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

Authors: Münevver Yildirim Yilmaz; Mehmet Bektaş
Journal: ISRN Geometry
 (2011)

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## 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.


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## Body

## 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].LetMn+p be an (n+p)-dimensional Hessian manifold of constant curvature c. Let Mn be an n-dimensional Riemannian manifold immersed in Mn+p. Let h be the second fundamental form of the immersion, and ξ the mean curvature vector. Denote by g the scalar product of Mn+p. If there exists a function λ on Mn such that(*)g(h(X,Y),ξ)=λg(X,Y)
for any tangent vector X,Y on Mn, then Mn is called a pseudo-umbilical submanifold of Mn+p. It is clear that λ≥0. If the mean curvature ξ=0 identically, then Mn is called a minimal submanifold of Mn+p.Every minimal submanifold ofMn+p is itself a pseudo umbilical submanifold of Mn+p. Cao [6] extended Bai’s well-known theorem to the case in which Mn+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.
LetMn be an n-dimensional compact pseudo-umbilical submanifold of (n+p)-dimensional Hessian manifold of constant Hessian sectional curvature c. Then
(1.1)∫Mn{p∑Rijkl2+2p∑Rij2-R2-n(3p+2n-2)c4R-n(3p-2n)H2R+2n2(1-n)pc4H2+2n3(1-n)c4H2+n3pH4-n4H4}*1≥n2(n-1)(n-p-1)c216VolMn,
where ∑Rijkl2 is the square length of the Riemannian curvature tensor, ∑Rij2 is the square length of the Ricci curvature tensor, R is the scalar curvature, and H is the mean curvature of Mn.We will use the same notation and terminologies as in [2] unless otherwise stated.LetMn+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,…,xn+p} with respect to D, that is, Ddxi=0.(i)
The Hessian metric,(1.2)gij=∂2u∂xi∂xj.(ii)
Letγ be a tensor field of type (1,2) defined by
(1.3)γ(X,Y)=∇XY-DXY,
where ∇ is the Riemannian connection for g. Then we have
(1.4)γjki=Γjki=12gir∂grj∂xk,γijk=12∂gij∂xk=12∂3u∂xi∂xj∂xk,γijk=γjik=γkji,
where Γjki are the Christoffel’s symbols of ∇.(iii)
Define a tensor fieldS of type (1,3)by
(1.5)S=Dγ
and call it the Hessian curvature tensor for (D,g). Then we have
(1.6)Sjkli=∂γjli∂xk,Sijkl=12∂4u∂xi∂xj∂xk∂xl-12grs∂3u∂xi∂xk∂xr-∂3u∂xj∂xl∂xs,Sijkl=Silkj=Skjil=Sjilk=Sklij.(iv)
The Riemannian curvature tensor for∇,
(1.7)Rjkli=γrkiγjlr-γrliγjkr,Rijkl=12(Sjikl-Sijkl)(see [2]).Definition 1.1.
LetSjlik 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
(1.8)ξ(ξ)ik=Sjlikξjl.
Then ξ is a symmetric operator, [2].Definition 1.2.
For a nonzero contravariant symmetric tensorξx of degree 2 at x, we set
(1.9)h(ξx)=〈ξ(ξx),ξx〉〈ξx,ξx〉
and call it the Hessian sectional curvature in the direction ξx, [2].Theorem 1.3.
Let(Mn+p,D,g) be a Hessian manifold of dimension ≥2. If the Hessian sectional curvature h(ξx) 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
(1.10)Sijkl=c2(gijgkl+gilgkj)(see [2]).Corollary 1.4.
If a Hessian manifold(Mn+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 constantc, a Hessian manifold with constant Hessian sectional curvature c. We now recall the following result due to Shima and Yagi [7]. Let (Mn+p,D,g) be a simply connected Hessian manifold. If g is complete, then (Mn+p,D,g) is isomorphic to (Ω,D̃,D̃2φ), where Ω is a convex domain in ℝn+p, D̃ is the canonical flat connection on ℝn+p, and φ is a smooth convex function on Ω.(A)
Casec=0It is obvious that the Euclidean space(ℝn+p,D̃,g=(1/2)D̃2{∑A=1n+p(xA)2}) is a simply connected Hessian manifold of constant Hessian sectional curvature 0 [1].(B)
Casec>0Theorem 1.5.
LetΩ be a domain in ℝn+p given by
(1.11)xn+p>c2∑A=1n+p-1(xA)2,
where c is a positive constant, and let φ be a smooth function on Ω defined by
(1.12)φ=-1clog⁡{xn+p-c2∑A=1n+p-1(xA)2}.
Then (Ω,D̃,g=D̃2φ) 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;
(1.13)H={(ξ1,…,ξn,ξn+p)∈Rn+p∣ξn+p>0},g=1(ξn+p)2{∑A=1n+p-1(dξA)2+4c(dξn+p)2}.(C)
Casec<0Theorem 1.6.
Letφ be a smooth function on ℝn+p defined by
(1.14)φ=-1clog⁡(∑A=1n+pe-cxA+1),
where c is a negative constant. Then (ℝn+p,D̃,g=D̃2φ) is a simply connected Hessian manifold of negative constant Hessian sectional curvature c. The Riemannian manifold (ℝn+p,g) is isometric to a domain of the sphere∑i=1n+p+1ξA2=-4/c defined by ξA>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 framese1,…,en+p in Mn+p such that restricted to Mn,e1,…,en are tangent to Mn. Let w1,…,wn+p be its dual frame field. Then the structure equations of Mn+p are given by(2.1)dwA=-∑wAB∧wB,wAB+wBA=0,dwAB=-∑wAC∧wCB+12∑R¯ABCDwC∧wD.
We restrict these forms to Mn, then we have (2.2)wα=0,wαi=∑hijαwj,hijα=hjiα,dwij=-∑wik∧wkj+12∑Rijklwk∧wl,Rijkl=-c4(δikδjl-δilδjk)+∑(hikαhjlα-hilαhjkα),
where Rijkl are the components of the curvature tensor of Mn.(2.3)dwαβ=-∑wαγ∧wγβ+12∑Rαβklwk∧wl,Rαβkl=∑(hikαhilβ-hikβjhilα).We call(2.4)h=∑hijαwiwjeα
the second fundamental form of the immersed manifold Mn. Denote by(2.5)S=∑(hijα)2
the square length of h, ξ=(1/n)∑trHαeα the mean curvature vector and H=(1/n)∑(trHα)2 the mean curvature of Mn, respectively. Here tr is the trace of the matrix Hα=(hijα). Now let en+1 be parallel to ξ. Then we have (2.6)tr⁡Hn+1=nH,tr⁡Hα=0,α≠n+1.
Let hijkα and hijklα denote the covariant derivative and the second covariant derivative of hijα, respectively, defined by (2.7)∑hijkαwk=dhijα-∑hikαwkj-∑hjkαwki-∑hijβwβα,∑hijklαwl=dhijkα-∑hijlαwlk-∑hilkαwlj-∑hljkαwli∑hijkβwβα.
Then we have(2.8)hijkα-hikjα=0,hijklα-hijlkα=∑himαRmjkl+∑hmjαRmikl-∑hijβRαβkl.
The Laplacian Δhijα of hijα is defined by Δhijα=∑hijkkα. By a direct calculation we have (2.9)12ΔS=∑(hijkα)2+∑hijαΔhijα=∑(hijkα)2+∑hijαhkkijα+∑hijαhmkαRmijk+∑hijαhmiαRmkjk-∑hijαhkiβRαβjk.


## 3. Proof of Theorem A

From (*) and (2.6), we have (3.1)g(h(ei,ej),Hen+1)=H2δij,
therefore, (3.2)∑hijαhkkijα=nHΔH,∑(hijkα)2≥∑(hiikn+1)2=n∑(ΔiH)2=n|∇H|2.
It is obvious that (3.3)12ΔH2=HΔH+|∇H|2
and, therefore, (3.4)∑(hijkα)2+∑hijαhkkijα≥n|∇H|2+nHΔH=12nΔH2.
On the other hand, from (2.2)(3.5)∑hijαhmkαRmijk=12(hijαhmkα-hmjαhikα)Rmijk=12∑{Rimjk+c4(δijδmk-δmjδik)}Rmijk=-12∑Rmijk2-c4R,∑hijαhmiαRmkjk=∑{(1-n)c4δmj+nH2δmj-Rmj}Rmj=-∑Rmj2+(1-n)c4R+nH2R.
From (2.3), we have(3.6)∑hijαhkiβRαβjk=∑hijαhljαhkiβhlkβ-∑hijαhlkαhikβhljβ
while(3.7)∑hijαhljαhkiβhlkβ=∑{(1-n)c4δil+nH2δil-Ril}2=n(1-n)2c216-2n2(1-n)c4H2-2(1-n)c4R-2nH2R+n3H4+Ril2.
Let (3.8)Sα=∑i,j(hijα)2,
then we have (3.9)S=∑αSα.
Since (3.10)S2=(∑αSα)2=∑αSα2+2∑α<βSαSβ,∑α<β(Sα-Sβ)2=(p-1)∑αSα2-2∑α<βSαSβ⩾0
it follows that (3.11)(p-1)∑αSα2⩾2∑α<βSαSβ=S2-∑αSα2,
that is (3.12)∑αSα2≥1pS2,
Since (3.13)∑i,j,l,k(hijαhlkα)2=∑i,j,l,k(∑βhikβhljβ)2=∑α,βhijαhijβhlkαhlkβ=∑α,β(∑i,jhijαhijβ)2≥∑α(∑i,jhijαhijα)2=∑αSα2
we have(3.14)∑α,βhijαhlkαhikβhljβ≥∑α,βhijαhlkαhikβhljβ-12∑i,j,l,k(∑αhijαhlkα)2-12∑i,j,l,k(∑βhikβhljβ)2+∑αSα2=-12∑{∑α(hijαhlkα-hikαhljα)}2+∑αSα2=-12∑{Riljk+c4(δijδlk-δikδlj)}2+∑αSα2≥-12∑Riljk2-2c4R-n(1-n)c216+1pS2.
From (2.9), (3.4)–(3.14), we have(3.15)12ΔS≥12nΔH2-∑Rijkl2-2∑Rij2-34cnR+3nH2R-n2(1-n)c216-2n2(1-n)c4H2-n3H4+1pS2.
Since Mn is compact and (3.16)S=n(1-n)c4+n2H2-R
we have(3.17)∫Mn{p∑Rijkl2+2p∑Rij2+34cnpR-3npH2R+n2(1-n)c216p+2n2(1-n)c4H2p+n3H4p-S2p∑Rijkl2∑Rijkl2c216}*1,S2=n2(1-n)2c216+n4H4+R2+2n3(1-n)H2c4-2n2H2R-2n(1-n)c4R,∫Mn{p∑Rijkl2+2p∑Rij2+34ncRp-3nH2Rp+n2(1-n)c216p+2n2(1-n)c4H2p+n3H4p-n2(1-n)2c216-n4H4-R2-2n3(1-n)H2c216+2n2H2R+2n(1-n)c4R}*1
and we have(3.18)∫Mn{p∑Rijkl2+2p∑Rij2-R2+n(3p-2n+2)c4R-n(3p-2n)H2R+2n2(1-n)pc4H2+2n3(1-n)c4H2+n3pH4-n4H4}*1≥n2(1-n)(n-p-1)c216Vol(Mn).Corollary 3.1.
LetMn be an n-dimensional compact pseudo-umbilical submanifold of (ℝn+p,D̃,g=(1/2)D̃2{(xA)2}). Then
(3.19)∫Mn{p∑Rijkl2+2p∑Rij2-R2-n(3p-2n)H2R+n3pH4-n4H4}*1≥0.Proof.
The Euclidean space(ℝn+p,D̃,g=(1/2)D̃2{∑A=1n+p(xA)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.
LetΩ be a domain in ℝn+p given by
(3.20)xn+p>c2∑A=1n+p-1(xA)2,
where c is a positive constant, and let φ be a smooth function on Ω defined by
(3.21)φ=-1clog⁡{xn+p-c2∑A=1n+p-1(xA)2}.
Let Mn be an n-dimensional compact pseudo-umbilical submanifold of (Ω,D̃,g=D̃2φ). Then Theorem A holds.Proof.
It is obvious that(Ω,D̃,g=D̃2φ) 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;
(3.22)H={(ξ1,…,ξn,ξn+p)∈Rn+p∣ξn+p>0},g=1(ξn+p)2{∑A=1n+p-1(dξA)2+4c(dξn+p)2}.
As a consequence of Theorem A, we conclude the proof.On the other hand let us defineφ as a smooth function on ℝn+p as follows (3.23)φ=-1clog⁡(∑A=1n+pe-cxA+1),
where c is a negative constant. Then (ℝn+p,D̃,g=D̃2φ) is a simply connected Hessian manifold of negative constant Hessian sectional curvature c. The Riemannian manifold (ℝn+p,g) is isometric to a domain of the sphere ∑i=1n+p+1ξA2=-4/c defined by ξA>0 for all A. Hence we acquire the following.Corollary 3.3.
Letφ be a smooth function on ℝn+p defined by
(3.24)φ=-1clog⁡(∑A=1n+pe-cxA+1),
where c is a negative constant and Mn be an n-dimensional compact pseudo-umbilical submanifold of (ℝn+p,D̃,g=D̃2φ). Then
(3.25)∫Mn{p∑Rijkl2+2p∑Rij2-R2+n(3p+2n-2)c4R-n(3p-2n)H2R-2n2(1-n)pc4H2-2n3(1-n)c4H2+n3pH4-n4H4∑}*1≥n2(n-1)(n-p-1)c216VolMn.


### 3.1. Applications in 3-Dimensional Spaces

Here we give some examples of the results indicated above.Example 3.4.
LetM2 be a 2-dimensional compact pseudo-umbilical surface of (ℝ3,D̃,g=(1/2)D̃2{(xA)2}). Then
(3.26)∫M2∑Rijkl2+2∑Rij2-R2-2H2(R+6H4)⪖0
also note that if the Ricci curvature tensor of the surface is given by Rij=Kgij, we may also compute the integral i terms of Gaussian curvature K.Example 3.5.
LetΩ be a domain in ℝ3 given by
(3.27)x3>c2[(x1)2+(x2)2],
where c is a positive constant, and let φ be a smooth function on Ω defined by
(3.28)φ=-1clog⁡{x3-c2[(x1)2+(x2)2]}.
Let M2 be a 2-dimensional compact pseudo-umbilical surface of (Ω,D̃,g=D̃2φ). Then
(3.29)∫M2∑Rijkl2+2∑Rij2-R2-52cR+2H2(R-3c-4H2)⪖0.Example 3.6.
Letφ be a smooth function on ℝ3 defined by c(3.30)φ=-1clog⁡(∑A=13e-cxA+1),
where c is a negative constant and M2 be a 2-dimensional compact pseudo-umbilical surface of (ℝ3,D̃,g=D̃2φ). Then
(3.31)∫M2∑Rijkl2+2∑Rij2-R2+32cR+2H2(R+3c-4H2)⪖0
and (ℝ3,g) is isometric to a domain of the sphere 𝕊3(-4/c).


## 3.1. Applications in 3-Dimensional Spaces

Here we give some examples of the results indicated above.Example 3.4.
LetM2 be a 2-dimensional compact pseudo-umbilical surface of (ℝ3,D̃,g=(1/2)D̃2{(xA)2}). Then
(3.26)∫M2∑Rijkl2+2∑Rij2-R2-2H2(R+6H4)⪖0
also note that if the Ricci curvature tensor of the surface is given by Rij=Kgij, we may also compute the integral i terms of Gaussian curvature K.Example 3.5.
LetΩ be a domain in ℝ3 given by
(3.27)x3>c2[(x1)2+(x2)2],
where c is a positive constant, and let φ be a smooth function on Ω defined by
(3.28)φ=-1clog⁡{x3-c2[(x1)2+(x2)2]}.
Let M2 be a 2-dimensional compact pseudo-umbilical surface of (Ω,D̃,g=D̃2φ). Then
(3.29)∫M2∑Rijkl2+2∑Rij2-R2-52cR+2H2(R-3c-4H2)⪖0.Example 3.6.
Letφ be a smooth function on ℝ3 defined by c(3.30)φ=-1clog⁡(∑A=13e-cxA+1),
where c is a negative constant and M2 be a 2-dimensional compact pseudo-umbilical surface of (ℝ3,D̃,g=D̃2φ). Then
(3.31)∫M2∑Rijkl2+2∑Rij2-R2+32cR+2H2(R+3c-4H2)⪖0
and (ℝ3,g) is isometric to a domain of the sphere 𝕊3(-4/c).



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