# Some New Characterizations of Real Hypersurfaces with Isometric Reeb Flow in Complex Two-Plane Grassmannians

Authors: Dehe Li; Bo Li; Lifen Zhang
Journal: Advances in Mathematical Physics
 (2023)

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## Abstract

In this note, we establish an integral inequality for compact and orientable real hypersurfaces in complex two-plane GrassmanniansG2ℂm+2, involving the shape operator A and the Reeb vector field ξ. 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 G2ℂm+2 with isometric Reeb flow can be presented.


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## Body

## 1. Introduction

The complex two-plane GrassmannianG2ℂm+2 consists of all complex two-dimensional linear subspaces in ℂm+2 and is of complex dimension 2m. It is known as the unique compact irreducible Hermitian symmetric space of rank two equipped with a Kähler structure J and a quaternionic Kähler structure J=SpanJ1,J2,J3 satisfying JJa=JaJ for a∈1,2,3 (cf. [1, 2]). When m=1, G2ℂ3 is isometric to the two-dimensional complex projective space ℂP28 with constant holomorphic sectional curvature 8, whereas when m=2, G2ℂ4 can be identified with the real Grassmannian manifold G2+ℝ6 of oriented two-dimensional linear subspace in ℝ6 (cf. [3, 4]). For the purpose of this note, we shall assume m≥3.LetM be a connected and orientable real hypersurface isometrically immersed in G2ℂm+2 for m≥3 and N be a unit normal vector field along M. Then, the almost contact structure vector field ξ defined by ξ=−JN is said to be the Reeb vector field. In particular, such a hypersurface is called Hopf if its shape operator A satisfies Aξ=αξ with α=gAξ,ξ, where g is the induced metric on M. Moreover, we denote by ξ1,ξ2,ξ3 the almost contact 3-structure vector fields, where ξa=−JaN for a∈1,2,3 and Ja is a canonical local basis of J. For the real hypersurface M in G2ℂm+2, there exist naturally two distributions, which we write as ξ=Spanξ and D⊥=Spanξ1,ξ2,ξ3, respectively.The study of real hypersurfaces inM∧mc 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 G2ℂm+2 for m≥3 such that both ξ and D⊥ are invariant under the shape operator A and they obtained the following well-known classification result:Theorem 1.
LetM be a connected real hypersurface in G2ℂm+2, m≥3. Then, both ξ and D⊥ are invariant under the shape operator of M if and only if one of the following holds:
(A)
M is an open part of a tube around a totally geodesic G2ℂm+1 in G2ℂm+2(B)
m is even, say m=2n, and M is an open part of a tube around a totally geodesic ℍPn in G2ℂm+2Since then, a number of interesting characterization results related to these Hopf hypersurfaces of typesA and B have been obtained. For instance, it was proved in [17] that a Hopf hypersurface in G2ℂm+2 for m≥3 must be the one of type B if and only if the Reeb vector ξ belongs to the orthogonal complement D of D⊥. In addition, more characterizations of real hypersurfaces of type B can be found in [18, 19]. On the other hand, for the real hypersurface M of type A, Berndt and Suh also gave a characterization under the assumption that the shape operator A commutes with the structure tensor ϕ, i.e., Aϕ=ϕA. This is equivalent to the condition that the Reeb flow on M is isometric, which means Lξg=0 with Lξ the Lie derivative L in the direction of ξ. More precisely, it can be stated as follows:Theorem 2.
LetM be a connected real hypersurface in G2ℂm+2, m≥3. Then, the Reeb flow on M is isometric if and only if M is an open part of a tube around a totally geodesic G2ℂm+1 in G2ℂm+2.In this note, motivated by this line, we establish an integral inequality for these compact real hypersurfaces inG2ℂm+2 for m≥3, and its equality case provides a new characterization of real hypersurfaces with isometric Reeb flow.Theorem 3.
LetM be a compact orientable real hypersurface in G2ℂm+2, m≥3. Then, in terms of the shape operator A and the Reeb vector field ξ of M, we have the following integral inequality of Simons’ type:
(1)∫M4m+HgAξ,ξ−A2dVM≤0,where H=trA denotes the mean curvature of M and · is the tensorial norm with respect to the induced metric g of M.
Moreover, the equality in (1) holds if and only if M has isometric Reeb flow and M is a tube around a totally geodesic G2ℂm+1 in G2ℂm+2.Then, for closed orientable real hypersurfaces inG2ℂm+2 with m≥3, Theorem 3 immediately gives the following rigidity theorem, involving the shape operator A and the Reeb vector field ξ.Corollary 4.
LetM be a compact orientable real hypersurface in G2ℂm+2, m≥3. If on M it holds
(2)A2−HgAξ,ξ≤4m,then it must be the case that A2−HgAξ,ξ=4m and M is a tube around a totally geodesic G2ℂm+1 in G2ℂm+2.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 G2ℂm+2m≥3 with isometric Reeb flow, the Reeb vector filed ξ belongs to the distribution D⊥ (cf. [22]).Remark 6.
According to Corollary4, it is known that, if A2≤4m holds on a closed minimal orientable real hypersurface M in G2ℂm+2 for m≥3, then A2=4m and M is a tube around a totally geodesic G2ℂm+1 in G2ℂm+2 with radius r=2/2arctan3/2m−1.


## 2. Preliminaries

In this section, we review some basic geometric properties of real hypersurfaces in complex two-plane GrassmanniansG2ℂm+2 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].LetM be a connected and oriented real hypersurface isometrically immersed in G2ℂm+2 equipped with a Kähler structure J and a quaternionic Kähler structure J not containing J. Denote by ∇¯ the Levi-Civita connection of the Riemannian metric g on G2ℂm+2 and by ∇ the Levi-Civita connection of the induced metric denoted still by g on M, respectively. Then, for a unit normal vector field N of M, the formulas of Gauss and Weingarten are given by
(3)∇¯XY=∇XY+gAX,YN,∇¯XN=−AX,where A denotes the shape operator of M and X,Y are tangent vector fields on M.For anyX∈TM, we can decompose JX∈TG2ℂm+2 as
(4)JX=ϕX+ηXN,JN=−ξ,ηX=gX,ξ,where ϕ denotes a tensor field of type (1) on M and η is the 1-form over M, corresponding to the Reeb vector field ξ. Then, the almost contact metric structure ϕ,ξ,η,g induced from J satisfies the following relations:
(5)ϕ2X=−X+ηXξ,ϕξ=0,gϕX,Y=−gX,ϕY,ηξ=1,gϕX,ϕY=gX,Y−ηXηY.ChooseJ1,J2,J3 to be a canonical local basis of J, and for a∈1,2,3, put
(6)JaX=ϕaX+ηaXN,JaN=−ξa,ηaX=gX,ξa.This induces a local almost contact metric 3-structureϕa,ξa,ηa,g on M satisfying
(7)ϕa2X=−X+ηaXξa,ϕaξa=0,gϕaX,Y=−gX,ϕaY,ηaξa=1,gϕaX,ϕaY=gX,Y−ηaXηaY.From the relationJaJa+1=Ja+2=−Ja+1Ja, we further obtain
(8)ϕaξa+1=ξa+2=−ϕa+1ξa,ϕaϕa+1X=ϕa+2X+ηa+1Xξa,ϕa+1ϕaX=−ϕa+2X+ηaXξa+1,where the index is taken modulo three.Noting thatJJa=JaJ, we have the relationships between these two almost contact metric structures ϕ,ξ,η,g and ϕa,ξa,ηa,g as below:
(9)ϕaϕX−ηXξa=ϕϕaX−ηaXξ,ϕξa=ϕaξ.Moreover, forX,Y,Z∈TM, the Gauss equation of M in G2ℂm+2 is given by
(10)RX,YZ=gY,ZX−gX,ZY+gϕY,ZϕX−gϕX,ZϕY−2gϕX,YϕZ+∑a=13gϕaY,ZϕaX−gϕaX,ZϕaY−2gϕaX,YϕaZ+∑a=13gϕaϕY,ZϕaϕX−gϕaϕX,ZϕaϕY−∑a=13ηYηaZϕaϕX−ηXηaZϕaϕY−∑a=13ηXgϕaϕY,Z−ηYgϕaϕX,Zξa+gAY,ZAX−gAX,ZAY.By contractingY and Z in (10), we have the following expression of the Ricci tensor of M:
(11)RicX,Y=4m+7gX,Y−3ηXηY+∑a=13ηaϕXηaϕY−3ηaXηaY−ηaξgϕX,ϕaY−ηaξηaYηX+HgAX,Y−gA2X,Y,where H=trA is the mean curvature of the real hypersurface M in G2ℂm+2.


## 3. Proof of Theorem3

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 typeA. It is stated as follows (cf. [1, 22]).Proposition 7.
LetM be the tube of radius 0<r<π/22 around the totally geodesic G2ℂm+1 in G2ℂm+2 for m≥3. Choose J1∈J to be the almost Hermitian structure such that JN=J1N. Then, M has three (if r=π/42) or four (otherwise) distinct constant principal curvatures. Their values and corresponding principal curvature spaces and multiplicities are given in Table 1. ℍξ denotes quaternionic span of the Reeb vector field ξ.Table 1
Principal curvatures and corresponding eigenspaces.
Principal curvatureEigenspaceMultiplicity22cot22rSpanξ112cot2rSpanξ2,ξ32−2tan2rXX⊥ℍξ,JX=J1X2m−20XX⊥ℍξ,JX=−J1X2m−2Next, as a crucial step towards the proof of Theorem3, 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.
LetM,g be a Riemannian manifold with Levi-Civita connection ∇. Then, for any tangent vector field X on M, it holds that
(12)div∇XX−divXX=RicX,X+12LXg2−∇X2−divX2,where LXg is the Lie derivative of g with respect to X and · denotes the length with respect to g.Proof.
Chooseei to be an orthonormal basis of M and assume that X=∑iXiei. Then, by adopting the usual notations for components of the covariant derivatives and the Riemannian curvature tensor, we have
(13)div∇XX=∑i,jX,ijXi,j=∑i,jX,ijjXi+∑i,jX,ijX,ji.
By using the Ricci identityX,kli=X,lki+∑jXjRjikl, we get
(14)∑i,jX,ijjXi=∑i,jX,jijXi+∑i,jXiXjRij=∑i,jX,jijXi+RicX,X.
Similarly, straightforward calculations give(15)divdivXX=∑i,jX,jjXi,i=∑i,jX,jijXi+∑i,jX,jjX,ii=∑i,jX,jijXi+divX2,12LXg2=∑i,jX,ijX,ji+∑i,jX,ji2=∑i,jX,ijX,ji+∇X2.
From the above calculations, the assertion (12) follows immediately.


### 3.1. Completion of the Proof of Theorem3.

With the help of the formulas of Gauss and Weingarten in (3), we derive from ∇¯J=0 and (4) that for X∈TM, it holds
(16)∇Xξ=−∇¯XJN−gAX,ξN=−∇¯XJN−J∇¯XN−gAX,ξN=JAX−gAX,ξN=ϕAX.Then, for the convenience of calculation, we chooseeii=14m−1 to be an orthonormal basis of TxM for any point x∈M. Using (16), by definition, we obtain
(17)Lξg2=∑i,j=14m−1g∇eiξ,ej+gei,∇ejξ2=∑i,j=14m−1gϕA−Aϕei,ej2,which is equivalent to
(18)Lξg2=ϕA−Aϕ2.From (5) and (16), it follows that
(19)∇ξ2=∑i=14m−1g∇eiξ,∇eiξ=∑i=14m−1gϕAei,ϕAei=A2−gA2ξ,ξ.Noting that the structure tensorϕ and the shape operator A of M is skew-symmetric and symmetric with respect to the induced metric g on M, respectively, from (16), we calculate
(20)divξ=∑i=14m−1g∇eiξ,ei=∑i=14m−1gϕAei,ei=0.Moreover, by virtue of (11), we have
(21)Ricξ,ξ=4m+1−4∑a=13ηaξ2+HgAξ,ξ−gA2ξ,ξ.Thus, by puttingX=ξ in (12) and applying equations (18)–(21), we derive from Lemma 8 that
(22)div∇ξξ=4m+1+HgAξ,ξ−A2+12ϕA−Aϕ2−4∑a=13ηaξ2.By the compactness ofM, we integrate the above equation and a divergence theorem for compact manifolds yields
(23)0=∫M4m+1+HgAξ,ξ−A2+12ϕA−Aϕ2−4∑a=13ηaξ2dVM≥∫M4m+HgAξ,ξ−A2dVM,where 0≤∑a=13ηaξ2≤1. In particular, the above equality holds if and only if Aϕ=ϕA and ∑a=13ηaξ2=1, or equivalently, Lξg=0 and ξ∈D⊥. Hence, the assertion follows from the combination of Theorem 2 and Remark 5.On the other hand, according to Table1 in Proposition 7, a direct calculation gives us that in a tube of radius 0<r<π/22, it holds
(24)A2=8cot222r+4cot22r+4m−1tan22r,H=22cot22r+22cot2r−22m−1tan2r,gAξ,ξ=22cot22r,which implies that A2=4m+HgAξ,ξ holds identically. In particular, it can be checked that the mean curvature H vanishes if and only if r=2/2arctan3/2m−1, which belongs to the interval 0,π/22 when m≥3.This completes the proof of Theorem3.


## 3.1. Completion of the Proof of Theorem3.

With the help of the formulas of Gauss and Weingarten in (3), we derive from ∇¯J=0 and (4) that for X∈TM, it holds
(16)∇Xξ=−∇¯XJN−gAX,ξN=−∇¯XJN−J∇¯XN−gAX,ξN=JAX−gAX,ξN=ϕAX.Then, for the convenience of calculation, we chooseeii=14m−1 to be an orthonormal basis of TxM for any point x∈M. Using (16), by definition, we obtain
(17)Lξg2=∑i,j=14m−1g∇eiξ,ej+gei,∇ejξ2=∑i,j=14m−1gϕA−Aϕei,ej2,which is equivalent to
(18)Lξg2=ϕA−Aϕ2.From (5) and (16), it follows that
(19)∇ξ2=∑i=14m−1g∇eiξ,∇eiξ=∑i=14m−1gϕAei,ϕAei=A2−gA2ξ,ξ.Noting that the structure tensorϕ and the shape operator A of M is skew-symmetric and symmetric with respect to the induced metric g on M, respectively, from (16), we calculate
(20)divξ=∑i=14m−1g∇eiξ,ei=∑i=14m−1gϕAei,ei=0.Moreover, by virtue of (11), we have
(21)Ricξ,ξ=4m+1−4∑a=13ηaξ2+HgAξ,ξ−gA2ξ,ξ.Thus, by puttingX=ξ in (12) and applying equations (18)–(21), we derive from Lemma 8 that
(22)div∇ξξ=4m+1+HgAξ,ξ−A2+12ϕA−Aϕ2−4∑a=13ηaξ2.By the compactness ofM, we integrate the above equation and a divergence theorem for compact manifolds yields
(23)0=∫M4m+1+HgAξ,ξ−A2+12ϕA−Aϕ2−4∑a=13ηaξ2dVM≥∫M4m+HgAξ,ξ−A2dVM,where 0≤∑a=13ηaξ2≤1. In particular, the above equality holds if and only if Aϕ=ϕA and ∑a=13ηaξ2=1, or equivalently, Lξg=0 and ξ∈D⊥. Hence, the assertion follows from the combination of Theorem 2 and Remark 5.On the other hand, according to Table1 in Proposition 7, a direct calculation gives us that in a tube of radius 0<r<π/22, it holds
(24)A2=8cot222r+4cot22r+4m−1tan22r,H=22cot22r+22cot2r−22m−1tan2r,gAξ,ξ=22cot22r,which implies that A2=4m+HgAξ,ξ holds identically. In particular, it can be checked that the mean curvature H vanishes if and only if r=2/2arctan3/2m−1, which belongs to the interval 0,π/22 when m≥3.This completes the proof of Theorem3.



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*Source: 2347915-2023-03-10.xml*