A Note on Pseudo-Umbilical Submanifolds of Hessian Manifolds with Constant Hessian Sectional Curvature

Abstract

The geometry of Hessian manifold, as a branch of statistics, physics, Kaehlerian, and affine differential geometry, is deeply fruitful and a new field for scientists. However, inspite of its importance submanifolds and curvature conditions of it have not been so well known yet. In this paper, we focus on the pseudo-umbilical submanifolds on Hessian manifold with constant Hessian sectional curvature and using sectional curvature conditions we obtain new results on it.

1. Introduction

A Riemannian metric on a flat manifold is called a Hessian metric if it is locally expressed by the Hessian of functions with respect to affine coordinate systems. The pair of (𝐷,𝑔) with flat connection 𝐷 and Hessian metric 𝑔 is called Hessian structure, and a manifold equipped with this structure is said to be a Hessian manifold. In [1, 2], Hirohiko Shima introduced Hessian sectional curvature and its relations with Kaehlerian manifold. He also proved theorems and gave important remarks on the spaceform of Hessian manifolds. In the light of these studies Bektaş et al. obtained some curvature conditions, results, and integral inequalities on this type of manifolds, [3–5].

Let 𝑀^𝑛+𝑝 be an (𝑛+𝑝)-dimensional Hessian manifold of constant curvature 𝑐. Let 𝑀^𝑛 be an 𝑛-dimensional Riemannian manifold immersed in 𝑀^𝑛+𝑝. Let ℎ be the second fundamental form of the immersion, and 𝜉 the mean curvature vector. Denote by 𝑔 the scalar product of 𝑀^𝑛+𝑝. If there exists a function 𝜆 on 𝑀^𝑛 such that𝑔(ℎ(𝑋,𝑌),𝜉)=𝜆𝑔(𝑋,𝑌)(∗) for any tangent vector 𝑋,𝑌 on 𝑀^𝑛, then 𝑀^𝑛 is called a pseudo-umbilical submanifold of 𝑀^𝑛+𝑝. It is clear that 𝜆≥0. If the mean curvature 𝜉=0 identically, then 𝑀^𝑛 is called a minimal submanifold of 𝑀^𝑛+𝑝.

Every minimal submanifold of 𝑀^𝑛+𝑝 is itself a pseudo umbilical submanifold of 𝑀^𝑛+𝑝. Cao [6] extended Bai’s well-known theorem to the case in which 𝑀^𝑛+𝑝 is pseudo-umbilical. The aim of the present work is to obtain this theorem for compact pseudo-umbilical submanifold of a Hessian manifold and also give some results and examples of it.

Theorem  A. Let 𝑀^𝑛 be an 𝑛-dimensional compact pseudo-umbilical submanifold of (𝑛+𝑝)-dimensional Hessian manifold of constant Hessian sectional curvature 𝑐. Then _𝑀𝑛𝑝𝑅^{2𝑖𝑗𝑘𝑙}+2𝑝𝑅^2𝑖𝑗−𝑅²−𝑛(3𝑝+2𝑛−2)𝑐4𝑅−𝑛(3𝑝−2𝑛)𝐻²+2𝑛₂𝑅(1−𝑛)𝑝𝑐4𝐻²+2𝑛³(1−𝑛)𝑐4𝐻²≥𝑛₂+𝑛³𝑝𝐻⁴−𝑛⁴𝐻⁴∗1(𝑛−1)(𝑛−𝑝−1)𝑐²16Vol𝑀^𝑛,(1.1) where ∑𝑅^{2𝑖𝑗𝑘𝑙} is the square length of the Riemannian curvature tensor, ∑𝑅^2𝑖𝑗 is the square length of the Ricci curvature tensor, 𝑅 is the scalar curvature, and 𝐻 is the mean curvature of 𝑀^𝑛.

We will use the same notation and terminologies as in [2] unless otherwise stated.

Let 𝑀^𝑛+𝑝 be a Hessian manifold with Hessian structure (𝐷,𝑔). We express various geometric concepts for the Hessian structure (𝐷,𝑔) in terms of affine coordinate system {𝑥¹,…,𝑥^𝑛+𝑝} with respect to 𝐷, that is, 𝐷𝑑𝑥^𝑖=0.(i)The Hessian metric, 𝑔_𝑖𝑗=𝜕²𝑢𝜕𝑥^𝑖𝜕𝑥^𝑗.(1.2)(ii)Let 𝛾 be a tensor field of type (1,2) defined by 𝛾(𝑋,𝑌)=∇_𝑋𝑌−𝐷_𝑋𝑌,(1.3) where ∇ is the Riemannian connection for 𝑔. Then we have 𝛾^𝑖𝑗𝑘=Γ^𝑖𝑗𝑘=12𝑔^𝑖𝑟𝜕𝑔_𝑟𝑗𝜕𝑥^𝑘𝛾_𝑖𝑗𝑘,=12𝜕𝜕𝑔𝑥^𝑖𝑗𝑘=12𝜕³𝑢𝜕𝑥^𝑖𝜕𝑥^𝑗𝜕𝑥^𝑘𝛾_𝑖𝑗𝑘,=𝛾_𝑗𝑖𝑘=𝛾_𝑘𝑗𝑖,(1.4) where Γ^𝑖𝑗𝑘 are the Christoffel’s symbols of ∇.(iii)Define a tensor field 𝑆 of type (1,3)by 𝑆=𝐷_𝛾(1.5) and call it the Hessian curvature tensor for (𝐷,𝑔). Then we have 𝑆^{𝑖𝑗𝑘𝑙}=𝜕𝛾^𝑖𝑗𝑙𝜕𝑥^𝑘𝑆_{𝑖𝑗𝑘𝑙},=12𝜕⁴𝑢𝜕𝑥^𝑖𝜕𝑥^𝑗𝜕𝑥^𝑘𝜕𝑥^𝑙−12𝑔^𝑟𝑠𝜕³𝑢𝜕𝑥^𝑖𝜕𝑥^𝑘𝜕𝑥^𝑟−𝜕³𝑢𝜕𝑥^𝑗𝜕𝑥^𝑙𝜕𝑥^𝑠𝑆_{𝑖𝑗𝑘𝑙},=𝑆_{𝑖𝑙𝑘𝑗}=𝑆_{𝑘𝑗𝑖𝑙}=𝑆_{𝑗𝑖𝑙𝑘}=𝑆_{𝑘𝑙𝑖𝑗}.(1.6)(iv)The Riemannian curvature tensor for ∇, 𝑅𝑅^{𝑖𝑗𝑘𝑙}_{𝑖𝑗𝑘𝑙}=𝛾^𝑖𝑟𝑘𝛾^𝑟𝑗𝑙−𝛾^𝑖𝑟𝑙𝛾^𝑟𝑗𝑘,=12𝑆^{𝑗𝑖𝑘𝑙}−𝑆_{𝑖𝑗𝑘𝑙}(1.7)(see [2]).

Definition 1.1. Let S^{𝑖𝑘𝑗𝑙} be a Hessian curvature tensor on a Hessian manifolds (𝑀,𝐷,𝑔). We define an endomorphism 𝜍 of the space of contravariant symmetric tensor fields of degree 2 by 𝜉(𝜉)^𝑖𝑘=𝑆^{𝑖𝑘𝑗𝑙}𝜉^𝑗𝑙.(1.8) Then 𝜉 is a symmetric operator, [2].

Definition 1.2. For a nonzero contravariant symmetric tensor 𝜉_𝑥 of degree 2 at 𝑥, we set ℎ𝜉_𝑥=𝜉𝜉^𝑥,𝜉_𝑥⟨𝜉_𝑥,𝜉_𝑥⟩(1.9) and call it the Hessian sectional curvature in the direction 𝜉_𝑥, [2].

Theorem 1.3. Let (𝑀^𝑛+𝑝,𝐷,𝑔) be a Hessian manifold of dimension ≥2. If the Hessian sectional curvature ℎ(𝜉_𝑥) depends only on 𝑥, then (𝑀,𝐷,𝑔) is of constant Hessian sectional curvature. (𝑀,𝐷,𝑔) is of constant Hessian sectional curvature 𝑐 if and only if 𝑆_{𝑖𝑗𝑘𝑙}=𝑐2𝑔^𝑖𝑗𝑔_𝑘𝑙+𝑔_𝑖𝑙𝑔_𝑘𝑗(1.10)(see [2]).

Corollary 1.4. If a Hessian manifold (𝑀^𝑛+𝑝,𝐷,𝑔) is a space of constant Hessian sectional curvature 𝑐, then the Riemannian manifold (𝑀,𝑔) is a space of constant sectional curvature −𝑐/4, [2].

From now on, we shall construct, for each constant 𝑐, a Hessian manifold with constant Hessian sectional curvature 𝑐. We now recall the following result due to Shima and Yagi [7]. Let (𝑀^𝑛+𝑝,𝐷,𝑔) be a simply connected Hessian manifold. If 𝑔 is complete, then (𝑀^𝑛+𝑝,𝐷,𝑔) is isomorphic to (Ω,𝐷,𝐷²𝜑), where Ω is a convex domain in ℝ^𝑛+𝑝, 𝐷 is the canonical flat connection on ℝ^𝑛+𝑝, and 𝜑 is a smooth convex function on Ω.(A)Case 𝑐=0

It is obvious that the Euclidean space (ℝ^𝑛+𝑝,𝐷,𝑔=(1/2)𝐷²{∑^{𝑛+𝑝𝐴=1}(𝑥^𝐴)²}) is a simply connected Hessian manifold of constant Hessian sectional curvature 0 [1].(B)Case 𝑐>0

Theorem 1.5. Let Ω be a domain in ℝ^𝑛+𝑝 given by 𝑥^𝑛+𝑝>𝑐2^{𝑛+𝑝−1}_𝐴=1𝑥^𝐴²,(1.11) where 𝑐 is a positive constant, and let 𝜑 be a smooth function on Ω defined by 𝜑=−1𝑐log𝑥^𝑛+𝑝−𝑐2^{𝑛+𝑝−1}_𝐴=1𝑥^𝐴².(1.12) Then (Ω,𝐷,𝑔=𝐷²𝜑) is a simply connected Hessian manifold of positive constant Hessian sectional curvature 𝑐. As Riemannian manifold (Ω,𝑔) is isometric to the hyperbolic space (𝐻(−𝑐/4),𝑔) of constant sectional curvature −𝑐/4; 𝑔𝐻==1𝜉¹,…,𝜉^𝑛,𝜉^𝑛+𝑝∈ℝ^𝑛+𝑝∣𝜉^𝑛+𝑝>0,𝜉^𝑛+𝑝²^{𝑛+𝑝−1}_𝐴₌₁𝑑𝜉^𝐴²+4𝑐𝑑𝜉^𝑛+𝑝².(1.13)

(C)Case 𝑐<0

Theorem 1.6. Let 𝜑 be a smooth function on ℝ^𝑛+𝑝 defined by 𝜑=−1𝑐log^𝑛+𝑝_𝐴=1𝑒^{−𝑐𝑥𝐴}+1,(1.14) where 𝑐 is a negative constant. Then (ℝ^𝑛+𝑝,𝐷,𝑔=𝐷²𝜑) is a simply connected Hessian manifold of negative constant Hessian sectional curvature 𝑐. The Riemannian manifold (ℝ^𝑛+𝑝,𝑔) is isometric to a domain of the sphere∑^{𝑛+𝑝+1𝑖=1}𝜉^2𝐴=−4/𝑐 defined by 𝜉_𝐴>0 for all 𝐴 [1].

For the proof of the theorems we refer to [1].

2. Local Formulas

We choose a local field of orthonormal frames 𝑒₁,…,𝑒_𝑛+𝑝 in 𝑀^𝑛+𝑝 such that restricted to 𝑀^𝑛,𝑒₁,…,𝑒_𝑛 are tangent to 𝑀^𝑛. Let 𝑤₁,…,𝑤_𝑛+𝑝 be its dual frame field. Then the structure equations of 𝑀^𝑛+𝑝 are given by𝑑𝑤𝑑_𝐴𝐵𝑤_𝐴=−𝑤_𝐴𝐵∧𝑤_𝐵,𝑤_𝐴𝐵+𝑤_𝐵𝐴=0,=−𝑤_𝐴𝐶∧𝑤_𝐶𝐵+12𝑅_{𝐴𝐵𝐶𝐷}𝑤_𝐶∧𝑤_𝐷.(2.1) We restrict these forms to 𝑀^𝑛, then we have 𝑑𝑤𝑤_𝛼𝑖𝑗=0,𝑤_𝛼𝑖=ℎ^𝛼𝑖𝑗𝑤_𝑗,ℎ^𝛼𝑖𝑗=ℎ^𝛼𝑗𝑖,=−𝑤_𝑖𝑘∧𝑤_𝑘𝑗+12𝑅^{𝑖𝑗𝑘𝑙}𝑅_{𝑖𝑗𝑘𝑙}𝑤_𝑘∧𝑤_𝑙,=−𝑐4𝛿^𝑖𝑘𝛿_𝑗𝑙−𝛿_𝑖𝑙𝛿_𝑗𝑘+ℎ^𝛼𝑖𝑘ℎ^𝛼𝑗𝑙−ℎ^𝛼𝑖𝑙ℎ^𝛼𝑗𝑘,(2.2) where 𝑅_{𝑖𝑗𝑘𝑙} are the components of the curvature tensor of 𝑀^𝑛.𝑑𝑤_𝛼𝛽=−𝑤_𝛼𝛾∧𝑤_𝛾𝛽+12𝑅^{𝛼𝛽𝑘𝑙}𝑅_{𝛼𝛽𝑘𝑙}𝑤_𝑘∧𝑤_𝑙,=ℎ^𝛼𝑖𝑘ℎ^𝛽𝑖𝑙−ℎ^𝛽𝑖𝑘𝑗ℎ^𝛼𝑖𝑙.(2.3)

We call ℎ=ℎ^𝛼𝑖𝑗𝑤_𝑖𝑤_𝑗𝑒_𝛼(2.4) the second fundamental form of the immersed manifold 𝑀^𝑛. Denote by𝑆=ℎ^𝛼𝑖𝑗²(2.5) the square length of ℎ, 𝜉=(1/𝑛)∑tr𝐻_𝛼𝑒_𝛼 the mean curvature vector and 𝐻=(1/𝑛)∑(tr𝐻_𝛼)² the mean curvature of 𝑀^𝑛, respectively. Here tr is the trace of the matrix 𝐻_𝛼=(ℎ^𝛼𝑖𝑗). Now let 𝑒_𝑛+1 be parallel to 𝜉. Then we have tr𝐻_𝑛+1=𝑛𝐻,tr𝐻_𝛼=0,𝛼≠𝑛+1.(2.6) Let ℎ^{𝛼𝑖𝑗𝑘} and ℎ^{𝛼𝑖𝑗𝑘𝑙} denote the covariant derivative and the second covariant derivative of ℎ^𝛼𝑖𝑗, respectively, defined by ℎ^{𝛼𝑖𝑗𝑘}ℎ_{𝛼𝑖𝑗𝑘𝑙}𝑤_𝑘=𝑑ℎ^𝛼𝑖𝑗−ℎ^𝛼𝑖𝑘𝑤_𝑘𝑗−ℎ^𝛼𝑗𝑘𝑤_𝑘𝑖−ℎ^𝛽𝑖𝑗𝑤_𝛽𝛼,𝑤_𝑙=𝑑ℎ^{𝛼𝑖𝑗𝑘}−ℎ^{𝛼𝑖𝑗𝑙}𝑤_𝑙𝑘−ℎ^{𝛼𝑖𝑙𝑘}𝑤_𝑙𝑗−ℎ^{𝛼𝑙𝑗𝑘}𝑤_𝑙𝑖ℎ^{𝛽𝑖𝑗𝑘}𝑤_𝛽𝛼.(2.7) Then we haveℎℎ^{𝛼𝑖𝑗𝑘}−ℎ^{𝛼𝑖𝑘𝑗}=0,_{𝛼𝑖𝑗𝑘𝑙}−ℎ^{𝛼𝑖𝑗𝑙𝑘}=ℎ^𝛼𝑖𝑚𝑅_{𝑚𝑗𝑘𝑙}+ℎ^𝛼𝑚𝑗𝑅_{𝑚𝑖𝑘𝑙}−ℎ^𝛽𝑖𝑗𝑅_{𝛼𝛽𝑘𝑙}.(2.8) The Laplacian Δℎ^𝛼𝑖𝑗 of ℎ^𝛼𝑖𝑗 is defined by Δℎ^𝛼𝑖𝑗=∑ℎ^{𝛼𝑖𝑗𝑘𝑘}. By a direct calculation we have 12Δ𝑆=ℎ^{𝛼𝑖𝑗𝑘}²+ℎ^𝛼𝑖𝑗Δℎ^𝛼𝑖𝑗=ℎ^{𝛼𝑖𝑗𝑘}²+ℎ^𝛼𝑖𝑗ℎ^{𝛼𝑘𝑘𝑖𝑗}+ℎ^𝛼𝑖𝑗ℎ^𝛼𝑚𝑘𝑅_{𝑚𝑖𝑗𝑘}+ℎ^𝛼𝑖𝑗ℎ^𝛼𝑚𝑖𝑅_{𝑚𝑘𝑗𝑘}−ℎ^𝛼𝑖𝑗ℎ^𝛽𝑘𝑖𝑅_{𝛼𝛽𝑗𝑘}.(2.9)

3. Proof of Theorem A

From (*) and (2.6), we have 𝑔ℎ𝑒_𝑖,𝑒_𝑗,𝐻_𝑒𝑛+1=𝐻²𝛿_𝑖𝑗,(3.1) therefore, ℎ^𝛼𝑖𝑗ℎℎ^{𝛼𝑘𝑘𝑖𝑗}=𝑛𝐻Δ𝐻,_{𝛼𝑖𝑗𝑘}²≥ℎ^{𝑛+1𝑖𝑖𝑘}²=𝑛Δ_𝑖𝐻²=𝑛||∇𝐻||².(3.2) It is obvious that 12Δ𝐻²=𝐻Δ𝐻+||∇𝐻||²(3.3) and, therefore, ℎ^{𝛼𝑖𝑗𝑘}²+ℎ^𝛼𝑖𝑗ℎ^{𝛼𝑘𝑘𝑖𝑗}≥𝑛||∇𝐻||²+𝑛𝐻Δ𝐻=12𝑛Δ𝐻².(3.4) On the other hand, from (2.2)ℎ^𝛼𝑖𝑗ℎ^𝛼𝑚𝑘𝑅_{𝑚𝑖𝑗𝑘}=12ℎ^𝛼𝑖𝑗ℎ^𝛼𝑚𝑘−ℎ^𝛼𝑚𝑗ℎ^𝛼𝑖𝑘𝑅_{𝑚𝑖𝑗𝑘}=12𝑅^{𝑖𝑚𝑗𝑘}+𝑐4𝛿^𝑖𝑗𝛿_𝑚𝑘−𝛿_𝑚𝑗𝛿_𝑖𝑘𝑅_{𝑚𝑖𝑗𝑘}=−12𝑅^{2𝑚𝑖𝑗𝑘}−𝑐ℎ4𝑅,_𝛼𝑖𝑗ℎ^𝛼𝑚𝑖𝑅_{𝑚𝑘𝑗𝑘}=(1−𝑛)𝑐4𝛿^𝑚𝑗+𝑛𝐻²𝛿_𝑚𝑗−𝑅_𝑚𝑗𝑅^𝑚𝑗=−𝑅^2𝑚𝑗+(1−𝑛)𝑐4𝑅+𝑛𝐻²𝑅.(3.5) From (2.3), we haveℎ^𝛼𝑖𝑗ℎ^𝛽𝑘𝑖𝑅_{𝛼𝛽𝑗𝑘}=ℎ^𝛼𝑖𝑗ℎ^𝛼𝑙𝑗ℎ^𝛽𝑘𝑖ℎ^𝛽𝑙𝑘−ℎ^𝛼𝑖𝑗ℎ^𝛼𝑙𝑘ℎ^𝛽𝑖𝑘ℎ^𝛽𝑙𝑗(3.6) whileℎ^𝛼𝑖𝑗ℎ^𝛼𝑙𝑗ℎ^𝛽𝑘𝑖ℎ^𝛽𝑙𝑘=(1−𝑛)𝑐4𝛿^𝑖𝑙+𝑛𝐻²𝛿_𝑖𝑙−𝑅_𝑖𝑙²=𝑛(1−𝑛)²𝑐²16−2𝑛²(1−𝑛)𝑐4𝐻²−2(1−𝑛)𝑐4𝑅−2𝑛𝐻²𝑅+𝑛³𝐻⁴+𝑅^2𝑖𝑙.(3.7) Let 𝑆_𝛼=_𝑖,𝑗ℎ^𝛼𝑖𝑗²,(3.8) then we have 𝑆=_𝛼𝑆_𝛼.(3.9) Since 𝑆²=_𝛼𝑆_𝛼²_𝛼<𝛽=_𝛼𝑆^2𝛼+2_𝛼<𝛽𝑆_𝛼𝑆_𝛽,𝑆_𝛼−𝑆_𝛽²=(𝑝−1)_𝛼𝑆^2𝛼−2_𝛼<𝛽𝑆_𝛼𝑆_𝛽⩾0(3.10) it follows that (𝑝−1)_𝛼𝑆^2𝛼⩾2_𝛼<𝛽𝑆_𝛼𝑆_𝛽=𝑆²−_𝛼𝑆^2𝛼,(3.11) that is _𝛼𝑆^2𝛼≥1𝑝𝑆²,(3.12) Since _𝑖_{,𝑗,𝑙,𝑘}ℎ^𝛼𝑖𝑗ℎ^𝛼𝑙𝑘²=_{𝑖,𝑗,𝑙,𝑘}_𝛽ℎ^𝛽𝑖𝑘ℎ^𝛽𝑙𝑗²=_𝛼,𝛽ℎ^𝛼𝑖𝑗ℎ^𝛽𝑖𝑗ℎ^𝛼𝑙𝑘ℎ^𝛽𝑙𝑘=_𝛼,𝛽_𝑖,𝑗ℎ^𝛼𝑖𝑗ℎ^𝛽𝑖𝑗²≥_𝛼_𝑖,𝑗ℎ^𝛼𝑖𝑗ℎ^𝛼𝑖𝑗²=_𝛼𝑆^2𝛼(3.13) we have_𝛼,𝛽ℎ^𝛼𝑖𝑗ℎ^𝛼𝑙𝑘ℎ^𝛽𝑖𝑘ℎ^𝛽𝑙𝑗≥_𝛼,𝛽ℎ^𝛼𝑖𝑗ℎ^𝛼𝑙𝑘ℎ^𝛽𝑖𝑘ℎ^𝛽𝑙𝑗−12_{𝑖,𝑗,𝑙,𝑘}_𝛼ℎ^𝛼𝑖𝑗ℎ^𝛼𝑙𝑘²−12_{𝑖,𝑗,𝑙,𝑘}_𝛽ℎ^𝛽𝑖𝑘ℎ^𝛽𝑙𝑗²+_𝛼𝑆^2𝛼=−12_𝛼ℎ^𝛼𝑖𝑗ℎ^𝛼𝑙𝑘−ℎ^𝛼𝑖𝑘ℎ^𝛼𝑙𝑗²+_𝛼𝑆^2𝛼=−12𝑅^{𝑖𝑙𝑗𝑘}+𝑐4𝛿^𝑖𝑗𝛿_𝑙𝑘−𝛿_𝑖𝑘𝛿_𝑙𝑗²+_𝛼𝑆^2𝛼≥−12𝑅^{2𝑖𝑙𝑗𝑘}−2𝑐4𝑅−𝑛(1−𝑛)1𝑐6²+1𝑝𝑆².(3.14) From (2.9), (3.4)–(3.14), we have12Δ𝑆≥12𝑛Δ𝐻²−𝑅^{2𝑖𝑗𝑘𝑙}−2𝑅^2𝑖𝑗−34𝑐𝑛𝑅+3𝑛𝐻²−𝑛₂𝑅(1−𝑛)𝑐²16−2𝑛²(1−𝑛)𝑐4𝐻²−𝑛³𝐻⁴+1𝑝𝑆².(3.15) Since 𝑀^𝑛 is compact and 𝑆=𝑛(1−𝑛)𝑐4+𝑛²𝐻²−𝑅(3.16) we have_𝑀𝑛𝑝𝑅^{2𝑖𝑗𝑘𝑙}+2𝑝𝑅^2𝑖𝑗+34𝑐𝑛𝑝𝑅−3𝑛𝑝𝐻²𝑅+𝑛²(1−𝑛)𝑐²+2𝑛₂16𝑝(1−𝑛)𝑐4𝐻²𝑝+𝑛³𝐻⁴𝑝−𝑆²𝑆∗1,₂=𝑛²(1−𝑛)²𝑐²16+𝑛⁴𝐻⁴+𝑅²+2𝑛³(1−𝑛)𝐻²𝑐4−2𝑛²𝐻²𝑅−2𝑛(1−𝑛)𝑐4𝑅,_𝑀𝑛𝑝𝑅^{2𝑖𝑗𝑘𝑙}+2𝑝𝑅^2𝑖𝑗+34𝑛𝑐𝑅𝑝−3𝑛𝐻²𝑅𝑝+𝑛²(1−𝑛)𝑐²+2𝑛₂16𝑝(1−𝑛)𝑐4𝐻²𝑝+𝑛³𝐻⁴𝑝−𝑛²(1−𝑛)²𝑐²16−𝑛⁴𝐻⁴−𝑅²−2𝑛³(1−𝑛)𝐻²𝑐²16+2𝑛²𝐻²𝑅+2𝑛(1−𝑛)𝑐4𝑅∗1(3.17) and we have_𝑀𝑛𝑝𝑅^{2𝑖𝑗𝑘𝑙}+2𝑝𝑅^2𝑖𝑗−𝑅²+𝑛(3𝑝−2𝑛+2)𝑐4𝑅−𝑛(3𝑝−2𝑛)𝐻²+2𝑛₂𝑅(1−𝑛)𝑝𝑐4𝐻²+2𝑛³(1−𝑛)𝑐4𝐻²≥𝑛₂+𝑛³𝑝𝐻⁴−𝑛⁴𝐻⁴∗1(1−𝑛)(𝑛−𝑝−1)𝑐²16Vol(𝑀^𝑛).(3.18)

Corollary 3.1. Let 𝑀^𝑛 be an 𝑛-dimensional compact pseudo-umbilical submanifold of (ℝ^𝑛+𝑝,𝐷,𝑔=(1/2)𝐷²{(𝑥^𝐴)²}). Then _𝑀𝑛𝑝𝑅^{2𝑖𝑗𝑘𝑙}+2𝑝𝑅^2𝑖𝑗−𝑅²−𝑛(3𝑝−2𝑛)𝐻²𝑅+𝑛³𝑝𝐻⁴−𝑛⁴𝐻⁴∗1≥0.(3.19)

Proof. The Euclidean space (ℝ^𝑛+𝑝,𝐷,𝑔=(1/2)𝐷²{∑^{𝑛+𝑝𝐴=1}(𝑥^𝐴)²}) is a simply connected Hessian manifold of constant Hessian sectional curvature 0. Taking into account of Theorem A, we conclude the corollary.

Corollary 3.2. Let Ω be a domain in ℝ^𝑛+𝑝 given by 𝑥^𝑛+𝑝>𝑐2^{𝑛+𝑝−1}_𝐴=1𝑥^𝐴²,(3.20) where 𝑐 is a positive constant, and let 𝜑 be a smooth function on Ω defined by 𝜑=−1𝑐log𝑥^𝑛+𝑝−𝑐2^{𝑛+𝑝−1}_𝐴=1𝑥^𝐴².(3.21) Let 𝑀^𝑛 be an 𝑛-dimensional compact pseudo-umbilical submanifold of (Ω,𝐷,𝑔=𝐷²𝜑). Then Theorem A holds.

Proof. It is obvious that (Ω,𝐷,𝑔=𝐷²𝜑) is a simply connected Hessian manifold of positive constant Hessian sectional curvature 𝑐. As Riemannian manifold (Ω,𝑔) is isometric to the hyperbolic space (𝐻(−𝑐/4),𝑔) of constant sectional curvature −𝑐/4; 𝑔𝐻==1𝜉¹,…,𝜉^𝑛,𝜉^𝑛+𝑝∈ℝ^𝑛+𝑝∣𝜉^𝑛+𝑝>0,𝜉^𝑛+𝑝²^{𝑛+𝑝−1}_𝐴₌₁𝑑𝜉^𝐴²+4𝑐𝑑𝜉^𝑛+𝑝².(3.22) As a consequence of Theorem A, we conclude the proof.

On the other hand let us define 𝜑 as a smooth function on ℝ^𝑛+𝑝 as follows 𝜑=−1𝑐log^𝑛+𝑝_𝐴=1𝑒^{−𝑐𝑥𝐴}+1,(3.23) where 𝑐 is a negative constant. Then (ℝ^𝑛+𝑝,𝐷,𝑔=𝐷²𝜑) is a simply connected Hessian manifold of negative constant Hessian sectional curvature 𝑐. The Riemannian manifold (ℝ^𝑛+𝑝,𝑔) is isometric to a domain of the sphere ∑^{𝑛+𝑝+1𝑖=1}𝜉^2𝐴=−4/𝑐 defined by 𝜉_𝐴>0 for all 𝐴. Hence we acquire the following.

Corollary 3.3. Let 𝜑 be a smooth function on ℝ^𝑛+𝑝 defined by 𝜑=−1𝑐log^𝑛+𝑝_𝐴=1𝑒^{−𝑐𝑥𝐴}+1,(3.24) where 𝑐 is a negative constant and 𝑀^𝑛 be an 𝑛-dimensional compact pseudo-umbilical submanifold of (ℝ^𝑛+𝑝,𝐷,𝑔=𝐷²𝜑). Then _𝑀𝑛𝑝𝑅^{2𝑖𝑗𝑘𝑙}+2𝑝𝑅^2𝑖𝑗−𝑅²+𝑛(3𝑝+2𝑛−2)𝑐4𝑅−𝑛(3𝑝−2𝑛)𝐻²−2𝑛₂𝑅(1−𝑛)𝑝𝑐4𝐻²−2𝑛³(1−𝑛)𝑐4𝐻²≥𝑛₂+𝑛³𝑝𝐻⁴−𝑛⁴𝐻⁴∗1(𝑛−1)(𝑛−𝑝−1)𝑐²16Vol𝑀^𝑛.(3.25)

3.1. Applications in 3-Dimensional Spaces

Here we give some examples of the results indicated above.

Example 3.4. Let 𝑀² be a 2-dimensional compact pseudo-umbilical surface of (ℝ³,𝐷,𝑔=(1/2)𝐷²{(𝑥^𝐴)²}). Then _𝑀2𝑅^{2𝑖𝑗𝑘𝑙}+2𝑅^2𝑖𝑗−𝑅²−2𝐻²𝑅+6𝐻⁴0𝑥00083⪖0(3.26) also note that if the Ricci curvature tensor of the surface is given by 𝑅_𝑖𝑗=𝐾𝑔_𝑖𝑗, we may also compute the integral 𝑖 terms of Gaussian curvature 𝐾.

Example 3.5. Let Ω be a domain in ℝ³ given by 𝑥³>𝑐2𝑥¹²+𝑥²²,(3.27) where 𝑐 is a positive constant, and let 𝜑 be a smooth function on Ω defined by 𝜑=−1𝑐log𝑥³−𝑐2𝑥¹²+𝑥²².(3.28) Let 𝑀² be a 2-dimensional compact pseudo-umbilical surface of (Ω,𝐷,𝑔=𝐷²𝜑). Then _𝑀2𝑅^{2𝑖𝑗𝑘𝑙}+2𝑅^2𝑖𝑗−𝑅²−52𝑐𝑅+2𝐻²𝑅−3𝑐−4𝐻²0𝑥00083⪖0.(3.29)

Example 3.6. Let 𝜑 be a smooth function on ℝ³ defined by 𝑐𝜑=−1𝑐log³_𝐴=1𝑒^{−𝑐𝑥𝐴}+1,(3.30) where 𝑐 is a negative constant and 𝑀² be a 2-dimensional compact pseudo-umbilical surface of (ℝ³,𝐷,𝑔=𝐷²𝜑). Then _𝑀2𝑅^{2𝑖𝑗𝑘𝑙}+2𝑅^2𝑖𝑗−𝑅²+32𝑐𝑅+2𝐻²𝑅+3𝑐−4𝐻²⪖0𝑥000830(3.31) and (ℝ³,𝑔) is isometric to a domain of the sphere 𝕊³(√−4/𝑐).