Abstract
In this note, we establish an integral inequality for compact and orientable real hypersurfaces in complex two-plane Grassmannians , involving the shape operator and the Reeb vector field . Moreover, this integral inequality is optimal in the sense that the real hypersurfaces attaining the equality are completely determined. As direct consequences, some new characterizations of the real hypersurfaces in with isometric Reeb flow can be presented.
1. Introduction
The complex two-plane Grassmannian consists of all complex two-dimensional linear subspaces in and is of complex dimension . It is known as the unique compact irreducible Hermitian symmetric space of rank two equipped with a Kähler structure and a quaternionic Kähler structure satisfying for (cf. [1, 2]). When , is isometric to the two-dimensional complex projective space with constant holomorphic sectional curvature , whereas when , can be identified with the real Grassmannian manifold of oriented two-dimensional linear subspace in (cf. [3, 4]). For the purpose of this note, we shall assume .
Let be a connected and orientable real hypersurface isometrically immersed in for and be a unit normal vector field along . Then, the almost contact structure vector field defined by is said to be the Reeb vector field. In particular, such a hypersurface is called Hopf if its shape operator satisfies with , where is the induced metric on . Moreover, we denote by the almost contact 3-structure vector fields, where for and is a canonical local basis of . For the real hypersurface in , there exist naturally two distributions, which we write as and , respectively.
The study of real hypersurfaces in initiated by Berndt and Suh [1, 5] is an attractive geometric topic, and many interesting results have been established in the last few decades; for details, see, e.g., [4, 6–16] and the references therein. In [1], Berndt and Suh considered the real hypersurfaces in for such that both and are invariant under the shape operator and they obtained the following well-known classification result:
Theorem 1. Let be a connected real hypersurface in , . Then, both and are invariant under the shape operator of if and only if one of the following holds: (A) is an open part of a tube around a totally geodesic in (B) is even, say , and is an open part of a tube around a totally geodesic in
Since then, a number of interesting characterization results related to these Hopf hypersurfaces of types and have been obtained. For instance, it was proved in [17] that a Hopf hypersurface in for must be the one of type if and only if the Reeb vector belongs to the orthogonal complement of . In addition, more characterizations of real hypersurfaces of type can be found in [18, 19]. On the other hand, for the real hypersurface of type , Berndt and Suh also gave a characterization under the assumption that the shape operator commutes with the structure tensor , i.e., . This is equivalent to the condition that the Reeb flow on is isometric, which means with the Lie derivative in the direction of . More precisely, it can be stated as follows:
Theorem 2. Let be a connected real hypersurface in , . Then, the Reeb flow on is isometric if and only if is an open part of a tube around a totally geodesic in .
In this note, motivated by this line, we establish an integral inequality for these compact real hypersurfaces in for , and its equality case provides a new characterization of real hypersurfaces with isometric Reeb flow.
Theorem 3. Let be a compact orientable real hypersurface in , . Then, in terms of the shape operator and the Reeb vector field of , we have the following integral inequality of Simons’ type:
where denotes the mean curvature of and is the tensorial norm with respect to the induced metric of .
Moreover, the equality in (1) holds if and only if has isometric Reeb flow and is a tube around a totally geodesic in .
Then, for closed orientable real hypersurfaces in with , Theorem 3 immediately gives the following rigidity theorem, involving the shape operator and the Reeb vector field .
Corollary 4. Let be a compact orientable real hypersurface in , . If on it holds then it must be the case that and is a tube around a totally geodesic in .
Remark 5. To obtain the inequality in (1), the key idea is to apply the classical Yano’s formula as it has been dealt with for hypersurfaces of other ambient spaces [20, 21]. Specifically, for such real hypersurfaces in with isometric Reeb flow, the Reeb vector filed belongs to the distribution (cf. [22]).
Remark 6. According to Corollary 4, it is known that, if holds on a closed minimal orientable real hypersurface in for , then and is a tube around a totally geodesic in with radius .
2. Preliminaries
In this section, we review some basic geometric properties of real hypersurfaces in complex two-plane Grassmannians and also derive some fundamental equations which shall be used in the proof of Theorem 3. More details can be found in the references [3, 23, 24].
Let be a connected and oriented real hypersurface isometrically immersed in equipped with a Kähler structure and a quaternionic Kähler structure not containing . Denote by the Levi-Civita connection of the Riemannian metric on and by the Levi-Civita connection of the induced metric denoted still by on , respectively. Then, for a unit normal vector field of , the formulas of Gauss and Weingarten are given by where denotes the shape operator of and are tangent vector fields on .
For any , we can decompose as where denotes a tensor field of type (1) on and is the 1-form over , corresponding to the Reeb vector field . Then, the almost contact metric structure induced from satisfies the following relations:
Choose to be a canonical local basis of , and for , put
This induces a local almost contact metric 3-structure on satisfying
From the relation , we further obtain where the index is taken modulo three.
Noting that , we have the relationships between these two almost contact metric structures and as below:
Moreover, for , the Gauss equation of in is given by
By contracting and in (10), we have the following expression of the Ricci tensor of : where is the mean curvature of the real hypersurface in .
3. Proof of Theorem 3
To complete the proof and obtain new characterizations, in this section, we first recall an important property related to the principal curvatures of such special real hypersurfaces of type . It is stated as follows (cf. [1, 22]).
Proposition 7. Let be the tube of radius around the totally geodesic in for . Choose to be the almost Hermitian structure such that . Then, has three (if ) or four (otherwise) distinct constant principal curvatures. Their values and corresponding principal curvature spaces and multiplicities are given in Table 1. denotes quaternionic span of the Reeb vector field .
Next, as a crucial step towards the proof of Theorem 3, we introduce the classical formula in Riemannian geometry, which was first proved by Yano [25] using tensor analysis. For reader’s convenience, we include a proof here (cf. [20, 21]).
Lemma 8. Let be a Riemannian manifold with Levi-Civita connection . Then, for any tangent vector field on , it holds that where is the Lie derivative of with respect to and denotes the length with respect to .
Proof. Choose to be an orthonormal basis of and assume that . Then, by adopting the usual notations for components of the covariant derivatives and the Riemannian curvature tensor, we have By using the Ricci identity , we get Similarly, straightforward calculations give From the above calculations, the assertion (12) follows immediately.
3.1. Completion of the Proof of Theorem 3.
With the help of the formulas of Gauss and Weingarten in (3), we derive from and (4) that for , it holds
Then, for the convenience of calculation, we choose to be an orthonormal basis of for any point . Using (16), by definition, we obtain which is equivalent to
From (5) and (16), it follows that
Noting that the structure tensor and the shape operator of is skew-symmetric and symmetric with respect to the induced metric on , respectively, from (16), we calculate
Moreover, by virtue of (11), we have
Thus, by putting in (12) and applying equations (18)–(21), we derive from Lemma 8 that
By the compactness of , we integrate the above equation and a divergence theorem for compact manifolds yields where . In particular, the above equality holds if and only if and , or equivalently, and . Hence, the assertion follows from the combination of Theorem 2 and Remark 5.
On the other hand, according to Table 1 in Proposition 7, a direct calculation gives us that in a tube of radius , it holds which implies that holds identically. In particular, it can be checked that the mean curvature vanishes if and only if , which belongs to the interval when .
This completes the proof of Theorem 3.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no competing interests.
Acknowledgments
This project was supported by grant of the NSFC (No. 11801011) and the Key Scientific Research Projects of Colleges and Universities of Henan Province (No. 23A110001).