It remains to describe the CR structures associated with the anti-involutionϕ3, which are all indefinite type and admitso∗(4)symmetry. To our knowledge, these models are new.
Whileso∗(4)is customarily defined via a skew-Hermitian formηonH2, whereHis the quaternions, we will instead focus on the special isomorphismso∗(4)∼=sl(2,R)×su(2). In doing so, we will work with a particularly simple representation of this Lie algebra and the choice ofηwill arise naturally from this. In contrast, if we were to fix a choice ofηfirst, the resulting realization ofso∗(4)could be quite complicated.
Recall thatHis the associativeR-algebra withR-basis 1,i,j,k, standard relationsij= k,i2 =j2=k2= −1, conjugation satisfyingq1q2=q2q1, and norm|q|2=qq=qq. Let SU(2)denote the unit quaternions, which act onHon the left. LetSL(2, R)denote 2×2 real
Table 4 Inequivalent Cartan hypersurfaces{[z] :(z,z)=β|(z,z)|} ⊂CP3\Q, where(z,z)=1z21+ · · · + 4z24withj= ±1. These complexify to the ILC D.6-3 model witha2= β92
Real form Signature of(·,·)|R4 β-range Levi-form type Anti-involution
O(4) + + ++ β >1 Definite ϕ(+2 1)
O(3,1) + + +− 0< β <1 Definite ϕ(−2 1)
β >1 Indefinite ϕ1
O(1,3) +− − − 0< β <1 Definite ϕ(+1)2
O(2,2) + + −− 0< β <1 Indefinite ϕ1
β >1 Definite ϕ(−2 1)
matrices with determinant±1, which act onR2. The groupS=SL(2,R)×SU(2)acts on the external tensor productR2⊗RH, and we identify this naturally withH2. This identification isS-equivariant if forq=
q1 q2
∈H2, we declare thatA∈SL(2, R)andq0∈SU(2)each act by multiplication on the left, with the latter identified with diag(q0,q0). (These actions commute.) WhileH2 is naturally arightH-vector space, it will be more important for us to consider it as aC-vector space by restricting this right action toC := {1s+it : s,t ∈ R} ⊂H. (This in particular distinguishes the imaginary uniti.) We will be interested in the 5-dimensionalS-orbits inCP3∼=PC(H2).
There exists anS-invariant skew-Hermitian form onH2(unique up to a real scaling) given byη(q, w)=q1w2−q2w1, and this is valued inIm(H). Letη(q,q)=ib+jμ, forb∈R andμ∈C. Givenλ∈C,η(qλ,qλ)= ¯λη(q,q)λ, hence(b, μ)→(b|λ|2, μλ2). Writing q=
z1+jz2
z3+jz4
forzj ∈C,
η(q,q)=z1z3+z2z4+j(z1z4−z2z3). (5.5) Note thatQ# = {[q] : Re(iη(q,q))= 0}is the flat (indefinite) CR structureIm(z1z3+ z2z4)=0, so we excludeb=0. We will also exclude theμ=0 case (see below), since this yields only 4-dimensionalS-orbits. Whenbμ=0, we have the complex scaling invariant γ = |μk|b ∈ R\{0}. Sinceb = −Re(iη(q,q))andμ = −jη(q,q)−bk, theS-orbits in CP3\Q#satisfy
Re(iη(q,q))= −γ|jη(q,q)k+Re(iη(q,q))|. (5.6) Equivalently, Im(z1z3 +z2z4) = γ|iRe(z1z3+z2z4)−(z1z4−z2z3)k|, which further simplifies to
Im(z1z3+z2z4)=γ
|Re(z1z3+z2z4)|2+ |z1z4−z2z3|2. (5.7) Let us find representatives for theS-orbits onCP3\Q#. Given 0 = q ∈ H2, we may swapq1 andq2 if necessary to assumeq1 =0, and then use SU(2)and a real rescaling to assumeq1 =1. Using
1 0
−Re(q2)1
∈ SL(2,R), we may assumeRe(q2)= 0. Using eiφ ∈SU(2)and the rightC-action bye−iφ, we can replaceq2 →eiφq2e−iφ, so chooseφ to normalizeq2 =is+jt, wheres,t ∈Rwitht ≤0. Finally, use diag(λ,1λ) ∈SL(2,R) and right multiplication by1λto normalizeq2to be of unit length. Thus, we haveq2 =iekθ,
for some fixed 0 ≤ θ < π. Forq = 1
iekθ
, we have η(q,q) = 2iekθ. When θ = π2 or 0, we have b = 0 or μ = 0, so we exclude these. The (S-equivariant) conjugation arising from the identificationH2 = C4, i.e.u+jv = ¯u+jv¯ (whereu, v ∈ C2), maps
. Thus, we can restrict toγ =cot(θ) >0, or equivalently require θ ∈ (0,π2). Hence, we can restrict to 0 < θ < π2, which is in 1-1 correspondence with multiplication byionH2. Defining
w1:=k, w2:=j−2 sin(θ)Y, w3:=H, w4:=i+2 cos(θ)Y, (5.8) Here we have distinguished the scalarifrom the Lie algebra elementi. The−i-eigenspacev/k forJis spanned by{v1,v2}modT, while the+i-eigenspacee/kis spanned by{v1,v2}modT. (We caution that this conjugation fixes each ofw1, . . . ,w4,Tand conjugates the scalars. It is distinct from the conjugation onH, and that associated withH2 =C4.) The ILC D.6-3 structure equations are satisfied if we take
ρ= In particular, a2= −γ92 <0 classifies the corresponding ILC D.6-3 structure.
6 Hypersurfaces of Winkelmann type
Generalizing (1.1), we say that a real hypersurface inC3is ofWinkelmann typeif in some holomorphic coordinate system(z1,z2, w), it is given by
Im(w+ ¯z1z2)=F(z1,z¯1), (6.1) whereFis an arbitrary real-valued analytic function. These all admit the symmetries
N1=z1∂z2, N2=∂z2+z1∂w, N3=i∂z2−i z1∂w, N4=∂w.
Table 5 CR structures underlying ILC N.6-2 models
Model ILC N.6-2 classification CR symmetries aside fromN1,N2,N3,N4 Im(w+ ¯z1z2)=(z1)α(¯z1)α¯
(α∈C\{−1,0,1,2}) b¯2=a2=(α+−(2α−1)1)(α−22)
∈C\{−4,12}
z1∂z1+(α+ ¯α−1)z2∂z2+(α+ ¯α)w∂w, i z1∂z1+i(α− ¯α+1)z2∂z2+i(α− ¯α)w∂w Im(w+ ¯z1z2)=exp(z1+ ¯z1) b2=a2= −4 i∂z1+i z2∂w,
∂z1+2z2∂z2+(2w−z2)∂w Im(w+ ¯z1z2)=ln(z1)ln(¯z1) b2=a2=12 z1∂z1−z2∂z2+2iln(z1)∂w,
i z1∂z1+i z2∂z2+2 ln(z1)∂w
They span an abelian Lie algebra that induces a (holomorphic) foliation of C3 by 2-dimensional holomorphic hypersurfacesz1=const outside of the singular setz1=0.
Complexifying (6.1), we get the following complex hypersurfaces inC3× ¯C3:
w=b−a1z2+z1a2+2i F(z1,a1). (6.2) Regardingw=w(z1,z2), we obtainw1:= ∂z∂w1 =a2+2i Fz1(z1,a1)andw2 :=∂z∂w2 = −a1. Differentiate with respect toz1andz2 once more and eliminate the parameters(a1,a2,b).
Making the variable changez2 → −z2, we arrive at the PDE system
w11=2i Fz1z1(z1, w2), w12=w22=0. (6.3) The harmonic curvature [9, (3.3)] is of type N ifFz1z1w2w2 =0. Let us consider the specific Winkelmann type hypersurfaces given in Table5.
Writing out (6.3) forF(z1,z¯1)=(z1)α(¯z1)α¯, we obtain
w11=2iα(α−1)(z1)α−2(w2)α¯, w12=w22=0. (6.4) Using a constant rescaling ofz2, and relabelling, we can bring (6.4) into the same form as that listed in [9, Table 1] for the N.6-2 models withμ= ¯αandκ=α−2. TheF(z1,z¯1)= exp(z1)exp(¯z1)andF(z1,¯z1)=ln(z1)ln(¯z1)cases are handled similarly. The corresponding ILC N.6-2 models haveμ=κ= ∞for the former and(μ, κ)=(0,−2)for the latter, c.f.
[9, Table 1].
As discussed in Sect.3, the N.6-2 models are described using(a,b)∈C2, with(a2,b2) being essential parameters. Using (4.15) with μ = ¯α andκ = α−2, we find that for α∈C\ {−1,0,1,2}:
b¯2=a2= −(2α−1)2 (α+1)(α−2) ∈C\
−4,1 2
.
(Theα= −1 andα =2 cases lead to the ILC N.8 model.) Theb2 =a2 ∈ {−4,12}cases were described in Example4.9. From Sect.4, theb2=a2∈Rcases are tubular (see Table7 for models).
Proposition 6.1 The hypersurfaces of Winkelmann type given in Table5are models for all real forms of the complex ILC N.6-2 structures. Their (6-dimensional) CR symmetry algebras are never transitive outside of these hypersurfaces.
Proof From Table 6,b2 = ¯a2 ∈ C are the parameter values that yield underlying CR structures, and in each case there is aunique structure. Thus, Table5 gives a complete classification as claimed.
For the second claim, let us fix a basepointo∈C3and suppose thatz1|o=c+di, where c,d ∈R. ThenN1−cN2−d N3+(c2+d2)N4vanishes ato. Thus, these symmetry algebras have at most 5-dimensional orbits at all points inC3.
7 Transitivity of the symmetry algebra
In this section, we prove Theorem1.3. According to [14, Cor.6.36], Sym(M)is transitive on an open subset ofC3if and only if Sym(Mc)is transitive on an open subset ofC3× ¯C3. Proposition 7.1 Suppose M ⊂ C3 is locally transitive, and let(s,k;e,v)be an algebraic model of the ILC structure(E,V)on Mc. ThenSym(Mc)is transitive on a non-empty open subset ofC3× ¯C3if and only if for some T ∈Int(s), we have:
e+T(v)=s. (7.1)
Proof Assume thats=Sym(Mc)is transitive on a non-empty open subset ofC3× ¯C3and fix a point(z,a)in this subset. Then the projection ofsonC3is transitive on an open subset of C3containingz∈C3. The isotropy subalgebra of this action atzis conjugate tovby means of some inner automorphismT1∈Int(s). Similarly, the projection ofstoC¯3is transitive at a∈ ¯C3with the isotropy subalgebra equal toT2(e)for someT2∈Int(s).
Note thatsis transitive at(z,a)if and only if the projection ofT1(v)toC¯3is transitive ata ∈C3, or, similarly, if the projection ofT2(e)toC3 is transitive atz ∈C3. Both these conditions are equivalent to the equalityT1(v)+T2(e)=s. ApplyingT2−1to both sides of this equality we gete+T(v)=s, whereT =T2−1T1.
Corollary 7.2 The symmetry algebras=Sym(Mc)is transitive on a non-empty open subset ofC3× ¯C3if and only if the set of∈Csatisfying:
e+exp(X)(v)=s is non-empty for any X ∈snot contained ine+v.
Proof FixX∈s\(e+v), and decomposesinto a direct sum of linear subspacesV1⊕CX⊕V2, whereV1⊂eandV2⊂v. It is well-known that the exponential map:
exp:V1×C×V2→Int(s), (X1, ,X2)→exp(X1)exp(X)exp(X2)
is locally biholomorphic in a neighborhood of 0. TakingT =exp(X1)exp(X)exp(X2)in (7.1) and multiplying both sides by exp(−X1)we get:
exp(−X1)(e)+exp(X)exp(X2)(v)=s But by construction exp(−X1)(e)=eand exp(X2)(v)=v.
Corollary 7.3 Ifshas a non-trivial center, thensis not transitive at any point ofC3× ¯C3. Proof Let Z be any central element ins. Since the subalgebrakis effective, thenZ ∈/ k. Recall that Levi non-degeneracy ofM⊂C3implies that the bilinear form:
2(e+v)/k→s/(e+v), (X+k)∧(Y+k)→ [X,Y] +(e+v)
is non-degenerate. Hence, it follows that Z does not lie ine+v. But it is obvious that exp(Z)(v)=vfor any∈C, andsis not transitive according to Corollary7.2.
This proposition implies that the local transitivity of Sym(Mc)onC3× ¯C3, and hence the local transitivity of Sym(M)onC3, is a property of the algebraic model(s,k;e,v)itself and does not depend on the particular realization of this model in local coordinates.
Transitivity off the hypersurface is well-known in the maximally symmetric and Winkel-mann hypersurface cases [23]. Consider all remaining cases. For N.7-2, D.6-1, and D.7,s has non-trivial center, so these are ruled out by Corollary7.3. (See also Example4.4where it is computed explicitly as a Lie algebra of vector fields onC3× ¯C3.)
The remaining cases have 6-dimensional symmetry, so it suffices to exhibit a relation amongst the symmetry vector fields. For N.6-1, tubular realizations are given in Table7and share the symmetriesN1 =i∂z2,N2 =i∂w,N3 =∂z2 +z1∂w, andN4 =i z1∂z2+iz221∂w. Lettingz1 =a+bi, we havea N1+a2+2b2N2−bN3−N4=0. The N.6-2 cases were similarly ruled out in Proposition 6.1. The D.6-2 realizations (see Table 8) share the symmetries N1 =i∂z2,N2 =i∂w,N3=∂z2+2z2∂w, andN4 =i z2∂z2+i z22∂w. Lettingz2 =c+di, we havecN1+(c2+d2)N2−d N3−N4=0. For D.6-3 (a2 =9), the symmetries of the tubular models in Table8have an obvious dependency. Finally, for D.6-3 (a2=9), see §5 where the orbits of Sym(M)are described explicitly.
Acknowledgements Open Access funding provided by UiT The Arctic University of Norway. The authors gratefully acknowledge the use of theDifferentialGeometrypackage inMaple. This project was supported by the Austrian Science Fund (FWF project M1884-N35 for D.T.), as well as the Tromsø Research Foundation.
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Appendix A: Representative admissible anti-involutions
In Table6, we classify all representative admissible anti-involutions (see Sect.3) for all non-flat 5-dimensional multiply-transitive complex ILC structures. Each anti-involution is expressed in the same basis used in [9, Tables 6–8].
There are ILC parameter redundancies resulting from certain basis changes [9, Table 13]:
• N.6-1, D.7, and D.6-3:a→ −a.
• N.6-2:a→ −aandb→ −bare independent redundancies.
• D.6-2: none.
These have no effect on anti-involutions except in the D.7 case: (ϕ1(1,2), ϕ2) → (ϕ1(2,1), ϕ2).
Table 6 Representative admissible anti-involutions
Model Representative anti-involution Parameter conditions Levi form type
Appendix B: Tubular hypersurfaces
In Tables7and8, we give the complete (local) classification of (non-flat) homogeneous tubular hypersurfaces inC3with non-degenerate Levi form, organized according to Petrov type. The third column classification is given in terms of our ILC classification and the anti-involutions presented in Table6. (No anti-involution is specified if there is a unique one.) We let = ±1 here, and writex =Re(z1),y =Re(z2), andu =Re(w). CR parameter redundancies are indicated, e.g.α∼ −α.
By Theorem4.8, the structures excluded from this list are D.6-3(a2 ∈ R\{0,9})and N.6-2 (b2=a2∈C\R). These are discussed in Sects.5and6respectively.
Remark B.1 Foru=yexp(x)+exp(αx)in the N.6-2 case,α=0,1 lead to the flat model, whileα= −1,2 give alternative descriptions of the N.8 model.
Remark B.2 In the first three D.7 cases,α= −1 yields the flat model. This is also true in the D.6-2 case whenα=0,1,2.
Appendix C: Loboda’s models
In Table9, we give a dictionary between Loboda’s classifications and our results. The first two series of examples describe all non-degenerate hypersurfaces with 7-dimensional symmetry algebra and indefinite [11] or definite [12] Levi form. The last seven rows correspond to hypersurfaces with 6-dimensional symmetry algebra and positive-definite Levi form found in [13].5All equations here use the notationzj =xj+i yjandw=u+iv.
Loboda’s models are not in the tubular form (4.2), but we find the corresponding complex ILC structures as in Sect.2.1(by replacing barred variables with parameters), see for instance Example2.2. We then proceed similarly as in the examples in Sect.4to identify these models in our classification.
5As noted in the Introduction, a D.6-1 real form is missing from Loboda’s list.
Table7RealaffinesurfacesandsymmetriesofcorrespondingtubularCRstructures:typeNcases RealaffinesurfaceF(x,y,u)=0Affinehom.?ClassificationCRsymsofF(Re(z1),Re(z2),Re(w))=0beyondi∂z1,i∂z2,i∂w u=xy+x4N.8
∂z2+z1∂w, iz1∂z2+i(z1)2 2∂w, z1∂z1+3z2∂z2+4w∂w, ∂z1−6(z1)2∂z2+(z2−2(z1)3)∂w, iz1∂z1+i(z2−2(z1)3)∂z2+i(z1z2−(z1)4 2)∂w u=xy+xln(x)N.7-2 sl(2,R)(V2⊕V0) ϕ(+1)
∂z2+z1∂w, iz1∂z2+i(z1)2 2∂w, z1∂z1−∂z2+w∂w, i(z1)2 2∂z1+i(w−z1)∂z2+iwz1∂w u=Xy+Xln(X), X=exp(2x)+1×N.7-2 su(2)(V2⊕V0) ϕ(−1)
cosh(z1)∂z1− 1 2exp(−z1)w+exp(z1) ∂z2+wsinh(z1)∂w, exp(−z1)∂z2+2cosh(z1)∂w, iexp(−z1)∂z2−2sinh(z1)∂w! , isinh(z1)∂z1+i 1 2exp(−z1)w−exp(z1) ∂z2+iwcosh(z1)∂w u=xy+xα (α∈R\{0,1,2,3,4})
N.6-1 a2=1−α α−4 ∈R\{−1,−1 4,0,1 2,2}
∂z2+z1∂w, iz1∂z2+i(z1)2 2∂w, z1∂z1+(α−1)z2∂z2+αw∂w u=xy+ln(x)N.6-1 a2=−1 4
∂z2+z1∂w, iz1∂z2+i(z1)2 2∂w, z1∂z1−z2∂z2+∂w u=xy+x2ln(x)N.6-1 a2=1 2
∂z2+z1∂w, iz1∂z2+i(z1)2 2∂w, z1∂z1+(z2−z1)∂z2+2w∂w
Table7continued RealaffinesurfaceF(x,y,u)=0Affinehom.?ClassificationCRsymsofF(Re(z1),Re(z2),Re(w))=0beyondi∂z1,i∂z2,i∂w u=xy+x3ln(x)×N.6-1 a2=2
∂z2+z1∂w, iz1∂z2+i(z1)2 2∂w, z1∂z1+(2z2−3 2(z1)2)∂z2+(3w−1 2(z1)3)∂w u=yexp(x)+exp(αx) (α∈R\{−1,0,1,2};α∼1−α)×
N.6-2 b2=a2=−(2α−1)2 (α+1)(α−2) ∈R\([−4,0)∪{1 2})
∂z1+(α−1)z2∂z2+αw∂w, exp(1 2z1)∂w+exp(−1 2z1)∂z2, iexp(1 2z1)∂w−iexp(−1 2z1)∂z2 ucos(x)+ysin(x)=exp(βx) (β∈R;β∼−β)×N.6-2 b2=a2=−4β2 β2+9∈(−4,0]∂z1−(βz2+w)∂z2+(z2−βw)∂w, sin(z1)∂z2−cos(z1)∂w, −icos(z1)∂z2−isin(z1)∂w u=xy+exp(x)N.6-2 b2=a2=−4
∂z2+z1∂w, iz1∂z2+i(z1)2 2∂w, ∂z1+z2∂z2+(w+z2)∂w u=yexp(x)−x2 2×N.6-2 b2=a2=1 2
∂z1−z2∂z2−z1∂w, exp(1 2z1)∂w+exp(−1 2z1)∂z2, iexp(1 2z1)∂w−iexp(−1 2z1)∂z2
Table8RealaffinesurfacesandsymmetriesofcorrespondingtubularCRstructures:typeDcases RealaffinesurfaceF(x,y,u)=0Affinehom.?ClassificationCRsymsofF(Re(z1),Re(z2),Re(w))=0beyondi∂z1,i∂z2,i∂w u=αln(x)+ln(y) (α∈R\{−1,0};α∼1 α)D.7 sl(2,R)×sl(2,R)×R a=3 4(α−1 α+1)∈R\{±3 4} ϕ(−1,−1) 1,|a|<3 4; ϕ(1,−1) 1,a>3 4; ϕ(−1,1) 1,a<−3 4
z1∂z1+α∂w, z2∂z2+∂w, i(z1)2∂z1+2iαz1∂w, i(z2)2∂z2+2iz2∂w u=αln(X)+ln(y), X=exp(2x)+1 (α∈R\{−1,0};α∼1 α)
×D.7 sl(2,R)×su(2)×R a=3 4(α−1 α+1)∈R\{±3 4} ϕ(1,1) 1,a<−3 4; ϕ(1,−1) 1,|a|<3 4; ϕ(−1,−1) 1,a>3 4
z2∂z2+∂w, i(z2)2∂z2+2iz2∂w, cosh(z1)∂z1+αexp(z1)∂w, isinh(z1)∂z1+iαexp(z1)∂w u=αln(X)+ln(Y), X=exp(2x)+1, Y=exp(2y)+1 (α∈R\{−1,0};α∼1 α)
×D.7 su(2)×su(2)×R a=3 4(α−1 α+1)∈R\{±3 4} ϕ(1,1) 1,|a|<3 4; ϕ(−1,1) 1,a>3 4; ϕ(1,−1) 1,a<−3 4
cosh(z1)∂z1+αexp(z1)∂w, cosh(z2)∂z2+exp(z2)∂w, isinh(z1)∂z1+iαexp(z1)∂w, isinh(z2)∂z2+iexp(z2)∂w u=αarg(ix+y)+ln(x2+y2) (α∈R;α∼−α)D.7 sl(2,C)R×R ϕ2,a=3 8iα z1∂z1+z2∂z2+∂w, z2∂z1−z1∂z2+α∂w, z2 1−z2 2 2∂z1+z1z2∂z2+(z1−αz2)∂w, z1z2∂z1−z2 1−z2 2 2∂z2+(αz1+z2)∂w
Table8continued RealaffinesurfaceF(x,y,u)=0Affinehom.?ClassificationCRsymsofF(Re(z1),Re(z2),Re(w))=0beyondi∂z1,i∂z2,i∂w u=y2+ln(x)D.7 Semisimplepart=sl(2,R) ϕ(,−1),a=3 4
∂z2+2z2∂w, z1∂z1+∂w, iz2∂z2+i(z2)2∂w, i(z1)2∂z1+2iz1∂w u=y2+ln(X), X=exp(2x)+1×D.7 Semisimplepart=su(2) ϕ(,1),a=3 4
∂z2+2z2∂w, iz2∂z2+i(z2)2∂w, cosh(z1)∂z1+exp(z1)∂w, isinh(z1)∂z1+iexp(z1)∂w xu=y2−xln(x)D.6.1 ϕ()z1∂z2+2z2∂w, 2z1∂z1+z2∂z2−2∂w, i(z1)2∂z1+iz1z2∂z2+i(−2z1+(z2)2)∂w u=y2+xα(x>0) (α∈R\{0,1,2})D.6-2 a=2 3(α+1 α)∈R\{2 3,4 3,1} ϕ(ρ),ρ=sgn[(a−2 3)(a−4 3)(a−1)]
∂z2+2z2∂w, iz2∂z2+i(z2)2∂w, z1∂z1+αz2 2∂z2+αw∂w u=y2+xln(x)D.6-2 ϕ(−),a=4 3∂z2+2z2∂w, iz2∂z2+i(z2)2∂w, z1∂z1+1 2z2∂z2+(z1+w)∂w u2+1x2+2y2=1 (1,2)∈{±(1,1),(1,−1)}D.6-3 a2=9 so(1,2)R3,ϕ1,(1,2)=(+1,−1); so(3)R3,ϕ(+1) 2,(1,2)=(+1,+1); so(1,2)R3,ϕ(−1) 2,(1,2)=(−1,−1) 1z2∂z1−2z1∂z2, w∂z1−1z1∂w, w∂z2−2z2∂w
Table9CorrespondencewithLoboda’smodels NumberSurfaceParameterOurclassification 7DIndef(2)v=(z1¯z2+z2¯z1)+(1+ε|z1|2)ln(1+ε|z1|2)ε=±1N.7-2,ϕ(−ε) 7DIndef(3)v=eiθln(1+z1¯z2)+e−iθln(1+z2¯z1)θ∈(−π 2,π 2)D.7,ϕ2,a=3i 4tan(θ) 7DIndef(4)v=ln(1−|z1|2)−bln(1−|z2|2)b∈(0,1)D.7,ϕ(1,−1) 1,a=3 41+b 1−b 7DIndef(5)v=ln(1+|z1|2)+bln(1−|z2|2)b∈(0,∞)D.7,ϕ(1,−1) 1,a=3 41−b 1+b 7DIndef(6)v=ln(1+|z1|2)−bln(1+|z2|2)b∈(0,1)D.7,ϕ(−1,1) 1,a=3 41+b 1−b 7DIndef(7)v=|z2|2+εln(1−ε|z1|2)ε=±1D.7,ϕ(ε,−ε) 1,a=3 4 7DDef(0.1)v=ln(1+|z1|2)+bln(1+|z2|2)b∈(0,1]D.7,ϕ(1,1) 1,a=3 41−b 1+b 7DDef(0.2)v=ln(1+|z1|2)−bln(1−|z2|2)b∈(0,1)D.7,ϕ(−1,−1) 1,a=3 41+b 1−b 7DDef(0.2)v=ln(1+|z1|2)−bln(1−|z2|2)b∈(1,∞)D.7,ϕ(1,1) 1,a=3 41+b 1−b 7DDef(0.3)v=ln(1−|z1|2)+bln(1−|z2|2)b∈(0,1]D.7,ϕ(−1,−1) 1,a=3 41−b 1+b 7DDef(0.4)v=|z2|2+εln(1+ε|z1|2)ε=±1D.7,ϕ(ε,ε) 1,a=3 4 6DDef(1)v=x2 2+(1+x1)α−1α∈(−∞,0)∪(1,2)D.6-2,ϕ(−1),a=2 3α+1 α 6DDef(1)v=x2 2+(1+x1)α−1α∈(2,∞)D.6-2,ϕ(1),a=2 3α+1 α 6DDef(2)v=x2 2−(1+x1)α+1α∈(0,1)D.6-2,ϕ(−1),a=2 3α+1 α 6DDef(3)v=x2 2+(1+x1)ln(1+x1)−D.6-2,ϕ(−1),a=4 3 6DDef(4),(5)(x2 1+x2 2)+u2=1=±1D.6-3,a2=9,ϕ() 2 6DDef(6)1+(|z1|2+|z2|2)+|w|2=c|1+z2 1+z2 2+w2|c>1,=±1D.6-3,a2=9 c2<9,ϕ() 2 6DDef(7)1+(|z1|2+|z2|2)−|w|2=c|1+z2 1+z2 2−w2|0<c<1,=±1D.6-3,a2=9 c2>9,ϕ(−) 2
Appendix D: Homogeneous 3-dimensional CR structures
It is well-known that all Levi non-degenerate real hypersurfaces inC2 admit at most an 8-dimensional symmetry algebra. Moreover, the submaximal symmetry dimension is 3 and É. Cartan gave a complete local classification of all such (homogeneous) models [4, bottom of p.70]. Here we outline how this classification can be alternatively derived from the well-known classification of (complex) 2nd order ODE that are homogeneous (in fact, simply-transitive, so the isotropy subalgebra is everywhere trivial) under point symmetries [15, Table 7].6The list is:
(A):u= 3(u2u)2 +u3
(B):u=6uu−4u3+c(u−u2)3/2, wherec∈C\{0}; c∼ −c (C):u=(u)γ, whereγ ∈C\{0,1,2,3}; γ ∼3−γ
(D):u=e−u
All parameters that yield equivalent models are indicated, e.g.γ ∼3−γ. Setp=ubelow.
Label Point symmetries Lie algebra structure
(A) e1=∂x, e2=x∂x−u∂u−2p∂p, e3=x2∂x−2xu∂u−(4x p+2u)∂p
[e1,e2] =e1 [e1,e3] =2e2 [e2,e3] =e3 (B) e1=∂x, e2=x∂x−u∂u−2p∂p,
e3=x2∂x−(2xu+1)∂u−(4x p+2u)∂p+c2e2−e1
[e1,e2] =e1 [e1,e3] =c2e1+2e2 [e2,e3] =2e1−2ce2+e3
(C) e1=(γ−1)x∂x+(γ−2)u∂u−p∂p e2=∂x, e3=∂u,
[e1,e2] = −(γ−1)e2 [e1,e3] = −(γ−2)e3 [e2,e3] =0
(D) e1=∂x, e2=∂u, e3=x∂x+(x+u)∂u+∂p
[e1,e2] =0 [e1,e3] =e1+e2 [e2,e3] =e2
For each model, pick a general pointo, identify the (1-dimensional) subalgebraseandv of the point symmetry algebrascorresponding to the line fieldsE=span{∂x+p∂u+f∂p} andV =span{∂p}ato, and then classify all anti-involutions ofsthat swapeandv. (This is tedious, but straightforward.) All representative such admissible anti-involutions are given in Table10.
6The ODEu= 3(2uu)2+cu3as listed in [15, Table 7] is flat whenc=0 (so 8 symmetries) and allc=0 are equivalent via scalings, so we normalizedc=1. Similar normalizations were done in the other cases.
Table 10 Anti-involutions associated to (non-flat) homogeneous 2nd order ODE
Label General point e v Anti-involutions swappingeandv
(A) x=0,u=1,
It is easy to recognize tubular CR structures from Table10. These arise from (C) and (D), since each admits a unique abelian subalgebraa(namely, span{e2,e3}and span{e1,e2} respectively) that is complementary to botheandv, and satisfies the properties given in Sect.4.
(None exists for (B) sinces∼=sl(2,C).) The listed anti-involutions preserveain these cases, and since dim(N(a)/a)=1, the base curve for the tubular CR hypersurface model is affine homogeneous. The classification (up to affine equivalence) of locally homogeneous curves in the affine plane consists of lines, quadrics, and the curves given below. (See Example 1 in [8], particularly p.32 there.) symmetries. On the other hand,u=xaandu=xln(x)lead to the CR models underlying (C) (withγ = a−2a−1 ∈R\{0,1,2,3}and anti-involutionϕ1) and (D). This is not so straightforward forx2+u2 =exp(barg(x+i u)), so we instead work abstractly and align the Lie algebra data. Letting
L1=(bz−2w)∂z+(bw+2z)∂w, L2=i∂z, L3=i∂w, we have
[L1,L2] = −bL2−2L3, [L1,L3] =2L2−bL3, [L2,L3] =0.
On the other hand, the anti-involutionϕ2from case (C) has real fixed point set spanned by E1=4i(e1−e2), E2=e2+e3, E3 =i(e2−e3). (5.2). Since the classified anti-automorphisms for (B) are very complicated, we proceed in a different manner. Note thatshas two real forms:sl(2,R)∼=so(2,1)andsu(2)∼=so(3). For each, a left-invariant CR structure is uniquely determined by a two-dimensional subspace C with[C,C] ⊂ C, and a complex structure J:C → C. The subspaceC is uniquely determined by its Killing perpC⊥. Insu(2),[C,C] ⊂C always. Insl(2,R),[C,C] ⊂C if and only ifC⊥is spanned by a nilpotent element, andCis conjugate to the subalgebra of upper-triangular matrices insl(2,R). (Exclude this last case.)
IdentifyingJwith a 2×2 real matrix satisfyingJ2= −1, we haveJ2−tr(J)J+det(J)= suchtexists. Thus,Jcan always be brought to the form
0 −α 1/α 0
. Note thatCis also stable under Ad 0 11 0!
. It induces the transformatione1 → −e1,e2 →e2and thus the parameter equivalenceα ∼ −α. Performing similar computations for other subspacesC, we get the following list of all algebraic models for case (B).
Real form
ofsl(2,C) Basis of C⊥ Basis of the contact planeC Complex structure Parameter equivalence
The first of these cases is treated in detail in [2]. The two others are their non-compact analogues.
To construct the local models, we proceed as in Sect.5.2and just give a summary here. can assume(v, w)=0. Taking(·,·)|non-degenerate, the following are normal forms for [z] ∈CP2\Q(which slightly differ from those given in Sect.5.2):
(1) is positive-definite and the signature of the scalar product(z,z)is(+++)or(++−):
z=(1,i y,0), where 0<y<1. Thenβ= 11−+yy22 >1.
(2) is indefinite: assuming signature(+ + −), we have:z=(1,0,i y), where 0<y=1.
Thenβ= 1−y1+y22 satisfies 0<|β|<1.
Matching these orbits with the above algebraic models is straightforward. Fix the affine chart (1,z2,z3)inCP2. For case (1), the orbit of O(3)or O(2,1)has 3-dimensional real tangent space inC2∼=T[z]CP2given by(1−y2)∂z2, ∂z3,i y∂z3. (The scalingz1∂z1+z2∂z2+z3∂z3
onC3induces a trivial action onCP2, so in the given chart, we can make the substitutions
∂z1 = −z2∂z2−z3∂z3intoZj k.) ThenC = ∂z3,i y∂z3, and multiplication byiis represented by
0 −y 1/y 0
in this basis. The range 0<y<1 matches with the first two cases in the table above (namely, sety=α). Case (2) is handled similarly.
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