RICCI CURVATURE OF SUBMANIFOLDS IN KENMOTSU SPACE FORMS

In 1999, Chen established a sharp relationship between the Ricci curvature and the squared mean curvature for a submanifold in a Riemannian space form with arbitrary codimension. Similar problems for submanifolds in complex space forms were studied by Matsumoto et al. In this paper, we obtain sharp relationships between the Ricci curvature and the squared mean curvature for submanifolds in Kenmotsu space forms.

1. Preliminaries.Let ( M, <, >) be a Hermitian manifold and denote by J the canonical almost complex structure on M. According to the behavior of the tangent bundle T M with respect to the action of J, we may distinguish two special classes of submanifolds M in M: (a) complex submanifolds, that is, J(T p M) = T p M, for all p ∈ M. (b) totally real submanifolds, that is, J(T p M) ⊂ T ⊥ p M, for all p ∈ M, where T p M (resp., T ⊥ p M) is the tangent (resp., the normal) vector space of M at p.Such submanifolds were defined and studied by Chen and Ogiue [4].
On the other hand, Yano and Ishihara [8] considered a submanifold M whose tangent bundle T M splits into a complex subbundle Ᏸ and a totally real subbundle Ᏸ ⊥ .Later, such a submanifold was called a CR-submanifold [1,2].Blair and Chen [1] proved that a CR-submanifold of a locally conformal Kaehler manifold is a Cauchy-Riemann manifold in the sense of Greenfield.
The first main result on CR-submanifolds was obtained by Chen [2]: any CRsubmanifold of a Kaehler manifold is foliated by totally real submanifolds (i.e., the totally real subbundle is involutive).
As nontrivial examples of CR-submanifolds, we can mention the (real) hypersurfaces of Hermitian manifolds.
2. Kenmotsu manifolds and their submanifolds.Tanno [7] has classified, into three classes, the connected almost contact Riemannian manifolds whose automorphisms groups have the maximum dimensions: (1) homogeneous normal contact Riemannian manifolds with constant φholomorphic sectional curvature; (2) global Riemannian products of a line or circle and a Kaehlerian space form; (3) warped product spaces L × f F , where L is a line and F a Kaehlerian manifold.
Kenmotsu [5] studied the third class and characterized it by tensor equations.Later, such a manifold was called a Kenmotsu manifold.
A (2m+1)-dimensional Riemannian manifold ( M, g) is said to be a Kenmotsu manifold if it admits an endomorphism φ of its tangent bundle T M, a vector field ξ, and a 1-form η, which satisfy: for any vector fields X, Y on M, where ∇ denotes the Riemannian connection with respect to g.
We denote by ω the fundamental 2-form of M, that is, ω(X, Y ) = g(φX, Y ), for all X, Y ∈ Γ (T M).It was proved that the pairing (ω, η) defines a locally conformal cosymplectic structure, that is, A Kenmotsu manifold with constant φ-holomorphic sectional curvature c is called a Kenmotsu space form and it is denoted by M(c).Then its curvature tensor R is expressed by (cf.[5]) Let M be a Kenmotsu manifold and M an n-dimensional submanifold tangent to ξ.For any vector field X tangent to M, we put where P X (resp., FX) denotes the tangential (resp., normal) component of φX.Then P is an endomorphism of tangent bundle T M and F is a normal bundle valued 1-form on T M. The equation of Gauss is given by for any vectors X, Y , Z, W tangent to M. We denote by H the mean curvature vector, that is, where {e 1 ,...,e n } is an orthonormal basis of the tangent space T p M, p ∈ M.
Also, we set h r ij = g h e i ,e j ,e r , g h e i ,e j ,h e i ,e j . (2.7) Let {e 1 ,...,e n } be an orthonormal basis of T p M. We put g 2 P e i ,e j . (2.8) By analogy with submanifolds in a Kaehler manifold, different classes of submanifolds in a Kenmotsu manifold were considered (cf.[6]).
A submanifold M tangent to ξ is said to be invariant (resp., anti-invariant [9] if there exists a pair of orthogonal differentiable distributions Ᏸ and Next, recall some notions introduced by Chen (see [3]).
Let L be a k-plane section of T p M and X a unit vector in L. We choose an orthonormal basis {e 1 ,...,e k } of L such that e 1 = X.
Define the Ricci curvature Ric L of L at X by where K ij denotes the sectional curvature of the 2-plane section spanned by e i , e j .We simply called such a curvature a k-Ricci curvature.
The scalar curvature τ of the k-plane section L is given by where L runs over all k-plane sections in T p M and X runs over all unit vectors in L.
Recall that for a submanifold M in a Riemannian manifold, the relative null space of M at a point p ∈ M is defined by (2.12) 3. Ricci curvature and squared mean curvature.Chen established a sharp relationship between the Ricci curvature and the squared mean curvature for submanifolds in real space forms (see [3]).
We prove similar inequalities for certain submanifolds of a Kenmotsu space form M(c). We will consider submanifolds M tangent to the Reeb vector field ξ.Theorem 3.1.Let M(c) be a (2m+1)-dimensional Kenmotsu space form and M an n-dimensional submanifold tangent to ξ.Then (i) for each unit vector X ∈ T p M orthogonal to ξ, Proof.Let X ∈ T p M be a unit tangent vector X at p. We choose an orthonormal basis e 1 ,...,e n = ξ, e n+1 ,...,e 2m+1 in T p M(c) such that e 1 ,...,e n are tangent to M at p, with e 1 = X.
Then, from the equation of Gauss, we have From (3.2), we get From the equation of Gauss, we find 2≤i<j≤n which is equivalent to (3.1).For (ii) assume that H(p) = 0. Equality holds in (3.1) if and only if Then h r 1j = 0, for all j ∈ {1,...,n}, r ∈ {n + 1,...,2m}, that is, X ∈ ᏺ p .For (iii) the equality case of (3.1) holds for all unit tangent vectors at p if and only if It follows that p is a totally geodesic point.The converse is trivial.
Corollary 3.2.Let M be an n-dimensional invariant submanifold tangent to ξ in a Kenmotsu space form M(c).Then, (i) for each unit vector X ∈ T p M orthogonal to ξ, we have (ii) a unit tangent vector X ∈ T p M orthogonal to ξ satisfies the equality case of (3.8) if and only if X ∈ ᏺ p ; (iii) the equality case of (3.8) holds identically for all unit tangent vectors orthogonal to ξ at p if and only if p is a totally geodesic point.
Proof.It is known that every invariant submanifold of a Kenmotsu space form is minimal (cf.[6]).
On the other hand, for any unit tangent vector X ∈ T p M orthogonal to ξ, we have Then, the inequality (3.1) implies (3.8).
Similarly, we can prove the following results.
Corollary 3.3.Let M be an n-dimensional anti-invariant submanifold tangent to ξ in a Kenmotsu space form M(c).Then, (i) for each unit vector X ∈ T p M orthogonal to ξ, we have (3.9) (ii) if H(p) = 0, then a unit tangent vector X ∈ T p M orthogonal to ξ satisfies the equality case of (3.9) if and only if X ∈ ᏺ p ; (iii) the equality case of (3.9) holds identically for all unit tangent vectors orthogonal to ξ at p if and only if p is a totally geodesic point.
Corollary 3.4.Let M be an n-dimensional contact CR-submanifold of a Kenmotsu space form M(c). Then (i) for each unit vector X ∈ Ᏸ p , In this section, we prove a relationship between the k-Ricci curvature and the squared mean curvature for submanifolds tangent to ξ in a Kenmotsu space form.Theorem 4.1.Let M(c) be a Kenmotsu space form and M an n-dimensional submanifold tangent to ξ.Then we have Proof.We choose an orthonormal basis {e 1 ,...,e n ,e n+1 ,...,e 2m+1 = ξ} at p such that e n+1 is parallel to the mean curvature vector H(p), and e 1 ,...,e n diagonalize the shape operator A n+1 .Then the shape operators take the forms From (3.2), we get On the other hand, since 0 we obtain which implies that We have from (4.3) which is equivalent to (4.1).
Using Theorem 4.1, we obtain the following.
Theorem 4.2.Let M(c) be a Kenmotsu space form and M an n-dimensional submanifold tangent to ξ.Then, for any integer k, 2 ≤ k ≤ n, and any point p ∈ M, we have Proof.Let {e 1 ,...,e n } be an orthonormal basis of T p M. Denote by L i 1 •••i k the kplane section spanned by e i 1 ,...,e i k .It follows from (2.9) and (2.10) that (4.9) Combining (2.11) and (4.9), we find In particular, we obtain the following.

Mathematical Problems in Engineering
Special Issue on Time-Dependent Billiards

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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: .10) From (4.1) and (4.10), we obtain (4.8).