© Hindawi Publishing Corp. SOME SUBMERSIONS OF CR-HYPERSURFACES OF KAEHLER-EINSTEIN MANIFOLD

The Riemannian submersions of a CR-hypersurface M of a Kaehler-Einstein manifold M˜ are studied. If M is an extrinsic CR-hypersurface of M˜, then it is shown that the base space of the submersion is also a Kaehler-Einstein 
manifold.


Introduction.
The study of the Riemannian submersions π : M → B was initiated by O'Neill [14] and Gray [9].This theory was very much developed in the last thirty five years.Besse's book [3,Chapter 9] is a reference work.Bejancu introduced a remarkable class of submanifolds of a Kaehler manifold that are known as CR-submanifolds (see [1,2]).On a CR-submanifold, there are two complementary distributions D and D ⊥ , such that D is J-invariant and D ⊥ is J-anti-invariant with respect to the complex structure J of the Kaehler manifold.The integrability of the anti-invariant distribution D was proved by Blair and Chen [4].
Recently, Kobayashi [10] considered the similarity between the total space of a Riemannian submersion and a CR-submanifold of a Kaehler manifold in terms of the distribution.He studied the case of a generic CR-submanifolds in a Kaehler manifold and proved that the base space is a Kaehler manifold.
In Section 3, we extend the result of Kobayashi to the general case of a CRsubmanifold.
In Section 4, we study a Riemannian submersion from an extrinsic hypersurface M of a Kaehler-Einstein manifold M onto an almost-Hermitian manifold B. In this case, we prove that the basic manifold is a Kaehler-Einstein manifold.If M is C n+1 , a standard example is the Hopf fibration S 2n+1 → CP n equipped with the canonical metrics.
For the basic formulas of Riemannian geometry, we use [11,12].

2.
Preliminaries.Let M be a complex m-dimensional Kaehler manifold with complex structure J and Hermitian metric •, • .Bejancu [2] introduced the concept of a CR-submanifold of M as follows: a real Riemannian manifold M, isometrically immersed in a Kaehler manifold M, is called a CR-submanifold of M if there exists on M a differentiable holomorphic distribution D and its x M, where T ⊥ x M is the normal space to M at x ∈ M for any x ∈ M. It is easily seen that each real orientable hypersurface of M is a CR-submanifold.The Riemannian metric induced on M will be denoted by the same symbol •, • .
Let ∇ (resp., ∇) be the operator of covariant differentiation with respect to the Levi-Civita connection on M (resp., M).The second-fundamental form B is given by for all E, F ∈ Γ (T M), where Γ (T M) is the space of differentiable vector field on M. We denote everywhere by Γ (τ) the space of differentiable sections of a vector bundle τ.
For a normal vector field N, that is, N ∈ Γ (T ⊥ M), we write where −L N E (resp., ∇ ⊥ E N) denotes the tangential (resp., normal) component of ∇E N.
Let µ be the orthogonal complementary vector bundle of Definition 2.1 (Kobayashi [10]).Let M be a CR-submanifold of a Kaehler manifold M. A submersion from a CR-manifold M onto an almost-Hermitian manifold is a Riemannian submersion π : M → M with the following conditions: (i) D ⊥ is the kernel of π * , (ii) π * : D x → T π(x) M is a complex isometry for every x ∈ M.This definition is given by Kobayashi for the case where µ is a null subbundle of T ⊥ M (see [10]).If JD ⊥ x = T ⊥ x M for any x ∈ M, we say that M is a generic CR-submanifold of M (Yano and Kon [15]).For example, any real orientable hypersurface of M is a generic CR-submanifold of M.
The vertical distribution of a Riemannian submersion is an integrable distribution.In our case, the distribution vertical is D ⊥ , which is integrable according to a theorem by Blair and Chen [4].
The sections of D ⊥ (resp., D) are called the vertical vector fields (resp., the horizontal vector fields) of the Riemannian submersion π : M → M .The letters U , V , W , and W will always denote vertical vector fields, and the letters X, Y , Z, and Z denote horizontal vector fields.For any E ∈ ᐄ(M), vE and hE denote the vertical and horizontal components of E, respectively.A horizontal vector field X on M is said to be basic if X is π -related to a vector field X on M .
It is easy to see that every vector field X on M has a unique horizontal lift X to M, and X is basic.
Conversely, let X be a horizontal vector field and suppose that X, Y x = X, Y y for all Y basic vector fields on M, for all x, y ∈ π −1 (x ), and for all x ∈ M .Then, the vector field X is basic.We have the following O'Neill's lemma (see [8,14]).Lemma 2.2.Let X and Y be basic vector fields on M.Then, they are satisfying the following: We recall that a Riemannian submersion π : (M, g) → (M ,g ) determines the fundamental tensor field T and A by the formulas for all E, F ∈ Γ (T M) (cf.O'Neill [14] and Besse [3]).
It is easy to prove that T and A satisfy ) for any U,V ∈ Γ (D ⊥ ) and X, Y ∈ Γ (D).Formula (2.4) means that the restriction of T to the integrable distribution D ⊥ is the second-fundamental form of the fiber submanifolds in M, and (2.5) measures the integrability of the distribution D.
3. Kaehler structure on the basic space M .From (2.1), we have for any X, Y ∈ Γ (D).
Here, we denote by h and v (resp., h and v) the canonical projections on D and D ⊥ (resp., µ and JD ⊥ ).Define a tensor field C on M as the vertical component v(∇ X Y ) of ∇ X Y (cf.Kobayashi [10]).The tensor field C is known to be a skew-symmetric tensor field defined by Kobayashi such that for all X, Y ∈ Γ (D).
Note that the tensor field C is the restriction of A to Γ (Ᏼ) × Γ (Ᏼ).
From Definition 2.1 and Lemma 2.2, we obtain that Jh∇ X Y (resp., h∇ X JY ) is a basic vector field and corresponds to J ∇ X Y (resp., ∇ X J Y ) for any basic vector fields X and Y on M.
On the Kaehler manifold M, we have Proof.From Lemma 2.2 and (3.4), we obtain that ∇ X J Y = J ∇ X Y , so that M is a Kaehler manifold.Remark 3.3.Proposition 3.1 is proved for generic CR-submanifolds of M (i.e., µ = 0) in [10].

Riemannian submersions from extrinsic hyperspheres of Einstein-Kaehler manifolds.
We recall that a totally umbilical submanifold M of a Riemannian manifold M is a submanifold whose first-fundamental form and second-fundamental form are proportional.
The extrinsic hyperspheres are defined to be totally umbilical hypersurfaces, having nonzero parallel mean-curvature vector field (cf.Nomizu and Yano [13]).Many of the basic results concerning extrinsic spheres in Riemannian and Kaehlerian geometry were obtained by Chen [5,6,7].
Let M be an orientable hypersurface in a Kaehler manifold M.Then, M is an extrinsic hypersphere of M if it satisfies for any vector fields E and F on M. Here, H denote the mean-curvature vector field of M. If we put k = H (where the norm • is, with respect to a scalar product, induced on every tangent space to M), then k is a nonzero constant function on the extrinsic hypersphere M. We denote by N the global unit normal vector field to M.Then, ξ = −JN is a global unit vector on M such that N = Jξ.Let D be the maximal J-invariant subspace (with respect to J) of the tangent space T p M for every p ∈ M. We see that M is a CR-hypersurface of M such that T M = D ⊕ D ⊥ , where D ⊥ is the one-dimensional anti-invariant distribution generated by the vector field ξ on M.
The anti-invariant distribution D ⊥ is integrable, and its leaves are totally geodesic in M (but not in M).
This is an easy consequence from Gauss and Weingarten's formulas of the leaves of D ⊥ in M.This means that O'Neill's tensor T vanishes on the fibres of the Riemannian submersion π : M → B.
The main result of this section is the following theorem.To prove Theorem 4.1, we need several lemmas.

Lemma 4.2. Following the assumptions of Theorem 4.1, then
for any horizontal vector X on M.
Proof.From Gauss's formula (2.1) and the umbilicality of M, we get ∇X ξ = ∇ X ξ for any vector field X on M.Then, we have On the other hand, M is a Kaehler manifold, so that ∇ commute with J: Consequently, for any horizontal vector fields on M.
Proof.We say that A X Y is a vertical vector field, hence Then, Lemma 4.4.Following the assumptions of Theorem 4.1, then where R and R are the curvature tensor on M and M, respectively.
Proof.We have the Gauss equation Using the umbilicality condition, we get (4.9).
Lemma 4.5.For any horizontal vector fields X and Y on M, Proof.For a Riemannian submersion with totally geodesic fibres, the following formula is known: On the other hand, the first term on the right part is skew-symmetric with respect to the vertical vector fields V and U. From (4.12) and (4.9), we obtain (4.11).
Proof of Theorem 4.1.For the horizontal vector fields X, Y , Z, and W on M, we have the following equation of O'Neill: (see [3,14]).By (4.9) and (4.11), we get the following formula that connects the curvature of M to the curvature of the Kaehler manifold M: Let (e 1 ,...,e p ; Je 1 ,...,Jl p ) be a local J-frame of basic vector fields for the horizontal distribution D.Then, (e 1 ,...,e p ; J e 1 ,...,J e p ) is a local J -frame if π star e i = e i on the Kaehler manifold B.
Using the above lemmas, from (4.14) by a straightforward calculation, we conclude that B is a Kaehler-Einstein manifold if M is a Kaehler-Einstein manifold.
Corollary 4.6.Let M be a complex-form space and M an orientable CRhypersurface of M.Then, the base space of submersion π : M → B is also a complex-form space.
Proof.The corollary follows by straightforward calculation making use of (4.14).
Example 4.7.Let S 2n+1 be the standard hypersphere in C n+1 .Then, S 2n+1 is an extrinsic hypersphere in C n+1 , and we have the Hopf fibration π : S 2n+1 → CP n equipped with the canonical metrics.

Call for Papers
This subject has been extensively studied in the past years for one-, two-, and three-dimensional space.Additionally, such dynamical systems can exhibit a very important and still unexplained phenomenon, called as the Fermi acceleration phenomenon.Basically, the phenomenon of Fermi acceleration (FA) is a process in which a classical particle can acquire unbounded energy from collisions with a heavy moving wall.This phenomenon was originally proposed by Enrico Fermi in 1949 as a possible explanation of the origin of the large energies of the cosmic particles.His original model was then modified and considered under different approaches and using many versions.Moreover, applications of FA have been of a large broad interest in many different fields of science including plasma physics, astrophysics, atomic physics, optics, and time-dependent billiard problems and they are useful for controlling chaos in Engineering and dynamical systems exhibiting chaos (both conservative and dissipative chaos).
We intend to publish in this special issue papers reporting research on time-dependent billiards.The topic includes both conservative and dissipative dynamics.Papers discussing dynamical properties, statistical and mathematical results, stability investigation of the phase space structure, the phenomenon of Fermi acceleration, conditions for having suppression of Fermi acceleration, and computational and numerical methods for exploring these structures and applications are welcome.
To be acceptable for publication in the special issue of Mathematical Problems in Engineering, papers must make significant, original, and correct contributions to one or more of the topics above mentioned.Mathematical papers regarding the topics above are also welcome.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/.Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/according to the following timetable:

Theorem 4 . 1 .
Let M be an orientable extrinsic hypersphere of an Kaehler-Einstein manifold M. If π : M → B is a CR-submersion of M on an almost-Hermitian manifold B, then B is an Kaehler-Einstein manifold.