A Basic Inequality for the Tanaka-Webster Connection

Abstract

For submanifolds tangent to the structure vector field in Sasakian space forms, we establish a Chen's basic inequality between the main intrinsic invariants of the submanifold (namely, its pseudosectional curvature and pseudosectional curvature on one side) and the main extrinsic invariant (namely, squared pseudomean curvature on the other side) with respect to the Tanaka-Webster connection. Moreover, involving the pseudo-Ricci curvature and the squared pseudo-mean curvature, we obtain a basic inequality for submanifolds of a Sasakian space form tangent to the structure vector field in terms of the Tanaka-Webster connection.

1. Introduction

One of the basic interests in the submanifold theory is to establish simple relationship between intrinsic invariants and extrinsic invariants of a submanifold. Gauss-Bonnet Theorem, Isoperimetric inequality, and Chern-Lashof Theorem are those such kind of study.

Chen [1] established a nice basic inequality-related intrinsic quantities and extrinsic ones of submanifolds in a space form with arbitrary codimension. Moreover, he studied the basic inequalities of submanifolds of complex space forms and characterize submanifolds when the equality holds.

In this paper, we introduce pseudosectional curvatures and pseudo-Ricci curvature for the Tanaka-Webster connection in a Sasakian space form. After then, we study basic inequalities for submanifolds of a Sasakian space form of a constant pseudosectional curvature and a pseudo-Ricci curvature in terms of the Tanaka-Webster connection.

2. Preliminaries

Let 𝑀 be an odd-dimensional Riemannian manifold with a Riemannian metric 𝑔 satisfying 𝜂(𝜉)=1,𝜑²(𝑋)(),()(𝑋)𝜂(𝑌).=−𝐼+𝜂⊗𝜉,𝜂=𝑔𝑋,𝜉𝑔𝜑𝑋,𝜑𝑌=𝑔𝑋,𝑌−𝜂 (2.1)

Then (𝜑,𝜉,𝜂,𝑔) is called the almost contact metric structure on 𝑀. Let Φ denote the fundamental 2 form in 𝑀 given by Φ(𝑋,𝑌)=𝑔(𝑋,𝜑𝑌) for all 𝑋,𝑌∈𝑇𝑀, the set of vector fields of 𝑀. If Φ=𝑑𝜂, then 𝑀 is said to be a contact metric manifold. Moreover, if 𝜉 is a Killing vector field with respect to 𝑔, and the contact metric structure is called a 𝒦-contact structure. Recall that a contact metric manifold is 𝒦-contact if and only if ∇_𝑋𝜉=−𝜑𝑋 (2.2) for any 𝑋∈𝑇𝑀, where ∇ is the Levi-Civita connection of 𝑀. The structure of 𝑀 is said to be normal if [𝜑,𝜑]+2𝑑𝜂⊗𝜉−0, where [𝜑,𝜑] is the Nijenhuis torsion of 𝜑. A Sasakian manifold is a normal contact metric manifold. In fact, an almost contact metric structure is Sasakian if and only if ∇_𝑋𝜑𝑌=𝑔(𝑋,𝑌)𝜉−𝜂(𝑌)𝑋 (2.3) for all vector fields 𝑋 and 𝑌. Every Sasakian manifold is a 𝒦-contact manifold.

Given a Sasakian manifold 𝑀, a plane section 𝜋 in 𝑇_𝑝𝑀 is called a 𝜑-section if it is spanned by 𝑋 and 𝜑𝑋, where 𝑋 is a unit tangent vector field orthogonal to 𝜉. The sectional curvature 𝐾(𝜋) of a 𝜑-section 𝜋 is called 𝜑-sectional curvature. If a Sasakian manifold 𝑀 has constant 𝜑-sectional curvature 𝑐,𝑀 is called a Sasakian space form, denoted by 𝑀(𝑐). (For more details, see [2]).

Now let 𝑀 be a submanifold immersed in (𝑀,𝜑,𝜉,𝜂,𝑔). We also denote by 𝑔 the induced metric on 𝑀. Let 𝑇𝑀 be the Lie algebra of vector fields in 𝑀 and 𝑇^⟂𝑀 the set of all vector fields normal to 𝑀. We denote by ℎ the second fundamental form of 𝑀 and by 𝐴_𝑣 the Weingarten endomorphism associated with any 𝑣∈𝑇^⟂𝑀. We put ℎ^𝑟_𝑖𝑗=𝑔(ℎ(𝑒_𝑖,𝑒_𝑗),𝑒_𝑟) for any orthonormal vector 𝑒_𝑖, 𝑒_𝑗∈𝑇𝑀 and 𝑒_𝑟∈𝑇^⟂𝑀. The mean curvature vector field 𝐻 is defined by 𝐻=(1/dim𝑀)trace(ℎ). 𝑀 is said to be totally geodesic if the second fundamental form vanishes identically.

From now on, we assume that the dimension of 𝑀 is 𝑛+1, and that of the ambient manifold 𝑀 is 2𝑚+1(𝑚≥2). We also assume that the structure vector field 𝜉 is tangent to 𝑀. Hence, if we denote by 𝐷 the orthogonal distribution to 𝜉 in 𝑇𝑀, we have the orthogonal direct decomposition of 𝑇𝑀 by 𝑇𝑀=𝐷⊕span{𝜉}. For any 𝑋∈𝑇𝑀, we write 𝜑𝑋=𝑇𝑋+𝑁𝑋, where 𝑇𝑋 (𝑁𝑋, resp.) is the tangential (normal, resp.) component of 𝜑𝑋. If 𝜑𝑀 is a 𝒦-contact manifold, (2.2) gives ℎ(𝑋,𝜉)=−𝑁𝑋, (2.4) for any 𝑋 in 𝑇𝑀. Given a local orthonormal frame {𝑒₁,…,𝑒_𝑛} of 𝐷, we can define the squared norms of 𝑇 and 𝑁 by ‖𝑇‖²=^𝑛_𝑖,𝑗=1𝑔𝑒_𝑖,𝑇𝑒_𝑗²,‖𝑁‖²=^𝑛_𝑖,𝑗=1𝑔𝑒_𝑖,𝑁𝑒_𝑗², (2.5) resepectively. It is easy to show that both ‖𝑇‖² and ‖𝑁‖² are independent of the choice of the orthonormal frames. The submanifold 𝑀 is said to be invariant if 𝑁 is identically zero, that is, 𝜑𝑋∈𝑇𝑀 for any 𝑋∈𝑇𝑀. On the other hand, 𝑀 is said to be an anti-invariant submanifold if 𝑇 is identically zero, that is, 𝜑𝑋∈𝑇^⟂𝑀 for any 𝑋∈𝑇𝑀.

3. The Tanaka-Webster Connection for Sasakian Space Form

The Tanaka-Webster connection [3, 4] is the canonical affine connection defined on a nondegenerate pseudo-Hermitian CR-manifold. Tanno [5] defined the Tanaka-Webster connection for contact metric manifolds by the canonical connection which coincides with the Tanaka-Webester connection if the associated CR-structure is integrable. We define the Tanaka-Webster connection for submanifolds of Sasakian manifolds by the naturally extended affine connection of Tanno's Tanaka-Webster connection. Now we recall the Tanaka-Webster connection ∇ for contact metric manifolds ∇_𝑋𝑌=∇_𝑋𝑌+𝜂(𝑋)𝜑𝑌+∇_𝑋𝜂(𝑌)𝜉−𝜂(𝑌)∇_𝑋𝜉, (3.1) for all vector fields 𝑋,𝑌∈𝑇𝑀. Together with (2.1), ∇ is written by ∇_𝑋𝑌=∇_𝑋𝑌+𝜂(𝑋)𝜑𝑌+𝜂(𝑌)𝜑𝑋−𝑔𝑌,𝜑(𝑋)𝜉. (3.2) Also, by using (2.1) and (2.3), we can see that ∇𝜂=0,∇𝜉=0,∇𝜑=0,∇𝑔=0. (3.3)

We define the Tanaka-Webster curvature tensor of 𝑅 (in terms of ∇) by 𝑅(𝑋,𝑌)𝑍=∇_𝑋∇_𝑌𝑍−∇_𝑌∇_𝑌𝑍−∇_[𝑋,𝑌]𝑍, (3.4) for all vector fields 𝑋,𝑌, and 𝑍 in 𝑀.

Let 𝑀(𝑐) be a Sasakian space form of constant sectional curvature 𝑐 and 𝑀 a submanifold of 𝑀(𝑐). Then, we have the following Gauss’ equation: 𝑅(𝑋,𝑌)𝑍=𝑐+34𝑔()(𝑌)𝜂(𝑍𝑔()(𝑋)𝜂(𝑍𝑌+𝑔()𝜂(𝑌)()𝜂(𝑋+𝑌,𝑍−𝜂)𝑋−𝑋,𝑍−𝜂)𝑋,𝑍−𝑔𝑌,𝑍)𝜉+2𝑔𝑋,𝜑𝑌𝜑𝑍𝑐+74𝑔𝑍,𝜑𝑌𝜑𝑋−𝑔𝑍,𝜑𝑋𝜑𝑌 (3.5) for any tangent vector fields 𝑋,𝑌,𝑍 tangent to 𝑀.

Let us define the connection ∇^∘ on 𝑀 induced from the Tanaka-Webster connection ∇ on 𝑀 given by ∇^∘_𝑋𝑌=∇^∘_𝑋𝑌+ℎ(𝑋,𝑌), (3.6) for any 𝑋,𝑌∈Γ(𝑇𝑀), where ℎ is called the lightlike second fundamental form of 𝑀 with respect to the induced connection ∇^∘. In the view of (3.2) and (3.6), ∇^∘_𝑋𝑌+ℎ(𝑋,𝑌)=∇_𝑋𝑌+ℎ(𝑋,𝑌)+𝜂(𝑋)𝜑𝑌+𝜂(𝑌)𝜑𝑋−𝑔𝑌,𝜑𝑋𝜉. (3.7) From (3.7), we obtain ∇^∘_𝑋𝑌=∇_𝑋𝑌+𝜂(𝑋)𝑇𝑌+𝜂(𝑌)𝑇𝑋−𝑔ℎ()()(𝑋)(𝑌)𝑌,𝜑𝑋𝜉,𝑋,𝑌=ℎ𝑋,𝑌+𝜂𝑁𝑌+𝜂𝑁𝑋, (3.8)(3.9) where 𝜑𝑋=𝑇𝑋+𝑁𝑋.

From (3.3), (3.8), and (3.9) it is easy to verify the following: ∇^∘𝜂=0,∇^∘𝜉=0,∇^∘𝜑=0,∇^∘𝑔=0. (3.10) Moreover, for the induced connection ∇, we have the following ∇_𝑋𝜉=−𝑇𝑋,ℎ(𝑋,𝜉)=−𝑁𝑋. (3.11) From the definition of 𝑅, together with (3.5), we have 𝑔𝑅^∘(𝑋,𝑌)𝑍,𝑊=𝑐+34𝑔()(𝑌)𝜂(𝑍𝑔()−𝑔()(𝑋)𝜂(𝑍𝑔()+𝑔()𝜂(𝑌)()𝜂(𝑋𝑔()𝑔+𝑌,𝑍−𝜂)𝑋,𝑊𝑋,𝑍−𝜂)𝑌,𝑊𝑋,𝑍−𝑔𝑌,𝑍)𝜉,𝑊+2𝑔𝑋,𝜑𝑌𝜑𝑍,𝑊𝑐+74𝑔𝑔𝑔+𝑍,𝜑𝑌𝜑𝑋,𝑊−𝑔𝑍,𝜑𝑋𝜑𝑌,𝑊𝑔ℎ(𝑋,𝑊),ℎ(𝑌,𝑍)−𝑔ℎ(𝑋,𝑍),ℎ(𝑌,𝑊), (3.12) for any 𝑋,𝑌,𝑍,𝑊∈𝑇𝑀.

For an orthonormal basis {𝑒₁,…,𝑒_𝑛+1} of the tangent space 𝑇_𝑝𝑀,𝑝∈𝑀, the pseudoscalar curvature 𝜏 at 𝑝 is defined by 𝜏=_𝑖<𝑗𝐾𝑒_𝑖∧𝑒_𝑗, (3.13) where 𝐾(𝑒_𝑖∧𝑒_𝑗) denotes the pseudosectional curvature of 𝑀 associated with the plane section spanned by 𝑒_𝑖 and 𝑒_𝑗 for the Tanaka-Webster connection ∇. In particular, if we put 𝑒_𝑛+1=𝜉_𝑝, then (3.13) implies that 2𝜏=_𝑖≠𝑗𝐾𝑒_𝑖∧𝑒_𝑗+2^𝑛_𝑖=1𝐾𝑒_𝑖∧𝜉. (3.14) Moreover, from (3.9), we have ℎ^𝑟_𝑖𝑗=ℎ_𝑖𝑗{},ℎ,𝑖,𝑗∈1,…,𝑛_{𝑟𝑖𝑛+1}=0,𝑗∈{1,…,𝑛+1}. (3.15) The pseudomean curvature vector field 𝐻 is defined by 𝐻=(1/dim𝑀)trace(ℎ). 𝑀 is said to be totally pseudogeodesic if the second fundamental ℎ form vanishes identically. From (2.5), (3.12) and (3.14), we obtain the following relationship between the pseudoscalar curvature and the pseudomean curvature of 𝑀, 2𝜏=(𝑛+1)²𝐻²−ℎ²+𝑛(𝑛−1)𝑐+34+3𝑐+134‖𝑇‖². (3.16)

We now recall the Chen's lemma.

Lemma 3.1 (see [6]). Let 𝑎₁,…,𝑎_𝑛,𝑐 be 𝑛+1(𝑛≥2) real numbers such that ^𝑛_𝑖=1𝑎_𝑖²=(𝑛−1)^𝑛_𝑖=1𝑎²_𝑖+𝑐. (3.17) Then, 2𝑎₁𝑎₂≥𝑐, with the equality holding if and only if 𝑎₁+𝑎₂=𝑎₃=⋯=𝑎_𝑛.

Let 𝑝∈𝑀 and let 𝜋 be a plane section of 𝑇_𝑝𝑀 which is generated by orthonormal vectors 𝑋 and 𝑌. We can define a function 𝛼(𝜋) of tangent space 𝑇_𝑝𝑀 into [0,1] by 𝛼(𝜋)=𝑔(𝑇𝑋,𝑌)², (3.18) which is well defined.

Now, we prove the following.

Theorem 3.2. Let 𝑀 be an (𝑛+1)-dimensional (𝑛≥2) submanifold isometrically immersed in a 𝑚-dimensional Sasakian space form 𝑀(𝑐) such that the structure vector field 𝜉 is tangent to 𝑀 in terms of the Tanaka-Wester connection ∇. Then, for each point 𝑝∈𝑀 and each plane section 𝜋⊂𝑇_𝑝𝑀, we have the following: 𝐾(𝜋)≤()𝜏−𝑛+1²(𝑛−1)𝐻2𝑛²+18()+𝑛+1)(𝑛+2)(𝑐+33𝑐+138‖𝑇‖²−3𝑐+134𝛼(𝜋). (3.19) Equality in (3.19) holds at 𝑝∈𝑀 if and only if there exist an orthonormal basis {𝑒₁,…,𝑒_𝑛+1} of 𝑇_𝑝𝑀 and an orthonormal basis {𝑒_𝑛+2,…,𝑒_𝑚} of 𝑇^⟂_𝑝𝑀 such that (a)  𝜋=Span{𝑒₁,𝑒₂} and (b) the shape operators 𝐴_𝑟=𝐴_𝑒𝑟, 𝑟=𝑛+2,…,𝑚, take the following forms: 𝐴=𝑎000−𝑎0000_𝑛−1,ℎ𝐴=^𝑟11ℎ^𝑟₁₂0ℎ^𝑟₁₂−ℎ^𝑟₁₁0000_𝑛−1,𝑟=𝑛+3,…,𝑚. (3.20)

Proof. Let 𝑀^𝑛+1 be a submanifold of 𝑀(𝑐). We introduce ()𝜌=2𝜏−𝑛+1²(𝑛−1)𝐻2𝑛²−(𝑛+1)(𝑛+2)𝑐+34−3𝑐+134‖𝑇‖². (3.21) Then, from (3.16) and (3.21), we get (𝑛+1)𝐻²=𝑛ℎ²2()+𝑛𝜌−𝑐+34. (3.22) Let 𝑝 be a point of 𝑀 and let 𝜋⊂𝑇_𝑝𝑀 be a plane section at 𝑝. We choose an orthonormal basis {𝑒₁,…,𝑒_𝑛+1} for 𝑇_𝑝𝑀 and {𝑒_𝑛+2,…,𝑒_𝑚} for 𝑇^⟂_𝑝𝑀 such that 𝑒_𝑛+1=𝜉, 𝜋= Span{𝑒₁,𝑒₂}, and the pseudomean curvature vector 𝐻 is parallel to 𝑒_𝑛+2. Then, from (3.22), we get ^𝑛+1_𝑖=1ℎ^𝑛+2_𝑖𝑖²=𝑛^𝑛+1_𝑖=1ℎ^𝑛+2_𝑖𝑖²+_𝑖≠𝑗ℎ^𝑛+2_𝑖𝑗²+^𝑚_{𝑟=𝑛+3𝑖,𝑗}ℎ^𝑟_𝑖𝑗²2()+𝜌−𝑐+34 (3.23) and so, by applying Lemma 3.1, we obtain 2ℎ^𝑛+2₁₁ℎ^𝑛+2₂₂≥_𝑖≠𝑗ℎ^𝑛+2_𝑖𝑗²+^𝑚_{𝑟=𝑛+3𝑖,𝑗}ℎ^𝑟_𝑖𝑗²2()+𝜌−𝑐+34. (3.24) On the other hand, from (3.12), we have 𝐾(𝜋)=ℎ^𝑛+2₁₁ℎ^𝑛+2₂₂−ℎ^𝑛+2₁₂²+^𝑚_𝑟=𝑛+3ℎ^𝑟₁₁ℎ^𝑟₂₂−ℎ^𝑟₁₂²𝑐+34+3𝑐+134𝑔²𝑒₁,𝜑𝑒₂. (3.25) Then, from (3.24) and (3.25), we get 𝐾(𝜋)=𝜌2+3𝑐+134𝑔²𝑒₁,𝜑𝑒₂+^𝑚_{𝑟=𝑛+2𝑗>2}ℎ^𝑟_1𝑗²+ℎ^𝑟_2𝑗²+12_{𝑖≠𝑗>2}ℎ^𝑛+2_𝑖𝑗²+12^𝑚_{𝑟=𝑛+3𝑖,𝑗>2}ℎ^𝑟_𝑖𝑗²+12^𝑚_𝑟=𝑛+3ℎ^𝑟₁₁+ℎ^𝑟₂₂²≥𝜌2+3𝑐+134𝛼(𝜋). (3.26)(3.27) Combining (3.21) and (3.27), the inequality (3.19) yields. If the equality in (3.19) holds, then the inequalities given by (3.24) and (3.27) become equalities. In this case, we have ℎ^𝑛+2_1𝑗=ℎ^𝑛+2_2𝑗=ℎ^𝑛+2_𝑖𝑗ℎ=0,𝑖≠𝑗>2,_𝑟1𝑗=ℎ^𝑟_2𝑗=ℎ𝑖𝑗^𝑟{}{},ℎ=0,𝑟∈𝑛+3,…,𝑚;𝑖,𝑗∈3,…,𝑛+1_𝑛+311+ℎ^𝑛+3₂₂=⋯=ℎ^𝑚₁₁+ℎ^𝑚₂₂=0. (3.28) Moreover, choosing 𝑒₁ and 𝑒₂ such that ℎ^𝑛+2₁₂=0, from (3.11), we also have the following ℎ^𝑛+2₁₁+ℎ^𝑛+2₂₂=ℎ^𝑛+2₃₃=⋯=ℎ^𝑛+2_𝑛+1𝑛+1=0. (3.29) Thus, with respect to the chosen orthonormal basis {𝑒₁,…,𝑒_𝑚}, the shape operators of 𝑀 take the forms.

We now define a well-defined function 𝛿_𝑀 on 𝑀 by using (inf𝐾)(𝑝)=inf{𝐾(𝜋)∣𝜋isaplanesection⊂𝑇_𝑝𝑀} in the following manner: 𝛿_𝑀=𝜏−inf𝐾. (3.30) If 𝑐=−13/3, then we obtain directly from (3.19) the following result.

Corollary 3.3. Let 𝑀 be an (𝑛+1)-dimensional (𝑛≥2) submanifold isometrically immersed in a m-dimensional Sasakian space form 𝑀(−13/3) such that the structure vector field 𝜉 is tangent to 𝑀 in terms of the Tanaka-Wester connection ∇. Then, for each point 𝑝∈𝑀 and each plane section 𝜋⊂𝑇_𝑝𝑀, we have the following: 𝛿_𝑀≤(𝑛+1)²(𝑛−1)𝐻2𝑛²−16(𝑛+1)(𝑛+2). (3.31) The equality in (3.31) holds if and only if 𝑀 is a anti-invariant submanifold with rank(𝑇)=2.

Proof. In order to estimate 𝛿_𝑀, we minimize ‖𝑇‖²−2𝛼(𝜋) in (3.19). For an orthonormal basis {𝑒₁,…,𝑒_𝑛+1} of 𝑇_𝑝𝑀 with 𝜋=Span{𝑒₁,𝑒₂}, we write ‖𝑇‖²−2𝛼(𝜋)=^𝑛+1_𝑖,𝑗=3𝑔²𝑒_𝑖,𝜑𝑒_𝑗+2^𝑛+1_𝑗=3𝑔²𝑒₁,𝜑𝑒_𝑗+𝑔²𝑒₂,𝜑𝑒_𝑗. (3.32) Thus, we see that the minimum value of ‖𝑇‖²−2𝛼(𝜋) is zero, provided that 𝜋=Span{𝑒₁,𝑒₂} is orthogonal to 𝜉, and span{𝜑𝑒_𝑗∣𝑗=3,…,𝑛} is orthogonal to 𝑇_𝑝𝑀. Thus we have (3.31) with equality case holding if and only if 𝑀 is anti-invariant such that rank(𝑇)=2.

4. A Pseudo-Ricci Curvature for Sasakian Space Form

We denote the set of unit vectors in 𝑇_𝑝𝑀 by 𝑇¹_𝑝𝑀 by 𝑇¹_𝑝𝑀=𝑋∈𝑇_𝑝𝑀∣𝑔(𝑋,𝑋)=1. (4.1) Let {𝑒₁,…,𝑒_𝑘}, 2≤𝑘≤𝑛, be an orthonormal basis of a 𝑘-place section Π_𝑘 of 𝑇_𝑝𝑀. If 𝑘=𝑛, then Π_𝑘=𝑇_𝑝𝑀, and if 𝑘=2, then Π₂ is a plane section of 𝑇_𝑝𝑀. For a fixed 𝑖∈{1,…,𝑘}, a 𝑘-pseudo-Ricci curvature of Π_𝑘 at 𝑒_𝑖, denoted by Ric_Π𝑘(𝑒_𝑖), is defined by [7] Ric_Π𝑘𝑒_𝑖=^𝑘_𝑗≠𝑖𝐾_𝑖𝑗, (4.2) where 𝐾_𝑖𝑗 is the pseudosectional curvature in terms of the Tanaka-Webster connection ∇ of the plane section spanned by 𝑒_𝑖 and 𝑒_𝑗. We note that an 𝑛-pseudo-Ricci curvature Ric_𝑇𝑝𝑀(𝑒_𝑖) is the usual pseudo-Ricci curvature of 𝑒_𝑖, denoted by Ric(𝑒_𝑖). Thus, for any orthonormal basis {𝑒₁,…,𝑒_𝑛+1} for 𝑇_𝑝𝑀 and for a fixed 𝑖∈{1,…,𝑛+1}, we have the following: Ric_𝑇𝑝𝑀𝑒_𝑖=Ric𝑒_𝑖=^𝑛+1_𝑗≠𝑖𝐾_𝑖𝑗. (4.3) The pseudoscalar curvature 𝜏(Π_𝑘) of the 𝑘-plane section Π_𝑘 is given by 𝜏Π_𝑘=_{1≤𝑖<𝑗≤𝑛+1}𝐾_𝑖𝑗. (4.4) The relative null spae of 𝑀 at 𝑝 is defined by [8] 𝒩_𝑝=𝑋∈𝑇_𝑝𝑀∣ℎ(𝑋,𝑌)=0,∀𝑌∈𝑇_𝑝𝑀. (4.5)

Theorem 4.1. Let 𝑀(𝑐) be a 𝑚-dimensional Sasakian space form and 𝑀 an 𝑛+1-dimensional submanifold tangent to 𝜉 with respect to the Tanaka-Webster connection ∇. Then, (i)for each unit vector 𝑋∈𝑇_𝑝𝑀 orthogonal to 𝜉, we have 4Ric(𝑋)≤(𝑛+1)²𝐻²+(𝑛−1)(𝑐+3)+(3𝑐+13)‖𝑇𝑋‖², (4.6)(ii)if 𝐻(𝑝)=0, then a unit tanget vector 𝑋∈𝑇_𝑝𝑀 orthogonal to 𝜉 satisfies the equality case of (4.6) if and only of 𝑋∈𝒩_𝑝. (iii)the equality case of (4.6) holds identically for all unit tangent vectors orthogonal to 𝜉 at 𝑝 if and only if 𝑝 is a totally pseudogeodesic point in terms of the Tanaka-Webster connection.

Proof. (i) Let 𝑋∈𝑇_𝑝𝑀 be a unit tangent vector at 𝑝, orthogonal to 𝜉. We choose an orthonormal basis {𝑒₁,…,𝑒_𝑛+1} for 𝑇_𝑝𝑀 and {𝑒_𝑛+2,…,𝑒_𝑚} for 𝑇^⟂_𝑝𝑀 such that 𝑒₁=𝑋 and 𝑒_𝑛+1=𝜉. Then, from (3.16), we have (𝑛+1)²𝐻²=2𝜏+ℎ²−𝑛(𝑛−1)𝑐+34−3𝑐+134‖𝑇‖². (4.7) From (4.7), we get (𝑛+1)²𝐻²=2𝜏+^𝑚_𝑟=𝑛+2ℎ²₁₁²+ℎ^𝑟₂₂+⋯+ℎ^𝑟_𝑛+1𝑛+1²+2_𝑖<𝑗ℎ^𝑟_𝑖𝑗²−2^𝑚_{𝑟=𝑛+22≤𝑖<𝑗≤𝑛}ℎ^𝑟_𝑖𝑖ℎ^𝑟_𝑗𝑗−𝑛(𝑛−1)𝑐+34−3𝑐+134‖𝑇‖²1=2𝜏+2^𝑚_𝑟=𝑛+2ℎ^𝑟₁₁+ℎ^𝑟₂₂+⋯+ℎ^𝑟_𝑛+1𝑛+1²+ℎ^𝑟₁₁−ℎ^𝑟₂₂−⋯−ℎ^𝑟_𝑛+1𝑛+1²+2^𝑚_{𝑛+2𝑖<𝑗}ℎ^𝑟_𝑖𝑗²−2^𝑚_{𝑟=𝑛+22≤𝑖<𝑗≤𝑛}ℎ^𝑟_𝑖𝑖ℎ^𝑟_𝑗𝑗−𝑛(𝑛−1)𝑐+34−3𝑐+134‖𝑇‖². (4.8) From (3.12), we have 𝐾_𝑖𝑗=^𝑚_𝑟=𝑛+2ℎ^𝑟_𝑖𝑖ℎ^𝑟_𝑗𝑗−ℎ^𝑟_𝑖𝑗²+𝑐+34+3𝑐+134𝑔²𝑒_𝑖,𝑇𝑒_𝑗, (4.9) and consequently _{2≤𝑖<𝑗≤𝑛+1}𝐾_𝑖𝑗=^𝑚_𝑟=𝑛+2ℎ^𝑟_𝑖𝑖ℎ^𝑟_𝑗𝑗−ℎ^𝑟_𝑖𝑗²+(𝑛−1)(𝑛−2)(𝑐+3)8+3𝑐+138‖𝑇‖²−2𝑇𝑒₁². (4.10) Substituting (4.10) into (4.8), one gets (𝑛+1)²𝐻²()≥2𝜏+𝑛+1²2𝐻²+2^𝑚_{𝑟=𝑛+2𝑗=2}ℎ^𝑟_1𝑗²−2_{2≤𝑖<𝑗≤𝑛+1}𝐾_𝑖𝑗−(𝑛−1)(𝑐+3)2−3𝑐+132𝑇𝑒₁². (4.11) Therefore, (𝑛+1)²2𝐻²(𝑋)−()≥2Ric𝑛−1)(𝑐+32−3𝑐+132‖𝑇𝑋‖², (4.12) which is equivalent to (4.6)
(ii) Assume that 𝐻(𝑝)=0. Equality holds in (4.6) if and only if ℎ^𝑟₁₂=⋯=ℎ^𝑟_1𝑛+1ℎ=0,_𝑟11=ℎ^𝑟₂₂+⋯+ℎ^𝑟_𝑛+1𝑛+1,𝑟∈{𝑛+2,…,𝑚}. (4.13) Then, ℎ^𝑟_1𝑗=0 for each 𝑗∈{1,…,𝑛+1},𝑟∈{𝑛+2,…,𝑚}, that is, 𝑋∈𝒩_𝑝.
(iii) The equality case of (4.6) holds for all unit tangent vectors at 𝑝 if and only if ℎ^𝑟_𝑖𝑗{},ℎ=0,𝑖≠𝑗,𝑟∈𝑛+2,…,𝑚_𝑟11+⋯+ℎ^𝑟_𝑛+1𝑛+1−2ℎ^𝑟_𝑖𝑖=0,𝑖∈{1,…,𝑛+1},𝑟∈{𝑛+2,…,𝑚}. (4.14) Since ℎ(𝑒_𝑖,𝑒_𝑛+1=𝜉)=0 from (3.10), 𝑝 is a totally pseudogeodesic point, and, hence, 𝜑(𝑇_𝑝𝑀)⊂𝑇_𝑝𝑀. The converse is trivial.

Corollary 4.2. Let 𝑀 be an 𝑛+1-dimensional invariant submanifold of a Sasakian space form 𝑀(𝑐). Then, (i)for each unit vector 𝑋∈𝑇_𝑝𝑀 orthogonal to 𝜉, we have 4Ric(𝑋)≤(𝑛−1)(𝑐+3)+(3𝑐+13). (4.15)(ii)A unit tanget vector 𝑋∈𝑇_𝑝𝑀 orthogonal to 𝜉 satisfies the equality case of (4.6) if and only if 𝑋∈𝒩_𝑝. (iii)The equality case of (4.6) holds identically for all unit tangent vectors orthogonal to 𝜉 at 𝑝 if and only if 𝑝 is a totally pseudogeodesic point in terms of the Tanaka-Webster connection.