Chen’s inequalities for submanifolds in (κ, μ)-contact space form with a semi-symmetric metric connection

Lemma 2.1

([16]). If U is a tangent vector field on Mⁿ, then H = H^′and σ = σ^′.

Let π ⊂ T_pMⁿ be a 2-plane section for any p ∈ Mⁿ and K(π) be the sectional curvature of Mⁿ associated to the semi-symmetric metric connection ▽. The scalar curvature τ associated to the semi-symmetric metric connection ▽ at p is defined by

Let L_k be a k-plane section of T_pMⁿ and {e₁, e₂,…, e_k} be any orthonormal basis of L_k. The scalar curvature τ(L_k) of L_k associated to the semi-symmetric metric connection ▽ is given by

We denote by (inf K)(p) = inf{K(π)∣π ⊂ T_pMⁿ, dimπ = 2}. In [1] B.-Y. Chen introduced the first Chen invariant δ_M(p) = τ(p) − (inf K)(p), which is certainly an intrinsic character of Mⁿ.

Suppose L is a k-plane section of T_pM and X is a unit vector in L, we choose an orthonormal basis {e₁, e₂,…, e_k} of L, such that e₁ = X. The Ricci curvature Ric_p of L at X is given by

where K_ij = K(e_i ∧ e_j). We call Ric_L(X) a k-Ricci curvature.

For each integer k, 2 ⩽ k ⩽ n, the Riemannian invariant Θ_k [2] on Mⁿ is defined by

where L is a k-plane section in T_pMⁿ and X is a unit vector in L.

Recently, T. Koufogiorgos [17] introduced the notion of (κ, μ)-contact space form, which contains the well known class of Sasakian space forms for κ = 1. Thus it is worthwhile to study relationships between intrinsic and extrinsic invariants of submanifolds in a (κ, μ) -contact space form with a semi-symmetric metric connection ▽.

A (2m+1)-dimensional differentiable manifold M̂ is called an almost contact metric manifold if there is an almost contact metric structure (φ, ξ, η, g) consisting of a (1, 1) tensor field φ, a vector field ξ, a 1-form η and a compatible Riemannian metric g satisfying

An almost contact metric structure becomes a contact metric structure if dη = Φ, where Φ(X, Y) = g(X, φY) is the fundamental 2-form of M̂.

In a contact metric manifold M̂, the (1, 1)-tensor field h defined by 2h = 𝓛_ξφ is symmetric and satisfies

where ▽^′ is a Levi-civita connection associated with the Riemannian metric g.

The (κ, μ)-nullity distribution of a contact metric manifold M̂ is a distribution

where κ and μ are constants. If ξ∈N(κ, μ), M̂ is called a (κ, μ)-contact metric manifold. Since in a (κ, μ)-contact metric manifold one has h² = (κ − 1)φ², therefore κ ⩽ 1 and if κ = 1 then the structure is Sasakian.

The sectional curvature K̂(X, φX) of a plane section spanned by a unit vector orthogonal to ξ is called a φ-sectional curvature. If the (κ, μ)-contact metric manifold M̂ has constant φ-sectional curvature C, then it is called a (κ, μ)-contact space form and it is denoted by M̂(C). The curvature tensor of M̂(C) is given by [17].

for all X, Y, Z ∈ 𝓧(M̂), where c+2κ = −1 = κ − μ if κ < 1.

For a vector field X on a submanifold M of a (κ, μ)-contact form M̂(c), let PX be the tangential part of φX. Thus, P is an endomorphism of the tangent bundle of M and satisfies g(X, PY) = −g(PX, Y) for all X, Y ∈ 𝓧(M). (φh)^TX and h^TX are the tangential parts of φhX and hX respectively. Let {e₁, e₂, …, e_n} be an orthonormal basis of T_pM. We set

Let π ⊂ T_pM be a 2-plane section spanned by an orthonormal basis {e₁, e₂}. Then, β(π) is given by

is a real number in [0, 1], which is independent of the choice of orthonormal basis {e₁, e₂}. Put

Then γ(π) and θ(π) are also real numbers and do not depend on the choice of orthonormal basis {e₁, e₂}, of course, γ(π) ∈ [0, 1]

Lemma 3.1

([20]). Let f(x₁, x₂, …, x_n) (n ≥ 3) be a function in Rⁿ defined by

If x₁+x₂+⋯+x_n = (n − 1)ε, then we have

with the equality holding if and only if x₁+x₂ = x₃ = ⋯ = x_n = ε.

Theorem 3.2

Let M ba an n-dimensional (n ≥ 3) submanifold of a (2m+1)-dimensional (κ, μ)-contact form M̂(c) endowed with a semi-symmetric metric connection▽such that ξ ∈ TM. For each 2-plane section π ⊂ T_pM, we have

If U is a tangent vector field to M, then the equality in(9)holds at p ∈ M if and only if there exists an orthonormal basis {e₁, e₂, …, e_n} of T_pM and an orthonormal basis {e_n+1, …, e_2m+1} ofTp⊥Msuch that

and the forms of shape operators A_r ≡ A_er (r = n+1,…, 2m+1) become

Proof

Let π ⊂ T_pM be a 2-plane section. We choose an orthonormal basis {e₁, e₂,…, e_n} for T_pM and {e_n+1,…, e_2m+1} for Tp⊥M such that π = Span{e₁, e₂}.

Setting X = W = e_i, Y = Z = e_j, i ≠ j, i, j = 1,…, n, and using (1), (2), (8), we have

From (10) we get

On the other hand, using (10) we have

From (11) and (12) it follows that

Let us consider the following problem:

where k^r is a real constant. From Lemma 3.1, we know

with the equality holding if and only if

From (13) and (14), we have

If the equality in (9) holds, then the inequalities given by (13) and (14) become equalities. In this case we have

Since U is a tangent vector field to M, we know σ^′ = σ. By choosing a suitable orthonormal basis, the shape operators take the desired forms.

The converse is easy to show. □

For a Sasakian space form M̂(c), we have κ = 1 and h = 0. So using Theorem 3.2, we have the following corollary.

Corollary 3.3

Let M be an n-dimensional (n ≥ 3) submanifold in a Sasakian space form M̂(c) endowed with a semi-symmetric metric connection such that ξ ∈ TM. Then, for each point p ∈ M and each plane section π ⊂ T_pM, we have

If U is a tangent vector field to M, then the equality in(16)holds at p ∈ M if and only if there exists an orthonormal basis {e₁, e₂, …, e_n} of T_pM and orthonormal basis {e_n+1, …, e_2m+1} ofTp⊥Msuch that

and the forms of shape operators A_r ≡ A_er (r = n+1,…, 2m+1) become

Since in case of non-Sasakian (κ, μ)-contact space form, we have κ < 1, thus c = −2κ − 1 and μ = κ+1. Putting these values in (9), we can have a direct corollary to Theorem 3.2.

Corollary 3.4

Let Let M be an n-dimensional (n ≥ 3) submanifold in a non-Sasakian (κ, μ)-contact space form M̂(c) with a semi-symmetric metric connection such that ξ ∈ TM. Then, for each point p ∈ M and each plane section π ⊂ T_pM, we have

If U is a tangent vector field to M, then the equality in(17)holds at p ∈ M if and only if there exists an orthonormal basis {e₁, e₂, …, e_n} of T_pM and orthonormal basis {e_n+1,…, e_2m+1} ofTp⊥Msuch that

and the forms of shape operators A_r ≡ A_er(r = n+1,…, 2m+1) become

Lemma 4.1

([20]). Let f(x₁, x₂, …, x_n) be a function in Rⁿ defined by

If x₁+x₂+ …+x_n = 2ε, then we have

with the equality holding if and only if x₁ = x₂+ … +x_n = ε.

Theorem 4.2

Let M be an n-dimensional (n ⩾ 2) submanifold of a (2m+1)-dimensional (κ, μ)-contact space form M̂(c) endowed with a semi-symmetric metric connection such that ξ ∈ TM. Then for each point p ∈ M,

1. For each unit vector X in T_pM, we have

   Ric(X)⩽n24∥H∥2+(n−1)(c+3)4+3(c−1)4∥PX∥2−c+3−4κ4[1+(n−2)η(X)2]+12[∥(φhX)T∥2−∥(hX)T∥2−g(φhX,X)trace((φh)T)+g(hX,X)trace(hT)]+(μ+n−3)g(hX,X)+[1+(μ−1)η(X)2]trace(hT)−(n−2)α(X,X)−λ.(18)

2. If H(p) = 0, a unit tangent vector X ∈ T_pM satisfies the equality case of(18)if and only if X ∈ N(p) = {X ∈ {T_pM|σ(X, Y) = 0, ∀Y ∈ T_pM}.

3. The equality of(18)holds identically for all unit tangent vectors if and only if either

   1. n ≠ 2, σijr = 0, i, j = 1, 2, …, n; r = n+1, …, 2m+1;

      or

   2. n = 2, σ11r=σ22r,σ12r=0,r=3,...,2m+1..

Proof

1. Let X ∈ T_pM be an unit vector. We choose an orthonormal basis e₁, …, e_n, e_n+1 …, e_2m+1 such that e₁, …, e_n are tangential to M at p with e₁ = X.

   Using (10), we have

   Ric(X)=(n−1)(c+3)4−c+3−4κ4[1+(n−2)η(X)2]+3(c−1)4∥PX∥2+12[∥(φhX)T∥2−∥(hX)T∥2+g(hX,X)trace(hT)−g(φhX,X)trace((φh)T)]+(μ+n−3)g(X,hX)+(1−η(X)2+μη(X)2)trace(hT)−(n−2)α(X,X)−λ+∑r=n+12m+1∑i=2n[σ11rσiir−(σ1ir)2]⩽(n−1)(c+3)4+3(c−1)4∥PX∥2−c+3−4κ4[1+(n−2)η(X)2]+12[∥(φhX)T∥2−∥(hX)T∥2−g(φhX,X)trace(φh)T+g(hX,X)trace(hT)]+(μ+n−3)g(hX,X)+[1−η(X)2+μη(X)2]trace(hT)−(n−2)α(X,X)−λ+∑r=n+12m+1∑i=2nσ11rσiir.(19)

   Let us consider the function f_r: Rⁿ → R, defined by

   fr(σ11r,σ22r,...,σnnr)=∑i=2nσ11rσiir.

   We consider the problem

   max{fr|σ11r+...+σnnr=kr},

   where k^r is a real constant. From Lemma 4.1, we have

   fr⩽(kr)24,(20)

   with equality holding if and only if

   σ11r=∑i=2nσiir=kr2.(21)

   From (19) and (20) we get

   Ric(X)⩽n24∥H∥2+(n−1)(c+3)4+3(c−1)4∥PX∥2−c+3−4κ4[1+(n−2)η(X)2]+12[∥(φhX)T∥2−∥(hX)T∥2−g(φhX,X)trace(φh)T+g(hX,X)trace(hT)]+(μ+n−3)g(hX,X)+[1+(μ−1)η(X)2]trace(hT)−(n−2)α(X,X)−λ.

2. For a unit vector X ∈ T_pM, if the equality case of (18) holds, from (19), (20) and (21) we have

   σ1ir=0,i≠1,∀r.σ11r+σ22r+...+σnnr=2σ11r,∀r.

   Since H(p) = 0, we know

   σ11r=0,∀r.

   So we get

   σ1jr=0,∀r.

   i.e X ∈ N(p)

   The converse is trivial.

3. For all unit vectors X ∈ T_pM, the equality case of (18) holds if and only if

   2σiir=σ11r+...+σnnr,i=1,...,n;r=n+1,...,2m+1.σijr=0,i≠j,r=n+1,...,2m+1.

   Thus we have two cases, namely either n ≠ 2 or n = 2.

   In the first case we have

   σijr=0,i.j=1,...,n;r=n+1,...,2m+1.

   In the second case we have

   σ11r=σ22r,σ12r=0,r=3,...,2m+1.

   The converse part is straightforward. □

Corollary 4.3

Let M be an n-dimensional (n ⩾ 2) submanifold in a Sasakian space form M̂(c) endowed with a semi-symmetric metric connection such that ξ ∈ TM. Then for each point p ∈ M and for each unit vector X ∈ T_pM, we have

Corollary 4.4

Let M be an n-dimensional (n ⩾2) submanifold in a non-Sasakian space form M̂(c) endowed with a semi-symmetric metric connection such that ξ∈ TM. Then for each point p ∈ M and for each unit vector X ∈ T_pM, we have

Theorem 4.5

Let M be an n-dimensional (n ⩾ 3) submanifold in a (2m+1)-dimensional (κ, μ)-contact space form M̂(c) endowed with a semi-symmetric connection such that ξ ∈ TM. Then we have

Proof

Let {e₁, …, e_n} be an orthonormal basis of T_pM. We denote by L_{i1, …, ik} the k-plane section spanned by e_i1, …, e_ik. From (4) and (5), it follows that

and