aa r X i v : . [ m a t h . DG ] M a y Complex Engel Structures
Zhiyong ZhaoMathematics DepartmentDuke University, Durham, NC 27708-0230, USAE-mail: [email protected]
Abstract
We study the geometry of Engel structures, which are 2-plane fields on 4-manifoldssatisfying a generic condition, that are compatible with other geometric structures.A complex Engel structure is an Engel 2-plane field on a complex surface for whichthe 2-planes are complex lines. We solve the equivalence problems for complex Engelstructures and use the resulting structure equations to classify homogeneous complexEngel structures. This allows us to determine all compact, homogeneous examples.Compact manifolds that support homogeneous complex Engel structures are diffeo-morphic to S ˆ SU p q or quotients of C , S ˆ SU p q , S ˆ G or H by co-compactlattices, where G is the connected and simply-connected Lie group with Lie algebra sl p R q and H is a solvable Lie group. Key words and phrases: complex Engel structure, structure equation, homoge-neous manifold. A distribution is a subbundle D Ă T M of the tangent bundle of a manifold M . We will considercertain distributions with special properties, for example, distributions with some integrabilityconditions. Given a distribution of rank n on a manifold M m , if, for each point of M , there exists acoordinate neighborhood U and local coordinates x , x , ¨ ¨ ¨ , x m such that BB x i , i “ , ¨ ¨ ¨ , n forms1 local basis for the distribution on U , then the distribution is said to be completely integrable .Besides complete integrability, one can consider partially integrable distributions. An extremecondition is to be nowhere integrable, i.e., for every p P M , there exist X, Y , sections of D ,such that r X, Y s p is not a section of D . For example, contact structures are nowhere integrabledistributions on odd dimensional manifolds. The study of contact structures [8] usually involvesinterplay between geometry, topology and dynamics. Contact structures play an important role inthe study of low-dimensional topology.We will study Engel structures , which are certain non-integrable distributions defined on 4-manifolds.We will see that, locally, all Engel structures are isomorphic but the global theory of Engel structuresis not trivial. A 4-manifold can carry many nonisomorphic Engel structures [13]. There are rela-tions between contact structures on 3-dimensional manfiolds and Engel structures. For example, V.Gershkovich [13] proved that each Engel manifold carries a canonical one-dimensional foliation andan Engel structure defines a contact distribution on any three-dimensional submanifold transversalto the canonical foliation.We will solve the equivalence problem for complex Engel structures and present the classificationof compact quotients of homogeneous complex Engel structures. Engel structures (to be definedbelow) can be characterized in terms of the derived system construction.
Proposition 1.1. [2] Given a Pfaffian system I , there exists a bundle map δ : I Ñ Λ p T ˚ M { I q that satisfies δω ” dω mod p I q for all ω P Γ p I q . Definition 1.1.
By Proposition 1.1, we have a bundle map δ . Set I p q “ ker δ and call I p q thefirst derived system . Continuing with this construction, we can get a filtration I p k q Ă ¨ ¨ ¨ Ă I p q Ă I p q Ă I p q “ I, defined inductively by I p k ` q “ p I p k q q p q .I p k q is called the kth derived system . Now we present the definition and characterization of Engel structures.
Definition 1.2 (Engel Structure) . Given a 4-manifold M and a Pfaffian system I Ă T ˚ M , an Engel structure is a sub-bundle D “ I K of the tangent bundle of M that satisfies: p q I is of rank , I p q is of rank 1 and I p q “ . A manifold endowed with an Engel structure D “ I K is calledan Engel manifold . Definition 1.3.
Given an ideal I generated by a Pffafian system I , a vector field ξ is calleda Cauchy characteristic vector field of I if ξ ⌟ I Ă I . At a point x P M , the set of Cauchycharacteristic vector fields is A p I q x “ t ξ x P T x M | ξ x ⌟ I x Ă I x u Ă I K and the retracting space or Cartan system is defined to be C p I q x “ A p I q K x Ă T ˚ x M .
By the definition of Engel structure I K , there is a canonical flag of sub-bundles0 Ă I p q Ă I Ă C p I p q q Ă T ˚ M .
V. Gershkovich [13] proved the following theorem which can also be found in [6].
Theorem 1.2.
If an orientable 4-manifold admits an orientable Engel structure, then it has trivialtangent bundle.
T. Vogel [1] proved the converse of the above theorem:
Theorem 1.3.
Every parallelizable 4-manifold admits an orientable Engel structure.
Thus for an orientable 4-manifold, parallelizability is equivalent to the existence of an orientableEngel structure. This is a global characterization of manifolds that support orientable Engel struc-tures. Locally, we have the following
Engel normal form [2], which implies that there is no localinvariant for Engel structures, i.e., all Engel structures are locally equivalent.
Theorem 1.4 (Engel normal form) . Let I be a Pfaffian system on M such that I K Ă T M is anEngel structure. Then every point of M has an open neighborhood U on which there exists localcoordinates p x, y , y , y q : U Ñ R such that I | U “ t dy ´ y dx, dy ´ y dx u .
3n this paper, we will consider complex Engel structures.
Definition 1.4.
Given a complex manifold p M, J, I q with an Engel structure D “ I K , if I K Ă T M is a complex line field, the Engel structure is called a complex Engel structure . Let J be the complex structure on the underlying manifold M . Choose a local J -complex coframing p ω , ω q on an open set U Ă M such that1. ω , ω are of J -type (1,0)2. ω “ I K on U Then p ω , ω q is called a for the Engel structure. Lemma 2.1.
The -adapted coframings are the sections of a G -structure on M , where G Ă GL p , C q is the 3-dimensional complex subgroup G “ a b c ¸ˇˇˇˇˇ a, b, c P C and a, c ‰ + . In the following analysis, we will denote the conjugate of ω , ω by ¯ ω , ¯ ω instead of ω , ω . Theorem 2.2.
A complex Engel structure has a canonical coframing p ω , ω q (i.e., an e -structure)such that dω ” ω ^ ¯ ω mod ω ,d p ω ` ¯ ω q ” ´ p ω ´ ¯ ω q ^ p ω ´ ¯ ω q mod ω ` ¯ ω . (1) Proof.
Since ω “ p ω , ω q is definedon an open set U Ă M up to the following change of coframing: ˜ ˆ ω ˆ ω ¸ “ ˜ a b c ¸ ˜ ω ω ¸ , where a, b, c are complex functions on U and ac ‰ U .4he Engel condition implies that dω ” A ω ^ ¯ ω mod ω , ¯ ω , where A ‰
0. Define ˆ ω “ A ω , then d ˆ ω “ d ` A ˘ ^ ω ` A dω ” A Aω ^ ¯ ω mod ω , ¯ ω ” ω ^ ¯ ω mod ω , ¯ ω . (2)Thus, after rescaling ω , we can arrange that A “
1. Then dω ” ω ^ ¯ ω mod ω , ¯ ω . (3)Such coframings will be said to be 1-adapted. They are the sections of a G -structure, where G Ă G is defined by c “ a ¯ a . The change of the coframing that preserves (3) is reduced to ˜ ˆ ω ˆ ω ¸ “ ˜ a b a ¯ a ¸ ˜ ω ω ¸ . (4)Now ω is defined up to a real multiple, so the real and imaginary parts of ω are uniquely definedup to a real multiple. Suppose the real part of ω spans I p q , the first derived system.By (3), the real part of ω spans I p q , the first derived system, i.e., d p ω ` ¯ ω q ” ω , ¯ ω . Since d p ω ` ¯ ω q is a real 2-form, there must be a complex function p such that d p ω ` ¯ ω q ” p p ω ´ ¯ p ¯ ω q ^ p ω ´ ¯ ω q mod ω ` ¯ ω . Because I p q “ p q , d p ω ` ¯ ω q ^ p ω ` ¯ ω q ı p ‰
0. By replacing p ω , ω q by ´ ´ p ω , p ¯ p ω ¯ , we can arrange p “ ´ . After this arrangement, a is fixed to be 1 in thetransformation. Now the coframing satisfies d p ω ` ¯ ω q ” ´ p ω ´ ¯ ω q ^ p ω ´ ¯ ω q mod ω ` ¯ ω . (5)5uch coframings will be said to be 2-adapted. They are the sections of a G -structure, where G Ă G is defined by a “
1. After setting a “
1, by (4), ω is unique and ω is unique modulo ω , i.e. a change of coframing that preserves (3) and (5) is reduced to ˜ ˆ ω ˆ ω ¸ “ ˜ b ¸ ˜ ω ω ¸ . Because ω is uniquely defined now, we can write dω ” ω ^ ¯ ω ` f ω ^ ¯ ω mod ω for some function f . Note that there is no ¯ ω ^ ¯ ω term since we assumed that p ω , ω q be of type(1,0), and the underlying almost complex structure is integrable.By adding a multiple of ω to ω , we can arrange f “ dω ” ω ^ ¯ ω mod ω . (6)After arranging this modification, the coframing satisfying (5) and (6) is completely determined:the original G -structure defines a canonical sub e-structure.Thus by a Theorem of Kobayashi [16], Corollary 2.3.
The symmetry group of a complex Engel structure acts freely on the underlyingconnected manifold.
Theorem 2.4.
The canonical coframing of a complex Engel structure satisfies dω “ ´p p ω ` p ω ` ¯ q ¯ ω ` ¯ q ¯ ω q ^ ω ´ p q ω ` ¯ r ¯ ω ` ¯ r ¯ ω q ^ ω dω “ p ω ´ ω q ^ ¯ ω ´ p p ω ` p ω ` ¯ p ¯ ω ` ¯ p ¯ ω q ^ ω (7) where p , p , q , q , r , r are complex functions. Thus, a complex Engel structure has fundamentalfunctional invariants.Proof. From Theorem 2.2, the structure equation can be writen as dω “ ω ^ ¯ ω ` p pω ` q ¯ ω ` r ¯ ω q ^ ω p, q, r . Therefore d p ω ` ¯ ω q ” p pω ` q ¯ ω ` r ¯ ω q ^ ω ` p ¯ p ¯ ω ` ¯ qω ` ¯ rω q ^ ¯ ω ” pp p ´ ¯ q q ω ` p q ´ ¯ p q ¯ ω q ^ ω mod p ω ` ¯ ω q . (8)But according to (5) d p ω ` ¯ ω q ” ´ p ω ´ ¯ ω q ^ p ω ´ ¯ ω q” ´p ω ´ ¯ ω q ^ ω mod p ω ` ¯ ω q . (9)By comparing (8) and (9), we find q “ ¯ p `
1. Thus dω “ ω ^ ¯ ω ` ¯ ω ^ ω ` p pω ` ¯ p ¯ ω ` r ¯ ω q ^ ω . Let α “ ´p pω ` ¯ rω q , then dω “ ω ^ ¯ ω ` ¯ ω ^ ω ´ p α ` ¯ α q ^ ω , (10)where α is a p , q -form, uniquely defined by (10). Let γ be a p , q´ form and β be any 1-form.The structure equation can be written as dω “ ´p α ` ¯ γ q ^ ω ´ β ^ ω ,dω “ ω ^ ¯ ω ` ¯ ω ^ ω ´ p α ` ¯ α q ^ ω . (11)Taking the exterior derivative of dω then yields ´ ¯ γ ^ ω ^ ¯ ω ` ω ^ ¯ β ^ ¯ ω ` ω ^ γ ^ ¯ ω ” ω . Recall that γ is a p , q -form, so¯ γ ^ ¯ ω ` ¯ β ^ ¯ ω ” ω , ω . Let γ “ q ω ` q ω , then β ” q ω mod ¯ ω , ω , ¯ ω . dω “ ´p p ω ` p ω ` ¯ q ¯ ω ` ¯ q ¯ ω q ^ ω ´ p q ω ` ¯ r ¯ ω ` ¯ r ¯ ω q ^ ω ,dω “ p ω ´ ω q ^ ¯ ω ´ p p ω ` p ω ` ¯ p ¯ ω ` ¯ p ¯ ω q ^ ω . (12) Remark 2.1.
We can take exterior derivatives of (12), and see that there are no further relationson p , p , q , q , r , r . All differential invariants of complex Engel structures are these six or theirderivatives with respective to the canonical coframing. In this section, we will classify homogeneous complex Engel structures. The group of diffeomor-phisms preserving a complex Engel structure also preserves its canonical coframing and hencepreserves its fundamental invariants. Thus, if it is homogeneous, then the invariants must be con-stant. Assume that the functions be constant and take exterior derivatives of dω , d ¯ ω , dω and d ¯ ω , and set all of these to be zero. This will yield quadratic equations on p, q, r . After solvingthese equations, which was done with the help of MAPLE, we arrive at the following theorem: Theorem 3.1 (Classification of Homogeneous Complex Engel Structures) . There are six distincttwo-parameter families of homogeneous structure equations of complex Engel structures. The con-stants p p , p , q , q , r , r q in equation (7) are listed as follows for the six cases: • Case C1: p a ` ib, , , , , q • Case C2: p ` ib, , ia, , , q • Case C3: p ´ ib, , ib, p b ` i qp ia ´ b q , b ´ ib, p b ` qp a ` ib qq • Case C4: p , a ´ ib, ib, a ´ ib, ´ a ´ ib, q • Case C5: p a p ´ ib q , p a ´ qp b ` i q , ib, ´ p b ` i q , ´ p b ` ab ` i p a ´ qqp´ b ` i q , a p´ ` bi qp ` b qq • Case C6: p p cos a ` i sin a qp b ` i q , p´ sin a ` b cos a ´ qp b ` i q , ib, ´ p ib sin a ` i cos a ´ b cos a ´ ib ` sin a ` qp b ` i q , p ` bi qp ib sin a ` i cos a ` b cos a ´ ib ´ sin a ´ q , p ` b qp´ ` ib qp ib sin a ` i cos a ` b cos a ´ ib ´ sin a ´ qq where a and b are real constants.Proof. First take the exterior derivatives of dω and d ¯ ω , which yield r “ p q ` p ´ q , ¯ r “ ¯ p ¯ q ` ¯ p ´ ¯ q . (13)Substituting these into exterior derivatives of dω and d ¯ ω yields q ` ¯ q “ q is pure imaginary.Set q “ iq , ¯ q “ ´ iq . (14)Substituting these relations into dω and d ¯ ω yields ℑ p “ q p p ` ¯ p ´ q , where ℑ p means the imaginary part of p .Now take the exterior derivatives of dω and d ¯ ω and set these to be zero. The equations arequadratic expressions in the coefficients of dω , dω . Then solve these quadratic equations. We getthe six different 2-parameter solutions, listed in the Theorem. We have proved the classification result for homogeneous complex Engel structures. Now we canclassify compact homogeneous complex Engel structures. We will prove the following theorem:
Theorem 4.1 (Classification of Compact Homogeneous Complex Engel Structures) . Let g be the4-dimensional Lie algebra of symmetry vector fields of a complex Engel structure. For the sixdistinct 2-parameter families of homogeneous complex Engel structures listed in Theorem 3.1, theresults about compactness in each case are listed as follows: • Case C1: If a “ and b “ , the Lie algebra is a 4-dimensional solvable Lie algebra. Thereexists a compact quotient that supports a homogeneous complex Engel structure if and only if “ and b “ . • Case C2: If there exists a complex number λ and a matrix A P SL p Z q such that1. b “ ´ a | λ | ‰
3. the eigenvalues of A are p λ ¯ λ q ´ , λ, ¯ λ
4. there exists k P Z such that ´ a log p| λ |q “ arg λ ` kπ then there exists a co-compact lattice Γ such that G { Γ supports a homogeneous complex Engelstructure. • Case C3:1. If a “ ´ , the Lie algebra is a solvable Lie algebra, and there exists a compact quotient.2. If a “ b ‰ , the Lie algebra is a solvable Lie algebra, but there does not exist a compactquotient.3. If p a ă ´ q or p ď a ă b q or p b “ and a ă q , the Lie algebra is R ˆ sl p , R q .There exists a compact quotient.4. If p´ ă a ă q or p a ą b q or p b “ and a ą q , the Lie algebra is R ˆ su p q . Thereexists a compact quotient. • Case C4: There is no compact quotient that supports a homogeneous complex Engel structure. • Case C5: If a “ , there exists a co-compact lattice Γ of a solvable Lie group G that G { Γ supports a homogeneous complex Engel structure. • Case C6: There is no compact quotients that supports a homogeneous complex Engel structureunless a “ ´ π ` kπ, k P Z and b “ . Under this condition, this is a special case of case C .In summary, compact quotients that support homogeneous complex Engel structures can occur incase C , case C , case C , and case C . We will prove the theorem in the following section by analyzing the structure equation for eachcase. If the structure equation is solvable, we will also provide a local coordinate system expressionof the coframing. 10
Proof of Theorem 4.1
In this case, p p , p , q , q , r , r q “ p a ` ib, , , , , q . The structure equation is dω “ ,dω “ p ω ´ ω q ^ ¯ ω ´ pp a ` ib q ω ` p a ´ ib q ¯ ω q ^ ω . (15)By (15), d p ω ^ ω ^ ¯ ω q “ ´p ´ a ` bi q ω ^ ¯ ω ^ ω ^ ¯ ω . (16)By Stokes’ Theorem, there is no compact example unless a “ and b “ dω “
0. By the complex Poincar´e Lemma, there exists a holomorphic function z locallyon the manifold such that ω “ dz . (17)Thus dω “ dz ^ d ¯ z ` r´p a ` ib q dz ` p ´ a ` ib q d ¯ z s ^ ω . We will find a local coordinate system for the coframing p ω , ω q in order to explicitly describe itsgroup of symmetries. Since the groups of symmetries are different for different a and b , we willconsider two cases: a ` ib “ If a ` ib “
1, i.e. a “ b “ d p ω ` ¯ zdz q “ ´ dz ^ p ω ` ¯ zdz q . (18)By the complex Frobenius Theorem, there exists a complex function f and a holomorphic function w locally on the manifold such that ω ` ¯ zdz “ f dw . (19)11ince p ω , ω q is a coframing, ω and ω are linearly independent. By comparing the local coordi-nate expressions (17) and (19), we know that f is nowhere zero on its defining domain. Substituting(19) into (18) yields df ^ dw “ ´ f dz ^ dw . Setting f “ e ´ z g for some function g ‰
0, we have dg ^ dw “
0. So the function g is a function of w only. ω ` ¯ zdz “ e ´ z g p w q dw . By defining ˜ w “ ş g p w q dw and dropping the tilde in the local coordinate, we get ω ` ¯ zdz “ e ´ z dw . Since p ω , ω q is a p , q -coframing, p z, w q can serve as a local holomorphic coordinate system in aneighborhood of the manifold. In the coordinate system of p z, w q , the coframing can be expressedas ω “ dz .ω “ ´ ¯ zdz ` e ´ z dw . (20)We consider the symmetry group of the coframing in these local coordinate. LetΓ “ t k ` ik |p z, w q Ñ p z, w ` k ` ik q , where k , k P Z u . Γ acts freely and discontinuously on the coframing in the local coordinate. So we can take a globalmodel for this complex Engel structure M “ C { Γ – R ˆ T . If a ` ib ‰
1, i.e. a ‰ b ‰ d ˆ ω ´ dz ´ a ` ib ˙ “ r´p a ` ib q dz ` p ´ a ` ib q d ¯ z s ^ ˆ ω ´ dz ´ a ` ib ˙ . (21)12y the complex Frobenius Theorem, there exists a complex function f and a holomorphic function w locally on the manifold such that ω ´ dz ´ a ` ib “ f dw . (22)Substituting (22) into (21) yields df ^ dw “ f r´p a ` ib q dz ` p ´ a ` ib q d ¯ z s ^ dw . By defining f “ e ´p a ` ib q z `p ´ a ` ib q ¯ z g for some function g ‰
0, we get dg ^ dw “
0. Thus ω ´ dz ´ a ` ib “ e ´p a ` ib q z `p ´ a ` ib q ¯ z g p w q dw . By defining ˜ w “ ş g p w q dw and dropping the tilde, we have ω ´ dz ´ a ` ib “ e ´p a ` ib q z `p ´ a ` ib q ¯ z dw . Since p ω , ω q is a coframing, p z, w q can serve as a local holomorphic coordinate system on a openset. The coframing can be written as ω “ dz ,ω “ dz ´ a ` ib ` e ´p a ` ib q z `p ´ a ` ib q ¯ z dw . We will analyze the symmetry group of the coframing. Define G a,b “ tp α, β q| ´ p a ` ib q α ` p ´ a ` ib q ¯ α “ kπi, where α, β P C and k P Z u , where G a,b acts on the local coordinate as p z, w q Ñ p z ` α, w ` β q . We will analyze the elementsof G a,b . Let α “ α ` iα , where α , α P R . We have p ´ a q α ` bα “ ,α “ kπ , (23)where k P Z . 13or different a, b , there exist three families of solution for α :1. if a ‰ , then α “ ´ bkπ ´ a , α “ kπ . DefineΓ “ " ˆˆ ´ bπ ´ a ` i π ˙ k, β ` iβ ˙ˇˇˇˇ k, β , β P Z * . We can get a non-compact quotient C { Γ – R ˆ S ˆ T that supports a homogeneous complex Engel structure.2. if a “ , b “
0, then α “ kπ . DefineΓ “ t p α ` i kπ, β ` iβ q| k, α , β , β P Z u . We get a compact quotient C { Γ that supports a homogeneous complex Engel structure.In this case, we can define θ “ ´ ω and θ “ ω ´ ω . The structure equation is dθ “ ,dθ “ p θ ´ ¯ θ q ^ θ . (24)Define θ “ α ` iβ and θ “ γ ` iδ for real parts and imaginary parts decomposition. Wehave dα “ ,dβ “ ,dγ “ ´ β ^ δ ,dδ “ β ^ γ . (25)Thus the Lie algebra is a 4-dimensional solvable Lie algebra g with nontrivial brackets: r X, Y s “ Z, r X, Z s “ ´ Y, where X, Y, Z P g . By the classification results in [15], the corresponding connected andsimply-connected Lie group has a co-compact lattice.14. if a “ , b ‰
0, then α “
0. DefineΓ “ t p α , β ` iβ q| α , β , β P Z u . Then we can get a non-compact quotient C { Γ – R ˆ S ˆ T . In summary, there exists a compact quotient of type C a “ and b “ Now, p p , p , q , q , r , r q “ p ` ib, , ia, , , q . Assume a ‰
0, otherwise, it is a special caseof C
1. The structure equation is dω “ ia ¯ ω ^ ω ,dω “ p ω ´ ω q ^ ¯ ω ´ “` ` ib ˘ ω ` ` ´ ib ˘ ¯ ω ‰ ^ ω . (26)By (26) d ` ω ` ` ´ ` i p a ´ b q ˘ ω ˘ “ ` ` ib ˘ p ¯ ω ´ ω q ^ ` ω ` ` ´ ` i p a ´ b q ˘ ω ˘ . By the complex Frobenius Theorem, there exist complex functions p and q , and holomorphicfunctions z and w such that ω “ pdz ,ω ` ´ ´ ` i p a ´ b q ¯ ω “ qdw . To calculate the function p , write ω “ α ` iβ , where α and β are the real and imaginary part of ω , respectively. From the structure equation dα “ ´ aα ^ β ,dβ “ . f, x, y such that α “ f dy, β “ dx and df ” af dx mod dy . Afterredefining y , we can write α “ e ax dy . Thus ω “ e ax d ` y ´ i a e ´ ax ˘ . (27)Define z “ y ´ i a e ´ ax as a local holomorphic coordinate ( Note: ℑ p z q ‰
0. So, according to thesign of a , we can restrict the definition of z to half of the complex plane), then ω “ ´ i a p z ´ ¯ z q dz . From the structure equation, we get dq ^ dw “ ´ ` ib ¯ i a p ¯ z ´ z q d p ¯ z ´ z q ^ qdw Thus dq ” ´ ` ib ¯ i a p ¯ z ´ z q d p ¯ z ´ z q q mod dw After redefining w , we can take q “ p ¯ z ´ z q ´ b ` i a . So in the local holomorphic coordinate system z, w , ω “ ´ i a p z ´ ¯ z q dz ,ω “ ´ ` p a ´ b q i ˆ p ¯ z ´ z q ´ b ` i a dw ` i a p z ´ ¯ z q dz ˙ . C d p ω ^ ¯ ω ^ ω q “ i p a ` b q ω ^ ¯ ω ^ ω ^ ¯ ω . (28)If a ` b ‰
0, the volume form is exact. By Stokes’ Theorem, there cannot be a compact quotientthat supports a homogeneous complex Engel structure when a ` b ‰
0. In the following, only16onsider a ` b “
0. Define θ “ ω . The structure equation is dθ “ ia ¯ θ ^ θ ,dθ “ ˆ ´ ia ˙ p ¯ θ ´ θ q ^ θ . (29)Let θ “ α ` iβ , θ “ γ ` iδ be real part and imaginary part decompositions. Then (29) is equivalentto dα “ ´ aα ^ β ,dβ “ ,dγ “ β ^ δ ´ aβ ^ γ ,dδ “ ´ β ^ γ ´ aβ ^ δ . Let X , X , X , X be left-invariant vector fields dual to the left-invariant forms α, β, γ, δ , respec-tively. Then the nontrivial brackets are r X , X s “ aX , r X , X s “ X ´ aX , r X , X s “ ´ aX ´ X . By [15], there exists a co-compact lattice for some a . We will calculate the conditions for theexistence of a co-compact lattice.Let f “ ´ a ` i , V “ X ` iX and W “ X ´ iX . The nontrivial brackets are r X , X s “ ´p f ` ¯ f q X , r X , V s “ f V , r X , W s “ ¯ f W . Since center of the Lie algebra g is trivial, we have an exact sequence0 Ñ g ad ÝÑ End p g q , where g ad ÝÑ End p g q is the adjoint representation. Let X “ xX ` yX ` zV ` ¯ zW be an element17f g . Then ad p X qp X , V, W, X q “ p X , V, W, X q »————– ´p f ` ¯ f q y p f ` ¯ f q x f y ´ f z f y ´ ¯ f ¯ z fiffiffiffiffifl . The connected and simply-connected Lie group corresponding to the Lie algebra g is G – $’’’’&’’’’% »————– y ´p f ` ¯ f q x y f z y ¯ f ¯ z fiffiffiffiffiflˇˇˇˇˇˇˇˇˇˇ x, y P R , y ą , z P C ,////.////- . Proposition 5.1.
If there exists a complex number λ and a matrix A P SL p Z q such that1. | λ | ‰
2. the eigenvalues of A are p λ ¯ λ q ´ , λ, ¯ λ
3. there exists k P Z such that ´ a log p| λ |q “ arg λ ` kπ then there exists a co-compact lattice Γ such that G { Γ supports a homogeneous complex Engelstructure.Proof. Let N Ă G be the subgroup N “ $’’’’&’’’’% »————– r z z fiffiffiffiffiflˇˇˇˇˇˇˇˇˇˇ r P R , z P C ,////.////- . (30)18t is easy to verify that N is a normal subgroup of G . Thus G { N – $’’’’&’’’’% »————– y ´p f ` ¯ f q y f y ¯ f
00 0 0 1 fiffiffiffiffiflˇˇˇˇˇˇˇˇˇˇ y P R , y ą ,////.////- is a quotient group. Let L “ x ~v , ~v , ~v y be a lattice of the normal subgroup N , to be determinedlater. We need to find a lattice L of G { N such that the lattice of the group G is L “ γ ~v ffˇˇˇˇˇ where γ P L , ~v P L + . By the multiplication rule of the group G , this is equivalent to γ~v P L for any γ P L and any ~v “ »——– v v v fiffiffifl P L . Hence we need to find a ij P Z and c ą c ‰ γ c ~v “ a ~v ` a ~v ` a ~v ,γ c ~v “ a ~v ` a ~v ` a ~v , (31) γ c ~v “ a ~v ` a ~v ` a ~v , where γ c is the linear transform with transformation matrix »——– c ´p f ` ¯ f q c f
00 0 c ¯ f fiffiffifl . Since x γ c ~v , γ c ~v , γ c ~v y will be a new basis for the lattice L , then A “ »——– a a a a a a a a a fiffiffifl P SL p Z q . »——– c ´p f ` ¯ f q c f
00 0 c ¯ f fiffiffifl “ p ~v , ~v , ~v q »——– a a a a a a a a a fiffiffifl p ~v , ~v , ~v q ´ (32)The eigenvalues of the matrix A should be c ´p f ` ¯ f q , c f and c ¯ f for some c ą
0. Assume the eigenvaluesare p λ ¯ λ q ´ , λ, ¯ λ and fix c ´ a ` i “ λ . (33)Thus c “ | λ | ´ a . (34)By (33), | λ | ˆ | λ | ´ a i “ | λ | ˆ e i arg λ . (35)Thus the eigenvalue λ and the parameter a satisfy ´ a log p| λ |q “ arg λ ` kπ (36)for certain k P Z .We will calculate an explicit condition on the existence of co-compact lattice. Since the eigenvaluesare p λ ¯ λ q ´ , λ, ¯ λ , the characteristic polynomial of the matrix A is p x ´ p λ ¯ λ q ´ qp x ´ λ qp x ´ ¯ λ q“ x ´ ´` λ ¯ λ ˘ ´ ` λ ` ¯ λ ¯ x ` ´ λ ` λ ¯ λ ˘ ´ ` ¯ λ ` λ ¯ λ ˘ ´ ` λ ¯ λ ¯ x ´ . (37)Since A P SL p Z q , there exist m, n P Z such that ` λ ¯ λ ˘ ´ ` λ ` ¯ λ “ m ,λ ` λ ¯ λ ˘ ´ ` ¯ λ ` λ ¯ λ ˘ ´ ` λ ¯ λ “ n . (38)Note if (38) satisfies, we can choose A “ »——– ´ n m fiffiffifl P SL p Z q .20efine p “ λ ` ¯ λ and q “ ` λ ¯ λ ˘ ´ . It is easy to verify that p and q are real numbers and q ě p .To be a co-compact lattice, 0 ă q ă
1. Then by (38), p ` q “ m ,pq ` q “ n . (39)Then the eigenvalue is λ “ p ` i b q ´ p . Remark 5.1.
There exist countably infinite families of solutions for p and q , that yield infinitelymany families of co-compact lattices. The co-compact lattices can be derived from solutions of (39). Now we give an example for some a such that there exists a compact quotient. Example 5.1.
Let »——– fiffiffifl P SL p Z q . Define s “ p ` ? q , then λ “ ´ s ´ s ` ? ` s ´ s ˘ i . We can calculate a by (36) and c by (34). Now p p , p , q , q , r , r q “ ´ ´ ib, , ib, p b ` i qp ia ´ b q , b ´ ib, p b ` qp a ` ib q ¯ Assume at least one of a or b is nonzero. Otherwise, it will be a special case of C a “ and b “
0. The structure equation is dω “ ´r´ ib ¯ ω ` p b ´ i qp´ ia ´ b q ¯ ω s ^ ω ´ „ p b ` i qp ia ´ b q ω ` p b ` ib q ¯ ω ` ˆ b ` ˙ p a ´ ib q ¯ ω ^ ω ,dω “ ω ^ ¯ ω ´ ˆ ´ ib ˙ p ω ´ ¯ ω q ^ ω . (40)21efine θ “ ω ` ` ´ ` ib ˘ ω . By (40), we have dθ “ “` ` ib ˘ p ¯ ω ` p a ´ ib q ¯ ω q ´ ` ` a ˘ ω ‰ ^ θ . (41)Since the symmetry groups are different for different parameters, we will consider the structureequation with different parameters:1. a “ ´ a “ b ‰ a ‰ ´ and a ‰ b a “ ´ The structure equation (41) reduces to d ´ ω ` ´ ´ ` ib ¯ ω ¯ “ ´ ` ib ¯´ ω ` ´ ´ ` ib ¯ ω ¯ (42) ^ ´ ω ` ´ ´ ` ib ¯ ω ¯ . Let ω ` ´ ´ ` ib ¯ ω “ α ` iβ , where α and β are real 1-forms. Then dα “ ´ bα ^ β ,dβ “ α ^ β . (43)Since d p α ` bβ q “
0, there exists function x such that α ` bβ “ dx . By the structure equation,we have dβ “ dx ^ β , that implies the existence of a function y such that β “ e x dy . Thus in termsof local coordinate x, y , α “ dx ´ be x dy ,β “ e x dy . So ω ` ˆ ´ ` ib ˙ ω “ e x d p´ e ´ x ´ by ` iy q . z “ ´ e ´ x ´ by ` iy . Then ω ` ´ ´ ` ib ¯ ω “ dz ´ ´ ` ib ¯ z ` ´ ´ ` ib ¯ z . By (40), we have dω “ ω ^ ¯ ω ´ ˆ ´ ib ˙ p ω ´ ¯ ω q ^ ω “ ´ ´ ` ib ¯ dz ^ ω ´ ´ ` ib ¯ z ` ´ ´ ` ib ¯ z ` ´ ´ ´ ` ib ¯ dz ^ ¯ ω ´ ´ ` ib ¯ z ` ´ ´ ` ib ¯ z ` dz ^ d ¯ z ˜´ ´ ` ib ¯ z ` ´ ´ ` ib ¯ z ¸ . (44) Lemma 5.2.
Assume x dz, ω y forms a Frobenius system. Then there exists a function f and aholomorphic function w such that ω “ dw ` f dz Proof.
Since x dz, ω y forms a Frobenius system of rank 2, there exist functions u, v, a, b such that dz “ adu ` bdv and du, dv are linearly independent . Since dz ‰
0, at least one of a, b is nonzero.Without loss of generality, assume b ‰
0. So dv “ b p dz ´ adu q . So v is a function of z and x andlocally we take z and u , instead of v and u , as local coordinates.By the complex Frobenius Theorem, there exist functions r p z, u q and s p z, u q such that ω “ r p z, u q du ` s p z, u q dz “ d ´ ż r p z, u q du ¯ ´ ´ ż B r p z, u qB z du ¯ dz ` s p z, u q dz “ d ´ ż r p z, u q du ¯ ` ” s p z, u q ´ ´ ż B r p z, u qB z du ¯ı dz . Since ω and dz are linearly independent, d ´ ş r p z, u q du ¯ ‰
0. Let w “ ´ ş r p z, u q du ¯ and f “ s p z, u q ´ ´ ş B r p z,u qB z du ¯ , then ω “ dw ` f dz . D “ ´ ` ib . By (44) and Lemma 5.2, we get df ^ dz “ dzDz ` ¯ D ¯ z ^ ˆ Ddw ´ ¯ Dd ¯ w ` d ¯ zDz ` ¯ D ¯ z ´ ¯ D ¯ f d ¯ z ˙ . Thus B f B w “ ´ DDz ` ¯ D ¯ z , B f B ¯ w “ ¯ DDz ` ¯ D ¯ z , B f B ¯ z “ ¯ D ¯ f ´ Dz ` ¯ D ¯ z Dz ` ¯ D ¯ z . (45)Let f “ f ` if , where f and f are real and imaginary parts of f , respectively. Let z “ x ` iy and w “ u ` iv , then (45) is equivalent to B f B u “ , B f B v “ , B f B u “ bx ` by , B f B v “ ´ x ` by , B f B x ´ B f B y “ f ` bf ´ x ` by x ` by , B f B y ` B f B x “ bf ´ f x ` by . (46)So f “ f p x, y q , f “ g p x, y q ` bux ` by ´ vx ` by . The equation is equivalent to B f B x ´ B g B y “ f ` bg ´ x ` by x ` by , B f B y ` B g B x “ bf ´ gx ` by . (47)24onsider the differential ideal I “ x θ , θ y , where θ “ df ´ pdx ´ qdy ,θ “ dg ` ˆ q ´ bf ´ gx ` by ˙ dx ´ ˜ p ´ f ` bg ´ x ` by x ` by ¸ dy . (48)and dθ “ ´ π ^ dx ´ π ^ dy ,dθ “ π ^ dx ´ π ^ dy , (49)where π ” dp mod p dx q and π ” dq mod p dy q . This system is involutive and its Cartancharacters are p s , s q “ p , q . So the solution depends on 2 functions of 1 variable.We will calculate the coframing in local coordinate for b “
0. Let F “ F p´ iz q , F “ G p i ¯ z q be twofunctions of one variable z and C be a constant. The general solution is of the following form f “ F ` F ` ˆ p z ` ¯ z q ` C ˙ z ` ¯ z ,f “ iF ´ iF ´ z ` ¯ z ˆ F ` F ` w ´ ¯ w i ˙ . Thus in the case b “
0, the coframing is ω “ p dw ` f dz q ´ dzz ` ¯ z ,ω “ dw ` f dz . In our original parametrization, z “ ´ e ´ x ´ by ` iy . So z ` ¯ z ă
0. Take a special form F “ F “
0. Then f p z, w q “ ´ p w ´ ¯ w q z ` ¯ z . The coframing can be written as ω “ ˆ dw ´ ` p w ´ ¯ w q z ` ¯ z dz ˙ ,ω “ dw ` ´ p w ´ ¯ w q z ` ¯ z dz .
25e will prove that there exist compact quotients that support homogeneous complex Engel struc-tures. Before proving this, we need to know the Lie algebra of the homogeneous complex Engelstructures.
Proposition 5.3.
There exists a basis p X , X , X , X q such that the nontrivial brackets of the Liealgebra are r X , X s “ X , r X , X s “ X , r X , X s “ ´ X . (50) Proof.
Recall that dα “ ´ bα ^ β ,dβ “ α ^ β . (51)Define ω “ γ ` iδ . dγ “ ´ bα ^ β ´ δ ^ β ` bδ ^ α ,dδ “ α ^ β ´ α ^ δ ´ bβ ^ δ . (52)Let e , e , e , e be left-invariant vector fields dual to the left-invariant forms α, β, γ, δ , respectively.Then by (52), the nontrivial brackets are r e , e s “ ´ be ` e ´ be ` e r e , e s “ ´ be ´ e (53) r e , e s “ e ´ be . We will consider 2 separate cases:1. b “ b ‰ b “
0, the nontrivial brackets are r e , e s “ e ` e r e , e s “ ´ e r e , e s “ e . X “ ´ e , X “ e , X “ e ` e , X “ e . Then r X , X s “ X , r X , X s “ X , r X , X s “ ´ X . (54)If b ‰
0, define ˜ e “ e ´ b e ´ b ` e ´ b e ˘ , ˜ e “ e ´ b e . The nontrivial brackets are r ˜ e , e s “ ´ b ˜ e ` ˜ e ` ` b ´ b ˘ e r ˜ e , ˜ e s “ ´ ` b ` b ˘ e r e , ˜ e s “ ´ b ˜ e Define ˜˜ e “ ˜ e ´ b ˜ e . Then the nontrivial brackets are r ˜˜ e , e s “ ´ b ˜˜ e ` ` b ´ b ˘ e r ˜˜ e , ˜ e s “ ´ ` b ` b ˘ e r e , ˜ e s “ ´ b ˜ e Define X “ ´ b ` b e , X “ ˜˜ e , X “ ˜ e , X “ b ` `` ´ b ´ b ˘ e ´ ` ´ b ` b ˘ e ˘ . Then r X , X s “ X , r X , X s “ X , r X , X s “ ´ X . (55)Thus the theorem is true for both cases.By [15], there exists a co-compact lattice when a “ ´ . Thus there exists a compact quotient oftype C a “ ´ . a “ b ‰ θ “ ω ` p b ` ib q ω ` b ` pp b ´ q ´ bi q ¯ ω ` b b ` p b p b ´ q ´ p b ´ q i q ¯ ω (56)27hen the coframing p θ, θ q satisfies dθ “ ,dθ “ ` ´ ` ib ˘ θ ^ θ . Thus there exists a holomorphic coordinate system p z, w q such that θ and θ can be written aslinear combinations of p dw, d ¯ w q and dz , respectively. There exists a function f p w, ¯ w q such that θ “ df ,θ “ e p´ ` ib q f dz . Thus ω “ e ´ ´ ` ib ¯ f dz ´ ` ´ ` ib ˘ ω . By (56), we have ´ b ´ ´ ib ¯ ¯ ω ` ´ b ` ¯ ω “ df ´ e p´ ` ib q f dz ´ b ` ´ p b ´ q ´ bi ¯ e p´ ´ ib q ¯ f d ¯ z . Let ω “ hdz ` gdw , where h and g are functions. By scaling, fix g “
1. Take f “ ´ b ` ¯ w ` ´ b ´ ´ ib ¯ ¯ w , then h “ ´ e p´ ` ib q « ´ b ` ¯ w ` ´ b ´ ´ ib ¯ ¯ w ff b ` . So ω “ ´ e p´ ` ib q rp b ` q w ` p b ´ ´ ib q ¯ w s b ` dz ` dw and ω “ e p´ ` ib q rp b ` q w ` p b ´ ´ ib q ¯ w s dz ´ ˆ ´ ` ib ˙ ω . Proposition 5.4.
There does not exist a compact quotient that supports a homogeneous complexEngel structure when a “ b ‰ . roof. After changing θ to ´ ´ ` ib ¯ θ , the structure equation is dθ “ ,dθ “ θ ^ θ . (57)Define θ “ α ` iβ, θ “ γ ` iδ as the real and imaginary parts decompositions. Then (57) isequivalent to dα “ ,dβ “ ,dγ “ α ^ γ ´ β ^ δ ,dδ “ α ^ δ ` β ^ γ . (58)Let ´ X , X , X , X be left-invariant vector fields dual to the left-invariant forms α, β, γ, δ , respec-tively. Then the nontrivial brackets are r X , X s “ X , r X , X s “ X , r X , X s “ ´ X , r X , X s “ X . The Lie algebra is a solvable Lie algebra, denoted by g , in [17]. According to the classificationresults of the existence of co-compact lattices for 4-dimensional solvable Lie groups in [15], theconnected and simply-connected Lie group corresponding to g , does not have a co-compactlattice. Therefore, there does not exist a compact quotient that supports a homogeneous complexEngel structure in this case. 29 .3.3 a ‰ ´ and a ‰ b Let ω “ α ` iβ and ω “ γ ` iδ . From the structure equation (40), we have dα “ ´ bα ^ β ´ b α ^ γ ` bα ^ δ ` p ab ´ b q β ^ γ ´ p a ` b q β ^ δ ´ b ˆ ` b ˙ γ ^ δ ,dβ “ ´ abα ^ γ ` aα ^ δ ` b β ^ γ ´ bβ ^ δ ´ a ˆ ` b ˙ γ ^ δ ,dγ “ β ^ p δ ´ bγ q ,dδ “ β ^ p α ´ γ ´ bδ q . (59)Define ˜ α “ ´ aα ` bβ ´ p a ´ b q γ ` abδ . Since at least one of a or b is not zero, ˜ α ‰ d ˜ α “
0. Let e , e , e , e be the dual vector fields of the 1-forms α, β, γ, δ , respectively. Remark 5.2.
From Lie theory, there is a split exact sequence [14] Ñ Rad p g q Ñ g Ñ g { Rad p g q Ñ , where Rad p g q is the radical ideal of g . Thus g – Rad p g q ‘ g { Rad p g q . Denote g “ g { Rad p g q . We will prove the following theorem: Theorem 5.5.
For the Lie algebra g corresponding to the structure equation(59), Rad p g q is 1-dimensional and g is simple 3-dimensional Lie algebra. Specially, • p a ă ´ q or p ď a ă b q or p b “ and a ă q , the Lie algebra is R ˆ sl p , R q . • p´ ă a ă q or p a ą b q or p b “ and a ą q , the Lie algebra is R ˆ su p q .Proof. We will prove this theorem by analyzing the result for the following 3 cases: • b “ α “ α ` aγ . Define ˜ β “ β, ˜ γ “ ´ α ` γ, ˜ δ “ δ , then d ˜ β “ ´ a ˜ γ ^ ˜ δ ,d ˜ γ “ ´p ` a q ˜ δ ^ ˜ β ,d ˜ δ “ ´ ˜ β ^ ˜ γ . (60)If a ą
0, we define ˜˜ β “ ? ` a ˜ β, ˜˜ γ “ ? a ˜ γ, ˜˜ δ “ ? a p ` a q ˜ δ . (60) is equivalent to d ˜˜ β “ ˜˜ γ ^ ˜˜ δ ,d ˜˜ γ “ ˜˜ δ ^ ˜˜ β ,d ˜˜ δ “ ˜˜ β ^ ˜˜ γ . Thus ˜˜ β, ˜˜ γ, ˜˜ δ are left-invariant forms of the Lie group SU p q . So if a ą
0, the manifold canbe taken as S ˆ SU p q .If a ă Rad p g q “ t e ` e u and g { Rad p g q “ t u “ e ´ a e , v “ e , w “ e u . Define H “ ´ ?´ a ` a u ,X “ ?´ av ` w ,Y “ ?´ aa p ` a q v ´ a p ` a q w . The nontrivial brackets are r H, X s “ X , r H, Y s “ ´ Y , r X, Y s “
H . p H, X, Y q forms a canonical basis for the Lie algebra sl p , R q . Since SL p R q has co-compactlattices [12], in this case, there exists a compact quotient that supports homogeneous complexEngel structure. • a “ b ‰
0. The radical ideal is
Rad p g q “ ! e ` e ` be ` b ) and g { Rad p g q “t u “ e , v “ e ´ e b , w “ e u . The nontrivial brackets are r u, v s “ bu ` w r u, w s “ ´ bu r v, w s “ ˆ b ´ ˙ u ` bv ` bw Define $&% A “ b ´ b , B “ b , C “ b , D “ b ` , E “ ´ b, F “ , s “ , if b ‰ ´ ,A “ b ` b , B “ ´ b , C “ b , D “ b ´ , E “ b, F “ , s “ ´ , if b ‰ . and H “ sv ,X “ Au ` Bv ` Cw ,Y “ Du ` Ev ` F w .
Then p H, X, Y q forms a canonical basis for the Lie algebra sl p , R q such that r H, X s “ X , r H, Y s “ ´ Y , r X, Y s “
H .
Since SL p R q has co-compact lattices [12], in this case there exists a compact quotient thatsupports a homogeneous complex Engel structure. • a ‰ , a ‰ ´ and b ‰ Rad p g q “ ! e ` e ` be ` b ) and g { Rad p g q “ " u “ e ` e b , v “ e ´ e a , w “ e ` a ´ b ab e * . r u, v s “ ´ ´ ab ` a ´ b ab u ` b w , r u, w s “ ´p´ ab ` a ´ b ` a q ˆ a u ` b v ˙ , r v, w s “ ´ b ` a ` ab ` b ab u ` ´ ab ` a ´ b ` a a v ´ ´ ab ` a ´ b ab w . After the proof of the following proposition, we will finish the proof of the theorem.
Proposition 5.6. If a ‰ , a ‰ ´ and b ‰ , there exists a compact quotient that supportsa homogeneous complex Engel structure.Proof. a ą b Define U “ c ´ ab ` a ´ b ` a ,V “ c a p´ ab ` a ´ b ` a q and A “ V a p´ ab U ` a U ´ b U ` b q , B “ V, C “ ´
U V,D “ ´ V a p´ ab U ` a U ´ b U ´ b q , E “ V, F “ U V,s “ bU . Then define H “ sv ,X “ Au ` Bv ` Cw , (61) Y “ Du ` Ev ` F w . p H, X, Y q forms a canonical basis for the Lie algebra su p q with the following nontrivialbrackets r H, X s “
Y , r H, Y s “ ´
X , r X, Y s “
H .
Since SU p q has co-compact lattices, there exists a compact quotient that supports a homo-geneous complex Engel structure. ă a ă b Define U “ c ´ ´ ab ` a ´ b ` a ,V “ c ´ a ´ ab ` a ´ b ` a and A “ ´ V a p´ ab U ` a U ´ b U ´ b q , B “ V, C “ U V,D “ V a p´ ab U ` a U ´ b U ` b q , E “ V, F “ ´
U V,s “ bU. Then define H “ sv ,X “ Au ` Bv ` Cw ,Y “ Du ` Ev ` F w . p H, X, Y q forms a canonical basis for the Lie algebra sl p , R q with nontrivial brackets r H, X s “ X , r H, Y s “ ´ Y , r X, Y s “
H .
Since SL p R q has co-compact lattices [12], there exists a compact quotient that supports ahomogeneous complex Engel structure. ´ ă a ă U “ c ´ ab ` a ´ b ` a ,V “ c a p´ ab ` a ´ b ` a q and A “ V a p´ ab U ` a U ´ b U ` b q , B “ V, C “ ´
U V,D “ ´ V a p´ ab U ` a U ´ b U ´ b q , E “ V, F “ U V,s “ bU . Then define H “ sv ,X “ Au ` Bv ` Cw ,Y “ Du ` Ev ` F w .
Note Y “ X . p H, X, Y q forms a canonical basis of the Lie algebra su p q with nontrivial35rackets r H, X s “
X , r H, X s “ ´
X , r X, X s “
H .
Since SU p q has co-compact lattices, there exists a compact quotient that supports a homo-geneous complex Engel structure. a ă ´ Define U “ c ´ ab ` a ´ b ` a ,V “ c ´ a p´ ab ` a ´ b ` a q and A “ V a p´ ab U ` a U ´ b U ` b q , B “ V, C “ ´
U V,D “ ´ V a p´ ab U ` a U ´ b U ´ b q , E “ V, F “ U V,s “ bU. Then define H “ sv ,X “ Au ` Bv ` Cw ,Y “ Du ` Ev ` F w . H, X, Y q forms a canonical basis of the Lie algebra sl p , R q with nontrivial brackets r H, X s “
Y , r H, Y s “ ´
X , r X, Y s “ ´
H .
Since SL p R q has co-compact lattices [12]), there exists a compact quotient that supports ahomogeneous complex Engel structure.Since there exists a co-compact lattice for each case, we have proved the theorem. Now p p , p , q , q , r , r q “ p , a ´ ib, ib, a ´ ib, ´ a ´ ib, q . The structure equation is dω “ ib ¯ ω ^ ω ´ p a ` ib q ¯ ω ^ ω ` p a ´ ib q ¯ ω ^ ω ,dω “ p ω ´ ω q ^ ¯ ω ´ p a ` ib q ¯ ω ^ ω . (62)It is easy to verify that d p ω ^ ¯ ω ^ ω q “ p ` bi q ω ^ ¯ ω ^ ω ^ ¯ ω ‰ . By Stokes’ Theorem, there is no compact quotient of type C ω and ω under different conditions for a and b .The symmetry groups of the coframing are different for different a and b . We define A “ B “ $&% ´ ˘ c ` ib, where c “ b ´ p b ` a q , if b ` a ď ´ ˘ ic ` ib, where c “ b p b ` a q ´ , if b ` a ą (63)By (62), d p Aω ` Bω q ^ p Aω ` Bω q “ . .4.1 a “ b “ dω “ ,dω “ p ω ´ ω q ^ ¯ ω . Since dω “
0, by the complex Poincar´e Lemma, there exists a holomorphic function z such that ω “ dz . So dω “ p dz ´ ω q ^ d ¯ z , that is equivalent to d p dz ´ ω q “ ´p dz ´ ω q ^ d ¯ z . By thecomplex Frobenius Theorem, there exists a function f and a holomorphic function w such that dz ´ ω “ f dw and df ^ dw “ f d ¯ z ^ dw . By modifying w , we can write dz ´ ω “ e ¯ z dw . So inthe local coordinate system p z, w q , the coframing is ω “ dz ,ω “ ´ e ¯ z dw ` dz . Take a discrete symmetry group of the coframingΓ “ tp α πi, β ` iβ q| α , β , β P Z u – Z . Γ acts on the local coordinate as p z, w q ÞÑ p z ` α πi, w ` β ` iβ q . This action keeps thecoframing invariant. The quotient can be taken as C { Γ – R ˆ S ˆ T . b ` a ă or b ` a ą Since the calculations are similar for these two cases, without loss of generality, assume b ` a ă .Define U “ ´ ` c ` ib, V “ ´ ´ c ` ib and θ “ ω ` U ω , θ “ ω ` V ω .Then the structure equation is dθ “ p ´ c ` ib q ¯ θ ^ θ ,dθ “ p ` c ` ib q ¯ θ ^ θ .
38y the complex Frobenius Theorem, there exist coordinates z and w and functions f and g , suchthat θ “ f dz, θ “ ¯ gd ¯ w . Since d p θ ^ ¯ θ q “ d p f gdz ^ dw q “
0, so there exists a function F p z, w q such that f g “ F p z, w q . Since dθ “ df ^ dz “ ´ ´ c ` ib ¯ F p z, w q dw ^ dz ,d ¯ θ “ dg ^ dw “ ´ ` c ` ib ¯ F p z, w q dz ^ dw ,f and g are both functions of z, w and f w “ ´ ´ c ` ib ¯ f g ,g z “ ´ ` c ` ib ¯ f g . Thus p log f g q wz “ ´ f w f ¯ z ` ´ g w g ¯ z “ ´ ´ c ` ib ¯ g z ` ´ g w g ¯ z “ ´ p ` ib q ´ c ¯ f g ` ´ g w g ¯ z , while p log f g q zw “ ´ f z f ¯ w ` ´ g z g ¯ w “ ´ f z f ¯ w ` ´ ` c ` ib ¯ f w “ ´ f z f ¯ w ` ´ p ` ib q ´ c ¯ f g. Since p log f g q wz “ p log f g q zw , ´ f z f ¯ w “ ´ g w g ¯ z . Since p log f q zw “ ´ f z f ¯ w and p log g q zw “ ´ g w g ¯ z , p log f q zw “ p log g q zw .
39o there exist functions A p z q and B p w q such that f p z, w q g p z, w q “ A p z q B p w q . So there exists a function G p z, w q such that θ “ A p z q G p z, w q dz, θ “ B p w q G p z, w q d ¯ w. After redefining z and w and the function f p z, w q , we can write θ “ f p z, w q dz, θ “ f p z, w q d ¯ w $&% f w “ ´ ´ c ` ib ¯ f f z “ ´ ` c ` ib ¯ f Thus there exists a constant C such that f p z, w q “ ´ ´ ` c ` ib ¯ z ` ´ ´ c ` ib ¯ w ` C .
After translating z or w , θ “ dz ´ ´ ` c ` ib ¯ z ` ´ ´ c ` ib ¯ w ,θ “ d ¯ w ´ ´ ` c ` ib ¯ ¯ z ` ´ ´ c ` ib ¯ ¯ w . Now change the notation from w to ¯ w , then θ “ dz ´ ´ ` c ` ib ¯ z ` ´ ´ c ` ib ¯ ¯ w ,θ “ dw ´ ´ ` c ` ib ¯ ¯ z ` ´ ´ c ` ib ¯ w .
40o the Engel structure can be defined on the complex 2-plane except two lines ´ ´ ` c ` ib ¯ z ` ´ ´ c ` ib ¯ ¯ w “ , ´ ´ ` c ` ib ¯ ¯ z ` ´ ´ c ` ib ¯ w “ . (64)Now we can get the local coordinate representation of ω and ω ω “ c ” p ` c ` ib q θ ` p´ ` c ` ib q θ ı ,ω “ c p θ ´ θ q . Let λ and µ be two complex constants. The symmetry group of the coframing is p z, w q Ñ p λz, ¯ λw q and p z, w q Ñ ˆ z ` ˆ ´ c ` ib ˙ µ, w ` ˆ ` c ` ib ˙ ¯ µ ˙ . b ` a “ The structure equation is dω “ ib ¯ ω ^ ω ´ ˆ ´ b ` ib ˙ ¯ ω ^ ω ` ˆ ´ b ´ ib ˙ ¯ ω ^ ω ,dω “ p ω ´ ω q ^ ¯ ω ´ ˆ ´ b ` ib ˙ ¯ ω ^ ω . Define θ “ ω ` ` ´ ` ib ˘ ω . Then dθ “ ´ ` ib ¯ ¯ θ ^ θ. θ “ α ` iβ be the real part and imaginary part decomposition. Then dα “ ´ bα ^ β ,dβ “ α ^ β, so d p α ` bβ q “
0. Thus there exists a real function x such that α ` bβ “ dx . Thus, dβ “ dx ^ β .This is equivalent to d p e ´ x β q “
0, that implies the existence of a real function y such that β “ e x dy .So θ “ dx ´ be x dy ` ie x dy. Let z “ ´ e ´ x ´ by ` iy . Then θ “ dz p´ ` ib q z ` p´ ´ ib q ¯ z . Recall that the structure equation is dω “ ˆ ` ib ˙ ¯ θ ^ ω ` ˆ ´ ` ib ˙ ¯ ω ^ θ,dω “ θ ^ ¯ θ ` ˆ ` ib ˙ ` θ ^ ¯ ω ` ¯ θ ^ ω ˘ . Let ω “ γ ` iδ be the real part and imaginary part decomposition. Since θ “ α ` iβ , the exteriorderivative of ω can be written as dγ “ α ^ γ ` β ^ δ ´ bα ^ δ ` bβ ^ γ,dδ “ . So there exists a real function u such that δ “ du . Since x γ, du y forms a Frobenius system, thereexist real functions p, q, v such that γ “ pdv ` qdu . Write dp “ p x dx ` p y dy ` p u du ` p v dv ,dq “ q x dx ` q y dy ` q u du ` q v dv . p and q satisfy p x “ p ´ x ´ by ,p y “ bp ´ x ´ by ,p u “ q v ,q x “ q ´ b ´ x ´ by ,q y “ ` bq ´ x ´ by . From the first two equations, we know that there exists a function C p u, v q such that p “ C p u,v q x ` by .From the last two equations, we know that there exists a function C p u, v q such that q “ p bx ´ y q` C p u,v q x ` by .From the third equation we know that B C B u “ B C B v . Thus d ´ ş C p u, v q dv ¯ “ C p u, v q dv ` C p u, v q du .Thus γ “ C p u, v q x ` by dv ` p bx ´ y q ` C p u, v q x ` by du “ C p u, v q dv ` C p u, v q dux ` by ` bx ´ yx ` by du “ d p ş C p u, v q dv q x ` by ` bx ´ yx ` by du. Now define p ş C p u, v q dv q as new v , then γ “ dvx ` by ` bx ´ yx ` by du. So ω “ γ ` iδ “ dvx ` by ` bx ´ yx ` by du ` idu “ x ` by r dv ` p b ` i qp x ` iy q du s“ x ` by r d p v ` p b ` i qp x ` iy q u q ´ p b ` i q u d p x ` iy qs . w “ v ` p b ` i qp x ` iy q u and redefine z “ x ` iy . The coframing can be written as ω “ x ` by r dw ´ p b ` i q u dz s“ ´ p´ ` ib q z ` p´ ´ ib q ¯ z ˆ „ dw ` ˆ ´ ib ˙ w ´ ¯ w p´ ` ib q z ` p´ ´ ib q ¯ z dz ,ω “ ´ ` ib p θ ´ ω q“ p´ ` ib qp´ ` ib q z ` p´ ´ ib q ¯ z ˆ „ dz ` dw ` ˆ ´ ib ˙ w ´ ¯ w p´ ` ib q z ` p´ ´ ib q ¯ z dz , where p z, w q is a local holomorphic coordinate system on the manifold. And ω “ λ, s, t be any real constants. The coframing is invariant under thefollowing local transformation p z, w q Ñ ˆ λz ` ˆ b ´ i ˙ t, λw ` s ˙ . We can take a discrete subgroup Γ such that M is locally biholomorphic to C { Γ – R ˆ T . Assume a ‰
0. Otherwise, this is a special case of homogeneous case C4, with a ` b “ . Thestructure equation is dω “ ˆ b ´ ´ ib ˙ ¯ ω ^ ω ` ˆ a ` ab ` i ˆ ab ` ab ˙˙ ¯ ω ^ ω ` ib ¯ ω ^ ω ` ˆ ´ a ´ ab ´ b ´ ib ˙ ¯ ω ^ ω ` ˆ a ´ ab ´ iab ˙ ω ^ ω ,dω “ ˆ ´ ` a ´ ab ` b ` i p ab ´ b q ˙ ¯ ω ^ ω ` ω ^ ¯ ω ` p´ a ` iab q ω ^ ω ` p´ ` a ` iab q ω ^ ¯ ω . θ “ ω ` ` ´ ` ib ˘ ω , that satisfies dθ “ ˆ ` ib ˙ ¯ θ ^ θ. (65)This structure equation is same as that of the case C4, with b ` a “ . But in case C5, a and b do not have to satisfy this relation.Let θ “ α ` iβ be the real and imaginary part decomposition. By (65), dα “ ´ bα ^ β,dβ “ α ^ β. Since d p α ` bβ q “
0, there exists a real function x such that α ` bβ “ dx . Thus dβ “ dx ^ β . Sothere exists a real function y such that β “ e x dy , that yields θ “ dx ´ be x dy ` ie x dy . Let z “ ´ e ´ x ´ by ` iy . Then θ “ dz p´ ` ib q z ` p´ ´ ib q ¯ z . The symmetry groups of the coframing are different for different parameters a and b . In thefollowing sections, we will consider the following cases:1. a ‰ ˘ a “ , b “ a “ , b ‰ a “ ´ .5.1 a ‰ ˘ Define A “ r ` is , where r “ ´ ` a ¯ p ` b q s “ b ´ ´ a ¯ p ` b q . After defining θ “ ω ` Aθ , the structure equation reduces to dθ “ a ´ ´ ` ib ¯ θ ^ θ ` p a ´ q ´ ` ib ¯ θ ^ ¯ θ ` ´ ` ib ¯ θ ^ ¯ θ . Let θ “ γ ` iδ and θ “ dx ` idyA p z q , where A p z q “ p´ ` ib q z ` p´ ´ ib q ¯ z “ ´ x ´ by is a real function.Then dγ “ ´ ´ aA p z q ` A p z q ¯ dx ^ γ ` A p z q dy ^ δ ´ abA p z q dy ^ γdδ “ bA p z q dx ^ γ ` ´ ´ abA p z q ` bA p z q ¯ dy ^ δ ´ aA p z q dx ^ δ. (66)By (66), x γ, dy y and x δ, dx y are two Frobenius systems, that implies the existence of functions p, q, r, s and u, v such that γ “ pdu ` qdy,δ “ rdv ` sdx. Write dp “ p x dx ` p y dy ` p u du ` p v dv,dq “ q x dx ` q y dy ` q u du ` q v dv,dr “ r x dx ` r y dy ` r u du ` r v dv,ds “ s x dx ` s y dy ` s u du ` s v dv.
46y (66), dγ “ dp ^ du ` dq ^ dy “ p x dx ^ du ` p y dy ^ du ` p v dv ^ du ` q x dx ^ dy ` q u du ^ dy ` q v dv ^ dy “ ´ x ´ by r p p ´ a q dx ^ du ` p q p ´ a q ´ s q dx ^ dy ` rdy ^ dv ´ abpdy ^ du s . Then the functions p and q must satisfy p y ´ q u “ ´ abp ´ x ´ by ,p x “ ´ a ´ x ´ by p,p v “ ,q v “ ´ r ´ x ´ by ,q x “ q p ´ a q ´ s ´ x ´ by . Thus there exists a function C p u, y q such that p “ p x ` by q a ´ C p u, y q . After redefining u and q , we can assume C p u, y q “
1. Then p “ p x ` by q a ´ . By (66), dδ “ dr ^ dv ` ds ^ dx “ r x dx ^ dv ` r y dy ^ dv ` r u du ^ dv ` s y dy ^ dx ` s u du ^ dx ` s v dv ^ dx “ ´ x ´ by ” bpdx ^ du ` p bq ` p ab ´ b q s q dx ^ dy ` p´ ab ` b q rdy ^ dv ´ ardx ^ dv ı . r and s must satisfy r x ´ s v “ ´ ar ´ x ´ by ,r y “ ´ ab ` b ´ x ´ by r,r u “ ,s y “ ´ bq ` p´ ab ` b q s ´ x ´ by ,s u “ ´ bp ´ x ´ by . (67)So there exists a function C p x, v q such that r “ p x ` by q a ´ C p x, v q . After redefining v and s ,we can assume C p x, v q “
1. Then r “ p x ` by q a ´ . Substituting this equation into (67), the equations are as follows: q u “ ´ b p x ` by q a ´ ,q v “ p x ` by q a ´ ,q x “ p a ´ q q ` sx ` by ,s v “ ´p x ` by q a ´ ,s u “ b p x ` by q a ´ ,s y “ bq ` p ab ´ b q sx ` by . This yields θ “ γ ` iδ “ p x ` by q a ´ p du ` idv q ` is ´ dx ´ i qs dy ¯ . (68)48o get a local holomorphic coordinate system, let q “ ´ s . Then q u “ ´ b p x ` by q a ´ ,q v “ p x ` by q a ´ ,q x “ p a ´ q qx ` by ,q y “ ´ p´ ab ` b q qx ` by . From these equations, we get q “ p ´ bu ` v qp x ` by q a ´ Substituting this equation into (68) yields θ “ γ ` iδ “ p pdu ` qdy q ` i p rdv ` sdx q“ pp x ` by q a ´ du ` p ´ bu ` v qp x ` by q a ´ dy q` i pp x ` by q a ´ dv ´ p ´ bu ` v qp x ` by q a ´ dx q“ p x ` by q a ´ d p u ` iv q ´ i p ´ bu ` v qp x ` by q a ´ d p x ` iy q . Define w “ u ` iv . We get the local coordinate representation of θ : θ “ p´ A p z qq a ´ „ dw ` „ i ` ˆ ´ bi ˙ w ´ ˆ ` bi ˙ ¯ w dzA p z q . Recall that θ “ ω ` ˆ ´ ` ib ˙ ω ,θ “ ω ` « ` ` a ˘ p ` b q ` i b ` ´ a ˘ p ` b q ff θ. p z, w q , the coframing can be written as ω “ ` ˆ ´ ` ib ˙ « ` ` a ˘ p ` b q ` i b ` ´ a ˘ p ` b q ff+ θ ´ ˆ ´ ` ib ˙ θ ω “ θ ´ « ` ` a ˘ p ` b q ` i b ` ´ a ˘ p ` b q ff θ. a “ , b “ p a, b q such thatlim a Ñ b Ñ b ´ ´ a ¯ p ` b q “ a ‰ applies to this case a “ , b “
0. The local coordinaterepresentation of the coframing is ω “ ˆ ` ib ˙ θ ´ ˆ ´ ` ib ˙ θ ω “ θ ´ θ. a “ , b ‰ dω “ ω ^ ¯ ω ` ˆ ´ ` ib ˙ ω ^ ω ` ˆ ´ ` ib ˙ ω ^ ¯ ω . (69)Substituting θ “ ω ` p´ ` ib q ω into (69) yields dω “ ” θ ´ ´ ´ ` ib ¯ ω ı ^ ” ¯ θ ´ ´ ´ ´ ib ¯ ¯ ω ı ` ´ ` ib ¯ θ ^ ω ` ´ ´ ` ib ¯ ω ^ ” ¯ θ ´ ´ ´ ´ ib ¯ ¯ ω ı “ θ ^ ¯ θ ` ´ ` ib ¯ θ ^ ¯ ω ` ´ ´ ` ib ¯ θ ^ ω . x ω , θ y is a Frobenius system. Write ω “ γ ` iδ . Then d p γ ` iδ q “ ´ iA p z q dx ^ dy ` iA p z q p dx ` idy q ^ p´ δ ` bγ q . Hence dγ “ ´ A p z q dy ^ p´ δ ` bγ q dδ “ ´ A p z q dx ^ dy ` A p z q dx ^ p´ δ ` bγ q . So x γ, dy y and x δ, dx y are two Frobenius systems. There exist functions p, q, r, s and u, v suchthat γ “ pdu ` qdyδ “ rdv ` sdx. From the structure equation, dγ “ dp ^ du ` dq ^ dy “ p x dx ^ du ` p y dy ^ du ` p v dv ^ du ` q x dx ^ dy ` q u du ^ dy ` q v dv ^ dy “ ´ A p z q dy ^ p´ rdv ´ sdx ` bpdu q . So p y ´ q u “ bpx ` by ,p x “ ,p v “ ,q v “ rx ` by ,q x “ sx ` by . (70)Since there are ambiguities for choosing p and u , by modifying u we can arrange p “ dδ “ dr ^ dv ` ds ^ dx “ r x dx ^ dv ` r y dy ^ dv ` r u du ^ dv ` s y dy ^ dx ` s u du ^ dx ` s v dv ^ dx “ ´ A p z q dx ^ dy ` A p z q dx ^ p´ rdv ` bpdu ` bqdy q . Then the functions r and s must satisfy r x ´ s v “ ´ r ´ x ´ by ,r y “ ,r u “ ,s y “ p x ` by q ` bx ` by q,s u “ ´ bp ´ x ´ by . (71)Since there is ambiguity for choosing r and v , by modifying v we can arrange r “ q u “ ´ bx ` by ,q v “ x ` by ,q x “ sx ` by ,s v “ ´ x ` by ,s y “ p x ` by q ` bx ` by q,s u “ bx ` by . Thus there exist functions C p x, y q and C p x, y q such that q “ A p z q p bu ´ v q ` C p x, y q s “ ´ A p z q p bu ´ v q ` C p x, y q . The functions C p x, y q and C p x, y q satisfy C x “ x ` by C ,C y “ bx ` by C ` p x ` by q . Choose C p x, y q “ ´ C p x, y q such that z is a local holomorphic coordinate. Then C x “ ´ x ` by C ,C y “ ´ bx ` by C ´ p x ` by q . Then C p x, y q “ ´ ln p x ` by q b p x ` by q . So ω “ γ ` iδ “ p pdu ` qdy q ` i p rdv ` sdx q“ d p u ` iv q ` ˆ A p z q p bu ´ v q ´ ln p x ` by q b p x ` by q ˙ d p y ´ ix q . Let w “ u ` iv . Then ω “ dw ´ i A p z q „´ b ` i ¯ w ` ´ b ´ i ¯ ¯ w ` b ln p´ A p z qq dz. Recall that θ “ ω ` ˆ ´ ` ib ˙ ω . So we can write ω as ω “ dzA p z q ´ ˆ ´ ` ib ˙ ω . .5.4 a “ ´ The structure equation is dω “ ´ ´ ` b ´ ib ¯ ¯ ω ^ ω ` ω ^ ¯ ω ` ´ ´ ib ¯ ω ^ ω ` ´ ´ ´ ib ¯ ω ^ ¯ ω “ θ ^ ¯ θ ` ´ ` ib ¯ θ ^ ¯ ω ´ ´ ` ib ¯ ω ^ ¯ θ ` ´ ´ ib ¯ θ ^ ω . Let ω “ γ ` iδ be the real and imaginary part decomposition. Then dγ “ A p z q dx ^ γ ` A p z q dy ^ δ ` bA p z q dy ^ γ,dδ “ ´ A p z q dx ^ dy ` A p z q dx ^ δ ` bA p z q dx ^ γ ` bA p z q dy ^ δ. So x γ, dy y and x δ, dx y are two Frobenius systems. So there exist functions p, q, r, s and u, v such that γ “ pdu ` qdy,δ “ rdv ` sdx. From the structure equation, dγ “ dp ^ du ` dq ^ dy “ p x dx ^ du ` p y dy ^ du ` p v dv ^ du ` q x dx ^ dy ` q u du ^ dy ` q v dv ^ dy “ A p z q r dx ^ p pdu ` qdy q ` dy ^ p rdv ` sdx q ` bdy ^ p pdu ` qdy qs . This yields p y ´ q u “ bpA p z q ,p x “ pA p z q ,p v “ ,q v “ ´ rA p z q ,q x “ q ´ sA p z q . (72)54ince there is ambiguity for choosing p and u , by modifying u we arrange p “ A p z q .From the structure equation, dδ “ dr ^ dv ` ds ^ dx “ r x dx ^ dv ` r y dy ^ dv ` r u du ^ dv ` s y dy ^ dx ` s u du ^ dx ` s v dv ^ dx “ ´ A p z q dx ^ dy ` A p z q r dx ^ p rdv ` sdx q ` bdx ^ p pdu ` qdy q ` bdy ^ p rdv ` sdx qs . The functions r and s must satisfy r x ´ s v “ rA p z q ,r y “ brA p z q ,r u “ ,s y “ A p z q ` ´ bq ` bsA p z q ,s u “ ´ bpA p z q . (73)Since there is ambiguity for choosing r and v , by modifying v we arrange r “ A p z q .(72) and (74) reduce to q u “ ´ b p x ` by q ,q v “ p x ` by q ,q x “ q ´ sA p z q ,s v “ ´ p x ` by q ,s y “ p x ` by q ` ´ bq ` bs ´p x ` by q ,s u “ b p x ` by q . C p x, y q and C p x, y q such that q “ A p z q p bu ´ v q ` C p x, y q (74)and s “ ´ A p z q p bu ´ v q ` C p x, y q . (75)The functions C p x, y q and C p x, y q satisfy C x “ C ´ C x ` by ,C y “ bC ´ bC x ` by ` p x ` by q . Choose C p x, y q “ ´ C p x, y q such that z is a local holomorphic coordinate. Then C x “ ´ x ` by C ,C y “ ´ bx ` by C ´ p x ` by q . We can solve C p x, y q C p x, y q “ ´ y p x ` by qp x ` by q . By (74), we get q “ A p z q p bu ´ v q ´ y p x ` by qp x ` by q . By (75), we get s “ ´ A p z q p bu ´ v q ` y p x ` by qp x ` by q . So ω “ γ ` iδ “ p pdu ` qdy q ` i p rdv ` sdx q“ A p z q d p u ` iv q ` ´ A p z q p bu ´ v q ´ y p x ` by qp x ` by q ¯ d p y ´ ix q . w “ u ` iv . Then ω “ A p z q dw ´ i A p z q ”´ b ` i ¯ w ` ´ b ´ i ¯ ¯ w ´ i p z ´ ¯ z qpp ´ ib q z ` p ` ib q ¯ z q ı dz. Recall that θ “ ω ` ˆ ´ ` ib ˙ ω , so we can write ω such that ω “ dzA p z q ´ p´ ` ib q ω C d p ω ^ ¯ ω ^ ω q “ p ´ a qp ` bi q ω ^ ¯ ω ^ ω ^ ¯ ω ‰ . If a ‰ , there is no compact quotient of type C
5. In the following analysis, we assume a “ .Recall that dθ “ ´ ` ib ¯ ¯ θ ^ θ,dω “ θ ^ ¯ θ ` ˆ ` ib ˙ θ ^ ¯ ω ` ˆ ´ ` ib ˙ θ ^ ω . (76)Let θ “ α ` iβ and ω “ γ ` iδ be the real and imaginary part decompositions. By (76), dα “ ´ bα ^ β,dβ “ α ^ β,dγ “ β ^ δ ´ bβ ^ γ,dδ “ ´ α ^ β ´ α ^ δ ` bα ^ γ. Let e , e , e , e be left-invariant vector fields dual to the left-invariant forms α, β, γ, δ , respectively.57he nontrivial brackets are r e , e s “ ´ be ` e ´ e , r e , e s “ be , r e , e s “ ´ e , r e , e s “ ´ be , r e , e s “ e . Make a basis change: ˜ e “ e ` be , ˜ e “ e ´ be . Then r e , ˜ e s “ , r e , ˜ e s “ . After dropping the tildes, the nontrivial brackets are r e , e s “ e ´ e , r e , e s “ ´ e , r e , e s “ e . Define ˜ e “ e ´ e . Then r e , ˜ e s “ ˜ e , r e , e s “ ´ e , r ˜ e , e s “ e . Then define X “ ´ e , X “ e , X “ ˜ e , X “ e . The nontrivial brackets are r X , X s “ X , r X , X s “ X , r X , X s “ ´ X .
58y the classification result of solvmanifolds [15], there exists a co-compact lattice Γ such that G { Γis compact and supports a homogeneous complex Engel structure.
In this case, the symmetry groups of the coframing are different for different parameters a and b .We will consider the following 2 cases:1. a “ ´ π ` kπ, k P Z .In this case, the constants p p , p , q , q , r , r q are p ´ ib, , bi, ib p b ` i q , ´ b p´ b ` i q , ´ b p´ b ` i qp b ` i q q . This is a special case of homogeneous case C3, with a “ b . Thereis no compact quotient that can support a homogeneous complex Engel structure unless b “ C a ‰ ´ π ` kπ, k P Z .By the structure equation, we get d p ¯ ω ^ ω ^ ¯ ω q “ ´ `` b ` i ˘ cos a ` ` ´ ` bi ˘ p sin a ` q ˘ ω ^ ¯ ω ^ ω ^ ¯ ω ,d p ω ^ ¯ ω ^ ¯ ω q “ ´ p´ sin a ` b cos a ´ q ` b ´ ` bi ˘ ω ^ ¯ ω ^ ω ^ ¯ ω . It is easy to verify that d p ¯ ω ^ ω ^ ¯ ω q “ d p ω ^ ¯ ω ^ ¯ ω q “ a “ ´ π ` kπ, k P Z .But this is contradictory to our assumption. Thus the volume form is exact in this case. ByStokes’ Theorem, there does not exist compact quotient that supports a homogeneous complexEngel structure. 59 eferences [1] Vogel, Thomas, Existence of Engel structures , Ann. of Math. (2), 169 (2009), 79–137.[2] Bryant, R. L. and Chern, S. S. and Gardner, R. B. and Goldschmidt, H. L. and Griffiths, P.A.,
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