A New Algebraic Inequality and Some Applications in Submanifold Theory

: We give a simple proof of the Chen inequality involving the Chen invariant δ ( k ) of submanifolds in Riemannian space forms. We derive Chen’s ﬁrst inequality and the Chen–Ricci inequality. Additionally, we establish a corresponding inequality for statistical submanifolds.


Introduction
One of the most important topics of research in the geometry of submanifolds in Riemanian manifolds is to establish sharp relationships between extrinsic and intrinsic invariants of a submanifold.
The most used intrinsic invariants are sectional curvature, scalar curvature and Ricci curvature. The main extrinsic invariant is the squared mean curvature.
There are well-known relationships between the above extrinsic and intrinsic invariants for a submanifold in a Riemannian space form: (generalized) Euler inequality, Chen-Ricci inequality, Wintgen inequality, etc.
In [1,2], B.-Y. Chen introduced a sequence of Riemannian invariants, which are known as Chen invariants. They are different in nature from the classical Riemannian invariants. B.-Y. Chen established optimal relationships between the squared mean curvature and Chen invariants for submanifolds in Riemannian space forms, known as Chen inequalities (see [2]). The proofs of these inequalities use an algebraic inequality, discovered by B.-Y. Chen in [1].
In the present paper, we give simple proofs of some Chen inequalities by using a different algebraic inequality.
Other Chen inequalities were proved in [3] by applying another inequality.
Let {e 1 , ..., e n } be an orthonormal basis of T p M. The scalar curvature τ at p is given by where K(e i ∧ e j ) is the sectional curvature of the plane section spanned by e i and e j . If X is a unit vector tangential to M at p, consider the orthonormal basis {e 1 = X, e 2 , ..., e n } of T p M. The Ricci curvature is defined by Let L be an r-dimensional subspace of T p M and {e 1 , ..., e r } an orthonormal basis of L, 2 ≤ r ≤ n. The the scalar curvature τ(L) of L is given by τ(L) = ∑ 1≤α<β≤r K(e α ∧ e β ).
In particular, for r = 2, τ(L) is the sectional curvature of L and for r = n, τ(T p M) = τ(p) is the scalar curvature of M at p.
We shall consider the Chen invariant δ(k), which is given by where L k is any k-dimensional subspace of T p M.

An Algebraic Inequality
In this section, we give an algebraic inequality and study its equality case. As an application, we get a simple proof of the Chen inequality for the invariant δ(k). Lemma 1. Let k, n be nonzero natural numbers, 2 ≤ k ≤ n − 1, and a 1 , a 2 , ..., a n ∈ R. Then Moreover, the equality holds if and only if ∑ k α=1 a α = a j , for all j ∈ {k + 1, ..., n}.
Proof. We prove this Lemma by using the Cauchy-Schwarz inequality. We have which implies the desired inequality. The equality holds if and only if we have equality in the Cauchy-Schwarz inequality, i.e., ∑ k α=1 a α = a j , for all j ∈ {k + 1, ..., n}.

Proof of the Chen Inequality for δ(K)
We apply Lemma 1 for obtaining a simple proof of the Chen inequality corresponding to the Chen invariant δ(k) for submanifolds in Riemannian space forms.
LetM(c) be an m-dimensional Riemannian space form of constant sectional curvature c. The Euclidean space E m , the sphere S m and the hyperbolic space H m are the standard examples.
Consider M an n-dimensional submanifold ofM(c) and denote by h the second fundamental form of M inM(c). The mean curvature vector H(p) at p ∈ M is defined by where {e 1 , ..., e n } is an orthonormal basis of T p M.
The submanifold M is called minimal if the mean curvature vector H(p) vanishes at any p ∈ M.
We recall the Gauss equation (see [4]): for all vector fields X, Y, Z, W tangential to M. Theorem 1. LetM(c) be an m-dimensional Riemannian space form of constant sectional curvature c and M an n-dimensional submanifold ofM(c). Then, for any 2 ≤ k ≤ n − 1, one has the following Chen inequality: Moreover, the equality holds at a point p ∈ M if and only if there exist suitable orthonormal bases {e 1 , ..., e n } ⊂ T p M and {e n+1 , ..., e m } ⊂ T ⊥ p M such that the shape operators take the forms Proof. Let p ∈ M, L ⊂ T p M be a k-dimensional subspace and {e 1 , ..., e k } be an orthonormal basis of L. We take {e 1 , ..., e k , e k+1 , ..., e n } ⊂ T p M and {e n+1 , ..., e m } ⊂ T ⊥ p M as orthonormal bases, respectively.
The Gauss equation implies Additionally, by the Gauss equation one has Then we get By using the algebraic inequality from the previous section, we obtain which implies the inequality to prove. If the equality case holds at a point p ∈ M, then we have equalities in all the inequalities in the proof, i.e., for any r ∈ {n + 1, ..., m}.
If we choose e n+1 parallel to H(p), then the shape operators take the above forms.
If k = 1, we derive Chen's first inequality: [1] LetM(c) be an m-dimensional Riemannian space form of constant sectional curvature c and M an n-dimensional submanifold ofM(c). Then one has Equality holds at a point p ∈ M if and only if, with respect to suitable orthonormal bases {e 1 , ..., e n } ⊂ T p M and {e n+1 , ..., e m } ⊂ T ⊥ p M, the shape operators take the following forms: Recall that δ(n − 1) = max Ric. Then, from Theorem 1 we deduce the Chen-Ricci inequality: It is known that T is a minimal hypersurface of S n+1 , but a non-minimal submanifold of E n+2 .

A Chen Inequality for Statistical Submanifolds
A statistical manifold is an m-dimensional Riemannian manifold (M, g) endowed with a pair of torsion-free affine connections∇ and∇ * , which satisfy for any X, Y, Z ∈ Γ(TM). The connections∇ and∇ * are called dual connections (see [6,7]), and it is easily seen that ∇ * * =∇. The pairing (∇, g) is said to be a statistical structure. If ∇ , g is a statistical structure onM m , then ∇ * , g is a statistical structure too [6,8].
Any torsion-free affine connection∇ onM always has a dual connection given bỹ where∇ 0 is the Levi-Civita connection onM. The dual connections are called conjugate connections in affine differential geometry (see [9]).
Denote byR andR * the curvature tensor fields of∇ and∇ * , respectively. They satisfy A statistical structure ∇ , g is said to be of constant curvature ε ∈ R if A statistical structure (∇, g) of constant curvature 0 is called a Hessian structure. The Equation (2) implies that if ∇ , g is a statistical structure of constant curvature ε, then ∇ * , g is also a statistical structure of constant curvature ε (obviously, if (∇, g) is Hessian, (∇ * , g) is also Hessian).
The dual connections are not metric, then we cannot define a sectional curvature in the standard way. A sectional curvature on a statistical manifold was defined by B. Opozda [10].
More precisely, if one considers p ∈M, π a plane section in T pM and an orthonormal basis {X, Y} of π, then a sectional curvature is defined bỹ which is independent of the choice of the orthonormal basis. Next, we consider a statistical manifold (M, g) and a submanifold M of dimension n ofM. Then (M, g| M ) is also a statistical manifold with the connection induced by∇ and induced metric g.
In Riemannian geometry, the fundamental equations are the Gauss and Weingarten formulae and the equations of Gauss, Codazzi and Ricci.
As usual, we denote by Γ T ⊥ M the set of the sections of the bundle normal to M. In our case, for any X, Y ∈ Γ(TM), according to [8], the corresponding Gauss formulae are∇ are symmetric and bilinear, called the imbedding curvature tensor (see [6,8]) of M inM for∇ and the imbedding curvature tensor of M inM for∇ * , respectively.
In [8], it was also proven that (∇, g) and (∇ * , g) are dual statistical structures on M.
Since h and h * are bilinear, there are linear transformations A ξ and A * ξ on TM defined by for any ξ ∈ Γ T ⊥ M and X, Y ∈ Γ(TM). Further (see [8]), the corresponding Weingarten formulae arẽ for any ξ ∈ Γ T ⊥ M and X ∈ Γ(TM). The connections ∇ ⊥ and ∇ * ⊥ are Riemannian dual connections with respect to the induced metric on Γ T ⊥ M . Let {e 1 , ..., e n } and {e n+1 , ..., e m } be orthonormal tangential and normal frames, respectively, on M. Then the mean curvature vector fields are defined by and for 1 ≤ i, j ≤ n and n + 1 ≤ α ≤ m. The Gauss equations for the dual connections∇ and∇ * , respectively, are given by (see [8]) Geometric inequalities for statistical submanifolds in statistical manifolds with constant curvature were obtained in [11].
In this section we prove the Chen inequality corresponding to the Chen invariant δ(k) for statistical submanifolds in statistical manifolds of constant curvature.
We consider an m-dimensional statistical manifoldM(ε) of constant curvature ε and an n-dimensional statistical submanifold M. Let p ∈ M and L be a k-dimensional subspace of T p M. Denote by {e 1 , ..., e k } an orthonormal basis of L, {e 1 , ..., e k , e k+1 , ..., e n } an orthonormal basis of T p M and {e n+1 , ..., e m } an orthonormal basis of T ⊥ p M, respectively.
The Gauss equation implies where h 0 is the second fundamental form of the Riemannian submanifold M. We denote by τ 0 the scalar curvature with respect to the Levi-Civita connection and byτ 0 = ∑ 1≤i<j≤nK0 (e i ∧ e j ).
We state the following result.
Theorem 2. Let M be an n-dimensional statistical submanifold of an m-dimensional statistical manifoldM(ε) of constant curvature. Then, for any p ∈ M and any k-plane section L of T p M, we have: ).