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Axioms 2019, 8(1), 6; doi:10.3390/axioms8010006
Review
Contact Semi-Riemannian Structures in CR Geometry: Some Aspects
Dipartimento di Matematica e Fisica “E. De Giorgi”, Universitá del Salento, Via Provinciale Lecce-Arnesano, 73100 Lecce, Italy
Received: 26 September 2018 / Accepted: 2 January 2019 / Published: 9 January 2019
Abstract
:There is one-to-one correspondence between contact semi-Riemannian structures and non-degenerate almost CR structures . In general, a non-degenerate almost CR structure is not a CR structure, that is, in general the integrability condition for is not satisfied. In this paper we give a survey on some known results, with the addition of some new results, on the geometry of contact semi-Riemannian manifolds, also in the context of the geometry of Levi non-degenerate almost CR manifolds of hypersurface type, emphasizing similarities and differences with respect to the Riemannian case.
Keywords:
contact semi-Riemannian structures; non-degenerate almost CR structures; tangent hyperquadric bundles; homogeneous non-degenerate CR three-manifolds; lie groups; levi-flat CR three-manifolds; bicontact metric structures; levi harmonicityMSC:
53D15; 32V05; 22E99; 53C07; 53C30; 53C25; 53C50; 53C15; 53C431. Introduction
Contact (semi-)Riemannian geometry and (almost) CR geometry are two fields of research that have been developed independently of each other, and with different motivations. However, the two theories are quite related to each other. We note that there is not a monograph dedicated to contact semi-Riemannian structures which emphasizes its connection with the non-degenerate almost CR structures.
We can say that the contact geometry begins with Sophus Lie (1872) when he introduced the notion of a contact transformation as a geometric tool to study systems of differential equations (we refer to H. Geiges [1] for an overview of the historical origins of contact geometry).
The study of contact manifolds from the Riemannian point of view was introduced in the 60’s of the last century by the Japanese school, with S. Sasaki as leader. From then, contact manifolds equipped with Riemannian metrics have been intensively studied. The odd dimensional spheres and the unit tangent sphere bundles of Riemannian manifolds are the most known examples of contact Riemannian manifolds.
The monograph of D.E. Blair [2] and the monograph of C. Boyer and K. Galicki [3] give a wide and detailed overview of the results obtained in this framework. Contact manifolds equipped with semi-Riemannian metrics were first introduced and studied by T. Takahashi [4], who focused on the Sasakian case. In particular, the author discussed the classification of Sasakian semi-Riemannian manifolds of constant -sectional curvature . The relevance in physics of contact semi-Riemannian structures was pointed out in K.L. Duggal [5] (see also H. Baum [6]). A systematic study of contact semi-Riemannian manifolds started with the paper of G. Calvaruso and D. Perrone [7] (see also [8]).
The paper of S.S. Chern and J. Moser [9] on the real hypersurfaces in complex manifolds, and the works by Tanaka [10] and S. Webster [11], have made an important contribution to the development of CR geometry (also in terms of pseudohermitian geometry). Then, (almost) CR structures have drawn a great amount of interest for their connection with several different research areas in both analysis and geometry (see the monograph of S. Dragomir and G. Tomassini [12] for a wide and detailed overview of CR structures).
If is a contact 1-form on an odd dimensional manifold and J is an almost complex structure, i.e., , defined on the contact distribution , such that the Levi form is a non-degenerate Hermitian form , then is said to be a non-degenerate almost CR structure. Different signatures of the Levi form correspond to different kind of geometries. There is one-to-one correspondence between contact semi-Riemannian structures and non-degenerate almost CR structures. In general, a non-degenerate almost CR structure is not a CR structure, that is, in general the integrability condition for is not satisfied. CR structures are considered mainly from a complex analytical point of view.
In this paper (which reflects the interests and knowledge of the author) we give a survey on some known results, with additions of some new result, on the geometry of contact semi-Riemannian manifolds, also in the context of the geometry of Levi non-degenerate almost CR manifolds of hypersurface type, emphasizing similarities and differences with respect to the Riemannian case. In particular, we explain the relationship between contact semi-Riemannian structures and non-degenerate pseudohermitian structures, describing also in some detail several important examples, like hypersurfaces of indefinite Kähler manifolds, and tangent hyperquadric bundles over semi-Riemannian manifolds.
The author believes that this paper will be useful especially to mathematician interested in contact Riemannian geometry, as developed for instance in D. Blair’s book [2], who want to have a comprehensive look at the main differences between the strictly pseudo-convex setting and the semi-Riemannian setting.
2. Contact semi-Riemannian Manifolds
2.1. Generality on Contact Semi-Riemannian Manifolds
A -dimensional manifold M is said to be a contact manifold if it admits a contact form, this is, a global 1-form such that . Given a contact form , there exists a unique vector field , called the characteristic vector field or the Reeb vector field, such that and . Furthermore, a semi-Riemannian metric g is said to be an associated metric (for the contact form ) if there exists a tensor of type such that
and so . In such a case, , or , is called contact semi-Riemannian structure, or contact pseudo-metric structure.
An associated semi-Riemannian metric satisfies
and thus its signature is either or , according to whether is time-like or space-like respectively.
More in general, an almost contact structure on a -dimensional manifold M is a triplet , where is a -tensor, a global vector field, and a 1-form, such that
To note that the properties , , and the fact that the endomorphism has rank , are deducible from and (cf. [2]). A semi-Riemannian metric g on M is said to be compatible with the almost contact structure if Equation (2) is satisfied, where . In such a case, is called almost contact semi-Riemannian structure. By Equations (2) and (3), and for any compatible metric. In particular, for (almost) contact semi-Riemannian manifolds, the characteristic vector field is either space-like or time-like, but can not be light-like. In the literature, (almost) contact Riemannian manifolds are also called (almost) contact metric manifolds.
In the sequel, we denote by ∇ the Levi-Civita connection of a semi-Riemannian manifold , by R the corresponding curvature tensor, taken with the sign convention , and by and r the Ricci tensor and the scalar curvature respectively.
Let be a contact semi-Riemannian -manifold. In contact semi-Riemannian geometry, the tensor
plays a fundamental role, where denotes the Lie derivative. This tensor is self-adjoint, moreover we have (cf. [7])
where is an arbitrary local pseudo-orthonormal basis on M, .
Change of the causal character of the Reeb vector field
There is a relationship between semi-Riemannian metrics of different signature associated to the same contact form . Let be a contact semi-Riemannian structure on M, with . If we consider the semi-Riemannian metric defined by
then Equation (1) easily implies that is a contact semi-Riemannian structure on M. However . Hence, the change of metric described by Equation (8) transforms an associated semi-Riemannian metric of signature into one of signature and conversely. In particular, there exists a one-to-one correspondence between contact Riemannian manifolds and contact Lorentzian manifolds.
Reversing the contact semi-Riemannian structure
Let be a contact semi-Riemannian structure on M. Then, the tensors
define another contact semi-Riemannian structure on M with . We shall say that the deformation of defined by Equation (13) reverses the contact semi- Riemannian structure.
Special contact semi-Riemannian manifolds are the following.
- Sasakian (semi-Riemannian) manifolds are contact semi-Riemannian manifolds whose almost contact structure is normal, that is, the almost complex structure J on defined by , where f is a real-valued function, is integrable, i.e., the Nijenhuis tensor , whereThe integrability condition is equivalent to the conditionMoreover, an almost contact semi-Riemannian manifold is a Sasakian manifold if and only ifIn the literature, Sasakian semi-Riemannian manifolds are also called pseudo-Sasakian manifolds. Pesudo-Sasakian manifolds can be also characterized by using cones on semi-Riemannian manifolds (see, for example, Refs. [3,6,13,14]). Let be a semi-Riemannian manifold. Consider equipped with the metric , . Then, is said to be the -cone on M. If is a pseudo-Sasakian structure on M, with , we put and define the tensor J on byJ is an almost complex structure on compatible with the metric : . Moreover, is the Kahler 2-form of : . In fact, since is a contact semi-Riemannian structure, for :Since is closed, is an almost pseudo-Kaehler structure. By using also the Sasakian condition one can show that J is parallel, that is, the structure on the cone is pseudo-Kaehler. Besides, the converse statement also holds. In other words, there is an one-to-one-correspondence beteween pseudo-Sasakian structures , with , on M, and pseudo-Kaehler structures on the -cone . Moreover, the pseudo-Sasakian manifold is Einstein (respectively, of constant sectional curvature) if and only if the corresponding -cone is Ricci-flat (respectively, flat).
- K-contact manifolds are contact semi-Riemannian manifolds whose Reeb vector field is a Killing vector field, or equivalently, . Any Sasakian semi-Riemannian manifold is K-contact and the converse also holds when .
- H-contact manifolds. The condition that be an eigenvector of the Ricci operator is a very natural condition in contact Riemannian geometry. Sasakian manifolds, K-contact manifolds, -spaces and locally -symmetric spaces satisfy this curvature condition. One of the more important interpretations of this condition is that of an H-contact manifold as introduced by the present author in [15]. Recall that on a Riemannian manifold , a unit vector field V is said to be a harmonic vector field if , where is the Sasaki metric (cf. Section 5.1), is a critical point for the energy functional restricted to maps defined by unit vector fields (see the recent monograph [16], and references therein). If is a semi-Riemannian manifold the same argument applies for vector fields of constant length (if is not light-like). The critical point condition which defines a harmonic vector field is: “ is collinear to V”, where is the so called rough Laplacian of V. H-contact semi-Riemannian manifolds are contact semi-Riemannian manifolds whose Reeb vector field is harmonic, besides we have that (see [15,17]): a contact semi-Riemannian manifold is H-contact if and only if ξ is a Ricci eigenvector. The class of H-contact semi-Riemannian manifolds extends the classes of Sasakian and K-contact semi-Riemannian manifolds. Results about the classification of H-contact Riemannian three-manifolds are given in [18] and in the recent paper of Cho [19].
Remark 1.
Sasakian structures, K-contact structures, and H-contact structures are preserved by the transformation Equation (8). In fact, the normality condition and the tensor do not depend on the metric, so that is Sasakian (respectively K-contact) if and only if is. Moreover, by using Equation (10), we have that is H-contact if and only if is H-contact.
A difference between the Riemannian case and the general semi-Riemannian one is the following: in both cases, from Equation (6), implies . But,
- K-contact Riemannian manifolds are characterized by the condition , since it implies tr and so, (because in the Riemannian case h is diagonalizable);
- in the semi-Riemannian case the condition does not imply . On the other hand, there exist contact semi-Riemannian manifolds for which tr but , and contact semi-Riemannian manifolds for which but (see Examples 3 and 5).
Example 1.
The hyperquadrics [4].
Consider the semi-Euclidean space with the standard indefinite Kaehler structure :
The pseudo-sphere
and the pseudo-hyperbolic space
are hyperquadrics of , both of dimension , of index and and of constant sectional curvature 1 and respectively. Such hyperquadrics have a canonical Sasakian semi-Riemannian structure induced by the indefinite Kaehler structure :
where or , and the characteristic vector field ξ is space-like and time-like respectively. In general, these tensors define an almost contact semi-Riemannian structure on an orintable non-degenerate hypersurface of an indefinite Kaehler manifold (see Proposition 9). Note that reversing the standard pseudo-Sasakian structure of the pseudo-hyperbolic space , we get a pseudo-Sasakian structure of constant sectional curvature 1, which identifies with the standard pseudo-Sasakian structure of the pseudo-sphere .
If we consider the Euclidean unit sphere , then is referred as the standard Hopf vector field on . A tangent vector field ξ on is said to be a Hopf vector field if for some orthogonal complex structure J on . By a result of G. Wiegmink, the Hopf vector fields on are precisely the unit Killing vector fields (cf., for example, Section 3.1 in [16]).
2.2. -Homothetic Deformations and Contact Lorentzian Manifolds
Let be a contact semi-Riemannian manifold of dimension , with . Then, it is easy to check that, for any real constant , the tensors
make up another contact semi-Riemannian structure on M, having the same contact distribution , called a -homothetic deformation (or, transverse homothety) of . Clearly, Equation (16) is the natural semi-Riemannian generalization of -homothetic deformations of a contact Riemannian structure, where one has and needs to assume so that is still Riemannian [20]. Notice that . In particular, , that is, -homothetic deformations preserve the causal character of the Reeb vector field. For , if g is of signature , then is of signature . The Ricci tensors and the scalar curvatures satisfy (see [7], Section 3):
Proposition 1.
The classes of Sasakian, K-contact and H-contact semi-Riemannian manifolds are invariant for a -homothetic deformation.
Recall that there is a canonical way to associate a contact Riemannian structure to a contact Lorentzian structure (and conversely). Let be a contact Lorentzian structure on a smooth manifold M, where the Reeb vector field is time-like. Then,
is a Riemannian metric, and is still compatible with the same contact structure . Moreover, in such case . Hence, is a contact Riemannian structure on M. We remark that
where is obtained by the -homothetic deformation of g for . Consequently, the Levi-Civita connection and curvature of can be easily deduced from the formulae valid for a general -homothetic deformation. In particular, if is the Levi-Civita connection of and ∇ is Levi-Civita connection of g, we have the following:
Taking into account that in the Lorentzian case the tensor h is diagonalizable, for a unit vector field , , from Equations (18) and (19) we have the following formulae
Moreover, a contact Lorentzian manifold is Sasakian (respectively K-contact, H-contact) if and only if the corresponding contact Riemannian manifold is so.
A contact semi-Riemannian manifold is called -Einstein if the Ricci tensor is given by
In particular, the Ricci tensor of the -Einstein K-contact Riemannian structure is given by
where the scalar curvature r is a constant when , and g is Einstein if and only if . Then, from Equations (20) and (21), the Ricci tensor of the corresponding Lorentzian K-contact structure is given by
where the scalar curvature is a constant when . Hence is -Einstein K-contact, and is Einstein if and only if .
In dimension three, every K-contact structure is automatically Sasakian and -Einstein, and thus by Equation (22) also every K-contact Lorentzian three-manifold is automatically Sasakian and -Einstein. Moreover, for a K-contact Lorentzian three-manifold, the scalar curvature and the -sectional curvature are related by .
Recall that a Lorentzian Sasakian manifold is Einsteinian if and only if the cone is Ricci-flat. Moreover, geometries of this type are interesting because they provide examples of twistor spinors on Lorentzian manifolds (see, for example, Ref. [6,14]). In particular, in [6] a twistorial characterization of Einstein Lorentzian Sasakian manifolds is given.
If (resp.) is Einstein K-contact, then (resp.) is -Einstein K-contact. Now, we see as the -Einstein Lorentzian K-contact structures are related to the Einstein Lorentzian Sasakian structures. Let be a Lorentzian K-contact structure on M with time-like, . For the new Lorentzian K-contact structure
from Equations (18) and (19) we have
If in addition is -Einstein, since , then the scalar curvature is a constant and the Ricci tensor of the new Lorentzian K-contact structure is given by
So, for any , the Lorentzian K-contact structure is -Einstein. Recall that the function is the so-called Webster scalar curvature of and (see Section 3.3). Now, if the scalar curvature of the -Einstein Lorentzian K-contact manifold satisfies , i.e., , then the Lorentzian K-contact structure obtained in correspondence to
is Einstein. Analogously, if is -Einstein K-contact and , i.e., , equivalently , the Riemannian K-contact structure obtained in correspondence to
is Einstein, and hence, when M is compact, is Sasakian-Einsten (cf. [21], Theorem A). If , that is , and M is compact, from Theorem 7.2 of [21], we get that the contact Riemannian structure which corresponds to the -Einstein Lorentzian K-contact structure is -Einstein Sasakian. So, summing up we get (see also [22], Section 5)
Theorem 1.
Let be an η-Einstein K-contact Lorentzian manifold of dimension . Then, the Webster scalar curvature is a constant and we have the following.
- If the Webster scalar curvature , i.e., , then there exists a transverse homothety whose resulting structure is Einstein-Lorentzian K-contact.
- If the Webster scalar curvature , i.e., , then there exists a transverse homothety whose resulting structure is Einstein Riemannian K-contact. If in addition M is compact, is Sasakian-Einsten and is η-Einstein Lorentzian-Sasakian.
- If the Webster scalar curvature , i.e., , and M is compact, then the structure is η-Einstein Lorentzian-Sasakian.
From this Theorem and Proposition 6.2 of [6], we get the following
Theorem 2.
([22]) Let be a simply connected η-Einstein Lorentzian Sasakian manifold of dimension and with scalar curvature , i.e., . Then, there exists a transverse homothety whose resulting Lorentzian manifold is a spin manifold. Moreover, there exists a twistor spinor ϕ which is an imaginary Killing spinor and the associated vector field (the Dirac current) is .
Any connected sum of admits a Lorentzian Sasaki-Einstein structure [23]. Now, we give the following.
Example 2.
Let Ω be a simply connected bounded domain in , equipped with the Kaehler structure of constant holomorphic sectional curvature . The corresponding the Kaehler form ω is closed and thus . Let be the natural projection, and t the coordinate on . We construct a Lorentzian-Sasakian structure on M like the Riemannian case (cf. [2], Ch.7)). We define , and define φ such that to be the horizontal lift of the complex structure J and zero in the vertical direction. Then, is an η-Einstein Lorentzian-Sasakian structure with ξ time-like. Moreover, the scalar curvature is given by , and hence . Then, for the resulting structure is Einstein-Lorentzian Sasakian.
2.3. Curvature of K-Contact (and Sasakian) Semi-Riemannian Manifolds
In the contact Riemannian case, the following curvature condition
characterizes the Sasakian structures. In the semi-Riemannian case any Sasakian manifold satisfies Equation (23), but there is not a proof for the conversely and we do not know examples of non-Sasakian contact semi-Riemannian manifolds which satisfy Equation (23). A partial result for this problem is given by the following (cf. [22]).
Theorem 3.
Let be a K-contact semi-Riemannian manifold. Then, M is Sasakian if and only if the curvature tensor R satisfies Equation (23).
Moreover, for a K-contact semi-Riemannian manifold, is an eigenvector of the Ricci operator Q: . In the Riemannian case, this condition holds in a stronger form: M is K-contact if and only if (cf. [2], Theorem 7.1 and Proposition 7.2). Always for a K-contact semi-Riemannian manifold, by (5), one gets . Then, for a non-degenerate plane section span, , the sectional curvature
Conversely, we have the following (cf. [22]).
Theorem 4.
If a semi-Riemannian manifold admits a Killing vector field ξ, , such that the sectional curvature of all nondegenerate plane sections containing ξ equals ε, then
is K-contact semi-Riemannian structure on M.
In the same paper [22], we proved
Theorem 5.
Any conformally flat K-contact semi-Riemannian manifold is Sasakian and of constant sectional curvature .
If is a locally symmetric contact semi-Riemannian three-manifold, then is either flat or of constant sectional curvature [7]. In particular, the pseudoEuclidean space and the universal covering of the pseudohyperbolic space , are the only three-dimensional symply connected symmetric contact Lorentzian manifolds. Note that the pseudo-sphere , which admits a symmetric contact semi-Riemannian structure , with and g of signature , is nothing but the pseudohyperbolic space with the reversed structure.
Now, we give the following new result
Theorem 6.
Any locally symmetric K-contact semi-Riemannian manifold is Sasakian and of constant sectional curvature ε.
Proof.
Moreover, for a locally symmetric semi-Riemannian manifold . Then we get
that is,
Replacing X by , we have
that is,
equivalently,
By using this last equation, we get
that is,
Replacing Z by , the above equation becomes
Now, let p be an arbitrary point and span be an arbitrary non-degenerate plane. Then, from Equation (25), we obtain
that is,
Therefore is a K-contact semi-Riemannian manifold of constant curvature . Then, by using Theorem 5, we conclude that the manifolds is also Sasakian. □
Remark 2.
Theorems 4–6, which include in particular the Lorentzian case, give results analogous to the Riemannian case (see [2]).
If is a semi-Riemannian manifold which admits a Killing vector field of constant length, , such that the sectional curvature of all non-degenerate plane sections containing equals c, then and , where if and if , satisfy the conditions of Theorem 4. Then, by using Theorems 4–6, we get the following (which extends Corollary 4.3 of [22]).
Theorem 7.
Let be a semi-Riemannian manifold whose admits a Killing vector field of constant length, , such that the sectional curvature of all non-degenerate plane sections containing equals c. Then, the following properties are equivalent
Numbered lists can be added as follows:
- (1)
- is conformally flat;
- (2)
- is locally symmetric;
- (3)
- is of constant sectional curvature .
2.4. Geometry of H-Contact Semi-Riemannian Manifolds
H-contact semi-Riemannian manifolds are related to the contact semi-Riemannian manifolds whose Reeb vector field is an infinitesimal harmonic transformation. Recall that a vector field V on a semi-Riemannian manifold is called an infinitesimal harmonic transformation (in short i.h.t.) if the one-parameter group of local transformations generated by V are local harmonic diffeomorphisms. Moreover, V is an i.h.t. if and only if (see [24,25]), where
for all tangent vector fields . With respect to a pseudo-orthonormal basis of , we have
where is the rough Laplacian. Thus, a vector field V is an i.h.t. if and only if .
Theorem 8.
Let be a contact semi-Riemannian manifold. Then, the following properties are equivalent:
- (1)
- ;
- (2)
- ξ is an infinitesimal harmonic transformation;
- (3)
- M is H-contact and tr.
In the Riemannian case, is equivalent to , and so ([26]).
In the semi-Riemannian case:
In fact, the following is an example of contact semi-Riemannian manifold where ξ is an i.h.t. but it is not Killing.
Example 3.
([17]) We consider the 5-dimensional connected Lie group G, whose Lie algebra admits a basis , such that
Consider the semi-Riemannian left-invariant metric g, for which is a pseudo-orthonormal basis with
Define the left-invariant tensors ξ, η, and φ on G putting
Remark 3.
The class of contact semi-Riemannian manifolds with ξ i.h.t. is invariant for -deformations. In fact, the class of H-contact semi-Riemannian manifolds is invariant and tr=tr.
The Lorentzian case. Let be a contact semi-Riemannian manifold, and the metric associated to described by Equation (8). Then, as remarked in Section 2, is H-contact if and only if is H-contact. In particular, there exists a one-to-one correspondence between H-contact Riemannian manifolds and H-contact Lorentzian manifolds. It follows that the class of H-contact Lorentzian manifolds is really large. To note that just like in the Riemannian case, for a contact Lorentzian manifold, with time-like one has if and only if [7]. Hence, using Equation (8) and the corresponding result valid in the Riemannian case ([26], [Theorem 4.1]), we have the following result.
Proposition 2.
Let be a contact Lorentzian manifold with ξ time-like. Then, the following properties are equivalent:
Remark 4.
Let be a Lorentzian manifold and V a unit time-like vector field on M. The space-like energy of V is defined as the integral of the square norm of the restriction of to the space-like distribution . A unit time-like vector field V, which is a critical point of the space-like energy, is called a spatially harmonic vector field. If V is a time-like unit geodesic vector field, then it is spatially harmonic if and only if it is a harmonic vector field ([16], Chapter 8 and [27]). On the other hand, the Reeb vector field of a contact semi-Riemannian manifold is geodesic. Thus, we have the following result [17]: A contact Lorentzian manifold, with ξ time-like, is H-contact if and only if ξ spatially harmonic.
Remark 5.
We note that the Reeb vector field of a three-dimensional contact Riemannian manifold defines a harmonic map from M to if and only if it is H-contact and , where are the nontrivial eigenvalues of tensor h [18]. The same characterization holds in the contact Lorentzian case (in fact, for the corresponding contact Lorentzian manifold we have ). Then, it is natural to ask which are the H-contact Lorentzian three-manifolds for which λ is a constant (equivalently, the Ricci eigenvalue related to ξ is constant). In the Riemannian case, it follows from the proof of Theorem 1.2 in [18] that a three-dimensional contact Riemannian manifold is H-contact with constant Ricci eigenvalue if and only if either it is Sasakian or is locally isometric to a unimodular Lie group G equipped with a non-Sasakian left-invariant contact metric structure. Then, a contact Lorentzian three-manifold is H-contact with constant Ricci eigenvalue (related to ) if and only if either it is Sasakian or is locally isometric to a unimodular Lie group G equipped with a non-Sasakian left-invariant contact Lorentzian structure. A complete classification of simply connected homogeneous contact Lorentzian three-manifolds will be given in Section 4.
Since the work of Hamilton and especially Perelman’s proof of the Poincaré conjecture, there has been considerable interest in the Ricci flow and its applications. For an introduction to Ricci flow we refer to the book of B. Chow and D. Knopf [28]. Ricci solitons have been intensively studied in recent years, particularly because of their relationship with the Ricci flow. For examples and more details on Ricci solitons in semi-Riemannian settings, we may refer for example to [29] and references therein. A Ricci soliton is a semi-Riemannian manifold , admitting a vector field V, such that
for some real constant . A Ricci soliton is said to be shrinking, steady, or expanding, according to whether , or , respectively. Clearly, an Einstein manifold, together with a Killing vector field, is a trivial solution of Equation (30). As proved in the paper [30], any Riemannian Ricci soliton is an infinitesimal harmonic transformation, and it is easily seen that the same argument applies to the semi-Riemannian case. By definition, a contact (semi-Riemannian) Ricci soliton is a contact semi-Riemannian manifold , for which Equation (30) is satisfied by . Since , from Equation (30) with , we have that the Reeb vector field of a contact semi-Riemannian manifold satisfies
So, a contact semi-Riemannian Ricci soliton is H-contact with constant Ricci eigenvalue. On the other hand, if is a contact semi-Riemannian Ricci soliton, then is an infinitesimal harmonic transformation. Hence, by Theorem 8, with and we get the following result [17]:
Theorem 9.
A -dimensional contact semi-Riemannian Ricci soliton is H-contact: , and it is either shrinking or expanding, according to the causal character of the Reeb vector field.
In Riemannian setting, the above Theorem yields a much stronger rigidity result. In fact, by Theorem 8 we have tr, that is . So, by using Equation (30), we have the following result (see [31] and also [22]).
Corollary 1.
A contact Riemannian manifold is a contact Ricci soliton if and only if it is K-contact and Einstein.
Recall the following result of C. Boyer and K. Galicki (see [21]): A compact K-contact Einstein manifold is Sasakian Einstein. Therefore, from Corollary 1 we get the following
Theorem 10.
A compact, contact Riemannian Ricci soliton is Sasakian Einstein.
Moreover, by Theorem 9 and Proposition 2, we deduce the following Lorentzian analogue of Corollary 1.
Corollary 2.
Let be a contact Lorentzian manifold with ξ time-like. Then, is a contact Ricci soliton if and only if it is Einstein and K-contact.
By Corollary 1, only trivial contact Ricci solitons occur in Riemannian settings. On the other hand, the above Theorem 9 specifies that semi-Riemannian Ricci solitons must be found among H-contact manifolds, but this does not exclude the existence of nontrivial contact semi-Riemannian Ricci solitons. As explicitly remarked in [17], the left-invariant contact semi-Riemannian structure described in Example 3 is H-contact (and is also an infinitesimal harmonic transformation), but not a contact Ricci soliton. Hence, the class of semi-Riemannian contact Ricci solitons is strictly included in the one of H-contact semi-Riemannian manifolds satisfying .
3. Non-Degenerate Almost CR Structures
Almost CR structures have drawn a great amount of interest for their connection with several different research areas in both analysis and geometry (Dragomir-Tomassini [12]). In this Section we will emphasize some aspects of their connection with the contact semi-Riemannia structures.
3.1. Generality on Almost CR Structures
Let M be a -dimensional manifold. An almost CR structure (of hypersurface type) on M is a pair where is a smooth real subbundle of rank of the tangent bundle (also called the Levi distribution), and is an almost complex structure: .
Example 4.
Any odd-dimensional Lie group G admits a left-invariant almost CR structure. If , is a basis of the Lie algebra of G and define and J by , then is a left invariant almost CR structure on G.
Starting from an almost contact structure , the pair defines a corresponding almost CR structure on M. It is a natural question to ask when an almost CR structure permits to reconstruct an almost contact structure , such that . The answer is given by the following result.
Proposition 3.
Let M denote an odd-dimensional manifold. An almost CR structure on M is induced by an almost contact structure if and only if M admits a (globally defined) vector field , transversal to at any point.
Proof.
If for some almost contact structure , then clearly the characteristic vector field satisfies the requirements above.
Conversely, suppose that is an almost CR structure, admitting a global vector field transversal to it. Then, it is enough to define , and by
for any vector field . It is then easy to check that is an almost contact structure, and . □
Given an almost CR structure, let be the subspace consisting of all pseudohermitian structures on M at . Then is (the total space of) a real line subbundle of the cotangent bundle and the pseudohermitian structures are the globally defined nowhere zero sections in E. If M is oriented then E is trivial i.e., (a vector bundle isomorphism). Therefore E admits globally defined nowhere vanishing sections, equivalently any orientable almost CR manifold admits a 1-form such that (cf., for example, Ref. [12] Section 1.1.2). On the other hand the existence of a (globally defined) vector field , transversal to at any point is equivalent to the existence of a 1-form such that . In fact, given with , we consider a Riemannian metric g on M (which is paracompact) and then define by for any vector field X. As such, is transversal to at any point because . Hence, by using Proposition 3, we get the following
Proposition 4.
On an orientable odd-dimensional manifold, the existence of an almost CR structure is equivalent to the existence of an almost contact structure.
Let be an almost CR manifold. Put
that is, (resp. ) is the eigenbundle of (the -linear extension of J to ) corresponding to the eigenvalue i (resp. ). Then the complexfication can be decomposed into direct sum of ()-distributions of :
Definition 1.
An almost CR structure is said to be aCR structureon M if (and hence also ) is (formally) integrable:
for any open set .
CR structures are considered mainly from a complex analytical point of view. It is easy to see that an almost CR structure is a CR structure if and only if the following two conditions hold:
Of course, if : Any almost CR structure is integrable (in dimension three, the integrability conditions are trivially satisfied). Moreover, if M is a real hypersurface of a complex manifold then the induced almost CR structure is CR (cf. [12], Proposition 1.1). Therefore, an integrable (codimension one) CR structure is often called CR structure of hypersurface type.
Remark 6.
Another way to define an almost CR structure is the following. Let M be a real -dimensional manifold. An almost CR structure on M is a complex subbbundle , of complex rank n, of the complexified tangent bundle such that where (overbars denote complex conjugates). The integer n is the CR dimension. An almost CR structure is integrable, and then is referred to as a CR structure, if yields for any open set . The Levi (or maximally complex) distribution is the real rank distribution on M given by . It carries the complex structure
Then, , i.e., is the eigenbundle of (the -linear extension of J to ) corresponding to the eigenvalue i. The pair is the real manifestation of .
Now, we want to compare the normality condition Equation (14) of an almost contact structure with the integrability conditions of the induced almost CR structure. We first give the following proposition.
Proposition 5.
Let be an almost CR structure on an odd-dimensional manifold M induced by an almost contact structure . Then,
- (1)
- is a CR structure if and only if for any X, Y , where
- (2)
- the tensor vanishes if and only if and ;
- (3)
- if g is a semi-Riemannian metric compatible with the almost contact structure , then
Proof.
We have to show that the condition is equivalent to the integrability conditions Equations (31) and (32). Since , we have
So implies . Then, by using , we get and thus , that is Equation (31). Replacing X by , we get , and so Equation (33) becomes
Thus implies the condition Equation (32). Conversely, if Equations (31) and (32) are satisfied, then is trivial.
(2) Recall that . So, if and only if for any . On the other hand, for , , and so means . Then, for : implies , that is, , and . Conversely, it is not difficult to see that and imply .
(3) In such a case, the condition is equivalent to the condition that is geodesic with respect to the Levi-Civita connection. In fact, for any : . □
Remark 7.
An almost contact structure satisfying the condition is called a natural almost contact structure. This class of almost contact structures has been introduced and studied in the paper [32]. We note that , so the condition that defines this structure is equivalent to the condition considered in Proposition 5. In particular, any contact semi-Riemannian manifold satisfies the condition , equivalently .
S. Ianus (cf. [2], Theorem 6.6 p. 92) proved that a normal almost contact manifold is a CR-manifold. The following Theorem completes this result.
Theorem 11.
Let be an almost CR structure on an odd-dimensional manifold M induced by an almost contact structure . Then, the almost contact structure is normal if and only if almost CR structure is integrable and the tensor . In particular, if , is normal if and only if is integrable and .
Proof.
By Proposition 5 we know that is a CR structure if and only the tensor S defined on by Equation (33) vanishes. For :
that is,
Moreover, for , from
and
we have
3.2. Non-Degenerate Almost CR Structures and Contact Semi-Riemannian Structures
We have already observed that the existence of an almost CR structure on an -dimensional manifold M induced by an almost contact structure is related to the existence of a 1-form such that (cf. Proposition 3).
Definition 2.
A pseudohermitian structure on an almost CR manifold is a 1-form θ such that and the Levi form , defined by
is Hermitian, that is, . In such a case, is called pseudohermitian almost CR structure on M.
It should be observed that, for any , the following are equivalent:
Then, we have
Proposition 6.
Let be an almost CR structure and θ an 1-form such that . Then, the following properties are equivalent:
(a) is Hermitian, that is, is pseudohermitian;
(b) is symmetric, that is, ;
(c) the partial integrability condition Equation (31) is satisfied.
In the case of an almost CR structure induced by an almost contact semi-Riemanian structure, we have:
Proposition 7.
Let be an almost CR structure induced by an almost contact semi-Riemannian structure . Then, is a pseudohermitian almost CR structure if and only if the tensor
is symmetric on , where ∇ is the Levi-Civita connection of g. In particular, if is a contact semi-Riemannian structure, or an almost α-coKähler structure, then and so it is symmetric.
Proof.
We show that the partial integrability condition Equation (31) is satisfied if and only if q is symmetric on .
If is a contact semi-Riemannian structure, follows from Equation (4).
If is an almost -coKähler structure, follows from (3.1) of [33]. □
Definition 3.
A pseudohermitian almost CR structure is said to be a non-degenerate (pseudohermitian) almost CR structure if the Levi form is, in addition, non-degenerate (equivalently, θ is a contact form, i.e., is a volume form).
In the sequel by a non-degenerate almost CR structure we will mean a non-degenerate pseudohermitian almost CR structure. So, a nondegenerate almost CR structure satisfies the partial integrability condition Equation (31). We remark that two pseudohermitian structures and on the same almost CR manifold, are related by
This gives , then and hence non degeneracy is a CR invariant.
Remark 8.
In dimension three any non-degenerate almost CR manifold is a CR manifold. If , there exist examples of non-degenerate almost CR manifolds which do not satisfy Equation (32), i.e., are not CR manifolds. For example, we have (cf. Theorem 25): If is a semi-Riemannian manifold, , the standard non-degenerate almost CR structure on the tangent hyperquadric bundle is integrable, i.e., it is a non-degenerate CR structure, if and only if has constant sectional curvature.
Let be a non-degenerate almost CR manifold. Let us extend J to an endomorphism of the tangent bundle by requesting that on and (T is the Reeb vector field of ). Then
and is an almost contact structure. In particular, . The Webster metric is the semi-Riemannian metric defined by
for any , where . Equivalently,
Since , and , then by Equation (1) we get that
is a contact semi-Riemannian structure on M. If we denote by the Webster metric with T space-like and by the Webster metric with T time-like, then
This fact agrees with the change of the causal character of the Reeb vector field (cf. Equation (8)). In particular, if is Riemannian, then is Lorentzian with T time-like.
Conversely, a contact semi-Riemannian structure defines a nondegenerate pseudohermitian almost CR structure given by
and is the corresponding Levi form which is nondegenerate and Hermitian, that is, Equation (31) is satisfied.
If the Levi-form is positive definite, the Webster metric (with ) is a Riemannian metric and “non-degenerate” is replaced by “strictly pseudo-convex”.
Thus, we have the following (cf. also [34], Proposition 2.1)
Proposition 8.
The notion of non-degenerate (resp. strictly pseudo-convex) almost CR structure is equivalent to the notion of contact semi-Riemannian (resp. Riemannian) structure . In particular, non-degenerate (resp. strictly pseudo-convex) CR structures correspond to contact semi-Riemannian (resp. Riemannian) structures satisfying the condition Equation (32).
• Some remarks
1. We note that the non-degeneracy is more natural in CR geometry with respect to strictly pseudo-convexity. In fact non-degeneracy is a CR invariant property, i.e., it is invariant under a transformation , where is a smooth function, while strictly pseudo-convexity is not a CR invariant property (if is positive definite and , then is negative definite). In particular, if is a non-degenerate almost CR structure, then for any real constant , is a non-degenerate almost CR structure. Moreover, the Webster metrics and are related, taking account that , by
that is,
This is related to the deformation Equation (16).
2. Let be a non-degenerate almost CR structure and the corresponding contact semi-Riemannian structure. Since
then we get that is still a contact semi-Riemannian structure with . This second structure is obtained by Equation (13), i.e., reversing the first contact semi-Riemannian structure.
3. Let be a contact 1-form. Then, there exists an associated metric for if and only if there exists an almost complex structure J on ker such that the Levi form is Hermitian.
A generalization of the basic results in pseudohermitian geometry to the case of a contact Riemannian manifold whose almost CR structure is not integrable was started by S. Tanno [20]. Results in this direction are given also in [35,36,37].
• Hypersurface of an indefinite Kaehler manifold.
The property (1) in Proposition 5 suggests to look the almost contact structure of a hypersurface of an indefinite Kaehler manifold. Let be an indefinite -Kaehler manifold (cf. [38] for definitions and examples). Suppose that M is an orientable non-degenerate real hypersurface of . Let N be a normal vector field, , that defines the orientation of M. Then, the tensors
define an almost contact semi-Riemannian structure. Moreover, we have (see, for example, Ref. [39])
where is the shape operator. Now, consider the almost CR structure induced . In particular, for , is a consequence of Equation (38)b. In fact,
Next, since , from Equation (33) we have
On the other hand, by Equation (38)a, we have for any , and so we get for any . Therefore, by 1) of Proposition 5, the almost CR structure is integrable. So, we proved the following
Proposition 9.
Let be an indefinite Kaehler manifold. Suppose that M is an orientable non-degenerate real hypersurface of . Then, the almost contact semi-Riemannian structure on M given by (37) defines a CR structure on M.
Now, we see when the almost contact semi-Riemannian structure defined by (37) is Sasakian. Suppose that this structure is Sasakian. Then, comparing Equation (15) with Equation (38)a, we get
and taking , we have
In particular, . Then, , and so from Equation (39) we obtain
Conversely, if A is given by Equation (40), by Equation (38)a we get Equation (15). Then we get the following (cf. [39], and [2] Theorem 6.15 in the Riemannian case).
Theorem 12.
By Proposition 9, the standard pseudohermitian almost CR structure of an orientable non-degenerate real hypersurface, that is, the one induced by Equation (37), is integrable. Then, Proposition 6 gives that is always a pseudohermitian CR structure. Moreover, by using Equation (38)b, i.e., , we have
Consequently, the condition is satisfied if and only if
Then, we have the following (cf. [2], Theorem 4.12, for the Riemannian case)
Theorem 13.
Let M be an orientable non-degenerate real hypersurface of an indefinite Kaehler manifold . Then, the almost CR structure induced on M is always a pseudohermitian CR structure. Moreover, it is a non-degenerate CR structure if and only if the shape operator satisfies Equation (42).
• Levi-flatness
The “opposite”of Levi non-degenerate is the following definition.
Definition 4.
A pseudohermitian almost CR structure is said to be Levi-flat, or Levi-degenerate, if the Levi form vanishes.
In particular, an almost contact structure on an odd-dimensional manifold M, with , defines a pseusdohermitian Levi-flat almost CR structure . If is a Levi-flat pseudohermitian almost CR manifold, then is zero on and the Frobenius Theorem shows that defines a foliation.
Examples of Levi-flat CR manifolds. First we recall some definitions which are related to the Levi-flat manifolds. An almost α-coKähler structure is an almost contact metric structure such that and for some real constant , where is the fundamental 2-form (see [33] and the references there in). In particular, gives an almost coKähler structure (equivalently, an almost cosymplectic structure following Goldberg-Yano [40]) and gives an almost -Kenmotsu structure. In any case, an almost -coKähler structure defines a Levi-flat pseudohermitian almost CR manifold.
In the case of an orientable non-degenerate real hypersurface of an indefinite Kaehler manifold , by using Equation (41), the standard pseudohermitian CR structure of M is Levi-flat, i.e., , if and only if on . On the other hand, if we consider the fundamental 2-form , we have (cf. [39]). Hence, an orientable non-degenerate real hypersurface of an indefinite Kähler manifold is almost coKähler if and only if .
Recently in the paper [33], see also [41], we proved that an orientable Riemannian three-manifold admits an almost -coKähler structure with g as a compatible metric if and only if M admits a foliation, defined by a unit closed 1-form, of constant mean curvature. Then, in the same paper we show that a simply connected homogeneous almost -coKähler three-manifold is either a Riemannian product of type , equipped with its standard coKähler structure, or it is a semidirect product Lie group equipped with a left invariant almost -coKähler structure. All the three-manifolds listed in this classification are examples of Levi-flat pseudohermitian CR three-manifolds.
• The embeddability
A natural difference between the class of CR manifolds and the class of almost CR manifolds is the question of embeddability. In fact, a question of principal interest in the theory of compact, -dimensional CR-manifolds is to understand when a given strictly pseudo-convex CR-structure can be realized by an embedding in . This question is only of interest in the three-dimensional case because a theorem of Boutet de Monvel [42] states that any strictly pseudo-convex CR-structure, on a compact -manifold, is realizable as an embedding in some , provided .
The global embedding problem in CR geometry in dimension 3 has received a lot of attention. In [43], Burns and Epstein considered perturbations of the standard CR structure on the three-sphere . They showed that a generic perturbation is non-embeddable and gave a sufficient condition for embeddability ([43], Theorem 5.3). In the same paper, they introduced the notion of stability for CR embeddings. Then Lempert [44] considered the problem of stability of CR embeddings of a compact three-dimensional CR manifold into , and proved that if a compact strictly pseudo-convex CR manifold admits a CR embedding into then this embedding is stable.
S. Chanillo, H. Chiu and P. Yang ([45,46]) discussed the relationship between the embeddability of three- dimensional closed strictly pseudo-convex CR manifolds and the positivity of the CR Paneitz operator and the CR Yamabe constant. In particular, they proved the embeddability into for some n when the CR Paneitz operator is non-negative and the CR Yamabe constant is positive.
3.3. The (Generalized) Tanaka-Webster Connection and the Pseudohermitian Torsion
Let be a non-degenerate almost CR manifold and the associated contact semi-Riemannian structure. The most convenient linear connection for studying is the so-called (generalized) Tanaka-Webster connection . This is the linear connection given by
for any , where ∇ is the Levi-Civita connection of . Equivalently, is defined by
where is the usual projection . The generalized Tanaka-Webster connection is due to Tanno [20] (though confined to the positive definite case). For a nondegenerate almost CR manifold, was considered in [47,48]. admits an axiomatic description similar to that of the ordinary Tanaka-Webster connection (cf. Tanaka [10]) except for the property . More precisely, is the unique linear connection obeying to the axioms
Here is the torsion tensor field of , and Q is the Tanno tensor, i.e.,
We note that and for any . Then, by the same proof given in [20], if and only if is integrable, that is:
and then is the ordinary Tanaka-Webster connection.
The pseudohermitian torsion of (introduced by Webster in the integrable case [11], see also [12], p. 26) is the vector valued 1-form on M defined by
and thus
and
Then, by using
(2) of Proposition 5 and Theorem 11, one gets:
Theorem 14.
Let be a non-degenerate almost CR manifold. Then,
- the Reeb vector field ξ is Killing with respect to Webster metric if and only if pseudohermitian torsion τ vanishes, equivalently ;
- the almost contact structure is normal, equivalently the Webster metric is Sasakian, if and only if the almost CR structure is integrable and the pseudohermitian torsion τ vanishes;
- a non-degenerate CR manifold is Sasakian if and only if .
Next, we recall that given a semi-Riemannian manifold , with the Levi-Civita connection, and a smooth nondegenerate distribution on , then is called minimal distribution if trace, where
and is the natural projection on . Moreover, the distribution is called totally geodesic if the symmetrized second fundamental form vanishes.
Consider the non-degenerate almost CR manifold . For the Levi distribution , the second fundamental form is given by , and by using Equation (4) we get
where is a local orthonormal basis. Since , we get . Moreover, by using Equation (4), the symmetrized second fundamental form is given by
Then, we get
Proposition 10.
([34]) For any non-degenerate almost CR manifold , the Levi distribution is minimal in , and it is totally geodesic if and only if the pseudohermitian torsion τ vanishes.
Now, we give some properties related to the pseudohermitian curvature of a non-degenerate almost CR manifold . Denote by the pseudohermitian curvature tensor, that is, the curvature tensor associated to the generalized Tanaka-Webster connection . Then, following Tanno [20,49], the pseudohermitian Ricci tensor and the pseudohermitian scalar curvature are defined by
The pseudohermitian scalar curvature is also called the (generalized) Tanaka-Webster scalar curvature [20]. In [48] we considered the following
Definition 5.
A nondegenerate almost CR manifold , , is said to be pseudo-Einstein if the pseudohermitian Ricci tensor is proportional to the Levi form, that is, , where .
In the case of a non-degenerate CR manifold with vanishing pseudohermitian torsion our definition of pseudo-Einstein structure coincides with the definition of J.M. Lee [50]. In general, the pseudo-Einstein condition does not imply that the pseudohermitian scalar curvature is constant, so such a structure is less rigid than an Einstein structure on a semi-Riemannian manifold [50]. Next, we show that this notion is related to the notion of -Einstein contact semi-Riemannian manifold given in Section 2.2.
Consider a non-degenerate almost CR manifold , , with pseudohermitian torsion . In this case, Equation (43) gives
and in particular
Then, the pseudohermitian Ricci tensor and the Tanaka-Webster scalar curvature are given by
where and r denote the Ricci tensor and the scalar curvature of the Webster metric . So, we get
Proposition 11.
([48]) Let be a non-degenerate almost CR manifold with pseudohermitian torsion . Then, the structure is pseudo-Einstein if and only if the corresponding semi-Riemannian contact structure is η-Einstein.
When the pseudohermitian torsion , there are other conditions on with an interesting meaning. Given an oriented, compact, contact manifold , denote by the set of all Riemannian metrics associated to the contact form and by the set of all almost CR structures J for which the Levi form is positive definite. By Proposition 8, the sets and can be identified.
• The condition . Tanno [20] considered the Dirichlet energy
defined for any . Then, he found the critical point condition ([20], Theorem 5.1)
We note that this condition has a tensorial character, so it holds also in the non compact case. The Dirichlet energy Equation (50) was studied by Chern and Hamilton [51] for compact contact three-manifolds as a functional defined on the set (there was an error in their calculation of the critical point condition, as was pointed out by Tanno). Moreover, since , the functional Equation (50) is equivalent to the functional studied in general dimension, for compact regular contact manifold, by Blair ([2], Section 10.3).
Now, since , where and is the pseudohermitian torsion, we have
Then, to consider the Dirichlet energy Equation (50) is equivalent to consider the following
defined on the set . Moreover, using the Tanaka-Webster connection given by Equation (43), we get
Thus, the critical point condition Equation (51) becomes
If is the tangent sphere bundle of a Riemannian manifold of constant curvature , equipped with its standard contact structure. Then, the standard associated Riemannian metric (cf. Section 5.1) satisfies the critical point condition , equivalently (cf. proof of Theorem 10.13 in [2]). Moreover, we recall the following (see [15,52]):
Theorem 15.
Let be a compact H-contact -manifold such that , i.e., g is critical for the Dirichlet energy . If is positive definite for some constant , then the first Betti number of M vanishes.
- The condition . This condition, equivalently , or also , is related to some interesting property. It is equivalent to the curvature property [34]:Recall that if M is an oriented compact manifold, by a classical result of Hilbert (see also Nagano [55]), a Riemannian metric g on M is a critical point of the integral of the scalar curvature, , as a functional on the set of all Riemannian metrics of the same total volume on M, if and only if g is an Einstein metric. Now, by using a result of [54], we get that a contact Riemannian three-manifold is -Einstein if and only if it is H-contact and satisfies the critical point condition (equivalently, ).
- The Chern-Hamilton functional. In CR geometry a natural functional is the the integral of the generalized Tanaka-Webster scalar curvature. For a strictly pseudo-convexity almost CR manifold, i.e., for a contact Riemannian manifold, the generalized Tanaka-Webster scalar curvature is given by (cf. [20])This is eight times the Webster scalar curvature W as defined by Chern and Hamilton [51] on three-dimensional contact manifolds. In the same paper, Chern and Hamilton proved, in dimension three, that the critical point condition for the functional
- An interpretation of the Tanaka-Webster scalar curvature. Recall that a contact form on a compact manifold M is called regular if its Reeb vector field is regular, i.e., any point of M has a neighborhood such that any integral curve of passing through the neighborhood passes through only once. In this case M is a principal -bundle over a symplectic manifold B whose fundamental 2-form has integral periods (a Hodge manifold). The corresponding fibration is known as the Boothby-Wang fibration [56]. Now, let be a compact simply connected regular Sasakian, -manifold. Then, the base of the Boothby-Wang fibration is a compact Kähler manifold of complex dimension n, with Kähler metric and fundamental 2-form satisfying (cf., for example, Ref. [57,58])Moreover, the scalar curvatures r, of and , respectively, are related byOn the other hand, in the Sasakian case, Equation (53) becomesSo, in this case, the Tanaka-Webster scalar curvature is the scalar curvature of the Kähler manifold base of the Boothby-Wang fibration. We note that a compact simply connected homogeneous Sasakian manifold is regular [57].
3.4. Contact Geometry of CR Manifolds
In this subsection we give a presentation of some results about the study of CR manifolds, i.e., the CR integrable case, from the point of view of contact geometry.
• The Olszak’s result in the semi-Riemannian setting
First rigidity results concerning non-degenerate almost CR manifold with the Webster metric of constant curvature were obtained in the Riemannian case by D.E. Blair and Z. Olszak. Blair [59] showed that a contact form does not admit any flat associated Riemannian metric in dimension . Then, Olszak [60] generalizing this result proved that if a contact Riemannian -manifold, , is of constant curvature , then the manifold is Sasakian and . In the semi-Riemannian case, we have ([7,8])
Theorem 16.
Let be a contact semi-Riemannian -manifold. If and is of constant sectional curvature κ, then and .
In terms of CR geometry, Theorem 16 becomes
Theorem 17.
Let be a -dimensional non-degenerate almost CR manifold, . If the Webster metric is of constant sectional curvature κ, then and the pseudohermitian torsion satisfies .
In particular, since , a non-degenerate almost CR structure does not admit any flat semi-Riemannian Webster metric in dimension , so Blair’s result also holds in the semi-Riemannian setting. But, there are examples of non-degenerate almost CR manifold with and . In fact we have the following (see [34] for details).
Example 5.
Consider the space and two smooth functions . We put , , and . Define the vector fields , on M by
Moreover, we define the 1-form , the vector field , the tensor φ by
and the semi-Riemannian metric g of signature by
Then defines a contact semi-Riemannian structure, and so a non-degenerate almost CR structure on M, with Levi distribution span. Moreover, we can construct a frame of vector fields on with null vector fields which satisfy
Therefore, . Moreover, if and only if . So, taking the functions such that , we obtain a non-degenerate almost CR structure with and . Moreover, this structure in general is not a CR structure. In fact, taking for example and , one gets that the integrability condition Equation (32) is satisfied if and only if .
If the almost CR structure is integrable, we get the Olszak’s result in the semi-Riemannian setting. In fact, we have the following.
Theorem 18.
([34]) Let be a non-degenerate CR manifold, , . If the Webster metric is of constant sectional curvature κ, then and the pseudohermitian torsion , i.e., the Webster metric is Sasakian.
Remark 9.
In dimension three, a non-degenerate CR manifold with the Webster metric locally symmetric (in particular, of constant sectional curvature) is either flat or of constant sectional curvature , and in the second case the metric is Sasakian [7].
• -spaces
We recall that a contact metric manifold is said to be a -space if its structure satisfies the -nullity condition, that is, the curvature tensor satisfies
for some . This condition, which is a generalization of the Sasakian condition Equation (23), defines an interesting class of contact metric manifolds introduced by Blair, Koufogiorgos and Papantoniou [61]. We note that this class is invariant for -deformations. A classification Theorem of the -spaces is given by E. Boechx [62]. Moreover, we recall the following (cf. [2], p. 124)
Theorem 19.
A -space M is a strictly pseudoconvex CR-manifold. Moreover, , and if the pseudohermitian torsion vanishes and thus the structure is Sasakian. If , the condition determines the curvature of M completely.
In particular, the Ricci operator Q and the scalar curvature r of a -dimensional -space M, , are given by
Then, -spaces are examples of H-contact manifolds. For a non-Sasakian space, Boeckx Boechx [62] introduced an invariant
and showed that for two non-Sasakian spaces , we have if and only if up to a -homothetic deformation, the two spaces are locally isometric as contact metric manifolds.
• Sasakian geometry by using a variational theory
In paper [63], Barletta and Dragomir built a variational theory of geodesics of the Tanaka-Webster connection on a strictly pseudoconvex CR manifold M. They obtained the first and second variation formulae for the Riemannian length of a curve in M and showed, in particular, that in general geodesics of admitting horizontally conjugate points do not realize the Riemannian distance. The paper also contained interesting results concerning the pseudohermitian sectional curvature , that is, the sectional curvature defined by the tensor
where is the pseudohermitian curvature tensor associated to the Tanaka-Webster connection , and is the Webster metric. For example they proved (cf. Theorems 4 and 5 of [63]) the following.
Theorem 20.
Let M be a a strictly pseudoconvex CR manifold.
- (1)
- If M has non-positive pseudohermitian sectional curvature, then it has no horizontally conjugate points.
- (2)
- If M, of CR dimension , has constant pseudohermitian sectional curvature, then it has vanishing pseudohermitian torsion () if and only if the Tanaka-Webster connection of M is flat.
• Almost contact structures belonging to a CR structure
Let be a CR structure on a odd-dimensional manifold M. We say that an almost contact structure belongs to the CR structure if and . Then, by Lemma 1.1 of [64], two almost contact structures and belong to the same CR structure if and only if
for some smooth function and vector field , where .
Denote by an almost contact structure belongs to a non-degenerate CR structure and satisfying the condition . Then, K.Sakamoto and Y. Takemura [64] proved the existence of a unique linear connection associated to . Moreover in [65], they obtained a curvature invariant of the pseudo-conformal geometry, that is, a tensor field invariant under the change of almost contact structures belonging to the same non-degenerate CR-structure. For the case of a normal almost contact structure the invariant tensor field is just the Bochner curvature tensor.
4. Homogeneous Non-Degenerate CR Three-Manifolds
The main purpose of this Section is to give a presentation of some results about homogeneous non-degenerate CR three-manifolds.
4.1. The Classification Theorem
Recall some definitions about the homogeneity.
A contact manifold is said to be homogeneous if there exists a (connected) Lie group G of diffeomorphisms acting transitively on M and leaving invariant. A contact semi-Riemannian manifold is said to be homogeneous if there exists a (connected) Lie group G of isometries acting transitively on M and leaving invariant, that is, for any there exists such that , and for every .
A map between almost CR manifolds is a CR map if
In particular a CR transformation is a diffeomorphism f such that
for any . If a CR transformation is also called CR automorphism.
Remark 10.
Typical examples of CR maps are got as traces of holomorphic maps of Kaehlerian manifolds on real hypersurfaces. Precisely, let be a Kaehlerian manifold. Any orientable real hypersurface admits a natural CR structure (cf. Proposition 9). If is another oriented real hypersurface in the Kaehlerian manifold and is a holomorphic map such that then is a CR map. The statements above hold true for traces of holomorphic maps among indefinite Kaehlerian manifolds [47]. A characterization of K-contact structures in terms of CR maps is presented in Theorem 32 of this paper.
Let and be pseudohermitian structures on the almost CR manifolds M and respectively. If is a CR map, then for some . In particular, a CR map f is called pseudohermitian map if for some . Also f is isopseudohermitian if .
Let be a pseudohermitian almost CR manifold. Denote by the group of all CR automorphisms such that .
In general the group is not a Lie group, but if M is a pseudohermitian, or a non-degenerate, CR manifold then is a Lie group (Ref. [12], p. 60; Ref. [66], p. 218) of dimension , where .
A non-degenerate, or pseudohermitian, CR manifold is said to be homogeneous non-degenerate, or pseudohermitian, CR manifold if there exists a Lie group acting transitively on M (cf. [12] p. 341), that is, for any there exists such that , and
If is a contact semi-Riemannian manifod and f a diffeomorphism of M, then
Note that two contact semi-Riemannian manifolds are called isomorphic (or equivalent) if there exists a isometry such that
By using Equation (56), we get the following
Proposition 12.
Let a non-degenerate CR manifold and the corresponding contact semi-Riemannian structure. Then,
In other words,
Recall that there is a canonical way to associate a contact Riemannian structure to a contact Lorentzian structure (and conversely). If is a contact Lorentzian structure on a smooth manifold M, , where the Reeb vector field is time-like, then (cf. Section 2.2 and also Equation (36))
is a contact Riemannian structure on M. The scalar curvatures and of g and are related by Equation (20):
Now, let be a non-degenerate CR three-manifold. Then, the Levi form is definite, and we can assume positive definite (if necessary, we change with ). Therefore, without loss in generality, in dimension three, we can consider either Lorentzian with T time-like or Riemannian. In particular, the Sasakian condition does not depend on the causal character of the Reeb vector field T. Moreover, for a non-degenerate CR three-manifold, the Tanaka-Webster scalar curvature is given by
So, the Tanaka-Webster scalar curvature does not depend on the causal character of the Reeb vector field T, i.e., . In fact,
If we consider the scalar torsion introduced by Chern and Hamilton in [51] in their study of contact Riemannian three-manifolds, since , we have
and thus
where W is the Webster scalar curvature as defined by Chern and Hamilton [51].
Since the Webster scalar curvature W and the scalar pseudohermitian torsion do not depend on the causal character of the Reeb vector field T, that is, they depend only on Levi form , then it is natural to consider these invariants in order to classify the homogeneous nondegenerate CR three-manifolds. More precisely, we consider the invariant W in the Sasakian case, and the invariant
in the non Sasakian case. Then, the classification Theorem of [67], can be reformulated in the following form.
Theorem 21.
A simply connected, homogeneous, non-degenerate CR three-manifold is a Lie group G equipped with a left-invariant non-degenerate CR structure. More precisely, one of the following cases occurs:
• Sasakian case (i.e., the pseudohermitian torsion ).
- (1)
- If G is unimodular, then it is
- (i)
- the Heisenberg group when ;
- (ii)
- the 3-sphere group , i.e., the special unitary group, when ;
- (iii)
- , the universal covering of the special linear group , when ;
- (2)
- If G is non-unimodular, then its Lie algebra is given by
• Non Sasakian case (i.e., the pseudohermitian torsion ).
- (1)
- If G is unimodular, then it is
- (i)
- when ;
- (ii)
- , universal covering of the Lie group of orientation-preserving rigid motions of Euclidean plane, when ;
- (iii)
- , the group of rigid motions of Minkowski plane, when ;
- (iv)
- when .
- (2)
- If G is non-unimodular, then its Lie algebra is given by
Remark 11.
In [67] we used the notation τ to denote . Moreover, in the same paper, in the statement of Theorem 21 we used the Riemannian Webster metric .
The structure on the unimodular Lie groups in Theorem 21 satisfy, in the Riemannian case, the -nullity condition Equation (54), that is, they are -space. Besides, we note that in the non Sasakian case the role played by the invariant p defined by Equation (57). Since, and , i.e., , then
which is the invariant of Boeckx defined by Equation (55).
4.2. Consequences of the Classification Theorem
In this subsection we examine some interesting consequences of the classification Theorem. A first immediate consequence is the following
Corollary 3.
The 3-sphere group is the only simply connected 3-manifold which admits a homogeneous non-degenerate CR structure with Webster scalar curvature .
In particular, the sphere is the only simply connected 3-manifold which admits a homogeneous Sasakian structure with Webster scalar curvature ( for the standard Sasakian structure).
Moreover, examining the proof of Theorem 21, more precisely by using (3.3) and (3.10) of [67], we get
Proposition 13.
The Lie group is the only simply connected 3-manifold which admits a homogeneous non-degenerate CR structure with flat Riemannian Webster metric. In such a case
In the Lorentzian case, one gets
Proposition 14.
([7]) The Lie group is the only simply connected 3-manifold which admits a homogeneous non-degenerate CR structure with flat Lorentzian Webster metric. In such a case
H. Geiges [68] proved that a compact 3-manifold admits a Sasakian structure if and only if it is diffeomorphic to a left invariant quotient of , the Heisenberg group or by a discrete group. As a consequence of Theorem 21 we have
Proposition 15.
The unimodular Lie groups , the Heisenberg group , , and the non-unimodular Lie group with Lie algebra defined by Equation (58), are the only simply connected three-manifolds which admit a homogeneous Sasakian structure.
Now, let be a homogeneous Sasakian structure on the sphere , with Webster scalar curvature . Since , then and , for , define a Sasakian structure on with of constant sectional curvature (cf. [52], Section 3). In particular, is isomorphic to the standard Sasakian structure [4]. Then, we can assume , and consequently we have
where , , is a Berger metric, that is, a metric defined as the canonical variation , , of the standard metric on , obtained deforming along the fibres of the Hopf fibration:
where denotes the standard Hopf vector field on . Therefore we get:
Proposition 16.
In the second part of Corollary 3, the Sasakian metric on is homothetic to a Berger metric.
Remark 12.
The main result of [51] says that any contact structure on a compact and orientable three-manifold has a contact form and a contact Riemannian metric whose Webster scalar curvature W is either a constant or is everywhere strictly positive. Now, if M is a compact Sasakian 3-manifold with Webster scalar curvature , then M admits a contact Riemannian structure of positive Ricci curvature [69]. If, in addition, M is simply connected, by a deep result of R.S. Hamilton [70], M is diffeomorphic to the sphere . However, this fact is not too surprising since a compact simply connected manifold which admits a nonsingular Killing vector field is diffeomorphic to (cf. [52] Section 4).
By the proof of Theorem 21, the Lie algebra of the lie group is defined by
where for we have the following possibilities: , , and . Moreover the Webster scalar curvature is given by . So, for we get . For the non-unimodular Lie group with Lie algebra defined by Equation (59), the the Webster scalar curvature is given by and so for . Then, we get the following (which corrects Corollary 3.3 of [67]).
Corollary 4.
The Heisenberg group , and the non-unimodular Lie group with its Lie algebra defined by Equation (59), are the only simply connected 3-manifolds which admit a homogeneous nondegenerate CR structure with Webster scalar curvature . In particular, the Heisenberg group is the only simply connected three-manifold which admits a non-degenerate CR structure with pseudohermitian torsion and Webster scalar curvature .
In Theorem 21, if we consider Lorentzian and denote by the corresponding scalar curvature, then in the Sasakian case (i.e., when ), the conditions , and are equivalent to , and , respectively. On the other hand, for a Lorentzian Sasakian three-manifold, when , the Lorentzian K-contact structure obtained by a -homothetic deformation in correspondence to is Einstein (see Section 2.2), and so of constant sectional curvature . Therefore, we get the following corollary which does not have a Riemannian counterpart.
Corollary 5.
The unimodular Lie group and the non-unimodular Lie group with Lie algebra defined by Equation (58), are the only simply connected three-manifolds which admit a homogeneous Lorentzian-Sasakian structure of constant sectional curvature .
• Homogeneous bi-contact metric three-manifolds
We close this subsection with a very short presentation of a recent notion introduced by the present author in [71]. H. Geiges and J. Gonzalo ([72,73]) introduced and studied the notion of taut contact circle on a three-manifold, that is, a pair of contact forms such that the 1-forms are contact forms with the same volume form for all . In the paper [71] we introduce a Riemannian approach to the study of taut contact circles on three-manifolds. A natural related notion, that we introduce, is the one of taut contact metric circle , that is, is a taut contact circle and g is a Riemannian metric associated to both the contact forms and . More in general, we introduce the notion of bi-contact metric structure , where is a pair of arbitrary contact forms and g is a Riemannian metric associated to both the contact forms and such that the same contact forms are orthogonal with respect to g, i.e., the corresponding Reeb vector fields are orthogonal. On the other hand, in the classical definition of three-contact metric structure , also called contact metric three-structure, we have three contact forms and a Riemannian metric g associated to the three contact forms, satisfying additional conditions that imply the orthogonality of the three forms with respect to g (see, for example, Ref. [2] Chapter 14 and [3] Chapter 13).
Moreover, a three-contact metric structure is three-Sasakian (see, for example, Ref. [2] p. 293, Theorem 14.1), and a three-Sasakian three-manifold is of constant sectional curvature (see, for example, Ref. [2] p. 294, Theorem 14.3). So, our definition of bi-contact metric structure seems more appropriate, at least in dimension three, in the sense that it is very less rigid. In particular, we characterize the existence of a taut contact metric circle and of a bi-contact metric structure on a three-manifold. Note that a taut contact metric circle is a bi-contact metric structure, but the converse is not true. Then, we give a complete classification of simply connected three-manifolds which admit a bi-H-contact metric structure. In particular, by using the classification given in Theorem 21, we get (cf. [71], Corollary 4.7).
Theorem 22.
A simply connected three-manifold admits a homogeneous bi-contact metric structure if and only if it is diffeomorphic to one of the following Lie groups: , , , .
4.3. Some Results in Arbitrary Dimension
Now we briefly recall some results, in arbitrary dimension, about contact homogeneity and spherical CR manifolds.
• D. E. Blair (see [2], p. 120) conjectured the non-existence of contact Riemannian manifolds having non positive sectional curvature, with the exception of the flat 3-dimensional case. In this direction, A. Lotta [74] got the following (as a consequence of a more general theorem and by using the classification given in Theorem 21).
Theorem 23.
The only simply connected homogeneous contact Riemannian (2n + 1)-manifold having non-positive sectional curvature is the Lie group endowed with a flat left invariant contact Riemannian structure.
• A contact Riemannian manifold is said to be a strongly locally φ-symmetric space if the reflections in the integral curves of the Reeb vector field are isometries. Examples of strongly locally -symmetric spaces include the non- Sasakian -manifolds (see [2], p. 146; more in general we refer to [2] Section 7.9 for a discussion on weakly and strongly locally -symmetric spaces). Boeckx and Cho in the paper [75] proved the following
Theorem 24.
Let M be a locally homogeneous contact Riemannian (2n + 1)-manifold. If M is strongly locally φ-symmetric, then it is a -space.
- Recently E.M. Correa [76] gives a new study on compact, -dimensional, homogeneous contact manifolds. More precisely, this paper contains:a description of contact structure for any compact homogeneous contact manifold;a description of G-invariant Sasaki-Einstein structure for any compact homogeneous contact manifold;a description of Calabi-Yau metrics on cones with compact homogeneous Sasaki-Einstein manifolds as link of isolated singularity;a description of crepant resolution of Calabi-Yau cones with certain compact homogeneous Sasaki-Einstein manifolds as link of isolated singularity.This study of homogeneous contact manifolds is based on the Kähler geometry of complex flag manifolds.
- The present author and L. Vanhecke [77] proved that a compact, simply connected, five-dimensional, homogeneous contact manifold M is diffeomorphic to or . In both cases the underlying homogeneous contact metric structure is Sasakian (and hence is a CR structure). This result is based on the fact that the contact structure is regular and the base B of the Boothby-Wang fibration is a compact simply connected homogeneous Kähler manifold of complex dimension two. In general, we note that every compact simply connected homogeneous contact manifold is a homogeneous Sasaki-Einstein manifold (Ref. [76], Remark 2.17).
- CR manifolds which are locally CR equivalent to the unit sphere , endowed with the standard CR structure as a real hypersurface of , are called spherical CR manifolds. In particular, non-degenerate CR manifolds with a vanishing Chern pseudoconformal curvature tensor are spherical ([12], p. 61). If M is a spherical CR manifold, Burns and Shnider ([79], Section 1) defined a development map , where is its universal cover. Moreover, they proved that if the group of CR automorphisms is transitive on M, then is a covering and is homogeneous domain in . Thus to classify the simply connected spherical homogeneous CR manifolds it suffices to classify homogeneous domain in ([79], Theorem 3.1). In particular, in dimension three, we have a list of five examples ([79], p. 229):
- (1)
- ;
- (2)
- ;
- (3)
- ;
- (4)
- ;
- (5)
- .
We note that domain (5) does not admit any compact quotients ([79], Proposition 5.5). - R. Lehmann and D. Feldmueller [80] proved that the only CR-structure (of hypersurface type) on , , which admits a transitive action of a Lie group of CR-transformations is the standard CR-structure. For all possible homogeneous CR-structures of hypersurface type are classified in [81] (cf. also [80], p. 524).
- G. Dileo and A. Lotta [82] studied spherical symmetric CR manifolds. A strictly pseudo-convex CR manifold M is said to be CR-symmetric if for each point there exists a CR-isometry such that and . In particular, they proved the following. Let M be a strictly pseudo-convex CR manifold, , with pseudohermitian torsion . Then, M is locally CR-symmetric if and only if the underlying contact metric structure satisfies the -nullity condition, that is, the curvature tensor satisfies Equation (54). In such a case M is spherical if and only if the Webster scalar curvature vanishes.
5. Geometry of Tangent Hyperquadric Bundles
The geometry of the unit tangent sphere bundle of a Riemannian manifold equipped with the Sasaki metric, and in particular with the standard contact Riemannian structure, has been studied by many authors. A motivation of this study depends of the fact that often properties of characterize the base manifold (see, for example, Blair’s book [2] Chapter 9, and from the point of view of the CR geometry Tanno [83]).
If is a semi-Riemannian manifold of index , the Sasaki metric induced on tangent hyperquadrics bundle , , is a semi-Riemannian metric of index if and the index is if . In such case we have few results about the the geometry of (see [84], Ref. [85] and more recently [48]). In this Section we discuss some results of [48] on the geometry of equipped with the standard non-degenerate almost CR structure.
5.1. The Standard Non-Degenerate Amost CR Structure on
Let be a semi-Riemannian manifold of index , . At any point of its tangent bundle , the tangent space of splits into the horizontal and vertical subspaces:
Each tangent vector can be written in the form , where are uniquely determined vectors.
The tangent bundle can be endowed in a natural way with a semi-Riemannian metric, the Sasaki metric G, depending only on the semi-Riemannian metric g. It is determined by
for any and for any . G is a semi-Riemannian metric of signature , and both and have index . There is also an almost complex structure J on given by
then the Sasaki metric G is Hermitian with respect to the almost complex structure J. We denote by the canonical vertical vector field on and by the geodesic flow on . They are defined by
The Liouville form on , defined by
satisfies the following (see Prop. 2 of [84], and [2] p. 139)
that is, is the fundamental 2-form, and so is an indefinite almost Kaehler manifold. Besides (see ([84], Proposition 3): J is integrable if and only if the semi-Riemannian manifold is locally isometric to the semi-Euclidean space .
Consider the tangent hyperquadric bundle
The vertical vector field is normal to in and along , moreover the geodesic flow ζ is tangent to . Any horizontal vector is tangent to , and a vertical vector is tangent to if and only if is orthogonal to . Consequently, the tangent space of , at a point , is given by
In general, the tangential lift of a vector field X is a vector field on defined by
The Sasaki metric on is the semi-Riemannian metric on induced from G, it is completely determined by the identities
for all and . Since the Sasaki metric on the tangent bundle is of signature , and is an orientable semi-Riemannian hypersurface of of sign , then
We now construct the standard non-degenerate almost CR structure on . The tangent hyperquadric bundle is an orientable non-degenerate hypersurface of the indefinite almost Kaehler manifold . Then, by the usual procedure, we construct the almost contact semi-Riemannian structure induced on , where
for and vector field on . Since for any vector fields on , if we rescale the structure tensors appropriately by
we get the standard contact semi-Riemannian structure on . In explicite form these tensors are given by
for any vector fields on M and , in particular . Hence, the corresponding standard non-degenerate almost CR structure on is defined by
Besides, the corresponding Levi form is defined by
for any , .
A natural problem for the standard non-degenerate almost CR structure on is to see when it is a CR structure. By using
- formulas which give the Levi-Civita connection of in terms of ∇ and R (the Levi-Civita connection and the curvature tensor of g);
- he generalized Tanaka-Webster connection associated to the standard non-degenerate almost CR structure , and
- a result of M.Dajczer - K.Nomizu [86] on the sectional curvatures of indefinite metrics, we get
Theorem 25.
Let be a semi-Riemannian manifold with . Then, the standard non-degenerate almost CR structure on is integrable, i.e., it is a non-degenerate CR structure if, and only if, has constant sectional curvature.
It is interesting to note that if is a semi-Riemannian manifold of constant sectional curvature , and of dimension , then is not integrable but, by Theorem 25, the standard non-degenerate almost CR structure on is integrable.
As a consequence of Theorems 18 and 25, we get
Corollary 6.
Let be a non-degenerate CR manifold, . Then, the standard non-degenerate almost CR structure of is integrable, i.e., is a non-degenerate CR structure, if and only if the Webster metric of M has constant sectional curvature and the pseudohermitian torsion vanishes.
Remark 13.
If we set then is abundle of nullcones, and in such a case we can not consider the standard contact semi-Riemannian structure on . In fact, also in this case the geodesic flow , for , is tangent to , but now it is a lightlike vector field while the Reeb vector field is never lightlike.
In the Riemannian case Y. Tashiro (see [2], Section 9.2) proved that the standard contact Riemannian structure on is K-contact, equivalently the geodesic flow is Killing, if and only if the Riemannian manifold has constant sectional curvature , and in such case the standard contact Riemannian structure on is Sasakian. Now, we consider the same question in the semi-Riemannian setting and in terms of CR geometry.
By using
- formulas for the pseudohermitian torsion of :
- a result of K. Nomizu [87] on the sectional curvatures of indefinite metrics;
we get the following
Theorem 26.
Let be a semi-Riemannian manifold of index ν, . Then, we have the following
- (i)
- The standard non-degenerate almost CR structure on has vanishing pseudohermitian torsion if and only if has constant sectional curvature .In such a case is a pseudo-Einstein CR structure, which is Sasakian, and the Ricci tensor and the pseudohermitian Ricci tensor are given by
- (ii)
- If , the pseudo-Einstein CR structure of (i) is Einstein, i.e., the Webster metric is Einstein, if and only if is a Lorentzian surface of constant curvature . In such a case, has constant sectional curvature .
Corollary 7.
Let be a semi-Riemannian manifold of index ν, . Then, the geodesic flow of is Killing if and only if M has constant sectional curvature ε.
5.2. Sasaki-Einstein and H-Contact Structures on
The geometry of a H-contact unit tangent sphere bundles when the base manifold is a Riemannian manifold has been extensively investigated (see, for example, Refs. [88,89,90,91,92]).
In the semi-Riemannian case, we have
Theorem 27.
Let be a semi-Riemannian manifold of constant sectional curvature c. Then, the standard contact semi-Riemannian structure on is H-contact. Moreover, the structure is η-Einstein if and only if either or , . In such a case, the Ricci tensor is given by
where or .
Remark 14.
Recall that η-Einstein, K-contact and Sasakian semi-Riemannian manifolds are H-contact. Now, we remark that:
- for , Theorem 27 gives examples of H-contact semi-Riemannian manifolds which are not K-contact;
- for and , Theorem 27 gives examples of H-contact semi-Riemannian manifolds which are not η-Einstein.
As a consequence of Theorems 25–27, we obtain the following result.
Theorem 28.
Let be a semi-Riemannian manifold, .
- (i)
- If , then the standard non-degenerate almost CR structure on is an η-Einstein CR structure if and only if has constant sectional curvature or .
- (j)
- If , then the standard non-degenerate almost CR structure on is an η-Einstein CR structure if and only if has constant sectional curvature . Moreover, in such case the structure is pseudo-Einstein and Sasakian.
In general, for a contact semi-Riemannian manifold, the Reeb vector field infinitesimal harmonic transformation does not imply K-contact (cf. Example 3). If is a semi-Riemannian n-manifold of constant sectional curvature c, by Theorem 27 the standard contact semi-Riemannian structure on is H-contact. Moreover, by using Equation (66), we get
Then, by using Equation (27), we get
Corollary 8.
Let be a semi-Riemannian manifold of constant sectional curvature c. Then, the Reeb vector field of the standard contact semi-Riemannian structure on is an infinitesimal harmonic transformation if and only if the pseudohermitian torsion vanishes (i.e., the structure is K-contact and ).
Now, given a K-contact semi-Riemannian structure on M, , consider the new K-contact semi-Riemannian structure defined by the -homothetic deformation (or transverse homothety) Equation (16). Then, by Equations (18) and (19), the Ricci tensors and the scalar curvatures of g and () are related by
In particular, if is -Einstein, that is,
where the scalar curvature r is a constant because , then the Ricci tensor of the new K-contact semi-Riemannian structure is given by
where is the Tanaka-Webster scalar curvature (see Equation (49)) of the -Einstein K-contact semi-Riemannian structure . So, for any , the new K-contact semi-Riemannian structure is also -Einstein. Moreover, if the scalar curvature r of the -Einstein K-contact semi-Riemannian manifold satisfies , that is, the Tanaka-Webster scalar curvature , then for the corresponding K-contact semi-Riemannian structure is Einstein. Then, we get
Proposition 17.
Let be an η-Einstein K-contact semi-Riemannian -manifold, . If the Tanaka-Webster scalar curvature satisfies , then there exists a unique transverse homothety whose resulting structure is an Einstein K-contact semi-Riemannian structure.
Remark 15.
Now, we apply the above considerations to the tangent hyperquadric bundle , where is a semi-Riemannian manifold of constant sectional curvature c, .
If , from Theorem 26 we know that is Einstein if and only if , i.e., is a surface of constant curvature . In such a case, is of constant sectional curvature .
If and , by using Equation (68), is Einstein if and only if is a flat Lorentzian surface. In such a case, is a flat contact Lorentzian 3-manifold.
In general, if , Theorem 26 gives that is is -Einstein Sasakian with constant scalar curvature
where . Then, by the -homothetic deformation Equation (16) with , we get (cf, also Proposition 17) the following
Theorem 29.
Let be a semi-Riemannian manifold, , of constant sectional curvature ε. Then, the tangent hyperquadric bundle admits a Sasaki-Einstein structure given by:
where is the standard contact semi-Riemannian structure.
Remark 16.
We note that for the two structures and coincide, on the other hand for the structure is Einstein.
Example 6.
Consider the pseudo-sphere and the pseudo-hyperbolic space . Then, the tangent pseudo-sphere bundle and the tangent pseudo-hyperbolic bundle admit a Sasaki-Einstein structure.
If we assume that is Lorentzian (), then is Lorentzian. Then, from Theorem 26, we get
Theorem 30.
Let be a n-dimensional Lorentzian manifold. Then the standard nondegenerate almost CR structure on has vanishing pseudohermitian torsion if and only if has constant sectional curvature . Moreover, in such a case the corresponding standard contact Lorentzian structure is Sasakian η-Einstein with scalar curvature
As an application in relativity theory, we have the the following
Corollary 9.
Let be a complete simply connected 4-dimensional Lorentzian manifold. Then, the standard nondegenerate almost CR structure on has vanishing pseudohermitian torsion if and only if is isometric to the universal anti-de Sitter space .
Remark 17.
The standard examples of non Sasakian -spaces are the tangent sphere bundles of Riemannian space forms of constant curvature c different from (see, for instance, Ref. [2] Teorem 7.9). In such a case, the corresponding Boeckx invariant defined by Equation (55) is given by:
Therefore as c varies over the reals, assumes all the real values strictly greater than . Boeckx found examples of -spaces, for every value of the invariant , namely a two parameter family of Lie groups with a left-invariant contact metric structure (cf. [62], and [2] pp. 125–126).
More recently, E. Loiudice and A. Lotta [93] showed that the tangent hyperquadric bundles over Lorentzian space forms of constant curvature c different from , equipped with a strictly pseudoconvex CR structure, also provide non equivalent examples. For these space, the formula for the Boeckx invariant changes as follows:
where , , so that for , these examples cover all possible values of the Boeckx invariant in . This result makes E. Boeckx’s classification of -spaces in [62] more geometric. We note that in this case the Webster metric of is not the Lorentzian metric Equation (65) induced from the Sasaki metric of .
6. Levi Harmonicity on Non-Degenerate Almost CR Manifolds
The papers [47,94] are devoted to the study of a class of variational principles whose corresponding Euler-Lagrange equations are degenerate elliptic and generalize ordinary harmonic map theory in the spirit of sub-Riemannian geometry (cf. [95]) i.e., given a smooth map of (semi) Riemannian manifolds and one replaces the Hilbert-Schmidt norm of by the trace with respect to g of the restriction of to a given codimension one distribution on M (rather than applying the same construction to the full ). E. Barletta et al., Ref. [96], introduced pseudoharmonic maps from a nondegenerate CR manifold M endowed with a contact form into a Riemannian manifold . When is itself a non-degenerate CR manifold carrying the contact form a result in [96] describes pseudoharmonicity of CR maps . R. Petit [97] considered the following (pseudohermitian analog to the) second fundamental form
where is the Tanaka-Webster connection of M and is the pullback of the Levi-Civita connection of . The approach in [96] is to replace by an arbitrary linear connection on , consider the restriction of (69) to the Levi distribution , and take the trace with respect to the Levi form . Then f is called pseudoharmonic (with respect to the data ) if .
More recently, Dragomir and R. Petit et al., [98], studied contact harmonic maps, i.e., maps from a compact strictly pseudoconvex manifold M into a contact Riemannian manifold which are critical points of the functional
where is a contact form on M and .
J. Konderak & R. Wolak, Ref. [99], introduced transversally harmonic maps as foliated maps between foliated Riemannian manifolds satisfying a condition similar to the vanishing of the tension field in Riemannian geometry.
As a natural continuation of the ideas in [96], and following the ideas of B. Fuglede (who started the study of the semi-Riemannian case within harmonic map theory, cf. [100], and [101] pp. 427–455), in the papers [47,94], S. Dragomir and the present author introduced the concept of Levi harmonic map f from an almost contact semi-Riemannian manifold into a semi-Riemannian manifold , i.e., solutions of , where is the second fundamental form of f, and is the restriction of to the Levi distribution . Thus, we studied the Levi harmonicity for CR maps between two almost contact semi-Riemannian manifolds. This is perhaps the most general geometric setting (metrics are semi-Riemannian, in general the contact condition is not satisfied and the underlying almost CR structures are not integrable). In such a study, an important role is played by the notion of -condition:
for any . Moreover, as emphasized in [47], the class of almost contact semi-Riemannian manifolds obeying to Equation (70) is quite large. For instance, contact semi-Riemannian manifolds, orientable real hypersurfaces in an indefinite Kaehler manifold (with the induced almost contact semi-Riemannian structure) and quasi-cosimplectic manifolds (which contains cosymplectic and almost cosympletic manifolds) satisfy the -condition. Moreover, the -condition extends (cf. [94], Section 3) the so-called condition of Rawnsley [102]. Rawnsley in his paper introduced the condition in order to study the harmonicity of f-holomorphic maps between an almost Hermitian manifold with coclosed Kaehler form and a Riemannian manifold equipped with a f-structure. Moreover, there is the following characterization of a contact Riemannian manifold ([103], Theorem 3.2): an almost contact Riemannian manifold is a contact Riemannian manifold if and only the following conditions are satisfied: the tensor is symmetric, and
This last condition, for , implies the -condition.
In this Section we report some results of [47,94], for almost contact semi-Riemannian manifolds. Let be a real -dimensional almost contact semi-Riemannian manifold and a semi-Riemannian manifold. Let be a map and the pullback of by f. Let be the pullback of the Levi-Civita connection of i.e., the connection in the vector bundle induced by . If and are local coordinate systems on M and N such that then is locally described by
where denotes the natural lift of and are the Christoffel symbolds of . Let and be the almost CR structure underlying . The second fundamental form of f is given by
Here ∇ is the Levi-Civita connection of and the vector field is given by for any and . Next, let be the tension field defined by
where is the restriction of to . Note that the tension field , .
Definition 6.
Let an almost contact semi-Riemannian manifold and a semi-Riemannian manifold. A map is said to be Levi harmonic with respect to if .
In the case of a CR map between two almost contact semi-Riemannian manifolds, we have the following
Theorem 31.
([94]) Let and be two almost contact semi-Riemannian manifolds with . Then, for each CR map we have
where , is the pullback of by f, and is the operator formal adjoint of ∇.
Proof.
(sketch) Let be a -basis and let us set . Then one has
We consider the operator , the formal adjoint of ∇ (see for example [16], pp. 108–110), thus if S is a tensor of type , . Then, after some computations, we get
Moreover, as f is a CR map,
Therefore, we obtain
Corollary 10.
([47]) Let and be two almost contact semi-Riemannian manifolds, , satisfying the φ-condition. Then, for any CR map
If additionally is a contact semi-Riemannian manifold, then
where . Hence f is Levi harmonic if and only if is collinear to .
There are several examples of Levi harmonic maps between almost contact semi Riemannian manifolds. Here we report the following (cf. also [47,94]).
1.Invariant submanifolds and Levi harmonicity
Let M be a submanifold of a -dimensional almost contact Riemannian manifold . M is an invariant submanifold of if for any . Two extreme cases may be distinguished (cf. [104]) as:
- (I)
- is tangent to M (and then M is odd-dimensional i.e., ), or
- (II)
- is transverse to M (and then M is even-dimensional).When is a contact Riemannian manifold case II doesn’t occur (cf. [2], p. 122). Here we only consider case (I). Then M carries the induced almost contact Riemannian structure defined by
If is an almost contact Riemannian manifold satisfying the -condition, then the submanifold is an almost contact Riemannian manifold satisfying the -condition. Moreover the map is Levi harmonic. The mean curvature H of i is defined by , where is the second fundamental form of i. If additionally is geodesic then i is minimal.
To give an explicit example, let be a complete simply connected Sasakian manifold of constant -sectional curvature c. As well known is (up to an isometry) one of the Sasakian manifolds , or equipped with Sasakian structures of -sectional curvature , and respectively, where is a simply connected bounded domain. Then, is an invariant submanifold of (cf. [105], p. 328), hence the inclusion is Levi harmonic.
2.Levi harmonicity of Reeb vector fields
Let be a Riemannian manifold. Let be the standard contact Riemannian structure on the tangent sphere bundle which we denoted before by . Then, it is defined by Equations (63)–(65). In particular the Reeb vector field , . Let , consider the D-homothetic deformation
Then, Equation (78) is a g-natural contact Riemannian structure in the sense of [106], and is the g-natural metric on defined by the parameters , , and . In particular, is of Kaluza-Klein type ([106], p. 1196).
Suppose that is an almost contact Riemannian manifold. Then, by Theorem 6.1 in [106] we get that is a CR map if and only if on . Moreover, since is of Kaluza-Klein type, by Theorem 6.2 in [106], is a K-contact Riemannian structure if and only if
Besides, the differential of , at x, is given by
Thus
and
Summing up, by using also Corollary 10, we get
Theorem 32.
Let be an almost contact Riemannian manifold and let , , a D-homothetic deformation of the standard contact Riemannian structure of . Then, is K-contact structure if and only if ξ is geodesic and is a CR map.
In particular, if is K-contact structure, then is a Levi harmonic map for any ; moreover, ξ is a pseudohermitian map (isopseudohermitian for ).
Let be the unit sphere endowed with the canonical Sasakian structure , hence is the standard Hopf vector field on . Then
Corollary 11.
is a Levi harmonic map for any .
Remark 18.
About the harmonicity of Hopf vector fields, Han and Yim [107] proved that these fields, namely, the unit Killing vector fields, are the unique unit vector fields on the unit sphere which define harmonic maps from to , where is the Sasaki metric. In [108], as a consequence of a more general result, we got in particular that Han-Yim’s Theorem is invariant under a three-parameter deformation of the Sasaki metric on .
Finally, we give a short presentation of the variational treatment of Levi harmonicity. Let be a -dimensional almost contact Riemannian manifold and a Riemannian manifold. If is a relatively compact domain we set
for any . Then we obtain the following ([47], Theorem 6.1):
Theorem 33.
Let be a relatively compact domain. A map is a critical point for the energy functional defined by (79) if and only if If be an immersion and a critical point of , then f is Levi harmonic if and only if the Reeb field ξ is geodesic and divergence free.
Remark 19.
The many ramifications of harmonicity (subelliptic harmonic, contact harmonic, Levi harmonic, and pseudoharmonic maps) seem to indicate that the theory of harmonic maps has reached a stage of mannerism. However, the mentioned ramifications (to which one may add p-harmonic and exponentially harmonic maps, Gromov’s tangentially harmonic maps and harmonic maps from Finslerian manifolds (cf. references in [47])) are but a measure of the enormous success enjoyed by the theory.
7. Some Open Problems
Question 1.
(related to the Section 2.3) It is an open problem, to our knowledge, to find examples of non-Sasakian contact semi-Riemannian manifolds which satisfy Equation (23), or to give a proof that an arbitrary contact semi-Riemannian manifolds satisfying Equation (23) is Sasakian.
Question 2.
(related to the Section 2.4) In dimension , it is an open problem, to our knowledge, the existence of non trivial semi-Riemannian contact Ricci solitons.
Question 3.
(related to Section 3.2) Study the geometry of an almost contact (semi) Riemannian structure when defines a pseudohermitian structure.
Question 4.
(related to the Section 3.4) In dimension , it is an open problem to see if the Olszak’s result holds for a general non-degenerate almost CR manifold.
Question 5.
(related to the Section 4 and Definition 4) Let be a pseudohermitian CR structure on a simply connected three-manifold.
- If is homogeneous and non-degenerate, then Theorem 21 gives a complete classification.
- If is an arbitrary homogeneous Levi-flat pseudohermitian CR structure, we do not know a classification.So, a natural open problem is: Classify all simply connected three-manifolds which admit a homogeneous Levi-flat pseudohermitian CR structure.
Question 6.
In the Riemannian case, S.H. Chun et al. [90] proved the following: if is an Einstein Riemannian manifold, then the standard contact Riemannian manifold on is H-contact if and only if is 2-stein. Then, a natural open problem (related to the Section 5.2) is the following.
Let be an Einstein semi-Riemannian manifold. When is the standard non-degenerate almost CR structure on H-contact?
Question 7.
Another open problem (related to the Section 5.2) is the following. Let be a emi-Riemannian manifold (or an Einstein semi-Riemannian manifold). When is the Reeb vector field of (the standard non-degenerate almost CR structure on) an i.h.t.?
Funding
Author supported by funds of Universitá del Salento.
Acknowledgments
The author would like to thank the anonymous referees for their useful comments.
Conflicts of Interest
The author declares no conflict of interest.
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