left-invariant cr structures on 3-dimensional lie groups · connected 3-dimensional lie groups g i...

Left-invariant CR structures on 3-dimensional Liegroups

Gil Bor and Howard Jacobowitz

June 8, 2020

Gil Bor and Howard Jacobowitz Left-invariant CR structures on 3-dimensional Lie groups June 8, 2020 1 / 26

A CR structure on a 3-dimensional manifold M is a rank 2 subbundleD ⊂ TM together with an almost complex structure J on D

J : D → D, J2 = −Id .

Equivalently, a CR structure is a complex line subbundle V ⊂ TM ⊗ Csuch that V ∩ V = {0}.

D = {<V ,=V }

Notation: (M,V ) or (M, L).Usually assume L, L̄, [L, L̄] linearly independent (spc).Note: [L, L̄] ≡ 0 means that M is foliated by complex curves.f : M → C is a CR function if Lf = 0 for each section of V .


G a 3 dimensional Lie groupG its Lie algebra

G = TeG = {L : g∗Le = Lg}

(G , L) is a CR manifold provided

L ∩ L̄ = {0}.


ConjugationΦ : G → Aut(G )

Φ(g)h = ghg−1

induces a mapAd : G → Aut(G)

with derivative at the identity

ad : G→ G

given byad(X )Y = [X ,Y ].


Immersions into CP2

A complex line sub-bundle V ⊂ T (G )⊗ C determines a point inP(G⊗ C) = CP2, L→ [L] ∈ P(G⊗ C), and the orbit of L

{X : ∃g ∈ G , AdgL = X}

gives a map of G into CP2

φ : G → CP2

g → [AdgL]

with differential at the identity

φ∗X = [[X , L]].

φ is always a CR mapG → CP2.

φ is an immersion if

[X , L] = 0 for X real ⇒ X = 0.


Spherical and Aspherical

(M,V ) is spherical near a point if it is locally CR equivalent to thestandard CR structure on S3.

Theorem (Cartan 1932)

The local CR equivalence group of a 3 dimensional aspherical CR manifoldhas dimension at most 3.

Consider two left-invariant aspherical CR structures (Gi ,Vi ) on twoconnected 3-dimensional Lie groups Gi

Corollary

If the two CR structures are locally equivalent, then there exists a groupisomorphism G1 → G2 which extends this local CR equivalence.


Proof.1 Right-invariant vector fields generate left-translations. SoRGi⊂ AutCR(Gi ,Vi ).

2 Since the dimension are equal these groups coincide near the identity.

3 The local CR equivalence f maps RG1 → RG2 .

4 f preserves the group structure.


The most interesting examples: SL(2,R) and SU(2)

Basis for sl(2,R) [1 00 −1

],

[0 10 0

],

[0 01 0

]Basis for su(2) [

1 00 −1

],

[0 1−1 0

],

[0 ii 0

]Note

sl(2,C) = su(2)⊗ C.

andP(G⊗ C) = P(sl(2,C)) = CP2.


Geometry of P(sl(2,C))

Let L ∈ sl(2,C) be given as [a bc −a

].

The Killing form is

K (L, L) = a2 + bc = −detL

andK (L, L) = 0

defines a conic in C = CP1 ⊂ P(sl(2,C)). The polar line with respect to Cof a point [L] ∈ CP2 determines two points in C.

S = {{ζ1, ζ2} : ζi ∈ CP1}.


L→ [L] ∈ P(sl(2,C))→ S

is SL(2,C) equivariant.

P(sl(2,C))Adg−→ P(sl(2,C)))y y

S −→g

S

Classifying the orbits of [L] under SL(2,R) or SU(2) uses that

SL(2,R) preserves the hyperbolic distance in the upper half-plane.

SU(2) preserves the Euclidean distance in S2.


So the following are equivalent:

1 [L] and [L̃] are in the same AutG orbit.

2 d(ζ1, ζ2) = d(ζ̃1, ζ̃2).

3 The left-invariant CR structures (G , L) and (G , L̃) are CR equivalentunder a group automorphism.


Theorem

Let Vt ⊂ TCSL(2,R), t ∈ (−1, 1] be the left-invariant complex line bundlespanned at e by

Lt =

(i 1+t

2 t1 −i 1+t

2

)∈ sl2(C).

Then

1 Vt is a left-invariant CR structure for all t 6= 0.

2 V0 is a foliation by complex curves.

3 Vt is spherical if t = 1 or −3± 2√

2 and aspherical otherwise.

4 Every left-invariant CR structure on SL2(R) is CR equivalent to Vt

for a unique t.

5 The aspherical left-invariant CR structures Vt ,t ∈ (−1, 1) \ {0,−3 + 2

√2}, are pairwise non-equivalent, even locally.


The immersion results rely on a geometric observation. Recall the nullcone of the Killing form of sl(2,R)

C = {L : a2 + bc = 0}.

A left-invariant CR structure on SL(2,R) is called hyperbolic if the real2-plane spanned by {<L,=L} intersects C in two lines. It is called ellipticif the intersection is the origin.Note: One spherical structure is elliptic, the other is hyperbolic.


Theorem1 The elliptic left-invariant spherical CR structure on SL(2,R) (t = 1)

is realizable by the map(a bc d

)→(b + iad + ic

).

It is also realizable as the complement of S3 ∩ C1.

2 The hyperbolic spherical structure on SL(2,R) (t = −3 + 2√

2) isrealizable as the complement of S3 ∩R2.

3 The rest of the left-invariant CR structures on SL(2,R) are either4 : 1 covers, in the aspherical elliptic case 0 < t < 1, or 2 : 1 covers,in the aspherical hyperbolic case −1 < t < 0, of the orbits ofSL(2,R) in P(sl(2,C)).


Theorem

Let Vt ⊂ TCSU(2), t ≥ 1 be the left-invariant complex line bundlespanned at e ∈ SU(2) by

Lt =

(0 t − 1

t + 1 0

).

1 Vt is a left-invariant CR structure on SU(2) for all t ≥ 1.

2 Vt is spherical if and only if t = 1.

3 Every left-invariant CR structure on SU(2) is CR equivalent to Vt fora unique t.

4 The aspherical left-invariant CR structures Vt , t > 1, are pairwisenon-equivalent, even locally.

5 V1 is realized by the usual CR structure on S3 ⊂ C2. The asphericalstructures are realized as 4 : 1 covers of the adjoint orbits of SU(2) inP(sl(2,C)).


The Heisenberg Group

H =

1 a c

0 1 b0 0 1

: a, b, c ∈ R

and

h =

0 x z

0 0 y0 0 0

: a, b, c ∈ R

H is isomorphic to the group on the hyperquadric

{(z ,w) ∈ C2 : =w = |z |2}.


Theorem1 Every plane in h containing 0 0 1

0 0 00 0 0

gives a foliation of H by complex curves.

2 Every other plane is in the orbit of

L =

0 1 00 0 i0 0 0

under Aut(H).

3 L gives a spherical CR structure equivalent to the usual one on thehyperquadric.


ObservationLet Vt , t ∈ R be a smooth family of CR structures on some M3, such that

Vt ≡ Vs for s 6= 0, t 6= 0

butV1 6≡ V0.

Then V0 is a foliation of M by complex curves.Example

Lt =

0 i 10 0 t0 0 0

→o i 1

0 0 00 0 0

.

{Lt , L̄t} is not involutive for t 6= 0 but {L0, L̄0} is involutive.


The Euclidean Group

E =

cos θ sin θ u− sin θ cos θ v

0 0 1

e =

0 x y−x 0 z0 0 0


Theorem

Let V ⊂ TCE be the left-invariant line bundle whose value at e ∈ E isspanned by

L =

0 −i 1i 0 00 0 0

.

Then

1 Every spc left-invariant CR structure on E is CR equivalent to V byAut(E ).

2 V is a aspherical.

3 V is realized in P(e⊗ C) = CP2 by the adjoint orbit of [L]. This isCR equivalent to the real hypersurface [<(z1)]2 + [<(z2)]2 = 1 in C2.


Lt =

0 −it ait 0 bi0 0 0

is an aspherical CR structure for t 6= 0 but a foliation by complex curvesfor t = 0.


Cartan’s Moving Frames

H −→ SU(2, 1)yQ

dimSU(2, 1) = 8, dimH = 5.Maurer-Cartan form Θ : TSU(2, 1)→ su(2, 1)

Let M be a three dimensional (spc) CR structure.

H5 −→ B8yM3

Θ : TB ⊗ C→ su(2, 1)⊗ C.


Theorem

With each spc CR 3-manifold M there is canonically associated a bundleB → M with Cartan connection Θ : TB → su(2, 1), satisfying

1 The eight components of Θ are pointwise linearly independent.

2 (The CR structure equations) There exist functions R,S : B → Csuch that

dθ = iθ1 ∧ θ̄1 − θ ∧ (θ2 + θ̄2),

dθ1 = −θ1 ∧ θ2 − θ ∧ θ3,dθ2 = 2i θ1 ∧ θ̄3 + i θ̄1 ∧ θ3 − θ ∧ θ4,dθ3 = −θ1 ∧ θ4 − θ̄2 ∧ θ3 − R θ ∧ θ̄1,dθ4 = i θ3 ∧ θ̄3 − (θ2 + θ̄2)θ4 + (S θ1 + S̄ θ̄1) ∧ θ.

3 (Spherical structures) M is spherical if and only if R ≡ 0.

4 Any local CR diffeomorphism of CR manifolds f : M → M ′ liftsuniquely to an map f̃ : B → B ′ with f̃ ∗Θ′ = Θ.


φ, φ1 is an adapted coframe for the CR structure V on M if

1

φ is real.

2

φ(V ) = 0, φ1(V ) = 0.

3

φ ∧ φ1 ∧ φ̄1 6= 0.

4

dφ = i φ1 ∧ φ̄1.

Each adapted coframe has a unique lift to σ : M → B such that σ∗θ = φand σ∗θ1 = φ1.


Pulling back the other components of Θ we obtain

dφ = iφ1 ∧ φ̄1 − φ ∧ (φ2 + φ̄2),

dφ1 = −φ1 ∧ φ2 − φ ∧ φ3,dφ2 = 2i φ1 ∧ φ̄3 + i φ̄1 ∧ φ3 − φ ∧ φ4,dφ3 = −φ1 ∧ φ4 − φ̄2 ∧ φ3 − r φ ∧ φ̄1,dφ4 = i φ3 ∧ φ̄3 + (s φ1 + s̄ φ̄1) ∧ φ.

M is spherical if and only if the Cartan relative curvature invariant r isidentically 0.


Lemma

Let M be a manifold with a CR structure given by a coframe φ, φ1satisfying

dφ = iφ1 ∧ φ̄1,dφ1 = aφ1 ∧ φ̄1 + b φ ∧ φ1 + c φ ∧ φ̄1,

(1)

for some complex constants a, b, c . Then

r = ic(|a|2

3+

3ib

2).

If a left-invariant CR structure has a coframe satisfying (1) then a, b, care constants and so the structure is spherical if and only if

c(|a|2

3+

3ib

2) = 0.


left-invariant cr structures on 3-dimensional lie groups · connected 3-dimensional lie groups g i...

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