Applied Mathematics
Vol.09 No.02(2018), Article ID:82568,16 pages
10.4236/am.2018.92008
On Quaternionic 3 CR-Structure and Pseudo-Riemannian Metric
Yoshinobu Kamishima
Department of Mathematics, Josai University, Saitama, Japan
Copyright © 2018 by author and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY 4.0).
http://creativecommons.org/licenses/by/4.0/
Received: January 15, 2018; Accepted: February 20, 2018; Published: February 23, 2018
ABSTRACT
A CR-structure on a -manifold gives a conformal class of Lorentz metrics on the Fefferman -bundle. This analogy is carried out to the quarternionic conformal 3-CR structure (a generalization of quaternionic CR-structure) on a -manifold M. This structure produces a conformal class of a pseudo-Riemannian metric g of type on . Let be the geometric model obtained from the projective boundary of the complete simply connected quaternionic hyperbolic manifold. We shall prove that M is locally modeled on if and only if is conformally flat (i.e. the Weyl conformal curvature tensor vanishes).
Keywords:
Conformal Structure, Quaternionic CR-Structure, G-Structure, Conformally Flat Structure, Weyl Tensor, Integrability, Uniformization, Transformation Groups
1. Introduction
This paper concerns a geometric structure on -manifolds which is related with CR-structure and also quaternionic CR-structure (cf. [1] [2] ). Given a quaternionic CR-structure on a -manifold M, we have proved in [3] that the associated endomorphism on the 4n-bundle naturally extends to a complex structure on . So we obtain 3 CR-structures on M. Taking into account this fact, we study the following geometric structure on -manifolds globally.
A hypercomplex 3 CR-structure on a -manifold M consists of (positive definite) 3 pseudo-Hermitian structures on M which satisfies that
1) is a 4n-dimensional subbundle of TM such that
.
2) Each coincides with the endomorphism such that constitutes a hypercomplex structure on .
We call the pair also a hypercomplex 3 CR-structure if it is represented by such pseudo-Hermitian structures on M. A quaternionic CR- structure is an example of our hypercomplex 3 CR-structure. As Sasakian 3- structure is equivalent with quaternionic CR-structure, Sasakian 3-structure is also an example. Especially the -dimensional standard sphere is a hypercomplex 3 CR-manifold. The pair is the spherical homogeneous model of hypercomplex 3 CR-structure in the sense of Cartan geometry (cf. [4] ). First we study the properties of hypercomplex 3 CR-structure. Next we introduce a quaternionic 3 CR-structure on M in a local manner. In fact, let be a 4n-dimensional subbundle endowed with a quaternionic structure Q on a -manifold M. The pair is called quaternionic 3 CR-structure if the following conditions hold:
1) ;
2) M has an open cover each of which admits a hypercomplex 3 CR-structure such that:
a) ;
b) Each hypercomplex structure on generates a quaternionic structure Q on .
A -manifold equipped with this structure is said to be a quaternionic 3 CR-manifold. A typical example of a quaternionic 3 CR-manifold but not a hypercomplex 3 CR-manifold is a quaterninic Heisenberg nilmanifold. In this paper, we shall study an invariant for quaternionic 3 CR-structure on - manifolds.
Theorem A. Let be a quaternionic 3 CR-manifold. There exists a pseudo-Riemannian metric g of type on . Then the con- formal class is an invariant for quaternionic 3 CR-structure.
As well as the spherical quaternionic 3 CR homogeneous manifold , we have the pseudo-Riemannian homogeneous manifold which is a two-fold covering of the pseudo-Riemannian homogeneous manifold . The pair is a subgeometry of conformally flat pseudo-Riemannian homogeneous geometry where .
Theorem B. A quaternionic 3 CR-manifold M is spherical (i.e. locally modeled on ) if and only if the pseudo-Riemannian manifold is conformally flat, more precisely it is locally modeled on .
We have constructed a conformal invariant on -dimensional pseudo- conformal quaternionic CR manifolds in [3] . We think that the Weyl conformal curvature of our new pseudo-Riemannian metric obtained in Theorem A is theoretically the same as this invariant in view of Uniformization Theorem B. But we do not know whether they coincide.
Section 2 is a review of previous results and to give some definition of our notion. In Section 3 we prove the conformal equivalence of our pseudo-Riemannian metrics and prove Theorem A. In Section 4 first we relate our spherical 3 CR-homogeneous model and the conformally flat pseudo-Riemannian homogeneous model . We study properties of 3-dimensional lightlike groups with respect to the pseudo- Riemannian metric of type on . We apply these results to prove Theorem B.
2. Preliminaries
Let be a (4n + 3)-dimensional hypercomplex 3 CR-manifold. Put for one of α’s. By the definition, is a CR-manifold. Let be the canonical bundle over M (i.e. the -line bundle of complex -forms). Put which is a principal bundle: . Compare [ [5] , Section 2.2]. Fefferman [6] has shown that admits a Lorentz metric g for which the Lorentz isometries induce a lightlike vector field. We recognize the following definition from pseudo-Riemannian geometry.
Definition 1. In general if induces a lightlike vector field with respect to a Lorentz metric of a Lorentz manifold, then is said to be a lightlike group acting as Lorentz isometries. Similarly if each generator of is chosen to be a lightlike group, then we call also a lightlike group.
We recall a construction of the Fefferman-Lorentz metric from [5] (cf. [6] ). Let be the Reeb vector field for . The circle generates the vector field on . Define to be a 1-form on such that
(2.1)
In [ [5] , (3.4) Proposition] J. Lee has shown that there exists a unique real 1-form on . The explicit form of is obtained from [ [5] , (5.1) Theorem] in this case:
(2.2)
Here 1-forms are connection forms of such that
(2.3)
The function R is the Webster scalar curvature on M. Note from (2.2)
(2.4)
Normalize so that we may assume . Let denote the symmetric 2-form defined by . Since , it follows . The Fefferman-Lorentz metric for on is defined by
(2.5)
Here . Since is the Reeb field, . As , . On the other hand, by the definition. We have
(2.6)
Thus g becomes a Lorentz metric on in which is a lightlike group.
Theorem 2 ( [5] ). If , then .
3. Hypercomplex 3 CR-Structure
Our strategy is as follows: first we construct a pseudo-Riemannian metric locally on each neighborhood of by Condition I below and then sew these metrics on each intersection to get a globally defined pseudo-Riemannian metric on using Theorem 4. (See the proof of Theorem A.)
Suppose that is a hypercomplex 3 CR-manifold of dimension . Put . It is an -valued 1-form annihilating . In general, there is no canonical choice of annihilating . In [ [3] , Lemma 1.3] we observed that if is another -valued 1-form annihilating , then
(3.1)
for some -valued function on M. (Here is the quaternion conjugate.) If we put for a positive function u and , then such that the map represents a matrix function . If is a hypercomplex structure on for , then they are related as .
For each , we obtain a unique real 1-form on from Section 2 (cf. (2.2)). First of all we construct a pseudo-Riemannian metric on . In general is a nontrivial principal -bundle. It is the trivial bundle when we restrict to a neighborhood. So for our use we assume:
Condition I. is trivial as bundle, i.e. .
We construct a 1-form on as follows. Let generate , , of respectively. Obtained as in (2.2), we have ’s on each such that
We then extend to by setting
(3.2)
Since on ,
. Note that for any
,
(3.3)
On the other hand, we recall the following from [ [3] , Lemma 4.1].
Proposition 3. The following hold:
In particular is a positive definite invariant symmetric bilinear form on ;
Choose a frame field on such that with . Let be the dual frame to such that
(3.4)
Let be the Reeb field for respectively. There is a decomposition .
As before let be a symmetric 2-form. Define a pseudo-Riemannian metric on by
(3.5)
As in (2.6) it follows that , . If we note , letting , it follows . So
. As is positive definite from Proposition 3, g is a pseudo-Riemannian metric of type on .
Theorem 4. Let be the pseudo-Riemannian metric on corre- sponding to another -valued 1-form on M representing , i.e. , then .
We divide a proof according to whether or .
Proposition 5. If , then .
Proof. (Existence.) Suppose . We show the existence of such a 1-form for . Let be the frame on for . Then determines another frame . Since each generates the same as that of , note
(3.6)
Let be the frame on . Then the Reeb field for each is described as
(3.7)
. As on and from Proposition 3, there exists a matrix such that
(3.8)
Two frames , give the coframes , on respectively. Then the above Equations (3.6), (3.7), (3.8) determine the relations between coframes:
(3.9)
Moreover if we put
(3.10)
then (3.15) and (3.10) show that
for which
If is a symmetric matrix defined by
(3.11)
it is easily checked that .
Letting and , we define a pseudo- Riemannian metric
(3.12)
Then a calculation shows
(3.13)
(Uniqueness.) We prove the above is uniquely determined with respect to . Let be the coframe for where . We have a Fefferman-Lorentz metric on from (3.5) and (3.4) under Condition I:
(3.14)
(We take the coefficient for our use.) When , the coframe
will be transformed into a coframe such as
(3.15)
.
If is the corresponding metric on , then by Theorem 2 and there exists a unique 1-form such that
(3.16)
If we sum up this equality for ;
which yields
(3.17)
Compared this with (3.13) it follows
(3.18)
By uniqueness of , defined by (3.10) is a unique real 1-form with respect to .
Next put . The conjugate represents a
matrix . Then it follows
(3.19)
By our definition, a hypercomplex structure on satisfies that . A new hypercomplex structure on is described as
(3.20)
Differentiate (3.19) and restrict to (in fact, on ), using Proposition 3, a calculation shows
(3.21)
In particular, we have .
Proposition 6. If , then .
Proof. Let . Since is uniquely determined by and from (3.19), it implies that
(3.22)
Note that
(3.23)
By (3.21),
Proof of Theorem 4. Suppose where . It follows from Proposition 5 that . By Proposition 6, we have and hence . This finishes the proof under Condition I.
Proof of Theorem A
Proof. Let be a quaternionic 3 CR-manifold. Then M has an open cover where each admits a hypercomplex 3 CR-structure . Put which is an -valued 1-form on . Since we may assume that is homeomorphic to a ball (i.e. contractible), Condition I is satisfied for each , i.e. . Then we have a pseudo-Riemannian metric on for by Theorem 4. Suppose . By condition a) of 2) (cf. Introduction), . Then by the equivalence (3.1) there exists a function defined on such that
(3.24)
It follows from Theorem 4 that on . We may put which is a positive function defined on . By construction, it is easy to see that on . This implies that defines a 1-cocycle on M. Since is a fine sheaf as the germ of local continuous functions, note that the first cohomology . (Here is a chain complex of covers running over all open covers of M.) Therefore there exists a local function defined on each such that , i.e. on . We obtain that
Then we may define
(3.25)
so that g is a globally defined pseudo-Riemannian metric on . If another family represents the same quaternionic 3 CR-structure , then the same argument shows that on for some positive function. Hence the conformal class is an invariant for quaternionic 3 CR-structure. In particular, the Weyl curvature tensor is also an invariant. This completes the proof of Theorem A.
4. Model Geometry and Transformations
We introduce spherical 3 CR-homogeneous model and conformally flat pseudo-Riemannian homogeneous model equipped with pseudo-Riemannian metric of type and then characterize the lightlike subgroup in .
4.1. Pseudo-Riemannian Metric g0
Let us start with the quaternionic vector space endowed with the Her- mitian form:
(4.1)
The q-cone is defined by
(4.2)
When is viewed as the real vector space , denotes the full subgroup of preserving the bilinear form . Consider the commutative diagrams below. The image of the pair by the projection is the homogeneous model of conformally flat pseudo-Riemannian geometry in which is diffeomorphic to a quotient manifold . The identification gives a natural embedding which results a special geometry from .
As usual, the image of by is spherical quarter- nionic 3 CR-geometry .
(4.3)
We describe a pseudo-Riemannian metric on . Let be the product of unit spheres. For , so . Then induces a 2-fold covering for which is an isomorphism.
Let where we put . Choose such that . Denote by the orthogonal complement in with respect to . As , it follows such that
In particular, . Note that this decomposition does not depend on the choice of points and with . (see [3] , Theorem 6.1]). We define a pseudo-Riemannian metric on to be
(4.4)
Noting , and is positive definite, is a pseudo-Riemannian metric of type at each .
4.2. Conformal Group
It is known more or less but we need to check that acts on as conformal transformations with respect to and so does on .
For any , so . However does not necessarily belong to . Normalized , there is such that for some . Note . If is the right multiplication defined by , then there is the commutative diagram:
in which . As , we have for some , . Since and is equivariant, it follows
Similarly for for some , . As , a calculation shows
Hence acts as conformal transformation with respect to .
4.3. Conformal Subgroup
Let be the standard hypercomplex structure on defined by
Put as the associated quaternionic structure. Then leaves invariant Q. The full subgroup of preserving Q is isomorphic to , i.e. the intersection of with .
Let be a faithful representation. Then the subgroup preserves Q so it is contained in
which is a subgroup of .
4.4. Three Dimensional Lightlike Group
Choose and consider a representation restricted to . As we may assume that the semisimple group belongs to , this reduces to a faithful representation: such that
(4.5)
Here we may assume that are relatively prime without loss of generality, and either or 1. The element acts on as
(4.6)
where for . If X is the vector field induced by at , then it follows
(4.7)
Proposition 7. If is a faithful lightlike 1-parameter group, then it has either one of the forms:
(4.8)
Proof. Case (i) . from (4.7) so that . Since and we assume , it follows
As ’s are relatively prime, this implies
As a consequence . In this case note that such that .
Case (ii) . It follows from (4.7) that
Put , such that and . Calculate
(4.9)
This shows
Thus
(4.10)
On the other hand, we may assume in general
(ii-1). Suppose . As for , it implies . Since from (4.10), it follows . Again from (4.10), and so . Note that because . Thus . This implies .
(ii-2). Suppose . In this case . By (4.10), it follows that and , . Thus . This contradicts that nonzero ’s are relatively prime.
(ii-3). Suppose and . Again and so .
To complete the proof of the proposition we prove the following. Put such that .
Lemma 8. Case (ii-1) does not occur.
Proof. It follows from (4.7) that
(4.11)
Put Then As implies . On the other hand, the equation
shows Note that if is the canonical subset in then if and only if Since X is a nontrivial vector field on there is a point x in the open subset such that and thus on S, which contradicts that X is a lightlike vector field.
4.5. Proof of Theorem B
Applying Proposition 7 to a lightlike group we obtain:
Corollary 9. Let be a faithful representation which preserves the metric on . If is a lightlike group on , then either one of the following holds.
(4.13)
Let be as in (4.13). If is a map defined by , then for , ,
So the equivariant diffeomorphism induces a quotient equivariant diffeomorphism
(4.14)
We prove Theorem B of Introduction.
Proof. Suppose that the pseudo-Riemannian manifold is conformally flat. Let be the fundamental group and the universal covering of M. By the developing argument (cf. [7] ), there is a developing pair:
where is a conformal immersion such that for some positive function u on and is a holonomy homomorphism for which is equivariant with respect to .
By Corollary 9, if , then the normalizer of in is isomorphic to . In particular, where . We have the commutative diagram:
(4.15)
where and is an immersion which is - equivariant.
If from (4.13), then . Composed with , we have an equivariant diffeomorphism where . In each case taking the developing map either of (4.15) or , a quaternionic 3 CR-manifold M is spherical, i.e. uniformized with respect to .
Conversely recall is the standard quaternionic 3 CR-structure on equipped with the standard hypercomplex structure on . Suppose that is a spherical quaternionic 3 CR-structure on M with a quaternionic structure Q, then there exists a developing map such that
for some -valued function on with a lift of quaternionic 3 CR-structure . In particular, and .
Let be a pseudo-Riemannian metric on for which is a lift of and to respectively. Put . Let be a function for and such that
By the definition, recall . The induced quaternionic structure for is obtained as . Since , taking , we obtain
(4.16)
As , note that .
On the other hand, let be the pseudo-Riemannian metric on for , it follows from Theorem 4
(4.17)
Take the above element and let be a homomorphism defined by . Define a map which makes the diagram commutative. (Here p is the projection onto the left summand.)
(4.18)
where both and are isomorphisms such that
(4.19)
Recall from (3.5) that . (We write p more pre- cisely.) Consider the pull-back metric
(4.20)
Calculate the first and the second summand of (4.20) respectively.
(4.21)
(4.22)
Thus
Then it follows by the construction of (3.5) that is the corresponding pseudo-Riemannian metric for and so by (4.17). Therefore is conformally flat and so is .
Cite this paper
Kamishima, Y. (2018) On Quaternionic 3 CR-Structure and Pseudo-Riemannian Metric. Applied Mathe- matics, 9, 114-129. https://doi.org/10.4236/am.2018.92008
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