Outline
What is a free CR manifold?
Cartan connections
Computation
Free CR Distributions
Gerd Schmalz
University of New England
joint with J. Slovák, Brno
ANU, Canberra, 19 September, 2011
Gerd Schmalz, UNE
Free CR Distributions
Outline
What is a free CR manifold?
Cartan connections
Computation
What is a free CR manifold?
Cartan connections
Basic idea
Tanaka prolongation
Parabolic Geometry
First surprise
Second surprise
Computation
Gerd Schmalz, UNE
Free CR Distributions
Outline
What is a free CR manifold?
Cartan connections
Computation
To start with:
I
Let M be a 2n + n2 -dimensional real submanifold of Cn+n
Gerd Schmalz, UNE
Free CR Distributions
2
Outline
What is a free CR manifold?
Cartan connections
Computation
To start with:
2
I
Let M be a 2n + n2 -dimensional real submanifold of Cn+n
I
Induced structure: Dp = Tp M ∩ i Tp M - generically of
dimension 2n (complex dimension n) - assume it is 2n for all p.
Gerd Schmalz, UNE
Free CR Distributions
Outline
What is a free CR manifold?
Cartan connections
Computation
To start with:
2
I
Let M be a 2n + n2 -dimensional real submanifold of Cn+n
I
Induced structure: Dp = Tp M ∩ i Tp M - generically of
dimension 2n (complex dimension n) - assume it is 2n for all p.
I
Jp : Dp → Dp with Jp2 = − id.
Gerd Schmalz, UNE
Free CR Distributions
Outline
What is a free CR manifold?
Cartan connections
Computation
To start with:
2
I
Let M be a 2n + n2 -dimensional real submanifold of Cn+n
I
Induced structure: Dp = Tp M ∩ i Tp M - generically of
dimension 2n (complex dimension n) - assume it is 2n for all p.
I
Jp : Dp → Dp with Jp2 = − id.
I
Levi form Lp : Dp ∧ Dp → Tp M/Dp
ˆ η̂ in D
extend ξ, η ∈ Dx to sections ξ,
ˆ
project [ξ, η̂]p to Tp M/Dp
Gerd Schmalz, UNE
Free CR Distributions
Outline
What is a free CR manifold?
Cartan connections
Computation
Partial integrability
Compatibility: Lp (Jp ξ, Jp η) = Lp (ξ, η)
(equivalent to ‘partial integrability’/ Lp is the imaginary part of a
Hermitian form/ Lp is ‘totally real’ - follows from the involutivity
2
of TM ⊗ C) and T 1,0 Cn+n .)
Non-degeneracy: H(ξ, η) = L(Jξ, η) + i L(ξ, η) consists of n2
scalar hermitian forms that are linearly independent. (Notice: n2 is
the maximal possible such number.)
Gerd Schmalz, UNE
Free CR Distributions
Outline
What is a free CR manifold?
Cartan connections
Computation
More abstract setting:
M is a 2n + n2 dimensional manifold with rank 2n distribution D
and family of complex structure Jp at Dp .
Partial integrability and non-degeneracy, as above, are assumed.
Gerd Schmalz, UNE
Free CR Distributions
Outline
What is a free CR manifold?
Cartan connections
Computation
Basic idea
Tanaka prolongation
Parabolic Geometry
First surprise
Second surprise
Measure the ‘difference’ of a manifold with given geometric
structure from a homogeneous space with the same type of
structure:
Homogeneous space G → G /P, right action of P on G ,
left-invariant Maurer-Cartan form ωMC on G valued in g, structure
equation dωMC = − 21 [ωMC , ωMC ].
For M construct P-principal bundle G → M and g-valued 1-form ω
on G with structure equation ‘as close as possible’ to MC form
dω + 12 [ω, ω] = K .
ω and κ carry the information on the structure and invariants and
other geometric data can be derived from them.
Gerd Schmalz, UNE
Free CR Distributions
Outline
What is a free CR manifold?
Cartan connections
Computation
Basic idea
Tanaka prolongation
Parabolic Geometry
First surprise
Second surprise
Need to derive G , g, P from the geometric structure.
g− = g−2 ⊕ g−1 = Tp M/Dp ⊕ Dp
nilpotent graded Lie algebra with Lp as product, does not depend
on p.
Let g0 be the Lie algebra of derivations that preserve J on g−1 and
g = g−2 ⊕ g−1 ⊕ g0 ⊕ g1 ⊕ g2
the maximal prolongation to a graded Lie algebra. In this case it
stops at g2 with g = su(n + 1, n).
Gerd Schmalz, UNE
Free CR Distributions
Outline
What is a free CR manifold?
Cartan connections
Computation
Basic idea
Tanaka prolongation
Parabolic Geometry
First surprise
Second surprise
The grading is given by:
n
1
n














0
1
2
-1
0
1
-2
-1
0












n
1
n
with P corresponding to the non-negative grades.
Q = G /P is the manifold of complex n-planes in C2n+1 with
pseudohermitian structure of signature (n + 1, n).
Gerd Schmalz, UNE
Free CR Distributions
Outline
What is a free CR manifold?
Cartan connections
Computation
Basic idea
Tanaka prolongation
Parabolic Geometry
First surprise
Second surprise
Parabolic geometry: There is a unique Cartan connection with
∂ ∗ -closed curvature!
Existence of G → M and ω, κ = K (ω −1 ·, ω −1 ·) : G → Lin(Λ2 g− , g)
with ∂ ∗ κ = 0 follows from the general theory.
κ splits into homogeneous components.
Our aim: Compute the essential part of the curvature and give
geometric interpretation.
Method: Construct at the same time a section σ : M → G and the
restriction of ω and κ on the section, using their existence and
algebraic properties as determined by the Lie algebra cohomology.
Gerd Schmalz, UNE
Free CR Distributions
Outline
What is a free CR manifold?
Cartan connections
Computation
Basic idea
Tanaka prolongation
Parabolic Geometry
First surprise
Second surprise
Uniqueness of J
J is determined (up to conjugate) by D!
Proposition
Let J be the standard complex structure on Cn and R the space of
totally real skew symmetric bilinear forms on Cn . If J̃ is an
alternative complex structure such that all forms from R are
totally real with respect to J̃ then J̃ = ±J.
Gerd Schmalz, UNE
Free CR Distributions
Outline
What is a free CR manifold?
Cartan connections
Computation
Basic idea
Tanaka prolongation
Parabolic Geometry
First surprise
Second surprise
Proof. Let J̃ be a real endomorphism of Cn with J̃ 2 = − id and
such that any ω ∈ R is totally real with respect to J̃.
This is equivalent to J̃ω is symmetric for any skew symmetric ω
such that Jω is symmetric.
Let C2n be
of R2n = Cn with coordinates such
the complexification
i id
0
that J =
. Then Jω is symmetric for skew symmetric
0 − i id
0
B
ω if and only if ω =
for some n × n matrix B.
−B t 0
P Q
−QB t PB
2
Now let J̃ =
with J̃ = − id. Then J̃ω is
R S
−SB t RB
which is symmetric for any B if and only if Q = R = 0 and
P = −S = λ id. From J̃ 2 = − id we get λ = ± i, as required.
Gerd Schmalz, UNE
Free CR Distributions
Outline
What is a free CR manifold?
Cartan connections
Computation
Basic idea
Tanaka prolongation
Parabolic Geometry
First surprise
Second surprise
Integrability
For n > 2 there is no curvature that can be related to the
Nijenhuis tensor, hence all partially integrable structures are
automatically integrable.
Gerd Schmalz, UNE
Free CR Distributions
Outline
What is a free CR manifold?
Cartan connections
Computation
Choose {Xi , Xī = X̄i } a frame of D ⊗ C, Xi j̄ = −[Xi , Xj̄ ].
θi , θī , θ[i j̄] the dual coframe
Structure equations
dθr =fijrk̄ θi ∧ θ[j k̄] + fījr k̄ θī ∧ θ[j k̄] + fi rj̄k l̄ θ[i j̄] ∧ θ[k l̄] ,
θ[i j̄] ∧ θ[k l̄]
dθ[rs̄] =θr ∧ θs̄ + fijrs̄k̄ θi ∧ θ[j k̄] + fījrs̄k̄ θī ∧ θ[j k̄] + fi rs̄
j̄k l̄
For a section s : M → G we can pull back the Cartan connection
form to:
 i

ωj
ωi
ω[i j̄]
 j −2i Im tr ω i −ω 
s ∗ω =  ω
ī 
j
ω [i j̄]
Gerd Schmalz, UNE
−ω j̄
−ωīj̄
Free CR Distributions
Outline
What is a free CR manifold?
Cartan connections
Computation
We can choose the section s so that ω i = θi mod D ⊥ , then
ω i = θi + Cjik̄ ω [j k̄] ,
i
ωji = Aikj ω k + Bk̄j
ω k̄ mod D ⊥
for some A, B, C , which are determined by the condition that the
curvature term (−1, −1) → −1 is absent and the term
(−1, −2) → −2 is tracefree (which follows from the corresponding
Lie algebra cohomology).
A, B can be interpreted as the Christoffel symbols of a ‘partial
connection’ and C as a splitting of the frame.
Gerd Schmalz, UNE
Free CR Distributions
Outline
What is a free CR manifold?
Cartan connections
Computation
From the vanishing of (−1, −1) → −1 we obtain
1
1
Ai[rs] = (Ajrj δsi − Ajsj δri ) − (Br̄j j δsi − Bs̄jj δri ),
2
2
i
i
Crs̄
= Bs̄r
+ Ajsj δri − Bs̄jj δri .
and from the trace condition
j
i j̄
i j̄
j̄
i
i
P(rs)
t̄ = frs t̄ + A(rs) δt̄ + 2B(r̄ |t| δs) +
1
l δ j̄ δ i .
B(s̄|l
t̄| r )
n−2
the rest of A and B are determined.
Gerd Schmalz, UNE
Free CR Distributions
Outline
What is a free CR manifold?
Cartan connections
Computation
Theorem [S.-Slovák]. The trace-free part of the tensor Prsi j̄ t̄
computed above is the only fundamental invariant of the
non–degenerate free CR distribution D of rank n ≥ 3 on a manifold
of dimension 2n + n2 . Hence, the Cartan connection associated
with D is flat if and only if this tensor vanishes identically.
Gerd Schmalz, UNE
Free CR Distributions