Contact Semi-Riemannian Structures in CR Geometry : Some Aspects

There is one-to-one correspondence between contact semi-Riemannian structures (η, ξ, φ, g) and non-degenerate almost CR structures (H, θ, J). In general, a non-degenerate almost CR structure is not a CR structure, that is, in general the integrability condition forH1,0 := {X− i JX, X ∈ H} 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.


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 S 2n+1 and the unit tangent sphere bundles T 1 M 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 ϕ-sectional curvature κ = −3.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 θ is a contact 1-form on an odd dimensional manifold and J is an almost complex structure, i.e., J 2 = −I, defined on the contact distribution H = ker θ, such that the Levi form L θ = dθ(•, J•) is a non-degenerate Hermitian form H, then (θ, J) is said to be a non-degenerate almost CR structure.Different signatures of the Levi form L θ 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 H 1,0 := {X − iJX, X ∈ H} 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.

Generality on Contact Semi-Riemannian Manifolds
A (2n + 1)-dimensional manifold M is said to be a contact manifold if it admits a contact form, this is, a global 1-form η such that η ∧ (dη) n = 0. Given a contact form η, there exists a unique vector field ξ, called the characteristic vector field or the Reeb vector field, such that η(ξ) = 1 and dη(ξ, •) = 0. Furthermore, a semi-Riemannian metric g is said to be an associated metric (for the contact form η) if there exists a tensor ϕ of type (1,1) and so g(ξ, ξ) = ε = ±1.In such a case, (η, ξ, ϕ, g), or (η, g), is called contact semi-Riemannian structure, or contact pseudo-metric structure.
Special contact semi-Riemannian manifolds are the following.
Since Ω is closed, ( M, J, g) 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 (η, ξ, ϕ, g), with g(ξ, ξ) = ε, on M, and pseudo-Kaehler structures (J, g) on the ε-cone M.Moreover, the pseudo-Sasakian manifold is Einstein (respectively, of constant sectional curvature) if and only if the corresponding ε-cone M is Ricci-flat (respectively, flat).
• K-contact manifolds are contact semi-Riemannian manifolds (M, η, ξ, ϕ, g) whose Reeb vector field ξ is a Killing vector field, or equivalently, h = 0. Any Sasakian semi-Riemannian manifold is K-contact and the converse also holds when n = 1.

•
H-contact manifolds.The condition that ξ be an eigenvector of the Ricci operator is a very natural condition in contact Riemannian geometry.Sasakian manifolds, K-contact manifolds, (κ, µ)-spaces and locally ϕ-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 (M, g), a unit vector field V is said to be a harmonic vector field if V : (M, g) → (T 1 M, G), where G 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 (M, g) 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: " ∆V is collinear to V", where ∆V is the so called rough Laplacian of V. H-contact semi-Riemannian manifolds are contact semi-Riemannian manifolds whose Reeb vector field ξ 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 h = 1 2 L ξ ϕ do not depend on the metric, so that (M, η, g) is Sasakian (respectively K-contact) if and only if (M, η, ḡ) is.Moreover, by using Equation (10), A difference between the Riemannian case and the general semi-Riemannian one is the following: in both cases, from Equation (6), trh 2 = 0 implies Ric(ξ, ξ) = 2n.But,

•
K-contact Riemannian manifolds are characterized by the condition Ric(ξ, ξ) = 2n, since it implies trh 2 = 0 and so, h = 0 (because in the Riemannian case h is diagonalizable); • in the semi-Riemannian case the condition tr h 2 = 0 does not imply h = 0. On the other hand, there exist contact semi-Riemannian manifolds for which trh 2 = 0 but h = 0, and contact semi-Riemannian manifolds for which h 2 = 0 but h = 0 (see Examples 3 and 5).
Recall that there is a canonical way to associate a contact Riemannian structure to a contact Lorentzian structure (and conversely).Let (η, ξ, ϕ, g L ) be a contact Lorentzian structure on a smooth manifold M, where the Reeb vector field ξ is time-like.Then, g = g L + 2η ⊗ η is a Riemannian metric, and is still compatible with the same contact structure (η, ξ, ϕ).Moreover, in such case g(ξ, ξ) = −g L (ξ, ξ) = +1.Hence, (η, ξ, ϕ, g) is a contact Riemannian structure on M. We remark that where g −1 is obtained by the D-homothetic deformation of g for t = −1.Consequently, the Levi-Civita connection and curvature of g L can be easily deduced from the formulae valid for a general D-homothetic deformation.In particular, if ∇ is the Levi-Civita connection of g L 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 X ∈ ker η, hX = λX, 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 η-Einstein if the Ricci tensor is given by In particular, the Ricci tensor of the η-Einstein K-contact Riemannian structure (η, g) is given by where the scalar curvature r is a constant when n > 1, and g is Einstein if and only if r = 2n(2n + 1).
Then, from Equations ( 20) and ( 21), the Ricci tensor of the corresponding Lorentzian K-contact structure (η, g L ) is given by where the scalar curvature r L = r + 4n is a constant when n > 1. Hence (η, g L ) is η-Einstein K-contact, and g L is Einstein if and only if r L = −2n(2n + 1).
In dimension three, every K-contact structure (η, g) is automatically Sasakian and η-Einstein, and thus by Equation ( 22) also every K-contact Lorentzian three-manifold is automatically Sasakian and η-Einstein.Moreover, for a K-contact Lorentzian three-manifold, the scalar curvature r L and the ϕ-sectional curvature H L are related by r L = 2H L − 4.
Recall that a Lorentzian Sasakian manifold (M, g, η) is Einsteinian if and only if the cone M 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 (η, g) (resp.(η,g L )) is Einstein K-contact, then (η, g L ) (resp.(η, g)) is η-Einstein K-contact.Now, we see as the η-Einstein Lorentzian K-contact structures are related to the Einstein Lorentzian Sasakian structures.Let (η, g L ) be a Lorentzian K-contact structure on M with ξ time-like, dim from Equations ( 18) and (19) we have If in addition (η, g L ) is η-Einstein, since n > 1, then the scalar curvature r L is a constant and the Ricci tensor of the new Lorentzian K-contact structure ( η, gL ) is given by So, for any t > 0, the Lorentzian K-contact structure ( η, gL ) is η-Einstein.Recall that the function r = r L − 2n = r + 2n is the so-called Webster scalar curvature of (η, g) and (η, g L ) (see Section 3.3).Now, if the scalar curvature r L of the η-Einstein Lorentzian K-contact manifold (η, g L ) satisfies r L < 2n, i.e., r < 0, then the Lorentzian K-contact structure ( η = η t , gL = (g L ) t ) obtained in correspondence to Then, the Webster scalar curvature r is a constant and we have the following.
From this Theorem and Proposition 6.2 of [6], we get the following Theorem 2. ([22]) Let (M, η, ξ, g L ) be a simply connected η-Einstein Lorentzian Sasakian manifold of dimension 2n + 1 > 3 and with scalar curvature r L < 2n, i.e., r < 0.Then, there exists a transverse homothety whose resulting Lorentzian manifold (M, gL ) is a spin manifold.Moreover, there exists a twistor spinor φ which is an imaginary Killing spinor and the associated vector field V φ (the Dirac current) is ξ.
Any connected sum of S 2 × S 3 admits a Lorentzian Sasaki-Einstein structure [23].Now, we give the following.

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 4. If a semi-Riemannian manifold (M, g) admits a Killing vector field ξ, g(ξ, ξ) = ε = ±1, 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 κ = ε.
Proof.For a K-contact semi-Riemannian manifold, from Equations (4) and (5), we have Moreover, for a locally symmetric semi-Riemannian manifold ∇R = 0. Then we get Replacing X by ϕX, we have By using this last equation, we get Replacing Z by ϕZ, the above equation becomes Then, by using Equation (24), we obtain Now, let p be an arbitrary point and span(X p , Y p ) be an arbitrary non-degenerate plane.Then, from Equation (25), we obtain Therefore (M, η, ξ, ϕ, g) is a K-contact semi-Riemannian manifold of constant curvature ε.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 (M, g) is a semi-Riemannian manifold which admits a Killing vector field X 0 of constant length, g(X 0 , X 0 ) = c = 0, such that the sectional curvature of all non-degenerate plane sections containing X 0 equals c, then ξ = (1/εc)X 0 and g = εc g, where ε = +1 if c > 0 and ε = −1 if c < 0, 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 (M, g) be a semi-Riemannian manifold whose admits a Killing vector field X 0 of constant length, g(X 0 , X 0 ) = c = 0, such that the sectional curvature of all non-degenerate plane sections containing X 0 equals c.Then, the following properties are equivalent Numbered lists can be added as follows:

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 (M, g) 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 tr(L V ∇) = 0 (see [24,25]), where for all tangent vector fields X, Y.With respect to a pseudo-orthonormal basis {E 1 , . . ., E m } of (M, g), we have tr(L V ∇) where ∆ is the rough Laplacian.Thus, a vector field V is an i.h.t.if and only if ∆V = QV.Now, let (M, η, g, ξ, ϕ) be an arbitrary contact semi-Riemannian manifold.Then, we have (cf.[15,17]) Besides, by using Equation (6), we get from which we get the following (cf.[17]).
ξ is an i.h.t.if and only if M is K-contact.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 g admits a basis Consider the semi-Riemannian left-invariant metric g, for which {E 0 , E i , V i } is a pseudo-orthonormal basis with Define the left-invariant tensors ξ, η, and ϕ on G putting Then, the metric g described in Equation ( 28), together with tensors described in Equation ( 29), define a left-invariant contact semi-Riemannian structure (η, g, ξ, ϕ) on G.This contact semi-Riemannian structure is H-contact and satisfies trh 2 = 0 with h = 0 (more precisely, Remark 3. The class of contact semi-Riemannian manifolds with ξ i.h.t. is invariant for D-deformations.In fact, the class of H-contact semi-Riemannian manifolds is invariant and trh The Lorentzian case.Let (M, η, ξ, ϕ, g) be a contact semi-Riemannian manifold, and ḡ the metric associated to (η, ξ, ϕ) described by Equation (8).Then, as remarked in Section 2, (M, η, ξ, ϕ, ḡ) is H-contact if and only if (M, η, ξ, ϕ, g) 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 ξ time-like one has trh 2 = 0 if and only if h = 0 [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 (M, η, ξ, ϕ, g) be a contact Lorentzian manifold with ξ time-like.Then, the following properties are equivalent: ( Remark 4. Let (M, g) 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 ∇V to the space-like distribution V ⊥ .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 (M 3 , η, ξ, ϕ, g) defines a harmonic map from M to T 1 M if and only if it is H-contact and ξ(λ) = 0, where λ, −λ 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 h = h).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 ξ) 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 (M, g), admitting a vector field V, such that for some real constant µ.A Ricci soliton is said to be shrinking, steady, or expanding, according to whether µ > 0, µ = 0 or µ < 0, 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 (M, η, ξ, ϕ, g), for which Equation ( 30) is satisfied by V = ξ.Since (L ξ g)(ξ, X) = 0, from Equation ( 30) with V = ξ, we have that the Reeb vector field of a contact semi-Riemannian manifold satisfies Qξ = µξ.
So, a contact semi-Riemannian Ricci soliton is H-contact with constant Ricci eigenvalue.On the other hand, if (M, η, ξ, ϕ, g) is a contact semi-Riemannian Ricci soliton, then ξ is an infinitesimal harmonic transformation.Hence, by Theorem 8, Qξ = µξ with µ = 2nε = ±2n and we get the following result [17]: Theorem 9. A (2n + 1)-dimensional contact semi-Riemannian Ricci soliton is H-contact: Qξ = ±2nξ, 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 trh 2 = 0, that is h = 0. 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 (M, η, ξ, g, ϕ) be a contact Lorentzian manifold with ξ time-like.Then, (M, η, ξ, ϕ, g) 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 ξ 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 Qξ = ±2nξ.

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.

Generality on Almost CR Structures
Let M be a (2n + 1)-dimensional manifold.An almost CR structure (of hypersurface type) on M is a pair (H = H(M), J) where H is a smooth real subbundle of rank 2n of the tangent bundle TM (also called the Levi distribution), and J : H → H is an almost complex structure: Starting from an almost contact structure (η, ξ, ϕ), the pair (H = kerη, J = ϕ |H ) defines a corresponding almost CR structure on M. It is a natural question to ask when an almost CR structure (H, J) permits to reconstruct an almost contact structure (η, ξ, ϕ), such that (H = kerη, J = ϕ |H ).The answer is given by the following result.Proposition 3. Let M denote an odd-dimensional manifold.An almost CR structure (H, J) on M is induced by an almost contact structure (η, ξ, ϕ) if and only if M admits a (globally defined) vector field X 0 , transversal to H at any point.
Conversely, suppose that (H, J) is an almost CR structure, admitting a global vector field X 0 transversal to it.Then, it is enough to define ξ, η and ϕ by for any vector field X ∈ H.It is then easy to check that (η, ξ, ϕ) is an almost contact structure, and (H, J) = (kerη, ϕ |kerη ).
Given an almost CR structure, let E x ⊂ T * x (M) be the subspace consisting of all pseudohermitian structures on M at x ∈ M. Then E = x∈M E x is (the total space of) a real line subbundle of the cotangent bundle T * (M) and the pseudohermitian structures are the globally defined nowhere zero C ∞ sections in E. If M is oriented then E is trivial i.e., E ≈ M × R (a vector bundle isomorphism).Therefore E admits globally defined nowhere vanishing sections, equivalently any orientable almost CR manifold admits a 1-form θ such that kerθ = H (cf., for example, Ref. [12] Section 1.1.2).On the other hand the existence of a (globally defined) vector field X 0 , transversal to H at any point is equivalent to the existence of a 1-form θ such that kerθ = H.In fact, given θ with kerθ = H, we consider a Riemannian metric g on M (which is paracompact) and then define X 0 by g(X 0 , X) = θ(X) for any vector field X.As such, X 0 is transversal to H at any point because kerθ = H.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 (M, H, J) be an almost CR manifold.Put that is, H 1,0 (resp.H 0,1 ) is the eigenbundle of J C (the C-linear extension of J to H C = H ⊗ C) corresponding to the eigenvalue i (resp.−i).Then the complexfication H C can be decomposed into direct sum of (±i)-distributions of J C : Definition 1.An almost CR structure (M, H, J) is said to be a CR structure on M if H 1,0 (and hence also H 0,1 ) is (formally) integrable: for any open set U ⊂ M.
CR structures are considered mainly from a complex analytical point of view.It is easy to see that an almost CR structure (M, H, J) is a CR structure if and only if the following two conditions hold: Of course, if dim M = 3: 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 (H, J) 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 (2n + 1)-dimensional manifold.An almost CR structure on M is a complex subbbundle T 1,0 (M), of complex rank n, of the complexified tangent bundle T(M) ⊗ C such that T 1,0 (M) ∩ T 0,1 (M) = (0) where T 0,1 (M) = T 1,0 (M) (overbars denote complex conjugates).The integer n is the CR dimension.An almost CR structure T 1,0 (M) is integrable, and then T Then, T 1,0 (M) = {X − iJX : X ∈ H} = H 1,0 , i.e., T 1,0 (M) is the eigenbundle of J C (the C-linear extension of J to H ⊗ C) corresponding to the eigenvalue i.The pair (H, J) is the real manifestation of T 1,0 (M).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.
Proof.(1) We have to show that the condition S = 0 is equivalent to the integrability conditions Equations ( 31) and (32).Since (31).Replacing X by JX, we get ([X, JY] + [JX, Y]) ∈ H, and so Equation (33) becomes Thus S(X, Y) = 0 implies the condition Equation (32).Conversely, if Equations ( 31) and ( 32) are satisfied, then S = 0 is trivial. ( (3) In such a case, the condition L ξ η = 0 is equivalent to the condition that ξ is geodesic with respect to the Levi-Civita connection.In fact, for any X ∈ H: Remark 7.An almost contact structure (η, ξ, ϕ) satisfying the condition ξ ∈ ker dη 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 2dη(ξ, X) = L ξ η, so the condition that defines this structure is equivalent to the condition L ξ η = 0 considered in Proposition 5.In particular, any contact semi-Riemannian manifold satisfies the condition L ξ η = 0, equivalently ∇ ξ ξ = 0.
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 (H, J) be an almost CR structure on an odd-dimensional manifold M induced by an almost contact structure (η, ξ, ϕ).Then, the almost contact structure (η, ξ, ϕ) is normal if and only if almost CR structure (H, J) is integrable and the tensor h = 0.In particular, if L ξ η = 0, (η, ξ, ϕ) is normal if and only if (H, J) is integrable and L ξ J = 0.
Proof.By Proposition 5 we know that (H, J) is a CR structure if and only the tensor S defined on H by Equation (33) vanishes.For X, Y ∈ H: Moreover, for Therefore, from Equations ( 14), (34), and ( 35) we obtain that (η, ξ, ϕ) is normal if and only if almost CR structure (H, J) is integrable (i.e., S = 0) and the tensor h = 0.The second part follows from (3) of Proposition 5.

Non-Degenerate Almost CR Structures and Contact Semi-Riemannian Structures
We have already observed that the existence of an almost CR structure (H, J) on an (2n + 1)-dimensional manifold M induced by an almost contact structure is related to the existence of a 1-form θ such that kerθ = H (cf. Proposition 3).Definition 2. A pseudohermitian structure on an almost CR manifold (M, H, J) is a 1-form θ such that kerθ = H and the Levi form L θ , defined by It should be observed that, for any X, Y ∈ H, the following are equivalent: Then, we have Proposition 6.Let (H, J) be an almost CR structure and θ an 1-form such that kerθ = H.Then, the following properties are equivalent: In the case of an almost CR structure induced by an almost contact semi-Riemanian structure, we have: Proposition 7. Let (H = ker θ, J) be an almost CR structure induced by an almost contact semi-Riemannian structure (η = θ, ξ, ϕ, g).Then, (H, J, θ) is a pseudohermitian almost CR structure if and only if the tensor q := ϕ • ∇ξ − ∇ξ • ϕ is symmetric on H, where ∇ is the Levi-Civita connection of g.In particular, if (η = θ, ξ, ϕ, g) is a contact semi-Riemannian structure, or an almost α-coKähler structure, then q = 2h = L ξ ϕ 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 H.
Definition 3. A pseudohermitian almost CR structure (H, J, ϑ) is said to be a non-degenerate (pseudohermitian) almost CR structure if the Levi form L θ is, in addition, non-degenerate (equivalently, θ is a contact form, i.e., θ ∧ (dθ) n 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 θ and θ on the same almost CR manifold, are related by θ = λθ for some C ∞ function λ : M → R \ {0}.
, it is a non-degenerate CR structure, if and only if (M, g) has constant sectional curvature.
Let (M, H, J, θ) be a non-degenerate almost CR manifold.Let us extend J to an endomorphism ϕ of the tangent bundle by requesting that ϕ = J on H and ϕ(T) = 0 (T is the Reeb vector field of θ).Then and (θ, T, ϕ) is an almost contact structure.In particular, θ • ϕ = 0.The Webster metric is the semi-Riemannian metric g θ defined by for any X, Y ∈ H, where ε = ±1.Equivalently, is a contact semi-Riemannian structure on M. If we denote by g + θ the Webster metric with T space-like and by g − θ 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 g + θ is Riemannian, then g − θ is Lorentzian with T time-like.Conversely, a contact semi-Riemannian structure (η, ξ, ϕ, g) defines a nondegenerate pseudohermitian almost CR structure given by and L θ = g |H is the corresponding Levi form which is nondegenerate and Hermitian, that is, Equation ( 31) is satisfied.
If the Levi-form L θ is positive definite, the Webster metric g θ (with ε = 1) is a Riemannian metric and "non-degenerate" is replaced by "strictly pseudo-convex".
• 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 θ = λθ, where λ : M → R − {0} is a smooth function, while strictly pseudo-convexity is not a CR invariant property (if L θ is positive definite and θ = −θ, then L θ is negative definite).In particular, if (H, θ, J) is a non-degenerate almost CR structure, then for any real constant t = 0, (H, θ = tθ, J) is a non-degenerate almost CR structure.Moreover, the Webster metrics g θ and g θ are related, taking account that φ = ϕ, by This is related to the deformation Equation ( 16). 2. Let (H(M), J, θ) be a non-degenerate almost CR structure and (η = −θ, ξ = −T, ϕ, g = g θ ) the corresponding contact semi-Riemannian structure.Since then we get that (θ, ξ = T, ϕ, ḡ = −g θ ) is still a contact semi-Riemannian structure with ε = −ε.This second structure is obtained by Equation (13), i.e., reversing the first contact semi-Riemannian structure.
3. Let η be a contact 1-form.Then, there exists an associated metric for η if and only if there exists an almost complex structure J on H =kerη such that the Levi form L η = dη(•, J•) 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 ( M, J, ḡ) be an indefinite (2n + 2)-Kaehler manifold (cf.[38] for definitions and examples).Suppose that M is an orientable non-degenerate real hypersurface of M. Let N be a normal vector field, ḡ(N, N) = ε, 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 A = − ∇N is the shape operator.Now, consider the almost CR structure induced On the other hand, by Equation ( 38)a, we have (∇ X ϕ)Y = −εg(AX, Y)ξ for any X, Y ∈ H, and so we get S(X, Y) = 0 for any X, Y ∈ H. Therefore, by 1) of Proposition 5, the almost CR structure (H, J) is integrable.So, we proved the following Proposition 9. Let ( M, J, ḡ) be an indefinite Kaehler manifold.Suppose that M is an orientable non-degenerate real hypersurface of M.Then, the almost contact semi-Riemannian structure on M given by (37) defines a CR structure (H, J) 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 Y = ξ, we have In particular, Aξ = η(Aξ)ξ.Then, η(AX) = εg(AX, ξ) = εg(X, Aξ) = εg(X, η(Aξ)ξ) = η(Aξ)η(X), 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. Let ( M, J, ḡ) be an indefinite Kaehler manifold.Suppose that M is an orientable non-degenerate hypersurface of M.Then, the almost contact semi-Riemannian structure on M given by Equation (37) is Sasakian if and only if the shape operator is given by Equation (40).
By Proposition 9, the standard pseudohermitian almost CR structure (H, θ, J) of an orientable non-degenerate real hypersurface, that is, the one induced by Equation (37), is integrable.Then, Proposition 6 gives that (H, θ, J) is always a pseudohermitian CR structure.Moreover, by using Equation (38)b, i.e., ∇ξ = ϕA, we have Consequently, the condition dη = g(•, ϕ) 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 M.
Then, the almost CR structure (H, θ, J) 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 (H, J, θ) is said to be Levi-flat, or Levi-degenerate, if the Levi form L θ vanishes.
In the case of an orientable non-degenerate real hypersurface of an indefinite Kaehler manifold M, by using Equation (41), the standard pseudohermitian CR structure (H, θ, J) of M is Levi-flat, i.e., L θ = 0, if and only if ϕA = −Aϕ on H. On the other hand, if we consider the fundamental 2-form Φ, we have (dΦ)(X, Y, Z) = 0 (cf.[39]).Hence, an orientable non-degenerate real hypersurface of an indefinite Kähler manifold M is almost coKähler if and only if ϕA = −Aϕ.
Recently in the paper [33], see also [41], we proved that an orientable Riemannian three-manifold (M, g) admits an almost α-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 α-coKähler three-manifold is either a Riemannian product of type R × S 2 (k 2 ), equipped with its standard coKähler structure, or it is a semidirect product Lie group G = R 2 A R equipped with a left invariant almost α-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, (2n + 1)-dimensional CR-manifolds is to understand when a given strictly pseudo-convex CR-structure can be realized by an embedding in C m .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 (2n + 1)-manifold, is realizable as an embedding in some C m , provided n > 1.
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 S 3 .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 C 2 , and proved that if a compact strictly pseudo-convex CR manifold admits a CR embedding into C 2 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 C n for some n when the CR Paneitz operator is non-negative and the CR Yamabe constant is positive.

The (Generalized) Tanaka-Webster Connection and the Pseudohermitian Torsion
Let (M, H, J, θ) be a non-degenerate almost CR manifold and (η = −θ, ξ = −T, ϕ, g = g θ ) the associated contact semi-Riemannian structure.The most convenient linear connection for studying (M, H, J, θ) is the so-called (generalized) Tanaka-Webster connection ∇.This is the linear connection given by ∇X for any X, Y ∈ X(M), where ∇ is the Levi-Civita connection of g θ .Equivalently, ∇ is defined by where π is the usual projection π : TM → H.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 ∇ϕ = 0.More precisely, ∇ is the unique linear connection obeying to the axioms ∇η = 0, ∇ξ = 0, ∇g = 0, T(ξ, ϕX) Here T(X, Y) = ∇X Y − ∇Y X − [X, Y] is the torsion tensor field of ∇, and Q is the Tanno tensor, i.e., We note that Q(ξ, X) = Q(Y, ξ) = 0 and Q(Y, X) = (∇ X ϕ)Y − η ∇ X ϕY ξ for any X, Y ∈ H.Then, by the same proof given in [20], Q = 0 if and only if (H, J) is integrable, that is: ∇ϕ = 0 ⇐⇒ (H, J) is a CR structure, 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 τ on M defined by τX := T(T, X), and thus Then, by using Next, we recall that given a semi-Riemannian manifold ( M, ḡ), with ∇ the Levi-Civita connection, and a smooth nondegenerate distribution D : and ( ∇X Y) ⊥ is the natural projection on D ⊥ .Moreover, the distribution D is called totally geodesic if the symmetrized second fundamental form B s (X, Y) := (1/2) B(X, Y) + B(Y, X) vanishes.Consider the non-degenerate almost CR manifold (M, H(M), J, θ).For the Levi distribution H(M), the second fundamental form B(X, Y) is given by B(X, Y) = ε g(∇ X Y, ξ)ξ , and by using Equation (4) we get where E i is a local orthonormal basis.Since trace g (ϕh) = 0, we get trace g (B) = 0.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 (M, H, J, θ), the Levi distribution H(M) is minimal in (M, g θ ), 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 (M, H, J, θ).Denote by R 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 Ric and the pseudohermitian scalar curvature r are defined by Ric(X, Z) = Tr g R(X, •)Z, r = Tr g Ric.
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 η-Einstein contact semi-Riemannian manifold given in Section 2.2.
Consider a non-degenerate almost CR manifold (M, H, J, θ), dim M = (2n + 1), with pseudohermitian torsion τ = 0.In this case, Equation (43) gives and in particular Then, the pseudohermitian Ricci tensor Ric and the Tanaka-Webster scalar curvature r are given by where Ric and r denote the Ricci tensor and the scalar curvature of the Webster metric g θ .So, we get Proposition 11. ( [48]) Let (M, H, J, θ) be a non-degenerate almost CR manifold with pseudohermitian torsion τ = 0.Then, the structure (H, J, θ) is pseudo-Einstein if and only if the corresponding semi-Riemannian contact structure is η-Einstein.
When the pseudohermitian torsion τ = 0, there are other conditions on τ with an interesting meaning.Given an oriented, compact, contact manifold (M, η), denote by M(η) the set of all Riemannian metrics associated to the contact form η and by A(η) the set of all almost CR structures J for which the Levi form is positive definite.By Proposition 8, the sets M(η) and A(η) can be identified.
• The condition ∇ξ τ = 0. Tanno [20] considered the Dirichlet energy defined for any g ∈ M(η).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 A(η) (there was an error in their calculation of the critical point condition, as was pointed out by Tanno).Moreover, since Ric(ξ, ξ) = 2n − trh 2 = 2n − L ξ g 2 /4, the functional Equation ( 50) is equivalent to the functional L(g) = M Ric(ξ, ξ)dv studied in general dimension, for compact regular contact manifold, by Blair ([2], Section 10.3).Now, since L ξ g = 2g(ϕ•, h•) = −2g(τ•, •), where g = g θ and τ = ϕh is the pseudohermitian torsion, we have Then, to consider the Dirichlet energy Equation ( 50) is equivalent to consider the following defined on the set A(η).Moreover, using the Tanaka-Webster connection given by Equation ( 43), we get Thus, the critical point condition Equation ( 51) becomes ∇ξ τ = 0.
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, I(g) = M rdv, 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 η-Einstein if and only if it is H-contact and satisfies the critical point condition ∇ξ τ = 2ϕτ (equivalently, ∇ ξ τ = 0).

•
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 r 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 I w (g) = M rdv defined on the set A(η), is the vanishing of the pseudohermitian torsion τ.An alternate proof of this important result was given by the present author [54].Tanno [20] studied the functional I w (g) in arbitrary dimension.

•
An interpretation of the Tanaka-Webster scalar curvature.Recall that a contact form η on a compact manifold M is called regular if its Reeb vector field ξ is regular, i.e., any point of M has a neighborhood such that any integral curve of ξ passing through the neighborhood passes through only once.In this case M is a principal S 1 -bundle over a symplectic manifold B whose fundamental 2-form Ω has integral periods (a Hodge manifold).The corresponding fibration p : M → B = M is known as the Boothby-Wang fibration [56].Now, let (M, η, g) be a compact simply connected regular Sasakian, (2n + 1)-manifold.Then, the base of the Boothby-Wang fibration is a compact Kähler manifold of complex dimension n, with Kähler metric g and fundamental 2-form Ω satisfying (cf., for example, Ref. [57,58]) Moreover, the scalar curvatures r,r of (M, g) and (B, g), respectively, are related by r = r + 2n.
On the other hand, in the Sasakian case, Equation ( 53) becomes r = r + 2n.
So, in this case, the Tanaka-Webster scalar curvature r is the scalar curvature r of the Kähler manifold (B, g) base of the Boothby-Wang fibration.We note that a compact simply connected homogeneous Sasakian manifold is regular [57].

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 ≥ 5.Then, Olszak [60] generalizing this result proved that if a contact Riemannian (2n + 1)-manifold, n ≥ 2, is of constant curvature κ, then the manifold is Sasakian and κ = 1.In the semi-Riemannian case, we have ( [7,8]) Theorem 16.Let (M, η, g) be a contact semi-Riemannian (2n + 1)-manifold.If n ≥ 2 and (M, g) is of constant sectional curvature κ, then κ = ε = g(ξ, ξ) and h 2 = 0.
In particular, since ε = ±1, a non-degenerate almost CR structure does not admit any flat semi-Riemannian Webster metric in dimension ≥ 5, so Blair's result also holds in the semi-Riemannian setting.But, there are examples of non-degenerate almost CR manifold with τ 2 = 0 and τ = 0.In fact we have the following (see [34] for details).
Example 5. Consider the space M = R 5 (x 1 , x 2 , x 3 , x 4 , z) and two smooth functions α, β ∈ C ∞ (R 5 ).We put Moreover, we define the 1-form η = 2x 1 dx 3 + 2x 2 dx 4 + dz, the vector field ξ = X 5 = ∂ z , the tensor ϕ by and the semi-Riemannian metric g of signature (−, −, Then (ξ, ϕ, η, g) defines a contact semi-Riemannian structure, and so a non-degenerate almost CR structure on M, with Levi distribution H = ker η =span(X 1 , X 2 , X 3 , X 4 ).Moreover, we can construct a frame {E 1 , E 2 , E 3 , E 4 , ξ} of vector fields on R 5 with E i ∈ H null vector fields which satisfy Therefore, τ 2 = 0.Moreover, τ = 0 if and only if ∂ z (β − α) = 0. So, taking the functions α, β such that , we obtain a non-degenerate almost CR structure with τ 2 = 0 and τ = 0.Moreover, this structure in general is not a CR structure.In fact, taking for example X = E 1 and Y = E 3 , 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 (M, H, J, θ) be a non-degenerate CR manifold, dim M = 2n + 1, n ≥ 2. If the Webster metric g θ is of constant sectional curvature κ, then κ = ε = g θ (ξ, ξ) and the pseudohermitian torsion τ = 0, i.e., the Webster metric is Sasakian.Remark 9.In dimension three, a non-degenerate CR manifold with the Webster metric g θ locally symmetric (in particular, of constant sectional curvature) is either flat or of constant sectional curvature κ = ε = g(ξ, ξ), and in the second case the metric is Sasakian [7].
In particular, the Ricci operator Q and the scalar curvature r of a (2n + 1)-dimensional (κ, µ)-space M, κ < 1, are given by Then, (κ, µ)-spaces are examples of H-contact manifolds.For a non-Sasakian (κ, µ) space, Boeckx Boechx [62] introduced an invariant and showed that for two non-Sasakian (κ, µ) spaces M, M , we have I M = I M if and only if up to a D-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 K θ , that is, the sectional curvature defined by the tensor where R(X, Y)Z is the pseudohermitian curvature tensor associated to the Tanaka-Webster connection ∇, and g θ 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 n > 1, has constant pseudohermitian sectional curvature, then it has vanishing pseudohermitian torsion (τ = 0) if and only if the Tanaka-Webster connection of M is flat.
• Almost contact structures belonging to a CR structure Let (H, J) be a CR structure on a odd-dimensional manifold M. We say that an almost contact structure (θ, ξ, ϕ) belongs to the CR structure (H, J) if ker θ = H and J = ϕ |H .Then, by Lemma 1.1 of [64], two almost contact structures (θ, ξ, ϕ) and (θ , ξ , ϕ ) belong to the same CR structure (H, J) if and only if for some smooth function λ and vector field X 0 ∈ H, where ε = ±1.
Denote by (θ, ξ, ϕ) * an almost contact structure belongs to a non-degenerate CR structure (H, J) and satisfying the condition [ξ, H] ⊂ H.Then, K.Sakamoto and Y. Takemura [64] proved the existence of a unique linear connection associated to (θ, ξ, ϕ) * .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.

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.

The Classification Theorem
Recall some definitions about the homogeneity.A contact manifold (M, η) is said to be homogeneous if there exists a (connected) Lie group G of diffeomorphisms acting transitively on M and leaving η invariant.A contact semi-Riemannian manifold (M, η, g) is said to be homogeneous if there exists a (connected) Lie group G of isometries acting transitively on M and leaving η invariant, that is, for any x, y ∈ M there exists f ∈ G such that y = f (x), and In particular a CR transformation is a diffeomorphism f such that Remark 10.Typical examples of CR maps are got as traces of holomorphic maps of Kaehlerian manifolds on real hypersurfaces.Precisely, let M be a Kaehlerian manifold.Any orientable real hypersurface M ⊂ M admits a natural CR structure (cf.Proposition 9).If M ⊂ M is another oriented real hypersurface in the Kaehlerian manifold M and F : M → M is a holomorphic map such that F(M) ⊂ M then f ≡ F| M : M → M 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 θ and θ be pseudohermitian structures on the almost CR manifolds M and M respectively.
Let (M, H, θ, J) be a pseudohermitian almost CR manifold.Denote by P sh (M, θ) the group of all CR automorphisms f : M → M such that f * θ = θ.
In other words, Recall that there is a canonical way to associate a contact Riemannian structure to a contact Lorentzian structure (and conversely).If (η, ξ, ϕ, g L ) is a contact Lorentzian structure on a smooth manifold M, dim M = 2n + 1, where the Reeb vector field ξ is time-like, then (cf.Section 2.2 and also Equation ( 36)) is a contact Riemannian structure on M. The scalar curvatures r R and r L of g and g L are related by Equation ( 20): Now, let (M, H, θ, J) be a non-degenerate CR three-manifold.Then, the Levi form L θ is definite, and we can assume L θ positive definite (if necessary, we change θ with −θ).Therefore, without loss in generality, in dimension three, we can consider either g θ Lorentzian with T time-like or g θ Riemannian.
In particular, the Sasakian condition τ = 0 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 r = r + ε(2 + tr h 2 ), where ε = g θ (T, T).
So, the Tanaka-Webster scalar curvature r does not depend on the causal character of the Reeb vector field T, i.e., rL = rR .In fact, If we consider the scalar torsion L ξ g θ introduced by Chern and Hamilton in [51] in their study of contact Riemannian three-manifolds, since 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 τ do not depend on the causal character of the Reeb vector field T, that is, they depend only on Levi form L θ , 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 (M, H, θ, J) is a Lie group G equipped with a left-invariant non-degenerate CR structure.More precisely, one of the following cases occurs: Proposition 13.The Lie group E(2) is the only simply connected 3-manifold which admits a homogeneous non-degenerate CR structure with flat Riemannian Webster metric.In such a case W = +1/2.
In the Lorentzian case, one gets Proposition 14. ( [7]) The Lie group E(1, 1) is the only simply connected 3-manifold which admits a homogeneous non-degenerate CR structure with flat Lorentzian Webster metric.In such a case W = −1/2.
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 SU(2), the Heisenberg group H 3 or SL(2, R) by a discrete group.As a consequence of Theorem 21 we have Proposition 15.The unimodular Lie groups SU(2), the Heisenberg group H 3 , SL(2, R), 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 (η, g) be a homogeneous Sasakian structure on the sphere S 3 , with Webster scalar curvature W > 0. Since W = (r + 2)/8 > 0, then η = tη and g = tg + t(t − 1)η ⊗ η, for t = W, define a Sasakian structure on S 3 with g of constant sectional curvature +1 (cf.[52], Section 3).In particular, ( η, g) is isomorphic to the standard Sasakian structure (η 0 , g 0 ) [4].Then, we can assume ( η, g) = (η 0 , g 0 ), and consequently we have where g a = g 0 + a − 1 η 0 ⊗ η 0 , a = 1/W > 0, is a Berger metric, that is, a metric defined as the canonical variation g a , a > 0, of the standard metric g 0 on S 3 , obtained deforming g 0 along the fibres of the Hopf fibration: •) = aη 0 , g a (ξ 0 , ξ ⊥ 0 ) = 0, where ξ 0 denotes the standard Hopf vector field on S 3 .Therefore we get: Proposition 16.In the second part of Corollary 3, the Sasakian metric on S 3 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 ≤ 0 or is everywhere strictly positive.Now, if M is a compact Sasakian 3-manifold with Webster scalar curvature W > 0, 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 S 3 .However, this fact is not too surprising since a compact simply connected manifold which admits a nonsingular Killing vector field is diffeomorphic to S 3 (cf.[52] Section 4).
Corollary 4. The Heisenberg group H 3 , SL(2, R) 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 W = 0.In particular, the Heisenberg group H 3 is the only simply connected three-manifold which admits a non-degenerate CR structure with pseudohermitian torsion τ = 0 and Webster scalar curvature W = 0.
In Theorem 21, if we consider g θ Lorentzian and denote by r L the corresponding scalar curvature, then in the Sasakian case (i.e., when τ = 0), the conditions W = 0, W > 0, W < 0, and W = −α 2 /4 are equivalent to r L = 2, r L > 2, r L < 2, and r L = −2α 2 + 2 < 2, respectively.On the other hand, for a Lorentzian Sasakian three-manifold, when r L < 2, the Lorentzian K-contact structure ( η, g) obtained by a D-homothetic deformation in correspondence to t = (2 − r L )/8 = −W is Einstein (see Section 2.2), and so of constant sectional curvature −1.Therefore, we get the following corollary which does not have a Riemannian counterpart.
Corollary 5.The unimodular Lie group SL(2, R) 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 κ = −1.

•
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 (η 1 , η 2 ) such that the 1-forms η a = a 1 η 1 + a 2 η 2 are contact forms with the same volume form for all a ∈ S 1 .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 (η 1 , η 2 , g), that is, (η 1 , η 2 ) is a taut contact circle and g is a Riemannian metric associated to both the contact forms η 1 and η 2 .More in general, we introduce the notion of bi-contact metric structure (η 1 , η 2 , g), where (η 1 , η 2 ) is a pair of arbitrary contact forms and g is a Riemannian metric associated to both the contact forms η 1 and η 2 such that the same contact forms are orthogonal with respect to g, i.e., the corresponding Reeb vector fields ξ 1 , ξ 2 are orthogonal.On the other hand, in the classical definition of three-contact metric structure (η 1 , η 2 , η 3 , g), 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 +1 (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: SU(2), SL(2, R), E(2), E(1, 1).

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 E(2) 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 ϕ-symmetric spaces include the non-Sasakian (κ, µ)-manifolds (see [2], p. 146; more in general we refer to [2] Section 7.9 for a discussion on weakly and strongly locally ϕ-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 (κ, µ)-space.

•
Recently E.M. Correa [76] gives a new study on compact, (2n + 1)-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 S 5 or S 2 × S 3 .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 π : M → B 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).• D. V. Alekseevsky and A. Spiro [66,78] gave a classification of all compact, simply connected, (2n + 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 S 2n+1 , n > 1, which admits a transitive action of a Lie group of CR-transformations is the standard CR-structure.For S 3 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 x ∈ M there exists a CR-isometry σ : M → M such that σ(x) = x and (dσ) x|H x = −I d .In particular, they proved the following.Let M be a strictly pseudo-convex CR manifold, dim M > 3, with pseudohermitian torsion τ = 0.Then, M is locally CR-symmetric if and only if the underlying contact metric structure (η, ξ, ϕ, g) satisfies the (k, µ)-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.

Geometry of Tangent Hyperquadric Bundles
The geometry of the unit tangent sphere bundle T 1 M of a Riemannian manifold (M, g) 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 T 1 M 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 (M, g) is a semi-Riemannian manifold of index ν > 0, the Sasaki metric induced on tangent hyperquadrics bundle T ε M, ε = ±1, is a semi-Riemannian metric of index 2ν − 1 if ε = −1 and the index is 2ν if ε = +1.In such case we have few results about the the geometry of T ε M (see [84], Ref. [85] and more recently [48]).In this Section we discuss some results of [48] on the geometry of T ε M equipped with the standard non-degenerate almost CR structure.

The Standard Non-Degenerate Amost CR Structure on T ε M
Let (M, g) be a semi-Riemannian manifold of index ν, 0 ≤ ν ≤ n = dim M. At any point (x, u) of its tangent bundle TM, the tangent space of TM splits into the horizontal and vertical subspaces: Each tangent vector Z ∈ (TM) (x,u) can be written in the form Z = X h + Y v , where X, Y ∈ M x are uniquely determined vectors.
The tangent bundle TM 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 z = (x, u) ∈ TM and for any X, Y ∈ M x .G is a semi-Riemannian metric of signature (2ν, 2n − 2ν), and both H z and V z have index ν.There is also an almost complex structure J on TM given by then the Sasaki metric G is Hermitian with respect to the almost complex structure J.We denote by N (x,u) the canonical vertical vector field on TM and by ζ (x,u) the geodesic flow on TM.They are defined by The Liouville form β on TM, defined by β( X) z = G( Xz , ζ z ) = g x (π * z Xz , u), satisfies the following (see Prop. 2 of [84], and [2] that is, 2(dβ) is the fundamental 2-form, and so (TM, J, G) is an indefinite almost Kaehler manifold.Besides (see ( [84], Proposition 3): J is integrable if and only if the semi-Riemannian manifold (M, g) is locally isometric to the semi-Euclidean space R n ν .Consider the tangent hyperquadric bundle moreover the geodesic flow ζ is tangent to T ε (M, g).Any horizontal vector X h is tangent to T ε (M, g), and a vertical vector X v is tangent to T ε (M, g) if and only if X v is orthogonal to N z .Consequently, the tangent space of T ε (M, g), at a point z = (x, u) ∈ T ε (M, g), is given by In general, the tangential lift of a vector field X is a vector field on T ε (M, g) defined by The Sasaki metric on T ε (M, g) is the semi-Riemannian metric G on T ε (M, g) induced from G, it is completely determined by the identities for all z = (x, u) ∈ T ε (M, g) and X, Y ∈ M x .Since the Sasaki metric on the tangent bundle TM is of signature (2ν, 2n − 2ν), G(N , N ) = ε and T ε (M, g) is an orientable semi-Riemannian hypersurface of (TM, G) of sign ε, then the index of T −1 (M, g) is 2ν − 1 and the index of T 1 (M, g) is 2ν.
We now construct the standard non-degenerate almost CR structure on T ε (M, g).The tangent hyperquadric bundle T ε (M, g) is an orientable non-degenerate hypersurface of the indefinite almost Kaehler manifold (TM, J, G).Then, by the usual procedure, we construct the almost contact semi-Riemannian structure (ξ , η , ϕ , G) induced on T ε (M, g), where for z = (x, u) ∈ T ε (M, g) and X vector field on T ε (M, g).Since 2ε(dη )( X, Ỹ) = G( X, ϕ Ỹ) for any X, Ỹ vector fields on T ε (M, g), if we rescale the structure tensors appropriately by η = (1/2ε)η , ξ = 2εξ , ϕ = ϕ and ḡ = (1/4) G, Riemannian manifold (M, g) has constant sectional curvature +1, and in such case the standard contact Riemannian structure on T 1 M 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 T ε (M, g): In such a case (H, θ, J) is a pseudo-Einstein CR structure, which is Sasakian, and the Ricci tensor and the pseudohermitian Ricci tensor are given by (ii) If 0 < ν < n, the pseudo-Einstein CR structure of (i) is Einstein, i.e., the Webster metric is Einstein, if and only if (M, g) is a Lorentzian surface of constant curvature c = ε.In such a case, T ε (M, g) has constant sectional curvature c = ε.
Corollary 7. Let (M, g) be a semi-Riemannian manifold of index ν, 0 ≤ ν ≤ n = dim M.Then, the geodesic flow of T ε (M, g) is Killing if and only if M has constant sectional curvature ε.

Sasaki-Einstein and H-Contact Structures on T ε M
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 (M, g) be a semi-Riemannian manifold of constant sectional curvature c.Then, the standard contact semi-Riemannian structure (η, ξ, ϕ, ḡ) on T ε (M, g) is H-contact.Moreover, the structure is η-Einstein if and only if either c = ε or c = (n − 2)ε, n = dim M. In such a case, the Ricci tensor is given by where c = ε or c = (n − 2)ε.

Remark 14.
Recall that η-Einstein, K-contact and Sasakian semi-Riemannian manifolds are H-contact.Now, we remark that: As a consequence of Theorems 25-27, we obtain the following result.
Therefore as c varies over the reals, I T 1 M assumes all the real values strictly greater than > −1.Boeckx found examples of (κ, µ)-spaces, for every value of the invariant I ≤ −1, 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 T −1 M over Lorentzian space forms (M, g) of constant curvature c different from −1, 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 c ∈ R, c = −1, so that for c ≤ 0, these examples cover all possible values of the Boeckx invariant in (−∞, −1).This result makes E. Boeckx's classification of (κ, µ)-spaces in [62] more geometric.We note that in this case the Webster metric of T −1 M is not the Lorentzian metric Equation (65) induced from the Sasaki metric of TM.

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 f : M → M of (semi) Riemannian manifolds (M, g) and (M , g ) one replaces the Hilbert-Schmidt norm of d f by the trace with respect to g of the restriction of f * g to a given codimension one distribution H on M (rather than applying the same construction to the full f * g ).E. Barletta et al., Ref. [96], introduced pseudoharmonic maps f : M → M from a nondegenerate CR manifold M endowed with a contact form θ into a Riemannian manifold M .When M is itself a non-degenerate CR manifold carrying the contact form θ a result in [96] describes pseudoharmonicity of CR maps f : M → M .R. Petit [97] considered the following (pseudohermitian analog to the) second fundamental form where ∇ is the Tanaka-Webster connection of M and ∇ = f −1 ∇ is the pullback of the Levi-Civita connection ∇ of M .The approach in [96] is to replace ∇ by an arbitrary linear connection D on M , consider the restriction Π H β f of (69) to the Levi distribution H = ker θ, and take the trace with respect to the Levi form L θ .Then f is called pseudoharmonic (with respect to the data (θ, D )) if trace L θ Π H β f = 0.More recently, Dragomir and R. Petit et al.,[98], studied contact harmonic maps, i.e., C ∞ maps f : M → M from a compact strictly pseudoconvex CR manifold M into a contact Riemannian manifold M which are critical points of the functional where θ is a contact form on M and (d f ) H,H = pr H • f * : H → H . J. Konderak & R. Wolak, Ref. [99], introduced transversally harmonic maps as foliated maps f : (M, F , g) → (M , F , g ) 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 (M, η, ξ, ϕ, g) into a semi-Riemannian manifold (M , g ), i.e., C ∞ solutions of τ H ( f ) ≡ trace g Π H β f = 0, where β f is the second fundamental form of f , and Π H β f is the restriction of β f to the Levi distribution H = ker η.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 ϕ-condition: ∇ ϕX ϕX + ∇ X X = ϕ[ϕX, X], equivalently : (∇ X ϕ)ϕX = (∇ ϕX ϕ)X, (70) for any X ∈ H.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 ϕ-condition.Moreover, the ϕ-condition extends (cf.[94], Section 3) the so-called condition (A) of Rawnsley [102].Rawnsley in his paper introduced the condition (A) 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 ( This last condition, for Y = ϕX, X ∈ ker η, implies the ϕ-condition. In this Section we report some results of [47,94], for almost contact semi-Riemannian manifolds.Let (M, η, ξ, ϕ, g) be a real (2n + 1)-dimensional almost contact semi-Riemannian manifold and (M , g ) a semi-Riemannian manifold.Let f : M → M be a C ∞ map and f −1 T(M ) → M the pullback of T(M ) by f .Let ∇ = f −1 ∇ be the pullback of the Levi-Civita connection ∇ of (M , g ) i.e., the connection in the vector bundle f −1 T(M ) → M induced by ∇ .If (U, x i ) and (U , y α ) are local coordinate systems on M and N such that f (U) ⊂ V then ∇ is locally described by where ) denotes the natural lift of Y ∈ X(U ) and Γ γ αβ are the Christoffel symbolds of (M , g ).Let H = kerη and J = ϕ| H be the almost CR structure underlying (η, ξ, ϕ, g).The second fundamental form β f of f is given by Here ∇ is the Levi-Civita connection of (M, g) and the vector field f * X is given by ( f * X)(x) = ( f * x )X x ∈ T f (x) M for any x ∈ M and X ∈ X(M).Next, let τ H ( f ) ∈ C ∞ ( f −1 TM ) be the tension field defined by where Π H β f is the restriction of β f to H ⊗ H.Note that the tension field Definition 6.Let (M, η, ξ, ϕ, g) an almost contact semi-Riemannian manifold and (M , g ) a semi-Riemannian manifold.A C ∞ map f : M → M is said to be Levi harmonic with respect to H = kerη if τ H ( f ) = 0.
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 (1, 1), ∇ * S = −trace∇S.Then, after some computations, we get Moreover, as f is a CR map, If additionally (M, ϕ, ξ, η, g) is a contact semi-Riemannian manifold, then where ε = g(ξ, ξ).Hence f is Levi harmonic if and only if f * ξ is collinear to ξ .
Let S 2n+1 ⊂ C n+1 be the unit sphere endowed with the canonical Sasakian structure (η 0 , ξ 0 , ϕ 0 , g 0 ), hence ξ 0 is the standard Hopf vector field on S 2n+1 .Then Corollary 11. ξ 0 : (S 2n+1 , η 0 , g 0 ) → (T 1 S 2n+1 , ηt , ḡ0t is a Levi harmonic map for any t > 0. 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 S 3 which define harmonic maps from S 3 to (T 1 S 3 , ḡ0 ), where ḡ0 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 T 1 S 3 .
Finally, we give a short presentation of the variational treatment of Levi harmonicity.Let (M, η, ξ, ϕ, g) be a (2n + 1)-dimensional almost contact Riemannian manifold and (M , g ) a Riemannian manifold.
If Ω ⊂ M is a relatively compact domain we set for any f ∈ C ∞ (M, M ).Then we obtain the following ( [47], Theorem 6.1): Let Ω ⊂ M be a relatively compact domain.A C ∞ map f : M → M is a critical point for the energy functional E Ω : C ∞ (M, M ) → R defined by (79) if and only if τ H ( f ) = f * ∇ ξ ξ + div(ξ) ξ .
If f : M → M be an immersion and a critical point of E Ω , 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.

Problems Question 1 .Question 2 .Question 3 .
(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.(related to the Section 2.4) In dimension ≥ 5, it is an open problem, to our knowledge, the existence of non trivial semi-Riemannian contact Ricci solitons.(related to Section 3.2) Study the geometry of an almost contact (semi) Riemannian structure (η, ξ, ϕ, g) when η defines a pseudohermitian structure.Question 4. (related to the Section 3.4) In dimension ≥ 5, it is an open problem to see if the Olszak's result holds for a general non-degenerate almost CR manifold.
and r > 0, i.e., r > −2n, equivalently r L > 2n, the Riemannian K-contact structure (η t , g t ) obtained in correspondence to of Proposition 5 and Theorem 11, one gets: • the Reeb vector field ξ is Killing with respect to Webster metric g θ if and only if pseudohermitian torsion τ vanishes, equivalently L ξ J = 0; • the almost contact structure (ξ, ϕ, η) is normal, equivalently the Webster metric g θ 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 L ξ J = 0.
1)-dimensional, homogeneous non-degenerate CR manifolds M.This classification is based on a description of the maximal connected compact group of automorphisms of M. • CR manifolds which are locally CR equivalent to the unit sphere S 2n+1 , endowed with the standard CR structure as a real hypersurface of C n+1 , 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 f : M → S 2n+1 , where M is its universal cover.Moreover, they proved that if the group of CR automorphisms is transitive on M, then f : M → S 2n+1 is a covering and f ( M) is homogeneous domain D in S 2n+1 .Thus to classify the simply connected spherical homogeneous CR manifolds it suffices to classify homogeneous domain in S 2n+1 ([79], Theorem 3.1).In particular, in dimension three, we have a list of five examples ([79], p. 229): Let (M, g) be a semi-Riemannian manifold of index ν, 0 ≤ ν ≤ n = dim M.Then, we have the following(i)The standard non-degenerate almost CR structure (H, θ, J) on T ε (M, g) has vanishing pseudohermitian torsion if and only if (M, g) has constant sectional curvature c = ε.