# Quasi-Einstein Hypersurfaces of Complex Space Forms

Authors: Xuehui Cui; Xiaomin Chen
Journal: Advances in Mathematical Physics
 (2020)

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## Abstract

Based on a well-known fact that there are no Einstein hypersurfaces in a nonflat complex space form, in this article, we study the quasi-Einstein condition, which is a generalization of an Einstein metric, on the real hypersurface of a nonflat complex space form. For the real hypersurface with quasi-Einstein metric of a complex Euclidean space, we also give a classification. Since a gradient Ricci soliton is a special quasi-Einstein metric, our results improve some conclusions of Cho and Kimura.


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## Body

## 1. Introduction

Denote byM~n the complex space form, i.e., a complex n-dimensional Kähler manifold with constant holomorphic sectional curvature c. A complete and simple connected complex space form is complex analytically isometric to a complex projective space ℂPn if c>0, a complex hyperbolic space ℂHn if c<0, and a complex Euclidean space ℂn if c=0. The complex projective and complex hyperbolic spaces are called nonflat complex space forms and denoted by M~nc. Let M be a real hypersurface of a complex space form. In particular, if ξ is an eigenvector of shape operator A then M is called a Hopf hypersurface. Since there are no Einstein real hypersurfaces in M~nc ([1, 2]), a natural question is whether there is a generalization of an Einstein metric in the real hypersurface of M~nc. As a generalization of the Einstein metric, the first author studied the real hypersurfaces admitting Miao-Tam critical metrics of complex space forms (see [3]). A Ricci soliton is a Riemannian metric, which satisfies(1)12LVg+Ric−λg=0,where V and λ are the potential vector field and some constant, respectively. It is clear that a trivial Ricci soliton is an Einstein metric with V zero or Killing. When the potential vector field V is a gradient vector field, i.e., V=∇f, where f is a smooth function, then it is called a gradient Ricci soliton. Cho and Kimura [4, 5] proved that a Hopf hypersurface and a non-Hopf hypersurface in a nonflat complex space form do not admit a gradient Ricci soliton. Moreover, this is true when the gradient Ricci soliton is replaced by a compact Ricci soliton due to Perelman’s result ([6], Remark 3.2).As another interesting generalization of an Einstein metric, a quasi-Einstein metric has been considered (see [7, 8]). We call a triple M,g,f,m (a Riemannian manifold M,g with a function f on M) (m-)quasi-Einstein if it satisfies the equation(2)Ric+Hessf−1mdf⊗df=λg,for some λ∈ℝ, where m is a positive integer. Hessf denotes the Hessian of f. Notice that Equation (2) recovers the gradient Ricci soliton when m=∞. A quasi-Einstein metric is an Einstein metric if f is constant. We call a quasi-Einstein metric shrinking, steady, or expanding, respectively, when λ<0,λ=0 or λ>0. For a general manifold, quasi-Einstein metrics have been studied in depth, and some rigid properties and gap results were obtained (cf. [7, 9, 10]). On the other hand, we also notice that for the odd-dimensional manifold, Ghosh in [11] studied quasi-Einstein contact metric manifolds. As is well known that a real hypersurface of M~nc is a 2n−1-dimensional almost contact manifold, and a gradient Ricci solution is just a special quasi-Einstein metric with m=∞. From this observation, we are inspired to improve the results of [4] and study the quasi-Einstein condition for the real hypersurface of a complex space form.In this article, we first study the quasi-Einstein metric on Hopf hypersurfaces in complex space forms as well as a class of non-Hopf hypersurfaces in nonflat complex space forms.Theorem 1.
There are no quasi-Einstein Hopf real hypersurfaces in a nonflat complex space form.Theorem 2.
There are no quasi-Einstein ruled hypersurfaces in a nonflat complex space form.Remark 3.
Since a gradient Ricci soliton is a special quasi-Einstein metric withm=∞, Theorem 1 and Theorem 2 improve the results of [4].Also, we consider the real hypersurfaces with a quasi-Einstein metric of complex Euclidean spaceℂn as in [4]. We first suppose that M is a contact hypersurface of complex Euclidean space ℂn, i.e., ϕA+Aϕ=2σϕ, where σ>0 is a smooth function.Theorem 4.
LetM2n−1 be a complete contact hypersurface of complex Euclidean space ℂn. If M admits a quasi-Einstein metric, then M is a sphere S2n−1 or a generalized cylinder ℝn×Sn−1.For a general hypersurface of complex Euclidean spaceℂn, we obtain the following.Corollary 5.
LetM2n−1 be a complete real hypersurface with Aξ=0 of complex Euclidean space ℂn. If M admits a nonsteady quasi-Einstein metric, it is a hypersphere, hyperplane, or developable hypersurface.In order to prove these conclusions, we need to recall some basic concepts and related results inSome Basic Concepts and Related Results. In Proof of Theorem 1 and Proof of Theorem 2, we give, respectively, the proofs of Theorem 1 and Theorem 2, and the real hypersurface with a quasi-Einstein metric of complex Euclidean spaces is presented in Proofs of Theorem 4 and Corollary 5.


## 2. Some Basic Concepts and Related Results

LetM~n,g~ be a complex n-dimensional Kähler manifold and M be an immersed, without boundary, real hypersurface of M~n with the induced metric g. Denote by J the complex structure on M~n. There exists a local defined unit normal vector field N on M, and we write ξ=−JN by the structure vector field of M. An induced one-form η is defined by η·=g~J·,N, which is dual to ξ. For any vector field X on M, the tangent part of JX is denoted by ϕX=JX−ηXN. Moreover, the following identities hold:(3)ϕ2=−Id+η⊗ξ,η∘ϕ=0,ϕ∘ξ=0,ηξ=1,(4)gϕX,ϕY=gX,Y−ηXηY,(5)gX,ξ=ηX,where X,Y∈XM. By (3)–(5), we know that ϕ,η,ξ,g is an almost contact metric structure on M.Denote by∇,A the induced Riemannian connection and the shape operator on M, respectively. Then the Gauss and Weingarten formulas are given by(6)∇~XY=∇XY+gAX,YN,∇~XN=−AX,where ∇~ is the connection on M~n with respect to g~. Also, we have(7)∇XϕY=ηYAX−gAX,Yξ,∇Xξ=ϕAX.In particular,M is said to be a Hopf hypersurface if the structure vector field ξ is an eigenvector of A, i.e., Aξ=αξ, where α=ηAξ.From now on, we always assume that the holomorphic sectional curvature ofM~n is constant c. When c=0, M~n is a complex Euclidean space ℂn. When c≠0, M~n is a nonflat complex space form, denoted by M~nc; then, from (6), we know that the curvature tensor R of M is given by(8)RX,YZ=c4gY,ZX−gX,ZY+gϕY,ZϕX−gϕX,ZϕY+2gX,ϕYϕZ+gAY,ZAX−gAX,ZAY,and the shape operator A satisfies(9)∇XAY−∇YAX=c4ηXϕY−ηYϕX−2gϕX,Yξ,for any vector fields X,Y,Z on M. From (8), we get for the Ricci tensor Q of type 1,1:(10)QX=c42n+1X−3ηXξ+hAX−A2X,where h denotes the mean curvature of M (i.e., h=traceA). We denote S the scalar curvature of M, i.e., S=traceQ.Now we supposeM is an Hopf hypersurface. Differentiating Aξ=αξ covariantly gives(11)∇XAξ=Xαξ+αϕAX−AϕAX.Using (9), we obtain(12)∇ξAX=Xαξ+αϕAX−AϕAX+c4ϕX,for any vector field X. Since ∇ξA is self-adjoint, by taking the antisymmetry part of (12), we get the relation(13)2AϕAX−c2ϕX=Xαξ−ηX∇α+αϕA+AϕX.As the tangent bundleTM can be decomposed as TM=ℝξ⊕D, where D=X∈TM:X⊥ξ, the condition Aξ=αξ implies AD⊂D; thus, we can pick up X∈D such that AX=μX for some function μ on M. Then from (13), we obtain(14)2μ−αAϕX=μα+c2ϕX.If2μ=α, then c=−4μ2, which shows that M is locally congruent to a horosphere in ℂHn(see [12]).Next, we recall two important lemmas for a Riemannian manifold satisfying quasi-Einstein Equation (2).Lemma 6.
([11]) For a quasi-Einstein metric, the curvature tensor R can be expressed as(15)RX,Y∇f=∇YQX−∇XQY−λmXfY−YfX+1mXfQY−YfQX,for any vector fields X,Y on M.Lemma 7.
([7]) For a quasi-Einstein M2n−1,g,f,m, the following equations hold:(16)12∇S=m−1mQ∇f+1mS−2n−2λ∇f,(17)12ΔS−m+22mg∇f,∇S=−m−1mRic−S2n−1g2−m+2n−2m2n−1×S−2n−1λS−2n−22n−1m+2n−2λ.Applying Lemma6, we obtain the following.Lemma 8.
For a quasi-Einstein Hopf real hypersurfaceM2n−1 of a complex space form M~n, the following equation holds:(18)αϕA2+A2ϕ=α2+cAϕ+ϕA+h−α2cϕ.Proof.
ReplacingZ in (8) by ∇f, we have(19)RX,Y∇f=c4YfX−XfY+ϕYfϕX−ϕXfϕY+2gX,ϕYϕ∇f+AYfAX−AXfAY.
By Lemma6, we get(20)∇YQX−∇XQY+1mXfQY−YfQX=c4−λmYfX−XfY+c4ϕYfϕX−ϕXfϕY+2gX,ϕYϕ∇f+AYfAX−AXfAY.
Now making use of (10), for any vector fields X,Y, we first compute(21)∇YQX=c4−3∇YηXξ−3ηX∇Yξ+YhAX+h∇YAX−∇YAAX−A∇YAX=−3c4gϕAY,Xξ+ηXϕAY+YhAX+h∇YAX−∇YAAX−A∇YAX.
By (9), we thus obtain(22)∇XQY−∇YQX=−3c4gϕAX+AϕX,Yξ+ηYϕAX−ηXϕAY+XhAY−YhAX+hc4ηXϕY−ηYϕX−2gϕX,Yξ−∇XAAY+∇YAAX−c4ηXAϕY−ηYAϕX−2gϕX,YAξ.
SinceM is Hopf, i.e., Aξ=αξ, taking the product of (20) with ξ and using (22), we conclude that(23)−1mXfηQY−YfηQX+c4−λmYfηX−XfηY+αAYfηX−AXfηY−3c4gϕAX+AϕX,Y+αXhηY−YhηX−h−α2cgϕX,Y−g∇XAAY−∇YAAX,ξ=0.
Moreover, using (11), we compute(24)g∇XAAY−∇YAAX,ξ=gXαξ+αϕAX−AϕAX,AY−gYαξ+αϕAY−AϕAY,AX=αXαηY−YαηX+2αgϕAX,AY−gAϕAX,AY+gAϕAY,AX.
Substituting this into (23) and using (10), we arrive at(25)−1mc2n−1+hα−α2−c4+λmXfηY−YfηX+αAYfηX−AXfηY−3c4gϕAX+AϕX,Y+αXhηY−YhηX−h−α2cgϕX,Y−αXαηY−YαηX−2αgϕAX,AY+gAϕAX,AY−gAϕAY,AX=0.
Moreover, applying (13) in the above formula, we have(26)−1mc2n−1+hα−α2−c4+λmXfηY−YfηX+αAYfηX−AXfηY−c2gϕAX+AϕX,Y+αXhηY−YhηX−h−α2cgϕX,Y−α2XαηY−YαηX−αgϕAX,AY+g−12ηX∇α+12αAϕX,AY−g−12ηY∇α+12αAϕY,AX=0.
ReplacingX and Y by ϕX and ϕY, respectively, and using (13) again yield (18).


## 3. Proof of Theorem1

In this section, we assumec≠0. Let M2n−1 be a Hopf hypersurface of M~nc, i.e., Aξ=αξ, then α is constant due to [13], Theorem 2.1. We first consider α=0, i.e., Aξ=0, then Equation (18) implies(27)Aϕ+ϕA+hϕ=0.LetX∈D be a principle vector field corresponding to principle curvature μ, then from (27), we know that ϕX is also a principle vector field with principle curvature −h−μ. Thus, we see that the mean curvature h must be zero, i.e., Aϕ+ϕA=0, which implies c=0 by the result of [14]. Hence, we obtain the following:Proposition 9.
An Hopf hypersurface ofM~nc,c≠0 with Aξ=0 does not admit a quasi-Einstein metric.Next, we consider the case whereα≠0. If A has only one principle curvature α/2 in D, the mean curvature h=nα is constant. From (26), we can obtain(28)−1mc2n−1+hα−α2−c4+λmXfηY−YfηX+αAYfηX−AXfηY−c2gϕAX+AϕX,Y−n−12αcgϕX,Y=0.LettingX∈D such that AX=α/2X and taking Y=ϕX, we arrive at nc=0. It is impossible.Now chooseX∈D such that AX=μX with μ≠α/2, so from (18), we have(29)αμ2+μ~2=α2+cμ+μ~+h−α2c.Here we have usedAϕX=μ~ϕX with μ~=μα+c/2/2μ−α followed from (14).Moreover, insertingμ~=μα+c/2/2μ−α into Equation (29), we have(30)4αμ4−4c+2α2μ3+4αc+4α3−4hcμ2+4hcα−2α2c−c2μ+34αc2+α3c−hcα2=0.Now, we denote the roots of the polynomial byf1,f2,f3,f4; then, from the relation between the roots and coefficients, we obtain(31)f1+f2+f3+f4=c+2α2α,f1f2+f1f3+f1f4+f2f3+f2f4+f3f4=αc+α3−hcα,f1f2f3+f1f2f4+f2f3f4=−4hcα−2α2c−c24α,f1f2f3f4=3c2+4α2c−4hcα16.As the proof of [4], Lemma 4.2, we can also get the following lemma.Lemma 10.
The mean curvatureh is constant.Hence, takingY=ξ in (26), we conclude(32)θ∇f−ξfξ+ααξfξ−A∇f=0.where(33)θ≔−1mc2n−1+hα−α2−c4+λm.By taking the inner product of (32) with the principal vector X∈D, we obtain(34)αμ−θXf=0.Ifαμ−θ≠0, then ∇f=ξfξ. Differentiating this along any vector field Z gives(35)∇Z∇f=Zξfξ+ξfϕAZ.Sinced2f=0, i.e., g∇Z∇f,W=g∇W∇f,Z for any vector fields Z,W, it follows from (35) that(36)gZξfξ+ξfϕAZ,W=gWξfξ+ξfϕAW,Z.ReplacingZ and W by ϕZ and ϕW, respectively, implies(37)ξfϕAZ+AϕZ=0.This impliesξf=0 since ϕA+Aϕ=0 will yield c=0 ([14]). Thus, f is constant and M is Einstein, which is impossible. So αμ−θ=0, i.e., M has at most two distinct constant principal curvatures α,μ=θ/α. This shows that the scalar curvature S is constant.Using (32), we derive from (10) that(38)Q∇f=c42n+1∇f−3ξfξ+hA∇f−A2∇f=c42n+1∇f−3ξfξ+hα−μξfξ+hμ∇f−Aα−μξfξ+μ∇f=c42n+1+hμ−μ2∇f−α2−μ2+3c4−hα−μξfξ.Ifm≠1, by (16) we have(39)α2−μ2+3c4−hα−μξfξ=c42n+1+hμ−μ2+1m−1S−2n−2λ∇f,which, by taking the inner product with any vector field X∈D, yields(40)c42n+1+hμ−μ2+1m−1S−2n−2λ=0.Here, we have usedgX,∇f≠0 for some vector field X∈D. Otherwise, if gX,∇f=0 for all X∈D, then f is constant since ξf=0, which is impossible as before.Since the hypersurfaceM has two distinct constant principle curvatures (α of multiplicity 1 and μ of multiplicity 2n−2) it is easy to get that the mean curvature h=α+2n−2μ and the scalar curvature S=cn2−1+2α2n−2μ+2n−22n−3μ2.Furthermore, sinceA has only one eigenvalue μ=θ/α in D, we see from (14) that(41)2μ2−2αμ−c2=0.By (41), the scalar curvature S may be written as(42)S=n−1c24n−1+22n−1αμ.Using (41) again and h=α+2n−2μ, we thus have(43)θ=−1mc2n−1+hα−α2−c4+λm=−n−1mc2+2μα−c4+λm.Sinceμ−θ/α=0, we obtain(44)λ=m+2n−2μα+n−12c+mc4.Inserting (42) and (44) into (40), we derive from (41) the following:(45)0=c44n−2+2n−2αμ+n−1m−1c24n−1+22n−1αμ−2λ=c22n−1+2n−2αμ+n−1m−1c22n+1−2m−1μα−mc2=nc21+2n−2m−1,which leads to nc=0. The contradiction implies m=1.Since the scalar curvature is constant, by (16), we get S=2n−2λ. Because (42) and (44) still hold for m=1, if S=2n−2λ, we obtain(46)n−1c24n−1+22n−1αμ=2n−22n−1μα+2n−14c.This also yieldsnc=0.Summarizing the above discussion, we thus assert the following:Proposition 11.
A hypersurface withAξ=αξ,α≠0 in M~nc does not admit a quasi-Einstein metric.Using Proposition9 together with Proposition 11, we complete the proof of Theorem 1.


## 4. Proof of Theorem2

In this section, we study a class of non-Hopf hypersurfaces with quasi-Einstein metric of nonflat complex space forms. Letγ:I⟶M~nc be any regular curve. For t∈I, let M~tnc be a totally geodesic complex hypersurface through the point γt which is orthogonal to the holomorphic plane spanned by γ′t and Jγ′t. Write M=M~tnc:t∈I. Such a construction asserts that M is a real hypersurface of M~nc, which is called a ruled hypersurface. It is well known that the shape operator A of M is written as:(47)Aξ=αξ+βWβ≠0,AW=βξ,AZ=0foranyZ⊥ξ,W,where W is a unit vector field orthogonal to ξ, and α,β are differentiable functions on M. From (10), we have(48)Qξ=12n−1c−β2ξ,(49)QW=142n+1c−β2W,(50)QZ=142n+1cZforanyZ⊥ξ,W.From these equations, we know that the scalar curvatureS=n2−1c−2β2.First, we assumen≥3 and write(51)T1M=X∈TM:ηX=gX,W=gX,ϕW=0.We know that the following relations are valid (see [15], Eq.(22), (17)):(52)ϕWβ=β2+c4andXβ=0forallX∈T1M.On the other hand, the Codazzi Equation (9) implies that ∇ξAW−∇WAξ=c/4ϕW, and using (47), we get(53)∇ξAW−∇WAξ=∇ξAW−A∇ξW−∇WAξ+A∇Wξ=ξβ−Wαξ+β2ϕW−A∇ξW−WβW−β∇WW,which, by taking an inner product with W, yields Wβ=0. Thus, we have(54)∇β=β2+c/4ϕW+ξβξ.Furthermore, the following lemma holds:Lemma 12.
([15]) For all Z∈T1M, we have the following relations:(55)∇WϕW=c4β−βW,∇WW=β−c4βϕW,∇ZϕW=c4βZ,∇ZW=−c4βϕZ,∇ϕWW=0,∇ϕWϕW=0.ForZ∈T1M, from (54), we know Zβ=0. Putting Y=ξ and X=Z in (20), we have(56)∇ξQZ−∇ZQξ+1mZfQξ−ξfQZ=c4−λmξfZ−Zfξ+AξfAZ−AZfAξ.SinceZβ=0, we obtain(57)∇ZQξ−∇ξQZ=−c42n+1∇ξZ+Q∇ξZ.By (48) and (50), the inner product of (56) with ξ gives(58)Zf1m12n−1c−β2+c4−λm=0.Similarly, puttingX=Z and Y=W in (20), we obtain(59)Zf1m142n+1c−β2+c4−λm=0.The previous two formulas give(60)Zf=0.Now, puttingY=ξ and X=W in (20) yields(61)Wf1m12n−1c−β2+c4−λm−β2=0,ξfc4−λm+1m142n+1c−β2−β2=−ξβ2.Here, we have used (54) and g∇ξW,W=g∇ξW,ξ=0.Case 1.
1/m1/2n−1c−β2+c/4−λ/m−β2=0. Then, β is constant and β2=−c/4 by (54). Then(62)λ=142n−1+2mc.
Moreover, from (61), we have(63)ξf=0.
Thus, we may write(64)∇f=WfW+ϕWfϕW.
Form≠1, since S=n2−1/2c is constant, it follows from (16) that(65)Wf142n+1c−β2W+ϕWf142n+1cϕW=−1m−1S−2n−2λWfW+ϕWfϕW.
By the orthogonality ofϕW and W, we obtain(66)Wf142n+1c−β2+1m−1S−2n−2λ=0,ϕWf142n+1c+1m−1S−2n−2λ=0.
Becausem>1, by (62), a direct computation implies(67)Wf=ϕWf=0.
Form=1, it follows from (16) that ∇f=0 or S=2n−2λ, i.e.(68)n2−12c=12n−12n+1c.
This is impossible sinceM does not need to be an Einstein hypersurface as in introduction.Case 2.
1/m1/2n−1c−β2+c/4−λ/m−β2≠0. Thus, Wf=0 by (61). Now, letting X=ξ and Y=ϕW in (20) gives(69)ξfc4−λm+14m2n+1c=0,(70)ϕWf1m12n−1c−β2+c4−λm+ϕWβ2−142n+1cβ+12n−1c−β2β=0.
Meanwhile, takingX=ϕW and Y=W in (20) and applying Lemma 12, we obtain(71)ϕWf1m142n+1c−β2+c−λm+β2c4β−β+ϕWβ2=0.
Comparing (70) with (71) gives(72)ϕWf3m+3+4mβ=0.
On the other hand, by using (69), we follow from Equation (61) that(73)ξf=ξβ21+1/m.
This means that(74)∇f=−4mβ3m+1ϕW+ξβ21+1/mξ.
Hence, for anyX,Y∈TM,(75)HessfX,Y=g∇X∇f,Y=−4m3m+1XβgY,ϕW+βg∇XϕW,Y+mm+1Xξβ2ηY+ξβ2gϕAX,Y.
By Lemma12 and (54), we compute(76)HessfW,W=−4m3m+1c4−β2,HessfϕW,ϕW=−4m3m+1β2+c4.
On the other hand, using (49) and (50), it follows from Equation (2) that(77)HessfW,W=λ−142n+1c−β2,HessfϕW,ϕW=λ+1m4mβ23m+12−142n+1c.
Combining (76) with (77), we obtain(78)15m2+22m−9=0.
This equation has no solution form≥1.
For the casen=2, it is obvious that these equations including from (61) to (77) still hold; we thus complete the proof of Theorem 2.


## 5. Proofs of Theorem4 and Corollary 5

In this section, we assumec=0. Namely, M~n is a complex Euclidean space ℂn.(79)αμ2μ−α2=0.Proof of Theorem 4.
For a contact hypersurface, by [6], Lemma 3.1, we know thatM is Hopf and α=ηAξ is constant. Therefore, we find that Equation (30) holds and can be simply presented asThis shows thatμ is also constant, and further, the scalar curvature is constant. For c=0, Equation (32) becomes(80)λ−hα+α2m∇f−ξfξ+ααξfξ−A∇f=0.Taking an inner product of (80) with X∈D, then(81)λ−hα+α2m−αμXf=0.Next, we decompose two cases.Case 1.
λ≠mαμ+hα−α2. We find ∇f=ξfξ by (81). Then, M is a sphere as the proof of [6], Theorem 3.2.Case 2.
λ=mαμ+hα−α2. If α=0, then μ≠0; otherwise, M is totally geodesic, which is impossible. In this case, M is a generalized cylinder ℝn×Sn−1. Next, we assume α≠0, then μ=0 or μ=α by (79). If μ=0, M is ℝ2n−1, which fails to be1 a contact hypersurface. Thus, μ=α; M is a totally umbilical hypersurface. Consequently, it is a portion of a 2n−1-dimensional sphere. Moreover, since λ=m+2n−2α2>0, M is compact (see [16]).Proof of Corollary 5.
IfAξ=0, Formula (80) becomes(82)λmξfξ−∇f=0.
Whenλ≠0, we have ∇f=ξfξ. Thus, Equation (37) holds. By (13), AϕA=0; then, we get ξfA2ϕZ=0. Moreover, ξfA2Z=0 for any Z∈TM; then, either A=0 or ξf=0. If A=0, then Q=0; then, from (17), we find λ=0, which is a contradictory to the assumption. Thus, ξf=0, i.e., f is constant. That means that M is Einstein and the scalar curvature S=2n−1λ by quasi-Einstein Equation (2). We complete the proof by [17], Theorem 7.3.



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