Chen-Ricci Inequalities with a Quarter Symmetric Connection in Generalized Space Forms

One of the most basic problems in submanifold theory is to develop a simple relationship between the extrinsic invariants and the intrinsic invariants. The sectional curvature, the scalar curvature, and the Ricci curvature are the main intrinsic invariants while the squared mean curvature is the main extrinsic invariant. Chen obtained the following important bound of the Ricci curvature Ric in terms of the mean curvature H for Lagrangian submanifolds in complex space forms [1]:


Introduction
One of the most basic problems in submanifold theory is to develop a simple relationship between the extrinsic invariants and the intrinsic invariants. The sectional curvature, the scalar curvature, and the Ricci curvature are the main intrinsic invariants while the squared mean curvature is the main extrinsic invariant.
Chen obtained the following important bound of the Ricci curvature Ric in terms of the mean curvature H for Lagrangian submanifolds in complex space forms [1]: where c is the constant holomorphic sectional curvature of the complex space form. Further, he discussed the geometry of a Lagrangian submanifold satisfying the equality case of the inequality under the condition that the dimension of the kernel of the second fundamental form is constant. The inequality (1) is known as the Chen-Ricci inequality. This inequality attracted many researchers due to its geometric importance [2][3][4][5][6][7][8][9][10][11][12].
Deng [13] improved the above inequality as In [14], Deng further extended his result for Lagrangian submanifolds in quaternion space forms. In [15], Tripathi improved the inequality in the case of curvature-like tensors. In [6], Mihai and Radulescu obtained the same relation in Sasakian space forms using semisymmetric connection as As the curvature invariants are of great interest in theoretical physics (see [16]), the above studies motivate us to obtain a complete characterization of Lagrangian submanifold in generalized complex space form and a Legendrian submanifold in a generalized Sasakian space form.

Preliminaries
Let N be a Riemannian manifold and ∇ be a linear connection on N. Then, ∇ is said to be a semisymmetric connection if its torsion tensor T satisfies for a 1-form π, then the connection ∇ is called a semisymmetric connection [17]. Let g be a Riemannian metric on N. If ∇g = 0, then ∇ is called a semisymmetric metric connection on N. The semisymmetric metric connection ∇ on N is given by for any U, V on N, where ∇ denotes the Levi-Civita connection with respect to Riemannian metric g and Γ is a vector field. Further, ∇ is said to be a semisymmetric nonmetric connection if it satisfies Moreover, the linear connection ∇ on a Riemannian manifold N with Riemannian metric g is said to be a quarter-symmetric connection if its torsion tensor T is given by which satisfies such that π is a 1-form given by where Γ is a vector field and ϕ is a (1,1) tensor field. Then, we can define a special quarter-symmetric connection by where ψ 1 and ψ 2 are real constants.
The curvature tensor In the same way, we can also define the curvature tensor e R. Let are ð0, 2Þ tensors. Then, the curvature tensor of N is given by Let M be an m-dimensional submanifold in a Riemannian manifold N. Let ∇ and∇ be the induced quarter symmetric-metric connection and Levi-Civita connection, respectively, on M. Then, the Gauss formulas are whereζ is the second fundamental form that satisfies the relation where Γ ⊥ is the normal component of the vector field Γ on M.
Moreover, the equation of Gauss is defined by [19]

Characterization of Lagrangian Submanifold in Generalized Complex Space Form
A smooth manifold N endowed with an almost complex structure J and a Riemannian metric g that is compatible with J is called an almost Hermitian manifold. Further, for 2 Advances in Mathematical Physics the Levi-Civita connection ∇ if ∇J = 0, then an almost Hermitian manifold is said to be a Kaehler manifold. A Kaehler manifold of constant holomorphic curvature is called a complex space form. The curvature tensor of a complex space form is given by However, an almost Hermitian manifold N is called a generalized complex space form [20][21][22], denoted by Nð f 1 , f 2 Þ, if for all vector fields U, V, and Z on N, the Riemann- where f 1 and f 2 are smooth functions on N.
In fact, we have following fundamental result from Tricerri and Vanhecke [20].
Theorem 3 (see [20]). Let N be a connected almost Hermitian manifold with real dimension 2m > 6 and Riemannian curvature e R is of the form (18) such that f 2 is not identically zero. Then, N is a complex space form. From (13) and (18) Lemma 5 (see [13]). Let f 1 ðu 1 , u 2 , ⋯, u m Þ be a function on ℝ m defined by If u 1 + u 2 +⋯+u m = 2ma, then and the equality holds if and only if ð1/ðm + 1ÞÞu 1 where a is a constant.
Lemma 6 (see [13]). Let f 2 ðu 1 , u 2 , ⋯, u m Þ be a function on ℝ m defined by If u 1 + u 2 +⋯+u m = 4a, then and the equality holds if and only if u 1 = a and u 2 +⋯+ Let M m be an m-dimensional submanifold of an almost Hermitian manifold N. Then, M m is said to be totally real if Then, we have the following relations [23]: or equivalently, whereÃ k is the shape operator with respect to ∇ and Remark 7. A totally real submanifold which is of maximal dimension is known as the Lagrangian submanifold [24].
Definition 8 (see [25] for some functions μ and λ with respect to an orthonormal frame fe 1 , ⋯, e m g, where J is the complex structure of N 2m ð4cÞ.

Theorem 9.
Let M m be a totally real submanifold of maximal dimension mðm ≥ 2Þ in a connected complex space form Nð f 1 , f 2 Þ of dimension 2m with a quarter-symmetric metric connection such that the vector field Γ is tangent to M m .
for all s > 1. Therefore, using Lemma 6, we derive ζ 1 1j = ζ j 11 = mH j 4 = 0, for all j > 1, Advances in Mathematical Physics Further, Lemma 5 yields In (33), we see that RicðUÞ = Ricðe 1 Þ. In the same way, by deriving Ricðe 2 Þ and making use of the equality, we conclude that In consequence, we find We see that the equality holds for every unit tangent vectors. The above conclusion is also valid for ðζ s jk Þ. Thus, Then, the only possible nonzero entries for ðζ 2 jk Þ (resp., for ðζ s jk Þ) are On the other hand, if we substitute U = Z = e 2 and V = W = e 1 in (16) Using (46) and (47), we find Moreover, the equality case of (29) implies that Using the fact m ≠ 1, 2, by (48) and (49), it is easy to see that H 1 = 0. This implies that M m is a totally geodesic in N 2m ð4cÞ.
The above theorem gives the following results.

Corollary 10.
Let M m be a totally real submanifold of maximal dimension mðm ≥ 2Þ in a connected complex space form Nð f 1 , f 2 Þ of dimension 2m with a semisymmetric metric connection such that the vector field Γ is tangent to M m . Then, for any unit tangent vector U to M m Proof. Using the fact ψ 1 = ψ 2 = 1 together with Theorem 9, the result directly follows.☐ Remark 11. It is worthy to mention here that Corollary 10 together with Remark 4 is the main result for complex case of the paper [26].

Corollary 12.
Let M m be a totally real submanifold of maximal dimension mðm ≥ 2Þ in a connected complex space form Nð f 1 , f 2 Þ of dimension 2m with a semisymmetric nonmetric connection such that the vector field Γ is tangent to M m . Then, for any unit tangent vector U to M m Proof. Using the fact ψ 1 = 1 and ψ 2 = 0 together with Theorem 9, the result directly follows.☐

Characterization of Legendrian Submanifold in Generalized Sasakian Space Form
Let a ð2m + 1Þ-dimensional almost contact metric manifold N 2m+1 furnished with the almost complex structure ðφ, ξ, η , gÞ, where φ is a (1,1) tensor field, ξ is the structure vector field, η, the 1-form, and g is the Riemannian metric on N 2m+1 . Then, following relations hold good: It also follows from the above relations that for all vector fields U, V on N.
Let ðN, φ, ξ, η, gÞ be an almost contact metric manifold whose curvature tensor satisfies [27]  From (13) and (54), we have A submanifold M m of an almost contact manifold N 2n+1 normal to ξ is called a C-totally real submanifold. On such a submanifold, φ maps any tangent vector to M m at p ∈ M m into the normal space T ⊥ p M m . In particular, if n = m, i.e., M m has maximum dimension, then it is a Legendrian submanifold. For a Legendrian submanifold M m , if fe 1 , ⋯, e m g and fe m+1 = φe 1 , ⋯, e 2m = φe m , e 2m+1 = ξg be tangent orthonormal frame and normal orthonormal frame, respectively, on M m . One has or equivalently, whereÃ k is the shape operator with respect to ∇ and Definition 14 (see [28] for some functions μ and λ with respect to an orthonormal frame fe 1 , ⋯, e m g, where ϕ is the contact structure of N 2m+1 ð4cÞ.  Proof. As Γ is tangent to M m , we have Let us assume an orthonormal basis fe 1 = U, e 2 , ⋯, e m g ⊂ T p M m and fe m+1 = φe 1 , ⋯, e 2m = φe m , e 2m+1 = ξg ⊂ T ⊥ p M m at point p ∈ M m with unit vector U ∈ T p M m . Then, by 6 Advances in Mathematical Physics combining (16) and (55) and substituting U = W = e j and V = Z = e 1 and summing over j = 2, ⋯, m, we compute From (62) and (57), we deduce Putting and by using the fact mH 1 = ζ 1 11 + ζ 1 22 +⋯+ζ 1 mm together with the Lemma 5, we see that Thus, we derive which is the desired inequality (60).☐ Now, we discuss the equality cases.
for all s > 1. Therefore, using Lemma 6, we derive Further, Lemma 5 yields In (63), we see that RicðUÞ = Ricðe 1 Þ. In the same way, by deriving Ricðe 2 Þ and making use of the equality, we conclude that ζ s 2j = ζ 2 js = 0, for all s ≠ 2, j ≠ 2, s ≠ j: In consequence, we find We see that the equality holds for every unit tangent vectors. The above conclusion is also valid for ðζ s jk Þ. Thus, Then, the only possible nonzero entries for ðζ 2 jk Þ (resp., for ðζ s jk Þ) are Substituting U = Z = e 2 and V = W = e j , j = 2, ⋯, m in