Chen optimal inequalities of CR-warped products of generalized Sasakian space form

Our main objective of this paper is to derive the relationship between the main extrinsic invariant, and the contact CR δ-invariant (new intrinsic invariant) on a generic submanifold in trans-Sasakian generalized Sasakian space forms. Further, we find a lower bound of the squared norm of the mean curvature (main extrinsic invariant) in terms of a CR δ-intrinsic invariant, and the Laplacian of the warping function for CR-warped products in the same ambient space forms. We also investigate the classifications and triviality of connected, compact CR-warped product manifolds isometrically immersed into the trans-Sasakian generalized Sasakian space forms.


Introduction and motivations
It is well known that curvature invariants play the most fundamental role in Riemannian geometry. Curvature invariants provide the intrinsic characteristics of Riemannian manifolds which affect the behaviour in general of the Riemannian manifold. They are the main Riemannian invariants and the most natural ones. They are widely used in the field of differential geometry and in physics also. The innovative work of Kaluza-Klein in general relativity and string theory in particle physics has inspired the mathematicians and physicists to do work on submanifolds of (pseudo-)Riemannian manifolds. Intrinsic and extrinsic invariants are very powerful tools to study submanifolds of Riemannian manifolds.
Theorems which relate intrinsic and extrinsic curvatures invariant always play an important role in mathematical analysis and their applications to physical sciences. The Nash theorem was aimed for in the hope that if Riemannian manifolds could be regarded as Riemannian submanifolds, this would then yield the opportunity to use extrinsic help in the study of Riemannian geometry. There were several reasons why it is so difficult to apply Nash's theorem. The main reason for this is the lack of control over the extrinsic properties of the submanifolds by the known classical intrinsic invariants. In order to overcome the difficulties, one needs to establish general optimal relationships between the main extrinsic invariants and the new intrinsic invariants for submanifolds. Such invariants and inequalities have many nice applications in several areas in mathematics.
In the early 1990s, B.-Y. Chen came up with new types of Riemannian invariants, called δ-invariants (or Chen invariants) on Riemannian manifolds. The δ-invariants are not similar to the classical scalar and Ricci curvatures in nature because both of them are the total sum of sectional curvatures on Riemannian manifolds. In contrast, all of the non-trivial δ-invariants are derived from the scalar curvature by removing a definite amount of sectional curvatures. He considered the concept of δinvariants in order to find new necessary conditions for the existence of minimal immersions into a Euclidean space of an arbitrary dimension and to obtain applications of the celebrated Nash embedding theorem. J. A. Schouten, D. V. Dantzig and E. Kähler discovered an important class of the Riemannian manifold which is known as Kähler manifold. A. Kähler manifold is an almost Hermitian manifold B with almost complex structure J if J is parallel with respect to the Levi-Civita connection ∇ of g, that is, ∇J = 0. In [1], Chen defined the CR δ-invariant δ(D) on a CR-submanifold B in a Kähler manifold B as follows: where τ denotes the scalar curvature of B and τ (D) denotes the scalar curvature of the holomorphic distribution D of B. He [1] also established an inequality for anti-holomorphic warped product submanifold B = He showed that the equality sign of (2) holds at x ∈ B in if and only if there exists an orthonormal frame {e 2r+1 , . . . , e n } of D ⊥ x such that the second fundamental form h satisfies Falleh et al. [2,3] proved an optimal inequality for this CR δ-invariant on an anti-holomorphic submanifold B in where H 2 is the partial mean curvature vector of B, rank C (D) = r and rank(D ⊥ ) = k ≥ 2. They also proved that the equality sign holds identically in (3) if and only if the following three conditions are fulfilled: (a) the partial mean curvature vector The concept of Sasakian manifolds was introduced in the 1960s by S. Sasaki as an odd-dimensional analogous to Kähler manifolds. Recently, Mihai et al. [4] studied Chen's CR δ-invariant for an odd-dimensional contact CR-submanifold, called contact CR δ-invariant and obtained an optimal estimate for a (s where dim(D) = 2r + 1 and dim(D ⊥ ) = k. The equality case holds in (4) as discussed in [3]. On the other hand, Kenmotsu [5] studied the third class (that is, the warped product spaces R × f B 2 , where R is a line and B 2 is a Kaehlerian manifold) in Tanno's classification of connected almost contact metric manifolds whose automorphism group has a maximum dimension. He analyzed the properties of R × f B 2 and characterized it by tensor equations. Now-a-days such a manifold is known by Kenmotsu manifold. This is a branch of differential geometry with nice applications in mechanics of dynamical systems with time-dependent Hamiltonian, geometrical optics, thermodynamics and geometric quantization. Also, the study of submanifolds in Kenmotsu ambient spaces is a valuable subject in Kenmotsu geometry, which has been analyzed by many geometers. There is another interesting class of almost contact metric manifolds, namely cosymplectic manifold. This manifold can be locally treated as a product of a Kähler manifold with a circle or a line. Motivated by the above studies, it is interesting to obtain an optimal inequality for generic submanifolds in trans-Sasakian generalized Sasakian space forms involving the contact CR δ-invariant. More precisely, generalized Sasakian space form is included three structures, cosymplectic space form, Kenmotsu space form and Sasakian space form. In this regard, we presented the following results as follows.

Theorem 1.1: Let B be a generic submanifold of a trans-Sasakian generalized Sasakian space form
Moreover, the equality case of inequality (5) The next result follow the idea of a relationship between the squared norm of mean curvature and the contact CR δ-invariant on CR-warped product submanifolds into the generalized Sasakian space form, that is, included the Laplacian of the warping function. Hence, we prove the our result in following form.
The equality sign in (6) holds at x ∈ B in if and only if there exists an orthonormal frame {e 2r+1 , . . . , e n } of D ⊥ x such that the second fundamental form h satisfies Lastly, we provide some applications of the obtained inequality in the view of Nash's embedding theorem.
The paper as follows: In Section 2, we recall some preliminaries formulas and definitions related to our study. In Section 3, we prove a proposition for generic submanifold into generalized Sasakian space and also given the proof of our first main theorem. In Section 3, we give the definition of warped product manifold and presented the proof of second main theorem. In Section 4, we described some geometric applications of derived results by using compactness and connectedness.

Basic formulas and notations
An odd-dimensional smooth manifold B has an almost contact metric structure (φ, ξ , η, g) if there exist on B a tensor field φ of type (1, 1), a structure vector field ξ , a dual 1-form η and a Riemannian metric g such that [6] for any X, Y ∈ (TB). The covariant derivative of the tensor field φ is given by for any X, Y ∈ (TB). Let B be an odd-dimensional contact metric manifold with contact metric structure (φ, ξ , η, g). If the contact metric structure of B is normal, then B is said to have a Sasakian structure and B is known as a Sasakian manifold [6]. It is denoted by (B, φ, ξ , η, g). Then for a Sasakian manifold, we have [6] ( and for any X, Y ∈ (TB). An almost contact metric structure and for any X, Y ∈ (TB), where ∇ represents the Riemannian connection with respect to the g on B.
where d is an exterior differential operator. In this case, (B, φ, ξ , η, g) is said to be cosymplectic manifold.
It is proved that an almost contact metric structure (φ, ξ , η, g) on B is cosymplectic structure if and only if [7] ( and for any X, Y ∈ (TB). An almost contact metric structure (φ, ξ , η, g) on B is called a trnas-Sasakian structure [8] if (B × R, J, G) belongs to the class W 4 of the Gray-Hervella classification of almost Hermitian manifolds [9], where J is the almost complex structure on B × R defined by for any X ∈ (TB) and smooth functionsf on B × R and G is the product metric on B × R. This may be expressed by the following condition [8]: for some smooth functions α and β on B and this trans-Sasakian structure is termed as structure of type (α, β). Thus, a trans-Sasakian of type (0, 0) is cosymplectic, of type (0, β) is β-Kenmotsu and of type (α, 0) is α-Sasakian.

Remark 2.2:
The generalized Sasakian space form extends the concept of Sasakian, Kenmotsu and cosymplectic space forms.
Let us denotes the Lie algebra of vector fields in B is (TB). Then for a differentiable function f on a Riemannian manifold (B, g) of dimension s, the gradient of f, ∇f , is defined by for any X ∈ (TB). Here ∇ be the Levi-Civita connection on B. As a consequence, the squared norm of warping function f is defined as we have for a local orthonormal frame {e 1 , . . . , e s } on B. Also, the Laplacian f of f is given by Likewise, the Hessian Hess f of f is defined as Hess f (e i , e i ) = −trace(Hess f ).
The theory of submanifolds as follows: Let a Riemannian submanifold B be isometrically immersed in an odd-dimensional almost contact metric manifold (B, φ, ξ , η, g) with induced metric g. The Lie algebra of vector fields in B and the set of all vector fields normal to B are, respectively, given by (TB) and (T ⊥ B). Let ∇ be the induced connection on B. Then the Gauss and Weingarten formulas are, respectively, given below [11]: for any X, Y ∈ (TB) and N ∈ (T ⊥ B). Here h and A are the bilinear symmetric second fundamental form of B in B and the shape operator of B, respectively. Both of them are related as [11] g for any X, Y ∈ (TB) and N ∈ (T ⊥ B). We denote the Riemannian curvature tensor fields of B and B by R and R, respectively. Then the Gauss equation is given by [11] R(X, Y, Z, for any X, Y, Z, W ∈ (TB). We assume that dim For any X ∈ (TB) and N ∈ (T ⊥ B), respectively, we put [11] φX = PX + FX and φN = QN + CN, where PX and FX are the tangential and the normal components of φX, respectively. Similarly, QN and CN are the tangential and the normal components of φN, respectively. For their geometric relations, see [11].
A submanifold B of an almost contact metric manifold (B, φ, ξ , η, g) is said to be a contact CR-submanifold [11] of B if there exists a differentiable distribution D : x −→ D x ⊂ T x B such that D is invariant with respect to φ and the orthogonal complementary distribution D ⊥ is anti-invariant with respect to φ. The tangent bundle TB has the orthogonal decomposition TB = D ⊕ D ⊥ ⊕ {ξ }, where {ξ } is a 1-dimensional distribution which is spanned by ξ . A submanifold B of B is said to be invariant [11] if F ≡ 0, that is, φX ∈ (TB), and anti-invariant [11] if P ≡ 0, that is, φX ∈ (T ⊥ B), for any X ∈ (TB).
A submanifold B of an almost contact metric manifold (B, φ, ξ , η, g) is called a generic submanifold [12] is a hypersurface of B, then B is obviously a generic submanifold. A submanifold B of B has following classification according to [11] as.

Some consequences
By using Remark 2.2 (a), we summarize the immediate consequences of Theorem 1.1, given below:

Corollary 3.1: Let B be a generic submanifold of Sasakian space form. Then CR δ-invariant satisfying the inequality
Similarly, from (c) of Remark 2.2, we find that Corollary 3.3: Assume that B is a generic submanifold of cosymplectic space forms. Thus Moreover, the equality case of the above inequalities holds if and only if (a), (b) and (c) of Theorem 1.1 are fulfilled.

Warped product manifolds and CR-warped product submanifolds
The concept of warped product manifolds has many applications in physics. For instance, different models of space-time in general relativity are expressed in terms of warped geometry, and the Einstein field equations and modified field equations have many exact solutions as the warped products. In 1969, the idea of warped product manifolds had been initiated by Bishop and O'Neil [13] with manifolds of negative curvature. These manifolds are the most fruitful and natural generalization of Riemannian product manifolds. Several nice results are available in the literature (see [14] and the references therein).

Definition 4.1 ([13]):
Let (B 1 , g 1 ) and (B 2 , g 2 ) be two (pseudo)-Riemannian manifolds and f > 0 be a differentiable function on B 1 . Consider the product π 1 : B 1 × B 2 −→ B 1 and π 2 : B 1 × B 2 −→ B 2 . Then the warped product B = B 1 × f B 2 is the product manifold B 1 × B 2 equipped with the Riemannian structure g such that for any X, Y ∈ (T (u,v) B), u ∈ B 1 and v ∈ B 2 , where * is the symbol for the tangent maps. The function f is called the warping function of the warped product.
A warped product manifold is said to be trivial if its warping function is constant. In this case, the warped product manifold is a Riemannian product manifold. For the trivial warped product manifold B = B 1 × f B 2 , submanifolds B 1 and B 2 are totally geodesic and totally umbilical of B, respectively. Let dim(B) = s, dim(B 1 ) = k, and dim(B 2 ) = r. For unit vectors X 1 and X 2 tangent to B 1 and B 2 , the sectional curvature K(X 1 ∧ X 2 ) of the plane section spanned by X 1 and X 2 is for each j = r + 1, . . . , s. From [15], a warped product submanifold B = B 1 × f B 2 of a Kenmotsu (or cosymplectic) manifold (B, φ, ξ , η, g), where B 1 is a (2r + 1)dimensional invariant submanifold tangent to ξ and B 2 is a k-dimensional anti-invariant submanifold of B, is said to be a contact CR-warped product submanifold of B.

Lower bounds for CR-warped product submanifolds
Chen [16,17] introduced the notion of a CR-warped product manifold and studied CR-warped products in Kaehler manifold. Later, Hasegawa and Mihai [15] studied contact CR-warped products in Sasakian manifold. We prepare some lemmas for later use.
for each p = 2r + 1, . . . , 2r + k. Motivated by above result, we give the following proposition which has an important role to prove our main theorem.
where f is the Laplacian of the warping function f.
Proof: By similar arguments as in Proposition 3.1, we arrive at Since, φD ⊥ = T ⊥ B, then from (24) Hence, our assertion is derived. The proof of proposition is completed.

Conclusion remarks
The δ-invariants are very different in nature from the classical scalar and Ricci curvatures; simply due to the fact that both scalar and Ricci curvatures are total sum of sectional curvatures on a Riemannian manifold. In contrast, almost all of the δ-invariants are obtained from the scalar curvature by throwing away certain amount of sectional curvatures. Curvatures invariants also play key roles in physics. For instance, the magnitude of a force required to move an object at a constant speed, according to Newton's laws, a constant multiple of the curvature of the trajectory. The motion of a body in a gravitational field is determined, according to Einstein, by the curvatures of space time. Classically, among the Riemannian curvature invariants, people have been studying sectional, scalar and Ricci curvatures in great details (see [19] and references therein). In the present paper, we established a general optimal relationships between extrinsic invariant with the new intrinsic invariants on the submanifolds. Also some of previous results are generalized from this study. As we know generalized Sasakian space form is a class in almost contact metric manifold that generalized three classes, that is, cosymplectic space form, Kenmotsu space form and Sasakian space form see [20][21][22][23][24][25]. This is the main reason to study CR-warped product submanifold into the generalized Sasakian space form. All the above argument shows that the paper contains interesting applications in mathematical analysis and mathematical physics as well.

Disclosure statement
No potential conflict of interest was reported by the author(s).