Betti numbers of M and Z :
b i (M )=b i (Z )+b i ?1(Z )?b i ?2(Z )?b i ?3(Z )(0≤i ≤2n ?1),
b 2n (M )=2[b 2n ?1(Z )?b 2n ?3(Z )].
On the other hand,since P has positive scalar curvature,both P and its twistor space Z have zero odd Betti numbers,cf.[4].The Gysin sequence of the ?bration Z →P then yields:
b 2p (Z )=b 2p (P )+b 2p ?2(P )
Together with the previous found relations this implies:
Theorem 4.6.[32],[33]Let M be a compact hyperhermitian Weyl manifold satisfying the assumptions of Theorem 4.5.Then the following relations hold good:
b 2p (M )=b 2p +1(M )=b 2p (P )?b 2p ?4(P )(0≤2p ≤2n ?2),
b 2n (M )=0,
n ?1 k =1k (n ?k +1)(n ?2k +1)b 2k (M )=0.
18LIVIU ORNEA
(Poincar′e duality gives the correspondent of the?rst two equalities for2n+2≤2p≤4n).In particular b1(M)=1.Moreover,if n is even,M cannot carry any quaternion K¨a hler metric.
The last identity is obtained,by applying S.Salamon’s constraints on compact positive quaternion K¨a hler manifolds to the same diagram(cf[19]).
Remark4.9.We obtain in particular b2p?4(P)≤b2p(P)for0≤2p≤2n?2.Since any compact quaternion K¨a hler P with positive scalar curvature can be realized as the quaternion K¨a hler base of a compact quaternion Hermitian Weyl manifold M,this implies,in the positive scalar curvature case,the Kraines-Bonan inequalities for Betti numbers of compact quaternion K¨a hler manifolds(cf.[4]).
b1(M)=1is a much stronger restriction on the topology of compact quaternion Hermitian Weyl manifolds in the larger class of compact complex Vaisman(generalized Hopf)manifolds.For the latter, the only restriction is b1odd and the induced Hopf bundles over compact Riemann surfaces of genus g provide examples of Vaisman(generalized Hopf)manifolds with b1=2g+1for any g,cf.[45].
The properties b1=1and b2n=0have the following consequences:
Corollary4.8.Let(M,I1,I2,I3)be a compact hypercomplex manifold that admits a locally and non globally conformal hyperK¨a hler metric.Then none of the compatible complex structures J=a1I1+ a2I2+a3I3,a21+a22+a23=1,can support a K¨a hler metric.In particular,(M,I1,I2,I3)does not admit any hyperK¨a hler metric.
Let M be a4n-dimensional C∞manifold that admits a locally and non globally conformal hyperK¨a hler structure(I1,I2,I3,g).Then,for n even,M cannot admit any quaternion K¨a hler structure and,for n odd,any quaternion K¨a hler structure of positive scalar curvature.
4.3.4.Homogeneous compact hyperhermitian Weyl manifolds.In the complex case,a complete classi?ca-tion of compact homogeneous Vaisman manifolds is still lacking.By contrast,for compact homogeneous hyperhermitian Weyl manifolds a precise classi?cation may be obtained.
De?nition4.2.A hyperhermitian Weyl manifold(M,[g],H,D)is homogeneous if there exists a Lie group which acts transitively and e?ectively on the left on M by hypercomplex isometries.
The homogeneity implies the regularity of the canonical foliations:
Theorem4.7.[32]On a compact homogeneous hyperhermitian Weyl manifold the foliations D,V and B are regular and in the diagram4.5,N,Z,P are homogeneous manifolds,compatible with the respective structures.
Proof.Fix J∈H be a compatible complex structure on M.Then(M,g,J)is a homogeneous Vaisman manifold and by Theorem3.2in[46]we have the regularity of both the foliations V J and B.Therefore M projects on homogeneous manifolds Z J and N.In particular the projections of IαB on N are regular Killing vector?elds.Then Lemma11.2in[42]assures that the3-dimensional foliation spanned by the projections of I1B,I2B,I3B is regular.This,in turn,implies that P is a homogeneous manifold,thus D is regular on M.
On the other hand,a compact homogeneous3-Sasakian manifolds have been classi?ed in[11].We use this classi?cation together with Corollary4.7to derive:
Proposition4.12.[32]The class of compact homogeneous hyperhermitian Weyl manifolds coincides with that of?at principal S1-bundles over one of the3-Sasakian homogeneous manifolds:S4n?1,R P4n?1 the?ag manifolds SU(m)/S(U(m?2)×U(1)),m≥3,SO(k)/(SO(k?4)×Sp(1)),k≥7,the exceptional spaces G2/Sp(1),F4/Sp(3),E6/SU(6),E7/Spin(12),E8/E7.
The?at principal S1-bundles over P are characterized by having zero or torsion Chern class c1∈H2(P;Z)and classi?ed by it.The integral cohomology group H2of the3-Sasakian homogeneous mani-folds can be computed by looking at the long homotopy exact sequence
...→π2(K)→π2(G)→π2(G/K)→π1(K)→π1(G)→...
WEYL STRUCTURES ON QUATERNIONIC MANIFOLDS19 for the3-Sasakian homogeneous manifolds G/K listed above.Sinceπ2(G)=0for any compact Lie group G,one obtains the following isomorphisms(cf.[12]):
SU(m)
H2(
h)?1dh?d
20LIVIU ORNEA
The metric on M will now be the projection of( i h i h i and is denoted with g.Moreover, we shall assume the resulting4-dimensional foliation D to have compact leaves.We may state: Proposition4.14.[32]The quaternion Hopf manifold M=(H n?{0})/G endowed with the metric g is a compact quaternion Hermitian Weyl manifold.The leaves of the foliation D are integrable quaternion Hopf4-manifolds.The leaf space P=M/D is a quaternion K¨a hler orbifold quotient of H P n?1whose set of singular points is,generally,R P n?1?H P n?1.Moreover:
If G is one of the groups in Kato’s list(see Theorem4.2),then M is hyperhermitian Weyl,The leaves of D are integrable Hopf surfaces and P is H P n?1.
The result follows from the fact that the group G,being a discrete subgroup of GL(n,H)·Sp(1), preserves the quaternionic structure of the universal covering of M.The structure of the leaves was discussed in Proposition4.10.Note that GL(n,H)acts on the left and Sp(1)acts on the right on the quaternionic coordinates,hence the induced action of G on H P n?1?xes the points which can be represented in real coordinates.If G belongs to Kato’s list,then it is a subgroup of GL(n,H)and preserves the hyperhermitian structure of the covering,inducing the same structure on the leaves. Example4.3.[34],[32]For n=2,let G be the cyclic group generated by(h0,h1)→(2e2πi/3h0,2e4πi/3h1) and M=(H2?{0})/G.Here the leaf space P=M/D is a Z3quotient H P1.The leaves of D are standard Hopf surfaces S1×S3over the regular points of the orbifold P and are non-primary Hopf surfaces(S1×S3)/Z3over the two singular points of homogeneous coordinates[1:0]and[0:1]of P.
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接口测试概念
一：到底什么是接口? 一般来说接口有两种，一种是程序内部的接口，一种是系统对外的接口。 广义来说，客户端与后台服务间的协议；插件间通信的接口；模块间的接口；再小到一个类提供的方法；都可以理解为接口 系统对外的接口 如果我们要从网站或服务器上获取资源或信息，网站肯定不会把数据库共享给你，它只会给你提供一个写好的方法来获取数据，我们通过引用它提供的接口就能获取数据 程序内部的接口 它是方法与方法之间，模块与模块之间的交互，也是程序内部抛出的接口。比如一个web 项目，有登录、新增，修改，删除等等，那么这几个模块会有交互，会抛出一个接口，供内部系统进行调用 二：接口的组成有哪些? 一个完整的接口应该包含以下内容： 1.接口说明 2.调用的url 3.请求方法（get\post） 4.请求参数、参数类型、请求参数说明 5.返回参数说明 三：常见的接口类型
webService接口 它使用soap协议并通过http传输，请求报文和返回报文都是xml格式的，我们在测试的时候通过工具才能进行调用。可以使用的工具有SoapUI、jmeter http-api接口 它使用http协议，通过路径来区分调用的方法，请求报文都是key-value形式的，返回报文一般都是json串，有get和post等方法，这也是最常用的两种请求方式。可以使用的工具有postman、jmeter等 四：前端和后端 前端 咱们使用的网页，打开的网站，都是前端。包括Web页面的结构、Web的外观视觉表现以及Web层面的交互实现； 后端 我们在页面上进行操作的时候，这些业务逻辑、功能，比如说新增，修改，删除这些功能是由后端来实现的。后端更多的是与数据库进行交互去处理相应的业务逻辑。需要考虑的是如何实现功能、数据的存取、平台的稳定性与性能等 前端和后端通过接口进行交互。前端页面通过调用后端接口来实现功能、数据的存取，将数据展现在用户面前 五：接口测试的价值 1.更早发现问题 测试应该更早的介入到项目开发中，因为越早的发现bug，修复的成本越低。然而功能测试必须要等到系统提供可测试的界面才能对系统进行测试。而接口测试可以功能界面开发出来之前对系统进行测试。系统接口是上层功能的基础，接口测试可以更早更低成本的发现和解决问题。然而，在实际的开发过程中，开发人员并没有充足的时间去编写单元测试，并且他们往往对自己编写的代码迷之自信，不愿意花时间在编写单元测试上。这个时候接口测试的
基于白盒测试的Parlay_API接口测试方法设计
基于白盒测试的Parlay API接口测试方法设计 下一代网络(NGN)是可以提供语音、数据和多媒体等各种业务的综合开放的网络架构，可以支持快速业务部署以及第三方业务控制。NGN开放式业务提供的是一个分布式系统，为了实现第三方业务开发，业务结构应采用开放式接口控制技术，正在研究和开发的技术包括移动代理技术、主动网络技术和应用编程接口(API)技术。目前现实可行的是API技术。许多组织提出了开放业务平台的API，Parlay是其中最活跃、最有影响力的一个。 在Parlay组织成立后不久，3GPP和ETSI启动了3G系统UMTS的开放式业务架构的研究，称之为OSA。两者非常类似，最初的OSA标准就是由Parlay 1.2和2.1加上少量的3GPP 新增功能组成的。其后，两个组织决定从Parlay 3.0和OSA R5开始统一发布接口标准，命名为Parlay/OSA，这奠定了固定和移动NGN业务层融合的技术基础。两者的差别在于，Parlay 是单纯的接口标准，而OSA是一种业务结构，不仅包括业务接口，还包括体系结构以及Parlay 至移动网络协议，如MAP、CAP等的映射。 一、Pariay APl对业务的支持 Parlay API是一种基于分布式技术的、开放的、面向对象的下一代业务开发技术，它通过协议映射技术把底层网络的通信细节抽象成标准的API形式供业务开发者开发业务逻辑程序。它带来的好处是降低了业务开发的技术门槛，能使业务开发者更快捷地满足用户的个性化需要，提供丰富多彩的业务，为下一代网络的应用和发展提供最有效的驱动力。 Parlay APl是一个标准的接口，从而能够使第三方通过此接口利用运营商的基础网络提供丰富多彩的业务，例如统一消息业务、基于位置的业务、呼叫中心业务等，这些业务的业务逻辑都位于应用服务器上。 通过Parlay提供的第三方业务主要分为以下几类： ·通信类业务，如点击拨号、VoIP、点击传真、可视通话、会议电话，以及与位置相关的紧急呼叫业务等； ·消息类业务，如统一消息、短消息、语音信箱、E-mail、多媒体消息、聊天等； ·信息类业务，如新闻、体育、旅游、金融、天气、黄页、票务等各种信息的查询、订制、通知，以及基于位置的人员跟踪、找朋友等； ·娱乐类业务，如游戏、博彩、谜语、教育、广告等。 各类业务可以相对独立，也可以有机地结合，例如可以在查询信息时根据相应的信息进行支付类业务，而且各种娱乐可以通过不同的消息方式来表现(短消息、E-mail)，将娱乐与消息业务相结合。 框架服务器接口和业务能力接口是Parlay API定义的两类主要接口。业务逻辑程序通过Parlay网关中框架服务器接口的鉴权后，被授权接入规定的业务，然后使用框架服务器接口提供的业务能力发现和业务能力选择功能，通过签订在线业务能力使用协议，获得在框架服务器中注册的、满足业务需求的业务能力管理类接口引用。业务逻辑通过获得业务能力管理类接口引用就可以和其对应的业务能力接口进行通信，实现特定业务逻辑的呼叫控制、用户交互及计赞等功能。 Parlay标准定义的是控制底层网络资源的API，并非网络协议。两者的差别在于：协议面向具体的网络，由严格定义的一组消息和通信规则组成；API面向软件编程者，由一组抽象的操作或过程组成。在不同的网络中完成同样的功能所用的协议可能完全不同，但是所用的API则完全相同。这样，原来对通信网技术知之甚少的软件人员也可以利用Parlay接口自如地开发应用业务程序。 二、开放式业务接口Parlay API的测试 业务支撑环境是业务实现的重要环节，下一代网络的业务支撑环境主要包括应用服务
软件测试工具大全
软件测试工具汇总 一、工具汇总 1.免费工具 下表中针对WEB页面或B/S结构进行功能和性能测试的工具有： 开源功能自动化测试工具：PureTest，OpenSTA，Watir、Selenium、MaxQ、WebInject、Fitnesse 开源性能自动化测试工具：Jmeter、OpenSTA、DBMonster PureTest Minq公司功能测试商业 免费 本是业内商业自动化测试工具之一，如今PureTest已经免费。它专注于对WEB应用程序进行功能自动化测试，并即时对WEB页面元素进行检 测，对HTTP请求、响应进行诊断分析。 PureTest is an application which is primarily used to setup scenarios of tasks, execute and debug them. Even though it supports testing a variety of applications it is especially useful for debugging and snooping of web applications. PureTest includes a HTTP Recorder and Web Crawler which makes it useful for generic verification of HTTP requests and web content checking. The normal way to access web sites is via a browser; however, there are times when it is desirable to bypass the browser and access a site from a program, including: Debugging of HTTP requests and responses Automated web site testing The HTTP Recorder simplifies the process of capturing all requests that are exchanged between a browser and the web server. Then use PureTest to replay each request in order to carefully watch the HTTP data that is transferred on the wire (HTTP headers, request parameters, response headers and response content). The Web Crawler is useful to pro-actively verify the consistence of a static web structure. It reports various metrics, broken links and the structure of the crawled web. Test scenarios that be saved to file and later be repeated, to verify that you server applictaion works as expected. This can be done using the PureTest debugger in the grapical user interface, but also using a command line interface. PureLoad Minq公司负载压力测 试 商业 免费 PureLoad正是一款基于Java开发的网络负压测试工具，它的Script代码 完全使用XML，所以，这些代码的编写很简单，可以测试各种C/S程序， 如SMTP Server等。它的测试报表包含文字和图形并可以输出为HTML 文件。由于是基于Java的软件，所以，可以通过Java Beans API来增强 软件功能。
app测试工程师的基本职责模板
app测试工程师的基本职责模板 app测试工程师需要根据产品测试需求完成测试环境的设计与配置工作。下面是第一范文网小编为您精心整理的app测试工程师的基本职责模板。 app测试工程师的基本职责模板1 职责 1. 负责移动端(SDK)APP测试; 2. 理解产品需求，负责测试方案制定，根据设计文档，能独立编写用例，并进行相互评审; 3. 设计执行测试用例，编写测试报告; 4. 完成相关产品功能测试; 5. 跟踪测试问题，协助开发定位分析问题，持续跟踪bug修复情况; 6. 积极主动与项目经理、产品经理、开发团队、嵌入式开发团队沟通协作，保障项目顺利进行和推动问题解决。 任职资格 1. 本科及以上学历，2年以上iOS\Andriod APP测试经验，熟悉Objective-C/java等至少一种语言，熟悉iOS/Andriod SDK 测试工作，基本掌握Xcode/Android Studio等开发工具 ; 2. 做过APP自动化测试性能测试优先; 3. 熟悉测试理论方法;有过 BLE/NFC 项目测试经验优先;
4. 熟练掌握数据库操作，能够独立编写数据库语句优先; 5. 性格开朗有较强的沟通协调能力与表达能力; 6. 熟练掌握fiddler/postman等测试辅助工具。 app测试工程师的基本职责模板2 职责： 1、制定项目测试计划、测试方案，设计测试用例，执行测试等。 2、编写及设计功能及性能测试用例，并提交测试报告。 3、协助开发人员快速定位问题，并对产品提出建设性意见，提升产品用户体验。 4、对缺陷进行跟踪分析和报告，推动测试中发现的问题及时合理地解决。 5、完善相关测试文档，完成其它测试相关工作。 任职要求： 1、计算机、电子相关专业毕业，一年以上工作经验，对互联网有一定的了解。 2、熟悉软件、服务器、web、APP测试流程和方法，可以编写测试用例和相关文档。 3、良好沟通能力、愿意学习、比较细心的人。 4、诚实、认真。有良好团队合作精神。 app测试工程师的基本职责模板3 职责：
常用办公软件测试题汇编
常用办公软件测试题 一、综合部分 1．对于Office XP应用程序中的“保存”和“另存为”命令，正确的是___。 A．文档首次存盘时，只能使用“保存”命令 B．文档首次存档时，只能使用“另存为”命令 C．首次存盘时，无论使用“保存”或“另存为”命令，都出现“另存为”对话框 D．再次存盘时，无论使用“保存”或“另存为”命令，会出现“另存为”对话框 2．对于Office XP应用程序中的“常用”工具栏上的“新建”命令按钮和“文件”菜单下的“新建”命令项，不正确的是___。 A．都可以建立新文档 B．作用完全相同 C．“新建”命令按钮操作没有“模板”对话框，使用空白模板 D．“文件”后“新建”命令可打开“模板”对话框，可以选择不同的模板 3．不能在“另存为”对话框中修改文档的___。 A．位置B。名称 C．内容D。类型 4．Office XP应用程序中的“文件”菜单底端列出的几个文件名表示___。 A．用于切换的文件B。已打开的文件 C．正在打印的文件D。最近被该Office XP应用程序处理过的文件 5．在文本编辑状态，执行“编辑”到“复制”命令后，___。
A．被选定的内容复制到插入点 B．被选定的内容复制到剪贴板 C．被选定内容的格式复制到剪贴板 D．剪贴板的内容复制到插入点 6．当“编辑”菜单中的“剪切”和“复制”命令呈浅灰色而不能被选择时，表示___。A．选定的内容太长，剪贴板放不了 B．剪贴板里已经有信息了 C．在文档中没有选定任何信息 D．选定的内容三图形对象 7．Office XP应用程序中的工具栏可以___。 A．放在程序窗口的上边或下边 B．放在程序窗口的左边或右边 C．作为一个窗口放在文本编辑区 D．以上都可以 8．可以从___中选择Office XP应用程序中的命令。 A．菜单B。工具栏 C．快捷菜单D。以上都可以 9．Office XP应用程序中使用鼠标进行复制操作应___。 A．直接拖动B。按住
软件测试工具
摘要 随着信息科技的发展。数字科技的进步。人们对所使用的软件要求越来越严格，许多大型的软件公司对自己严发出来的软件要求也越来越严格，为了解决其中的BUG，软件测试行业开始在国内崛起。新兴的科技技术，带领着软件业开始飞速发展，产品趋于完美化，智能化，易用程度也大大的提高。 但是软件测试行业的形成是因为什么呢？许多人只知道软件测试，但是不知道其根本，它的源头是什么，它是怎么发展衍变的？ 本文在探讨软件测试技术的基础上，详细介绍了软件测试的发展，它的衍变过程。同时为大家介绍了多种系列的软件测试工具及它们各自的特点。为软件测试人员理清了测试思路，详细的划分了软件测试的种类。在阅读众多参考文献的情况下对于软件的安全的问题也进行了详细的阐述。最后详细介绍了一款基于主机的入侵检测的工具—PortSentry的安装，配置及使用方法。 关键词软件测试；发展；种类；工具
Abstract Along with information science and technology development.Digital science and technology progress.The people for the software request which uses are more and more strict, many large-scale software companies the software request which sends strictly to oneself more and more are also strict, in order to solve BUG, the software test profession starts in to rise domestically.The emerging technical technology, leads the software industry to start to develop rapidly, the product tends to the beautification, the intellectualization, easy to use the degree also big enhancement. But is the software test profession formation because of what? Many people only know the software test, but did not know its basic, what is its source, how is it develops evolves? This article in the discussion software test technology foundation, introduced in detail the software tests the development, it evolves the process.Meanwhile introduced many kinds of series software testing tool and they respective characteristic for everybody.Tested the personnel for the software to clear off the test mentality, the detailed division software has tested type.Has also carried on the detailed elaboration in the reading multitudinous reference situation regarding the software security question.Finally introduced one section in detail based on the main engine invasion examination tool - PortSentry installment, the disposition and the application method. Keywords software test，development，kind,tool
软件测试用例实例非常详细汇总
1、兼容性测试 在大多数生产环境中，客户机工作站、网络连接和数据库服务器的具体硬件规格会有所不同。客户机工作站可能会安装不同的软件例如，应用程序、驱动程序等而且在任何时候，都可能运行许多不同的软件组合，从而占用不同的资源。测试目的 配置说明操作系统系统软件外设应用软件结果 服务器Window2000(S) WindowXp Window2000(P) Window2003 用例编号TestCase_LinkWorks_WorkEvaluate 项目名称LinkWorks 模块名称WorkEvaluate模块 项目承担部门研发中心-质量管理部 用例作者 完成日期2005-5-27 本文档使用部门质量管理部 评审负责人 审核日期 批准日期 注：本文档由测试组提交，审核由测试组负责人签字，由项目负责人批准。历史版本： 版本/状态作者参与者起止日期备注
1.1. 疲劳强度测试用例 强度测试也是性能测试是的一种，实施和执行此类测试的目的是找出因资源不足或资源争用而导致的错误。如果内存或磁盘空间不足，测试对象就可能会表现出一些在正常条件下并不明显的缺陷。而其他缺陷则可能由于争用共享资源（如数据库锁或网络带宽）而造成的。强度测试还可用于确定测试对象能够处理的最大工作量。 测试目的 测试说明 前提条件连续运行8小时,设置添加10用户并发 功能1 2小时 4小时 6小时 8小时 功能1 2小时 4小时 6小时 8小时 一、功能测试用例 此功能测试用例对测试对象的功能测试应侧重于所有可直接追踪到用例或业务功能和业务规则的测试需求。这种测试的目标是核实数据的接受、处理和检索是否正确，以及业务规则的实施是否恰当。主要测试技术方法为用户通过GUI（图形用户界面）与应用程序交互，对交互的输出或接受进行分析，以此来核实需求功能与实现功能是否一致。
微服务接口测试中的参数传递
微服务接口测试中的参数传递 这是一个微服务蓬勃发展的时代。在微服务测试中，最典型的一种场景就是接口测试，其目标是验证微服务对客户端或其他微服务暴露的接口是否能够正常工作。对于最常见的基于Restful风格的微服务来说，其对外暴露的接口就是HTTP端点(Endpoint)。 这种情况下，完成微服务接口测试的主要方式就是构造并发送HTTP请求消息给微服务，然后接收并验证微服务回复的HTTP响应消息。在这个过程中，最基础的工作是正确构造HTTP请求消息。 一条HTTP请求消息中，包含各种各样的参数。了解HTTP请求参数的类型，对于我们正确构造HTTP请求消息十分重要。接下来，我们就一起看看HTTP请求消息中可能包含哪些类型的参数，以及它们各自的特点。 路径参数(path parameter)。在HTTP中，URL是一个很基本的概念，它表示的是服务端资源的路径，供客户端寻址和访问。URL一般是常量字符串，但在有些情况下，URL 中某些部分是可变的。路径参数就是URL中可变的部分，其描述方式为{参数名}。例如，路径/blogs是不变的，而路径/blogs/{id}是可变的，其中可变的id就是路径参数。 路径参数一般用来指定集合中的某个具体元素。例如，服务端可能有许多blogs，而/blogs/{id}表示的就是某一篇具有特定id的blog。路径参数的特点如下：一个URL中可以包含多个路径参数。 在传递路径参数时，直接将{参数名}替换成具体的值，例如/blogs/123456。 路径参数是必填的，不是选填的。 查询参数(query parameter)。和路径参数相同的是，查询参数也是URL的一部分，通常用来对资源进行排序或过滤。除此之外，它们有许多不同点:
软件测试常用术语表
第119贴【2004－10－12】：常见测试术语一 Acceptance Testing－－可接受性测试 一般由用户/客户进行的确认是否可以接受一个产品的验证性测试。 actual outcome－－实际结果 被测对象在特定的条件下实际产生的结果。 Ad Hoc Testing－－随机测试 测试人员通过随机的尝试系统的功能，试图使系统中断。algorithm－－算法 （1）一个定义好的有限规则集，用于在有限步骤内解决一个问题；（2）执行一个特定任务的任何操作序列。 algorithm analysis－－算法分析 一个软件的验证确认任务，用于保证选择的算法是正确的、合适的和稳定的，并且满足所有精确性、规模和时间 方面的要求。 Alpha Testing－－Alpha测试 由选定的用户进行的产品早期性测试。这个测试一般在可控制的环境下进行的。 analysis－－分析 （1）分解到一些原子部分或基本原则，以便确定整体的特性；（2）一个推理的过程，显示一个特定的结果是假 设前提的结果；（3）一个问题的方法研究，并且问题被分解为一些小的相关单元作进一步详细研究。 anomaly－－异常 在文档或软件操作中观察到的任何与期望违背的结果。
application software－－应用软件 满足特定需要的软件。 architecture－－构架 一个系统或组件的组织结构。 ASQ－－自动化软件质量（Automated Software Quality） 使用软件工具来提高软件的质量。 assertion－－断言 指定一个程序必须已经存在的状态的一个逻辑表达式，或者一组程序变量在程序执行期间的某个点上必须满足的 条件。 assertion checking－－断言检查 用户在程序中嵌入的断言的检查。 audit－－审计 一个或一组工作产品的独立检查以评价与规格、标准、契约或其它准则的符合程度。 audit trail－－审计跟踪 系统审计活动的一个时间记录。 Automated Testing－－自动化测试 使用自动化测试工具来进行测试，这类测试一般不需要人干预，通常在GUI、性能等测试中用得较多。 第120贴【2004－10－13】：常见测试术语二 Backus-Naur Form－－BNF范式 一种分析语言，用于形式化描述语言的语法 baseline－－基线
常用工具软件测试题及答案
一、判断题 1. Realone Player不支持多节目连续播放。（N） 2. 网际快车可以上传和下载文件。（N） 3. 天网防火墙的拦截功能是指数据包无法进入或出去。（Y） 4. SnagIt可以捕获DOS屏幕，RM电影和游戏等画面。（Y） 5. Adobe Acrobat Reader可以解压缩文件。（N） 6. 金山词霸2002支持Windows XP,但不支持office XP系统。（N） 7. 在用Ner-Burning Room刻录CD音乐时，若误将数据文件从本地资源管理器中拖入刻录机虚拟资源管理器中时，该文件将被添加到音乐CD中。（N） 8. Symantec Ghost 可以实现数据修复。（N） 9. Easy Recovery 可以恢复任何被从硬盘上删除的文件。（N） 10. Ctrem软件具有防发呆功能。（Y） 二．选择题（每小题2分，共40分） 1、下列不属于金山词霸所具有的功能的是：（C ） A、屏幕取词 B、词典查词 C、全文翻译 D、用户词典 2、东方快车提供了（C ）种语言翻译。 A、1种 B、2种 C、3种 D、4种 3、:Vintual CD 中的Creat按钮的功能为（B ） A、编辑映像文件 B、创建光盘的映像文件 C、映像文件的显示方式 D、将映像文件插入虚拟光驱 4、下列哪一个软件属于光盘刻录软件（A ） A、Nero-Buring Room B:Virtual CD C: DAEMON Tools D:Iparmor 5、下列不属于媒体播放工具的是（D ） A、Winamp B、超级解霸 C、Realone Player D:WinRAR 6、下列媒体播放器可以自由截取单个画面或整段电影的是非曲直（B ） A、Winamp B、超级解霸 C、Realone Player D、音频解霸 7、下列哪一个不是网际快车为已下载的文件设置的缺省创建类别（ D） A、软件 B、游戏和mp3 C、驱动程序 D、电影 8、CuteFTP具有网际快车不具备的功能是（ A） A、上传文件 B、下载文件 C、断点续传 D、支持多线程下载 9、如果在天网防火墙的ICMP规则中输入（ B）则表示任何类型代码都符合本规则。 A、254 B、255 C、256 D、253 10、Norton Antivirus的安全扫描功能包括（D ） ①自动防护②电子邮件扫描③禁止脚本④全面系统扫描 A、①②③ B、①②④ C、①③④ D、①②③④ 11、ACDSee不能对图片进行下列哪种操作（C ） A、浏览和编辑图像 B、图片格式转换 C、抓取图片 D、设置墙纸和幻灯片放映 12、SnagIt捕获的图片可被存为下列哪些格式（D ） ①BMP ②PCX ③TGA ④RSB A、①②③ B、①②④ C、①②③④ D、①② 13、WinRAR不可以解压下列哪些格式的文件（ D）
接口自动化测试框架设计
IAT框架设计 1 背景 1.1项目背景 在移动平台服务端接口测试覆盖度为零的情况下，根据服务端接口的特点，以及升级更新的速度较快等，需要开发此框架来实施服务端接口的自动化测试。 1.2接口测试 接口测试属于灰盒测试范畴，通常不需要了解接口底层的实现逻辑，但需要测试人员能够使用代码的方式来调用接口。接口测试主要用例测试接口的功能以及接口返回数据的正确性。根据接口测试的复杂度接口测试分为两种。即单一接口测试，以及多接口组合功能测试。由于接口测试是通过代码调用的方式完成，而且接口测试与前端 UI 属于松耦合（或无耦合）因此通过自动化手段将极大提高测试效率以及回归测试的复用率。本文中提到的接口测试主要是指基于 http,https ，rpc 协议的 web 接口。 1.3 适用性分析 移动平台大部分以 http 接口方式提供服务，通过前台 App 调用接口方式实现功能。同时大部分接口功能，以及表现形式稳定，对于前台变化敏感度较低。基于上述接口测试的特点，认为移动平台项目非常适合接口层级的自动化测试。 2 IAT 框架 2.1IAT 介绍 IAT 是 Interface Automation Testing 的简称。通过热插拔的方式支持 http,rpc,soap 类协议的 web 接口测试。框架支持单一接口，多接口组合测试，支持用户通过自定义方法实现精确验证结果的需求。 2.2框架特点 提供多种接口测试方式。即单一接口测试，多接口业务流程测试。目前多见的为单一接口的测试。根 据用户需求不同，不同的接口测试方式，用例开发难易度不同。用例开发门槛低，用户只需要将接口用例 数据填入格式化文件即可自动通过工具生成用例。对于高级需求，框架提供自定义配置包括数据构造，精 确匹配测试结果等。框架对于不同域名下的相同接口支持自定义配置，只需要简单修改测试平台配置即 可轻松将用例
软件测试学习工具大全
软件测试学习工具大全 软件测试是时下新兴的热门IT专业，很多朋友都有转行其中的意愿。然而，工欲善其事，必先利其器。想要学会软件测试，不了解软件测试学习工具可不行。下面，就让小编带你了解一下，千锋软件测试学习工具有哪些。 Test Platform软件测试平台，简称TP，是业界唯一的对软件测试全过程进行支撑的软件测试学习工具。 业界已有的软件测试工具基本上都局限在测试执行阶段，只能支撑测试执行阶段的活动，而测试分析、测试设计、测试实现这三个前期阶段的活动缺乏有效的测试工具支撑，直接影响了软件测试的完整性和充分性，从而影响最终研发的软件质量。
软件测试学习工具引入缺陷分析模型 在业界首先将各种有效的缺陷分析模型引入到该软件平台中，包括ODC分析、Gompertz分析、Rayleigh分析、四象限分析、缺陷注入分析、DRE/DRM 等工程方法，帮助管理者建立软件研发过程的质量基线、测试能力基线，并帮助管理者将项目实际缺陷、能力数据和基线数据进行对比分析，发现软件过程中的改进点，判断测试是否可以退出、软件是否可以发布，并对软件中残留缺陷数进行预测； 利用理论框架分析 建立了测试分析和设计的理论框架和一整套工程方法，能够很好的支撑测试的辅助分析和设计； 建立测试跟踪关系 建立“开发需求项－>测试项－>测试子项－>测试用例－>缺陷”的测试跟踪关系，能够及时的反应开发需求和设计的变更对测试的影响范围，保证软件的一致性和测试的充分性，从而保证软件的质量； 使用TestPlatform 能够全面的管理软件质量工作，具有高度的集成性，一款TestPlatform能够完成多款其他各类的相关质量管理工具集成在一起才能完成的软件质量管理工作。它集成了需求跟踪、静态测试、动态测试、测试人员管理、测试环境管理、测试计划管理、测试用例管理、缺陷管理、缺陷分析等软件质量相关的流程。那
标准云听测试报告
2.7.4标准云听测试总结报告 测试人员：***
目录 1引言 (3) 1.1编写目的 (3) 1.2背景 (3) 1.3用户群 (3) 1.4定义 (3) 1.5 测试对象 (4) 1.6 测试阶段 (4) 1.7 测试工具 (4) 1.8 参考资料 (4) 2测试概要 (4) 2.1进度回顾 (5) 2.2测试执行 (5) 2.3 测试用例 (5) 2.3.1 功能性 (5) 2.3.2 易用性 (5) 3测试环境 (6) 4 测试结果 (6) 4.1 Bug 趋势图 (6) 4.2 Bug 严重程度 (7) 4.3 BUG分类统计占比 (8) 5测试结论 (9) 5.1功能性 (9) 5.2易用性 (9) 5.3可靠性 (10) 5.4兼容性 (10) 5.5安全性 (10) 6 分析摘要 (10) 6.1 建议 (10) 7度量 (11) 7.1 资源消耗 (11) 8典型缺陷引入原因分析 (11)
1引言 1.1编写目的 编写标准云听测试报告主要目的罗列如下： 1．通过对测试结果的分析，得到对软件质量的评估 2．分析测试的过程，产品，资源，信息，为以后制定测试计划提供参考3．评估测试执行和测试计划是否符合 4．分析系统存在的缺陷，为修复和预防bug 提供建议 1.2背景 客户需求 1.3用户群 主要使用者： (1) 电台主播(主持人) (2) 频道负责人 (3) 媒体负责人 (4) 电台听众 1.4定义 1.出现以下缺陷，定义为致命bug (1级) ： (1) 系统出现闪退、崩溃； (2) 系统无响应，处于死机状态，需要其他人工修复系统才可复原；’ (3) 操作某个功能出现报错或者返回异常错误； (4) 进行某个操作（增加、修改、删除等）后，出现报错或者返回异常错误； (5) 实现功能和需求不符等； 2.出现以下缺陷，定义为严重(功能)bug (2级) ： (1) 当对必填字段进行校验时，未输入必输字段，出现报错或者返回异常错误 (2) 系统定义不能重复的字段输入重复数据后，出现报错或者返回异常错误 (3) 系统刷新加载不正常，不能正确显示； (4) 显示信息与配置信息不一致等； 3.出现以下缺陷，定义为一般bug(3级)： (1) 显示问题； (2) 提示问题；
常用通讯测试工具使用
常用通讯测试工具 鉴于很多MCGS用户和技术人员对通讯测试工具并不很熟悉，本文档将针对实际的测试情况，对串口、以太网通讯调试过程中所涉及到的常用的测试软件进行相关的讲解。 1. 串口测试工具： 串口调试工具：用来模拟上下位机收发数据的串口工具，占用串口资源。如：串口调试助手，串口精灵，Comm等。 串口监听工具：用来监听上下位机串口相关操作，并截获收发数据的串口工具。不占用串口资源。如：PortMon，ComSky等。 串口模拟工具：用来模拟物理串口的操作，其模拟生成的串口为成对出现，并可被大多数串口调试和监听软件正常识别，是串口测试的绝好工具。如：Visual Serial Port等。 下面将分别介绍串口调试助手、Comm、PortMon和Visual Serial Port的使用。
1.1. 串口调试助手： 为最常用的串口收发测试工具，其各区域说明及操作过程如下： 串口状态 打开/关闭串口 十六进制/ASCII 切换 串口数据 接收区 串口参数 设置区 串口数据 发送区 串口收发计数区 发送数据功能区 保存数据功能区 操作流程如下： ? 设置串口参数（之前先关闭串口）。 ? 设置接收字符类型（十六进制/ASCII 码） ? 设置保存数据的目录路径。 ? 打开串口。 ? 输入发送数据（类型应与接收相同）。 ? 手动或自动发送数据。 ? 点击“保存显示数据”保存接收数据区数据到文件RecXX.txt。 ? 关闭串口。 注：如果没有相应串口或串口被占用时，软件会弹出“没有发现此串口”的提示。
1.2. PortMon 串口监听工具： 用来监听上下位机串口相关操作，并截获收发数据的串口工具。不占用串口资源， 但在进行监听前，要保证相应串口不被占用，否则无法正常监听数据。 连接状态 菜单栏 工具栏 截获数据显示区 PortMon 设置及使用： 1). 确保要监听的串口未被占用。 如果串口被占用，请关闭相应串口的应用程序。比如：要监视MCGS 软件与串口1设备通讯，应该先关闭MCGS 软件。 说明：PortMon 虽不占用串口资源，但在使用前必须确保要监听的串口未被占用，否则无法进行监视。 2). 运行PortMon，并进行相应设置。 ? 连接设置： 在菜单栏选择“计算机(M)”->“连接本地(L)”。如果连接成功，则连接状态显示为“PortMon 于\\计算机名(本地)”。如下图：
自动化概述
一、概述 1.1 什么是自动化测试 自动化测试是把以人为驱动的测试行为转化为机器执行的一种过程。通常，在设计了测试用例并通过评审之后，由测试人员根据测试用例中描述的规程一步步执行测试，得到实际结果与期望结果的比较。在此过程中，为了节省人力、时间或 硬件资源，提高测试效率，便引入了自动化测试]的概念。 提高测试效率，保证产品质量 1.自动化测试完全取代手工测试 2.自动化测试一定比手工测试厉害，更加高大上 3.自动化可以发掘更多的bug 二、自动化层次模型 2.1 单元自动化测试 1.主要是针对于类、方法的测试。
2.此阶段测试效益最大。 3.常见测试框架：Junit 、TestNG、Unittest。 1、节省了测试成本 根据数据模型推算，底层的一个程序BUG可能引发上层的8个左右BUG，而且 底层的BUG更容易引起全网的死机；接口测试能够提供系统复杂度上升情况下的低成本高效率的解决方案。 2、接口测试不同于单元测试 接口测试是站在用户的角度对系统接口进行全面高效持续的检测。 3、效益更高 将接口测试实现为自动化和持续集成，当系统复杂度和体积越大，接口测试的成本就越低，相对应的，效益产出就越高。 4.常见工具 httpUnit （接口框架）、postman（接口调试工具）。 1、界面元素测试 2、面向用户，测试工作占比大 3、robot framework ，selenium，appium
三、自动化测试框架模型 3.1 线性测试## 独立功能测试，流水线执行 模块复用（如登录模块） 参数化 关键字封装（QTP、selenium） 1.需求变动不频繁 2.项目周期足够长 3.项目需要重复回归测试
软件测试个人总结及小结
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