In this note, we establish an integral inequality for compact and orientable real hypersurfaces in complex two-plane Grassmannians G₂(ℂ^m+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 G₂(ℂ^m+2) with isometric Reeb flow can be presented.

1. Introduction

The complex two-plane Grassmannian G₂(ℂ^m+2) consists of all complex two-dimensional linear subspaces in ℂ^m+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 satisfying JJ_a = J_aJ for a ∈ {1, 2, 3} (cf. [1, 2]). When m = 1, G₂(ℂ³) is isometric to the two-dimensional complex projective space ℂP²(8) with constant holomorphic sectional curvature 8, whereas when m = 2, G₂(ℂ⁴) can be identified with the real Grassmannian manifold of oriented two-dimensional linear subspace in ℝ⁶ (cf. [3, 4]). For the purpose of this note, we shall assume m ≥ 3.

Let M be a connected and orientable real hypersurface isometrically immersed in G₂(ℂ^m+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 = g(Aξ, ξ), where g is the induced metric on M. Moreover, we denote by {ξ₁, ξ₂, ξ₃} the almost contact 3-structure vector fields, where ξ_a = −J_aN for a ∈ {1, 2, 3} and {J_a} is a canonical local basis of . For the real hypersurface M in G₂(ℂ^m+2), there exist naturally two distributions, which we write as [ξ] = Span{ξ} 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 G₂(ℂ^m+2) for m ≥ 3 such that both [ξ] and are invariant under the shape operator A and they obtained the following well-known classification result:

Theorem 1. Let M be a connected real hypersurface in G₂(ℂ^m+2), m ≥ 3. Then, both [ξ] and 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 G₂(ℂ^m+1) in G₂(ℂ^m+2)
(B)
m is even, say m = 2n, and M is an open part of a tube around a totally geodesic ℍPⁿ in G₂(ℂ^m+2)

Since then, a number of interesting characterization results related to these Hopf hypersurfaces of types (A) and (B) have been obtained. For instance, it was proved in [17] that a Hopf hypersurface in G₂(ℂ^m+2) for m ≥ 3 must be the one of type (B) if and only if the Reeb vector ξ belongs to the orthogonal complement of . 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 with the Lie derivative in the direction of ξ. More precisely, it can be stated as follows:

Theorem 2. Let M be a connected real hypersurface in G₂(ℂ^m+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 G₂(ℂ^m+1) in G₂(ℂ^m+2).

In this note, motivated by this line, we establish an integral inequality for these compact real hypersurfaces in G₂(ℂ^m+2) for m ≥ 3, and its equality case provides a new characterization of real hypersurfaces with isometric Reeb flow.

Theorem 3. Let M be a compact orientable real hypersurface in G₂(ℂ^m+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)

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 G₂(ℂ^m+1) in G₂(ℂ^m+2).

Then, for closed orientable real hypersurfaces in G₂(ℂ^m+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. Let M be a compact orientable real hypersurface in G₂(ℂ^m+2), m ≥ 3. If on M it holds

(2)

then it must be the case that ‖A‖² − Hg(Aξ, ξ) = 4m and M is a tube around a totally geodesic G₂(ℂ^m+1) in G₂(ℂ^m+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 G₂(ℂ^m+2)(m ≥ 3) with isometric Reeb flow, the Reeb vector filed ξ belongs to the distribution (cf. [22]).

Remark 6. According to Corollary 4, it is known that, if ‖A‖² ≤ 4m holds on a closed minimal orientable real hypersurface M in G₂(ℂ^m+2) for m ≥ 3, then ‖A‖² = 4m and M is a tube around a totally geodesic G₂(ℂ^m+1) in G₂(ℂ^m+2) with radius .

2. Preliminaries

In this section, we review some basic geometric properties of real hypersurfaces in complex two-plane Grassmannians G₂(ℂ^m+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].

Let M be a connected and oriented real hypersurface isometrically immersed in G₂(ℂ^m+2) equipped with a Kähler structure J and a quaternionic Kähler structure

not containing J. Denote by

the Levi-Civita connection of the Riemannian metric g on G₂(ℂ^m+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)

where A denotes the shape operator of M and X, Y are tangent vector fields on M.

For any X ∈ TM, we can decompose JX ∈ TG₂(ℂ^m+2) as

(4)

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)

Choose {J₁, J₂, J₃} to be a canonical local basis of

, and for a ∈ {1, 2, 3}, put

(6)

This induces a local almost contact metric 3-structure (ϕ_a, ξ_a, η_a, g) on M satisfying

(7)

From the relation J_aJ_a+1 = J_a+2 = −J_a+1J_a, we further obtain

(8)

where the index is taken modulo three.

Noting that JJ_a = J_aJ, we have the relationships between these two almost contact metric structures (ϕ, ξ, η, g) and (ϕ_a, ξ_a, η_a, g) as below:

(9)

Moreover, for X, Y, Z ∈ TM, the Gauss equation of M in G₂(ℂ^m+2) is given by

(10)

By contracting Y and Z in (10), we have the following expression of the Ricci tensor of M:

(11)

where H = trA is the mean curvature of the real hypersurface M in G₂(ℂ^m+2).

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 (A). It is stated as follows (cf. [1, 22]).

Proposition 7. Let M be the tube of radius around the totally geodesic G₂(ℂ^m+1) in G₂(ℂ^m+2) for m ≥ 3. Choose to be the almost Hermitian structure such that JN = J₁N. Then, M has three (if ) 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 curvature	Eigenspace	Multiplicity
	Span{ξ₁}	1
	Span{ξ₂, ξ₃}	2
	{X\|X⊥ℍξ, JX = J₁X}	2m − 2
0	{X\|X⊥ℍξ, JX = −J₁X}	2m − 2

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 (M, g) be a Riemannian manifold with Levi-Civita connection ∇. Then, for any tangent vector field X on M, it holds that

(12)

where

is the Lie derivative of g with respect to X and ‖·‖ denotes the length with respect to g.

Proof. Choose {e_i} to be an orthonormal basis of M and assume that X = ∑_iXⁱe_i. Then, by adopting the usual notations for components of the covariant derivatives and the Riemannian curvature tensor, we have

(13)

By using the Ricci identity , we get

(14)

Similarly, straightforward calculations give

(15)

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 X ∈ TM, it holds

(16)

Then, for the convenience of calculation, we choose

to be an orthonormal basis of T_xM for any point x ∈ M. Using (16), by definition, we obtain

(17)

which is equivalent to

(18)

From (5) and (16), it follows that

(19)

Noting that the structure tensor ϕ 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)

Moreover, by virtue of (11), we have

(21)

Thus, by putting X = ξ in (12) and applying equations (18)–(21), we derive from Lemma 8 that

(22)

By the compactness of M, we integrate the above equation and a divergence theorem for compact manifolds yields

(23)

where

. In particular, the above equality holds if and only if Aϕ = ϕA 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

(24)

which implies that ‖A‖² = 4m + Hg(Aξ, ξ) holds identically. In particular, it can be checked that the mean curvature H vanishes if and only if

, which belongs to the interval

when m ≥ 3.

This completes the proof of Theorem 3.

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).

Open Research

Data Availability

No data were used to support this study.

References

1 Berndt J. and Suh Y. J., Real hypersurfaces in complex two-plane Grassmannians, Monatshefte für Mathematik. (1999) 127, no. 1, 1–14, https://doi.org/10.1007/s006050050018, 2-s2.0-0002211195.
10.1007/s006050050018
Web of Science® Google Scholar
2 Hwang D. H., Lee H., and Woo C., Semi-parallel symmetric operators for Hopf hypersurfaces in complex two-plane Grassmannians, Monatshefte für Mathematik. (2015) 177, no. 4, 539–550, https://doi.org/10.1007/s00605-015-0778-8, 2-s2.0-84937974226.
10.1007/s00605-015-0778-8
Web of Science® Google Scholar
3 Suh Y. J., Real hypersurfaces in complex two-plane Grassmannians with harmonic curvature, Journal de Mathématiques Pures et Appliquées. (2013) 100, no. 1, 16–33, https://doi.org/10.1016/j.matpur.2012.10.010, 2-s2.0-84878346959.
10.1016/j.matpur.2012.10.010
Web of Science® Google Scholar
4 Wang Y., Nonexistence of Hopf hypersurfaces in complex two-plane Grassmannians with GTW parallel normal Jacobi operator, The Rocky Mountain Journal of Mathematics. (2019) 49, no. 7, 2375–2393, https://doi.org/10.1216/RMJ-2019-49-7-2375.
10.1216/RMJ-2019-49-7-2375
Web of Science® Google Scholar
5 Berndt J. and Suh Y. J., Hypersurfaces in noncompact complex Grassmannians of rank two, International Journal of Mathematics. (2012) 23, no. 10, article 1250103, https://doi.org/10.1142/S0129167X12501030, 2-s2.0-84869455643.
10.1142/S0129167X12501030
Web of Science® Google Scholar
6 Choe S. H. and Suh Y. J., Real minimal hypersurfaces in complex two-plane Grassmannians, Acta Mathematica Hungarica. (2011) 131, no. 1-2, 174–192, https://doi.org/10.1007/s10474-010-0019-5, 2-s2.0-79952573928.
10.1007/s10474-010-0019-5
Web of Science® Google Scholar
7 Choi Y. S. and Suh Y. J., Real hypersurfaces with η-parallel shape operator in complex two-plane Grassmannians, Bulletin of the Australian Mathematical Society. (2007) 75, no. 1, 1–16, https://doi.org/10.1017/S0004972700038934.
10.1017/S0004972700038934
Web of Science® Google Scholar
8 Jeong I., Kimura M., Lee H., and Suh Y. J., Real hypersurfaces in complex two-plane Grassmannians with generalized Tanaka-Webster Reeb parallel shape operator, Monatshefte für Mathematik. (2013) 171, no. 3-4, 357–376, https://doi.org/10.1007/s00605-013-0475-4, 2-s2.0-84881614544.
10.1007/s00605-013-0475-4
Web of Science® Google Scholar
9 Jeong I., Lee H., and Suh Y. J., Real hypersurfaces in complex two-plane Grassmannians with D^⊥-parallel shape operator, Results in Mathematics. (2013) 64, no. 3-4, 331–342, https://doi.org/10.1007/s00025-013-0317-7, 2-s2.0-84888022263.
10.1007/s00025-013-0317-7
Web of Science® Google Scholar
10 Lee R. H. and Loo T. H., Hopf hypersurfaces in complex Grassmannians of rank two, Results in Mathematics. (2017) 71, no. 3-4, 1083–1107, https://doi.org/10.1007/s00025-016-0601-4, 2-s2.0-84986252200.
10.1007/s00025-016-0601-4
Web of Science® Google Scholar
11 Lee H., Pak E., and Suh Y. J., Hopf hypersurfaces in complex two-plane Grassmannians with generalized Tanaka–Webster D-parallel shape operator, Acta Mathematica Sinica. (2017) 33, no. 1, 61–70, https://doi.org/10.1007/s10114-016-4738-6, 2-s2.0-84989201925.
10.1007/s10114-016-4738-6
Web of Science® Google Scholar
12 Lee H., Woo C., and Suh Y. J., Quadratic Killing normal Jacobi operator for real hypersurfaces in complex Grassmannians of rank 2, Journal of Geometry and Physics. (2021) 160, article 103975, https://doi.org/10.1016/j.geomphys.2020.103975.
10.1016/j.geomphys.2020.103975
Web of Science® Google Scholar
13 Li D. and Zhai S., Real hypersurfaces in complex Grassmannians of rank two, Mathematics. (2021) 9, no. 24, https://doi.org/10.3390/math9243238.
10.3390/math9243238
PubMed Web of Science® Google Scholar
14 Wang Y., Some recurrent normal Jacobi operators on real hypersurfaces in complex two-plane Grassmannians, Universitatis Debreceniensis. (2019) 95, no. 3-4, 307–319, https://doi.org/10.5486/PMD.2019.8405.
10.5486/PMD.2019.8405
Web of Science® Google Scholar
15 Wang Y., Hopf hypersurfaces in complex two-plane Grassmannians with GTW Killing shape operator, Zurnal matematiceskoj fiziki, analiza, geometrii. (2022) 18, no. 2, 286–297, https://doi.org/10.15407/mag18.02.286.
10.15407/mag18.02.286
Google Scholar
16 Wang Y., Structure Lie operator on real hypersurfaces of complex two-plane Grassmannians, Colloquium Mathematicum. (2022) 170, no. 2, 315–320, https://doi.org/10.4064/cm8558-1-2022.
10.4064/cm8558-1-2022
Web of Science® Google Scholar
17 Lee H. and Suh Y. J., Real hypersurfaces of type B in complex two-plane Grassmannians related to the Reeb vector, Bulletin of the Korean Mathematical Society. (2010) 47, no. 3, 551–561, https://doi.org/10.4134/BKMS.2010.47.3.551, 2-s2.0-78049303129.
10.4134/BKMS.2010.47.3.551
Web of Science® Google Scholar
18 Suh Y. J., Real hypersurfaces in complex two-plane Grassmannians with commuting shape operator, Bulletin of the Australian Mathematical Society. (2003) 68, no. 3, 379–393, https://doi.org/10.1017/S0004972700037795.
10.1017/S0004972700037795
Web of Science® Google Scholar
19 Suh Y. J., Real hypersurfaces of type B in complex two-plane Grassmannians, Monatshefte für Mathematik. (2006) 147, no. 4, 337–355, https://doi.org/10.1007/s00605-005-0329-9, 2-s2.0-33645029309.
10.1007/s00605-005-0329-9
Web of Science® Google Scholar
20 Hu Z. and Xing C., A new centroaffine characterization of the ellipsoids, Proceedings of the American Mathematical Society. (2021) 149, no. 8, 3531–3540, https://doi.org/10.1090/proc/15504.
10.1090/proc/15504
Web of Science® Google Scholar
21 Hu Z. and Yin J., New characterizations of real hypersurfaces with isometric Reeb flow in the complex quadric, Colloquium Mathematicum. (2021) 164, no. 2, 211–219, https://doi.org/10.4064/cm8075-12-2019.
10.4064/cm8075-12-2019
Web of Science® Google Scholar
22 Berndt J. and Suh Y. J., Real hypersurfaces with isometric Reeb flow in complex two-plane Grassmannians, Monatshefte für Mathematik. (2002) 137, no. 2, 87–98, https://doi.org/10.1007/s00605-001-0494-4, 2-s2.0-0036774505.
10.1007/s00605-001-0494-4
Web of Science® Google Scholar
23 Loo T. H., Semi-parallel real hypersurfaces in complex two-plane Grassmannians, Differential Geometry and its Applications. (2014) 34, 87–102, https://doi.org/10.1016/j.difgeo.2014.03.011, 2-s2.0-84899844011.
10.1016/j.difgeo.2014.03.011
Web of Science® Google Scholar
24 Suh Y. J., Real hypersurfaces in complex two-plane Grassmannians with parallel Ricci tensor, Proceedings of the Royal Society of Edinburgh: Section A Mathematics. (2012) 142, no. 6, 1309–1324, https://doi.org/10.1017/S0308210510001472, 2-s2.0-85012500938.
10.1017/S0308210510001472
Web of Science® Google Scholar
25 Yano K., On harmonic and Killing vector fields, Annals of Mathematics. (1952) 55, no. 1, 38–45, https://doi.org/10.2307/1969418.
10.2307/1969418
Web of Science® Google Scholar

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

Abstract

1. Introduction

2. Preliminaries

3. Proof of Theorem 3

3.1. Completion of the Proof of Theorem 3.

Conflicts of Interest

Acknowledgments

Open Research

Data Availability

References

References

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

Abstract

1. Introduction

2. Preliminaries

3. Proof of Theorem 3

3.1. Completion of the Proof of Theorem 3.

Conflicts of Interest

Acknowledgments

Open Research

Data Availability

References

References

Related

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