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.

1. Introduction

Denote by the complex space form, i.e., a complex -dimensional Kähler manifold with constant holomorphic sectional curvature . A complete and simple connected complex space form is complex analytically isometric to a complex projective space if , a complex hyperbolic space if , and a complex Euclidean space if The complex projective and complex hyperbolic spaces are called nonflat complex space forms and denoted by . Let be a real hypersurface of a complex space form. In particular, if is an eigenvector of shape operator then is called a Hopf hypersurface. Since there are no Einstein real hypersurfaces in ([1, 2]), a natural question is whether there is a generalization of an Einstein metric in the real hypersurface of . 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 satisfieswhere and are the potential vector field and some constant, respectively. It is clear that a trivial Ricci soliton is an Einstein metric with zero or Killing. When the potential vector field is a gradient vector field, i.e., , where 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 (a Riemannian manifold with a function on ) (-)quasi-Einstein if it satisfies the equationfor some , where is a positive integer. denotes the Hessian of . Notice that Equation (2) recovers the gradient Ricci soliton when . A quasi-Einstein metric is an Einstein metric if is constant. We call a quasi-Einstein metric shrinking, steady, or expanding, respectively, when or . 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 is a -dimensional almost contact manifold, and a gradient Ricci solution is just a special quasi-Einstein metric with . 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 with , Theorem 1 and Theorem 2 improve the results of [4].

Also, we consider the real hypersurfaces with a quasi-Einstein metric of complex Euclidean space as in [4]. We first suppose that is a contact hypersurface of complex Euclidean space , i.e., , where is a smooth function.

Theorem 4. Let be a complete contact hypersurface of complex Euclidean space . If admits a quasi-Einstein metric, then is a sphere or a generalized cylinder .

For a general hypersurface of complex Euclidean space , we obtain the following.

Corollary 5. Let be a complete real hypersurface with of complex Euclidean space . If 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 in Some 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.

Let be a complex -dimensional Kähler manifold and be an immersed, without boundary, real hypersurface of with the induced metric . Denote by the complex structure on . There exists a local defined unit normal vector field on , and we write by the structure vector field of . An induced one-form is defined by , which is dual to . For any vector field on , the tangent part of is denoted by . Moreover, the following identities hold:where . By (3)–(5), we know that is an almost contact metric structure on .

Denote by the induced Riemannian connection and the shape operator on , respectively. Then the Gauss and Weingarten formulas are given bywhere is the connection on with respect to . Also, we have

In particular, is said to be a Hopf hypersurface if the structure vector field is an eigenvector of , i.e., , where

From now on, we always assume that the holomorphic sectional curvature of is constant . When , is a complex Euclidean space . When , is a nonflat complex space form, denoted by ; then, from (6), we know that the curvature tensor of is given byand the shape operator satisfiesfor any vector fields on . From (8), we get for the Ricci tensor of type :where denotes the mean curvature of (i.e., ). We denote the scalar curvature of , i.e.,

Now we suppose is an Hopf hypersurface. Differentiating covariantly gives

Using (9), we obtainfor any vector field . Since is self-adjoint, by taking the antisymmetry part of (12), we get the relation

As the tangent bundle can be decomposed as , where , the condition implies ; thus, we can pick up such that for some function on . Then from (13), we obtain

If , then which shows that is locally congruent to a horosphere in (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 can be expressed asfor any vector fields on .

Lemma 7. ([7]) For a quasi-Einstein , the following equations hold:

Applying Lemma 6, we obtain the following.

Lemma 8. For a quasi-Einstein Hopf real hypersurface of a complex space form , the following equation holds:

Proof. Replacing in (8) by , we haveBy Lemma 6, we getNow making use of (10), for any vector fields , we first computeBy (9), we thus obtainSince is Hopf, i.e., , taking the product of (20) with and using (22), we conclude thatMoreover, using (11), we computeSubstituting this into (23) and using (10), we arrive atMoreover, applying (13) in the above formula, we haveReplacing and by and , respectively, and using (13) again yield (18).

3. Proof of Theorem 1

In this section, we assume . Let be a Hopf hypersurface of , i.e., , then is constant due to [13], Theorem 2.1. We first consider , i.e., , then Equation (18) implies

Let be a principle vector field corresponding to principle curvature , then from (27), we know that is also a principle vector field with principle curvature . Thus, we see that the mean curvature must be zero, i.e., , which implies by the result of [14]. Hence, we obtain the following:

Proposition 9. An Hopf hypersurface of with does not admit a quasi-Einstein metric.

Next, we consider the case where . If has only one principle curvature in , the mean curvature is constant. From (26), we can obtain

Letting such that and taking , we arrive at . It is impossible.

Now choose such that with , so from (18), we have

Here we have used with followed from (14).

Moreover, inserting into Equation (29), we have

Now, we denote the roots of the polynomial by ; then, from the relation between the roots and coefficients, we obtain

As the proof of [4], Lemma 4.2, we can also get the following lemma.

Lemma 10. The mean curvature is constant.

Hence, taking in (26), we concludewhere

By taking the inner product of (32) with the principal vector , we obtain

If , then Differentiating this along any vector field gives

Since , i.e., for any vector fields , it follows from (35) that

Replacing and by and , respectively, implies

This implies since will yield ([14]). Thus, is constant and is Einstein, which is impossible. So , i.e., has at most two distinct constant principal curvatures . This shows that the scalar curvature is constant.

Using (32), we derive from (10) that

If , by (16) we havewhich, by taking the inner product with any vector field , yields

Here, we have used for some vector field . Otherwise, if for all , then is constant since , which is impossible as before.

Since the hypersurface has two distinct constant principle curvatures ( of multiplicity and of multiplicity ) it is easy to get that the mean curvature and the scalar curvature

Furthermore, since has only one eigenvalue in , we see from (14) that

By (41), the scalar curvature may be written as

Using (41) again and , we thus have

Since , we obtain

Inserting (42) and (44) into (40), we derive from (41) the following:which leads to The contradiction implies .

Since the scalar curvature is constant, by (16), we get . Because (42) and (44) still hold for , if , we obtain

This also yields .

Summarizing the above discussion, we thus assert the following:

Proposition 11. A hypersurface with in does not admit a quasi-Einstein metric.

Using Proposition 9 together with Proposition 11, we complete the proof of Theorem 1.

4. Proof of Theorem 2

In this section, we study a class of non-Hopf hypersurfaces with quasi-Einstein metric of nonflat complex space forms. Let be any regular curve. For , let be a totally geodesic complex hypersurface through the point which is orthogonal to the holomorphic plane spanned by and . Write . Such a construction asserts that is a real hypersurface of , which is called a ruled hypersurface. It is well known that the shape operator of is written as:where is a unit vector field orthogonal to , and are differentiable functions on . From (10), we have

From these equations, we know that the scalar curvature .

First, we assume and write

We know that the following relations are valid (see [15], Eq.(22), (17)):

On the other hand, the Codazzi Equation (9) implies that and using (47), we getwhich, by taking an inner product with , yields . Thus, we have

Furthermore, the following lemma holds:

Lemma 12. ([15]) For all , we have the following relations:

For , from (54), we know . Putting and in (20), we have

Since , we obtain

By (48) and (50), the inner product of (56) with gives

Similarly, putting and in (20), we obtain

The previous two formulas give

Now, putting and in (20) yields

Here, we have used (54) and .

Case 1. . Then, is constant and by (54). ThenMoreover, from (61), we haveThus, we may writeFor , since is constant, it follows from (16) thatBy the orthogonality of and , we obtainBecause , by (62), a direct computation impliesFor , it follows from (16) that or , i.e.This is impossible since does not need to be an Einstein hypersurface as in introduction.

Case 2. . Thus, by (61). Now, letting and in (20) givesMeanwhile, taking and in (20) and applying Lemma 12, we obtainComparing (70) with (71) givesOn the other hand, by using (69), we follow from Equation (61) thatThis means thatHence, for any ,By Lemma 12 and (54), we computeOn the other hand, using (49) and (50), it follows from Equation (2) thatCombining (76) with (77), we obtainThis equation has no solution for .
For the case , it is obvious that these equations including from (61) to (77) still hold; we thus complete the proof of Theorem 2.

5. Proofs of Theorem 4 and Corollary 5

In this section, we assume . Namely, is a complex Euclidean space .

Proof of Theorem 4. For a contact hypersurface, by [6], Lemma 3.1, we know that is Hopf and is constant. Therefore, we find that Equation (30) holds and can be simply presented as

This shows that is also constant, and further, the scalar curvature is constant. For , Equation (32) becomes

Taking an inner product of (80) with , then

Next, we decompose two cases.

Case 1. . We find by (81). Then, is a sphere as the proof of [6], Theorem 3.2.

Case 2. . If , then ; otherwise, is totally geodesic, which is impossible. In this case, is a generalized cylinder . Next, we assume , then or by (79). If , is , which fails to be1 a contact hypersurface. Thus, ; is a totally umbilical hypersurface. Consequently, it is a portion of a -dimensional sphere. Moreover, since , is compact (see [16]).

Proof of Corollary 5. If , Formula (80) becomesWhen , we have . Thus, Equation (37) holds. By (13), ; then, we get . Moreover, for any ; then, either or . If , then ; then, from (17), we find , which is a contradictory to the assumption. Thus, , i.e., is constant. That means that is Einstein and the scalar curvature by quasi-Einstein Equation (2). We complete the proof by [17], Theorem 7.3.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The second author is supported by the Natural Science Foundation of Beijing, China (Grant No. 1194025) and supported partially by the Science Foundation of China University of Petroleum at Beijing (No. 2462020XKJS02 and 2462020YXZZ004).