Abstract
We investigate relationships between Lie groupoids and generalized almost contact manifolds. We first relate the notions of integrable Jacobi pairs and contact groupoids on generalized contact manifolds, and then we show that there is a one to one correspondence between linear operators and multiplicative forms satisfying Hitchin pair. Finally, we find equivalent conditions among the integrability conditions of generalized almost contact manifolds, the condition of compatibility of source, and target maps of contact groupoids with contact form and generalized contact maps.
1. Introduction
A groupoid is a small category in which all morphisms are invertible. More precisely, a groupoid consists of two sets, and , called arrows and objects, respectively, with maps called source and target. It is equipped with a composition defined on the subset , (see Figure 1), an inclusion map of objects and an inverse map . For a groupoid, the following properties are satisfied: , , , , and whenever both sides are defined, , . Here, we have used and instead of , and . Generally, a groupoid is denoted by the set of arrows . A topological groupoid is a groupoid whose set of arrows and set of objects are both topological spaces whose structure maps are all continuous and are open maps.
A Lie groupoid is a groupoid whose set of arrows and set of objects are both manifolds whose structure maps are all smooth maps and are submersions. The latter condition ensures that and -fibres are manifolds. One can see from the above definition that the space of composable arrows is a submanifold of . We note that the notion of Lie groupoids was introduced by Ehresmann [1]. Relations among Lie groupoids, Lie algebroids, and other algebraic structures have been investigated by many authors [2–6].
On the other hand, Lie algebroids were first introduced by Pradines [7] as infinitesimal objects associated with the Lie groupoids. More precisely, a Lie algebroid structure on a real vector bundle on a manifold is defined by a vector bundle map , the anchor of , and an -Lie algebra bracket on , satisfying the Leibnitz rule for all , where is the Lie derivative with respect to the vector field , where denotes the set of sections in .
In [8], Hitchin introduced the notion of generalized complex manifolds by unifying and extending the usual notions of complex and symplectic manifolds. Later, such manifolds have been studied widely by Gualtieri. He also introduced the notion of generalized Kähler manifold [9]. On the other hand, the concept of generalized almost contact structure on odd-dimensional manifolds has been studied in [10–12].
Recently, Crainic [13] showed that there is a close relationship between the equations of a generalized complex manifold and a Lie groupoid. More precisely, he obtained that the complicated equations of such manifolds turn into simple structures for Lie groupoids.
In this paper, we investigate relationships between the complicated equations of generalized contact structures and Lie groupoids. We showed that the equations of such manifolds are useful to obtain equivalent results on a contact groupoid. The paper is organized as follows. In Section 2, we gather main definitions and results used in the other sections. In Section 3, we first state necessary and sufficient conditions for generalized almost contact structure to be integrable, and then we obtain a relation between integrable Jacobi pairs and contact groupoids defined on generalized manifolds. Moreover, we observe that there is a close relationship between -tensors satisfying certain conditions in terms of tensor fields defined on generalized manifolds and multiplicative forms. Finally, we find one to one correspondence among generalized contact map, source and target maps, and the conditions of a generalized contact structure to be integrable.
2. Preliminaries
In this section, we give basic facts of Jacobi geometry, Lie groupoids and Lie algebroids. We first recall notions of contact manifold and contact groupoid from [5]. A contact manifold is a smooth (odd-dimensional) manifold with 1-form such that . is called the contact form of . Let be a Lie groupoid on and a form on Lie groupoid ; then is called -multiplicative if where , , are the canonical projections and , is a function [14]. A contact groupoid over a manifold is a Lie groupoid over together with a contact form on such that is -multiplicative. We recall that multiplicative of a 2-form is defined by
We now recall the notion of Jacobi manifolds. A Jacobi manifold is a smooth manifold equipped with a bivector field and a vector field such that where denotes the Schouten bracket. In this case, defines a bracket on , which is called Jacobi bracket and is given, for all , by The Jacobi bracket endows with a local Lie algebra structure in the sense of Kirillov [15].
We now give a relation between Lie algebroid and Lie groupoid; more details can be found in [16]. Given a Lie groupoid on , the associated Lie algebroid has fibres , for any . Any extends to a unique right-invariant vector field on , which will be denoted by the same letter . The usual Lie bracket on vector fields induces the bracket on , and the anchor is given by .
Given a Lie algebroid , an integration of is a Lie groupoid together with an isomorphism . If such a exists, then it is said that is integrable. In contrast with the case of Lie algebras, not every Lie algebroid admits an integration. However, if a Lie algebroid is integrable, then there exists a canonical source-simply connected integration , and any other source-simply connected integration is smoothly isomorphic to . From now on, we assume that all Lie groupoids are source-simply connected.
We now recall the notion of form (infinitesimal multiplicative form) on a Lie algebroid [17], which will be useful when we deal with relations between Lie groupoids and Lie algebroids. An form on a Lie algebroid is a bundle map satisfying the following properties: (i)(ii)
for , where and denotes the usual pairing between a vector space and its dual.
If is a Lie algebroid of a Lie groupoid , then a closed multiplicative 2-form on induces an form of by For the relationship between form and closed 2-form, we have the following.
Theorem 1 (see [17]). If is an integrable Lie algebroid and if is its integration, then is a one to one correspondence between closed multiplicative 2-forms on and IM forms of .
Finally, in this section, we give brief information on the notion of generalized geometry; details can be found in [9]. A central idea in generalized geometry is that should be thought of as a generalized tangent bundle to manifold . If and denote a vector field and a dual vector field on , respectively, then we write (or ) as a typical element of . The Courant bracket of two sections of is defined by where , , and denote exterior derivative, Lie derivative, and interior derivative with respect to , respectively. The Courant bracket is antisymmetric, but it does not satisfy the Jacobi identity. Here, we use the notations and , which are defined as , for any 1-forms and , 2-form and bivector field , and vector fields and . Also we denote by the bracket on the space of 1-forms on defined by
3. Lie Groupoids and Generalized Contact Structures
In this section, we first give a characterization for generalized contact structures to be integrable; then we obtain certain relationships between generalized contact manifolds and contact groupoids. We recall a generalized almost contact pair and then a generalized almost contact structure.
Definition 2 (see [12]). A generalized almost contact pair on a smooth odd-dimensional manifold consists of a bundle endomorphism of and a section of such that where , for any . Since has a matrix form, where is a -tensor, is a bivector field, is a -form, and is dual of , one sees that a generalized almost contact pair is equivalent to a quintuplet , where is a vector field, a 1-form.
Definition 3 (see [12]). A generalized almost contact structure on is an equivalent class of such pairs .
We now present two examples of generalized almost contact manifolds.
Example 4 (see [11]). An -dimensional smooth manifold has an almost contact structure if it admits a tensor field of type , a vector field , and a -form satisfying the following compatibility conditions: Associated with any almost contact structure, we have an almost generalized contact structure by setting
Example 5 (see [11]). On the three-dimensional Heisenberg group , we choose a basis and let be a dual frame. For , where and for some real number , we define for any real numbers . We also define Then, is a family of generalized almost contact structures.
Given a generalized almost contact pair , we define The endomorphism is linearly extended to the complexified bundle . It has three eigenvalues, namely, , , and . The corresponding eigenbundles are , and , where and are the complex vector bundles of rank generated with and , respectively. Define
Definition 6 (see [11]). Consider a generalized almost contact pair and let be its associated maximal isotropic subbundle. One says that the generalized almost contact pair is integrable if the space of sections of is closed under the Courant bracket; that is, . In this case, the generalized almost contact pair is simply called a generalized contact pair. A generalized contact structure is an equivalence class of generalized contact pairs.
In the sequel, we give necessary and sufficient conditions for a generalized almost contact structure to be integrable in terms of the above tensor fields. We note that the following result was stated in [12].
Theorem 7. A generalized almost contact pair corresponding to the quintuplet is integrable if and only if the following relations are satisfied: (C1)(C2)(C3)(C4)(C5)where
We note that if (11) is a generalized contact structure, then is also a generalized contact structure. is called the opposite of . In this paper, we denote a generalized contact manifold endowed with by .
As an analogue of a Hitchin pair on a generalized complex manifold, a Hitchin pair on a generalized almost contact manifold is a pair consisting of a contact form and a -tensor with the property that and commute (i.e., ). We note that since a generalized almost contact structure is equivalent to a generalized almost complex structure on , the bivector field of the generalized almost contact structure is not nondegenerate in general. But we emphasize that we are putting this condition for restricted case.
Lemma 8. Let be a generalized almost contact manifold. If is a nondegenerate bivector field on , is the inverse 2-form (defined by ), and satisfies (20), then if and only if .
Proof. For , we apply to (20) and using the dual structure , we have We obtain Since (26) holds, for all and , we get In a similar way, one can get the converse.
From now on, when we mention a nondegenerate bivector field , we mean it is nondegenerate on . We note that if is the inverse 2-form of , nondegenerate on implies that is also nondegenerate on .
We say that 2-form is the twist of Hitchin pair . Note that in this case is neither an almost contact structure nor torsion() free.
Lemma 9. Let be an almost contact manifold. and commute if and only if .
Proof. We will only prove the sufficient condition. We have Since is dual contact structure, we get Substituting by , and using contact structure property, Hence, we obtain which shows that and commute. The converse is clear.
Next, we see that (C1) is satisfied automatically when one chooses as the 2-form which is the inverse of defined by .
Lemma 10. Let be a nondegenerate bivector on a generalized almost contact manifold , and the inverse 2-form (defined by ). Then satisfies (C1).
Proof. Since is a closed form, it is obvious due to [13, Lemma 2.7].
Definition 11 (see [14]). The Lie algebroid of the Jacobi manifold is , with the anchor given by and the bracket The associated groupoid, is called contact groupoid of the Jacobi manifold . We say that is integrable as a Jacobi manifold if the associated algebroid is integrable (or, equivalently, if is smooth).
Thus, we have the following result which shows that there is a close relationship between the condition (C1) and a contact groupoid.
Theorem 12. Let be a generalized almost contact manifold and a contact form. There is a 1-1 correspondence between (i)integrable Jacobi pair on (i.e., is satisfying (C1), integrable),(ii)contact groupoids over .
Proof. Since is a contact pair and satisfies (C1), then is an integrable Jacobi pair [18]. From Definition 11, one sees that a contact groupoid is obtained from an integrable Jacobi pair.
The converse is clear.
We now give the conditions for (C2) in terms of and .
Lemma 13. Let be a generalized almost contact manifold and a 2-form. Given a nondegenerate bivector on (i.e., ) and a map , then and satisfy (C2) if and only if and commute.
We now give a correspondence between generalized contact structures with nondegenerate and Hitchin pairs .
Proposition 14. There is a one to one correspondence between generalized contact structures given by (11) with nondegenerate and Hitchin pairs such that . In this correspondence, is the inverse of , and is the twist of the Hitchin pair .
Proof. Since is Hitchin pair, and are closed. By using the following equation (see [13]):
we get
Since , we derive
Applying to (37), then we get
By assumption, we have
Putting this equation into (38), we obtain
which is the second equation of (C3). Now we show that . From (26), we obtain
Hence, we have
From definition of twist, we get
This equation is the first equation of (C4). Now, we will obtain
which is second equation of (C4). Writing the equation as
and since , then we should find
A straightforward computation shows that
Using (35), then we get
Since , applying to (48), we have
In a similar way, we can obtain (C5).
The converse is clear from Lemmas 10 and 13.
We note that, similar to 2-forms, given a Lie groupoid , a -tensor is called multiplicative [13] if for any and any , such that is tangent to at , so is , and Let be a contact manifold. Then it is easy to see that there is a one to one correspondence between -tensors commuting with and 2-forms on . On the other hand, it is easy to see that (C2) is equivalent to the fact that is an form on the Lie algebroid associated Jacobi structure . Thus, from the above discussion, Lemma 13 and Theorem 1, one can conclude with the following theorem.
Theorem 15. Let be a generalized almost contact manifold. Let be an integrable Jacobi structure on and a contact groupoid over . Then there is a natural 1-1 correspondence between (i)-tensors on satisfying (C2),(ii)multiplicative -tensors on with the property that is a Hitchin pair.
We recall the notion of generalized contact map between generalized contact manifolds. This notion is similar to the generalized holomorphic map given in [13].
Let , , be two generalized contact manifolds, and let be the components of in the matrix representation (11). A map is called generalized contact if and only if maps into , into , and into , and .
We now state and prove the main result of this paper. This result gives equivalent assertions between the condition (C3), twist of , and contact maps for a contact groupoid over .
Theorem 16. Let be a generalized almost contact manifold and an induced contact groupoid over with the induced multiplicative -tensor. Assume that satisfy (C1), (C2) with integrable . Then, for a 2-form on , the following assertions are equivalent. (i)(C3) is satisfied,(ii),(iii) is a generalized contact map; condition of generalized contact map on is ; this condition on is .
Proof. : define , such that is the twist of and . We know from Theorem 1 that closed multiplicative 2-form on vanishes if and only if form ; that is, , such that , . This case can be applied for forms with higher degree; that is, 3-form vanishes if and only if .
Since and are closed, from (35) we get . Putting , we obtain
Since , we have
On the other hand, we obtain
If we take in (53) for , we get
On the other hand, from [17], we know that
Differentiating (55), we obtain
Using (56) in (54), we get
In a similar way, we see that
Since , then . Hence, . Thus, we obtain
Using (51) in (59), we derive
On the other hand, it is clear that . Thus, we obtain
Since , we get
Since Jacobi structure is integrable, it defines a Lie algebroid whose anchor map is . Let us use instead of in (60) and (62); then we get
Since , , from (63) we have
that is, .
Since the above equation holds for all nondegenerate , we get
Then, we arrive at
On the other hand, from (64) we obtain
Thus, we get
Then (i)(ii) follows from (67) and (69).
: says that is compatible with 2-form . Also, it is clear that and bivectors are compatible because is a contact groupoid. We will check the compatibility of and -tensors. From compatibility condition of and , we get and .
For all , , and , we have
which is equivalent to
Since and such that , we get
Since this equation holds for all , . Using ,
which shows that . Thus, proof is completed.
Conflict of Interests
The author declares that she has no conflict of interests.