Real hypersurfaces in the complex quadric with Reeb parallel structure Jacobi operator

TL;DR

This paper classifies Hopf real hypersurfaces in complex quadrics with Reeb parallel structure Jacobi operator, providing explicit curvature tensor formulas and a complete classification for dimensions m ≥ 3.

Contribution

It introduces explicit formulas for the curvature tensor and Jacobi operator, and fully classifies Hopf hypersurfaces with Reeb parallel structure Jacobi operator in complex quadrics.

Findings

Complete classification of Hopf hypersurfaces with Reeb parallel Jacobi operator in Q^m for m ≥ 3.

Explicit formulas for the Riemannian curvature tensor of hypersurfaces in complex quadrics.

Derivation of the structure Jacobi operator and its covariant derivative.

Abstract

In this paper, we first introduce the full express of the Riemannian curvature tensor of a real hypersurface $M$ in complex quadric $Q^{m}$ from the equation of Gauss. Next we derive a formula for the structure Jacobi operator $R_{\xi}$ and its derivative under the Levi-Civita connection of $M$. We give a complete classification of Hopf real hypersurfaces with Reeb parallel structure Jacobi operator, $\nabla_{\xi}R_{\xi} =0$, in the complex quadric $Q^{m}$, $m \geq 3$.

Equations

$$T_{[z]}{Q}^m=V(A_{\rho })\oplus JV(A_{\rho }),$$

$$\begin{array}{rl}\bar{R}(X,Y)Z& =g(Y,Z)X-g(X,Z)Y+g(JY,Z)JX-g(JX,Z)JY\\ & -2g(JX,Y)JZ+g(AY,Z)AX\\ & -g(AX,Z)AY+g(JAY,Z)JAX-g(JAX,Z)JAY.\end{array}$$

$$U=\cos (t)Z_1+\sin (t)J{Z}_2$$

$$\begin{split}{\bar{R}}(X,Y)Z&=R(X,Y)Z-g(SY,Z)SX+g(SX,Z)SY\\ &\quad+\big{\{}g((\nabla_{X}S)Y,Z)-g((\nabla_{Y}S)X,Z)\big{\}}N,\end{split}$$

$$\begin{array}{rl}& g(R(X,Y)Z,W)-g(SY,Z)g(SX,W)+g(SX,Z)g(SY,W)\\ & =g(Y,Z)g(X,W)-g(X,Z)g(Y,W)+g(JY,Z)g(JX,W)\\ & -g(JX,Z)g(JY,W)-2g(JX,Y)g(JZ,W)+g(AY,Z)g(AX,W)\\ & -g(AX,Z)g(AY,W)+g(JAY,Z)g(JAX,W)-g(JAX,Z)g(JAY,W),\end{array}$$

$$\begin{array}{rl}& g(({\nabla }_{X}S)Y,Z)-g(({\nabla }_{Y}S)X,Z)\\ & =\eta (X)g(JY,Z)-\eta (Y)g(JX,Z)-2\eta (Z)g(JX,Y)\\ & +g(AY,Z)g(AX,N)-g(AX,Z)g(AY,N)\\ & +\eta (AX)g(JAY,Z)-\eta (AY)g(JAX,Z)\end{array}$$

$$AX=BX+\rho (X)N$$

$$\rho (X)=g(AX,N)=g(X,AN)=g(X,AJ\xi )=g(JX,A\xi ).$$

$$\begin{array}{rl}& R(X,Y)Z-g(SY,Z)SX+g(SX,Z)SY\\ & =g(Y,Z)X-g(X,Z)Y+g(JY,Z)\varphi X-g(JX,Z)\varphi Y-2g(JX,Y)\varphi Z\\ & +g(AY,Z)BX-g(AX,Z)BY+g(JAY,Z)\varphi BX\\ & -g(JAY,Z)\rho (X)\xi -g(JAX,Z)\varphi BY+g(JAX,Z)\rho (Y)\xi \end{array}$$

$$\begin{array}{rl}& ({\nabla }_{X}S)Y-({\nabla }_{Y}S)X\\ & =\eta (X)\varphi Y-\eta (Y)\varphi X-2g(JX,Y)\xi \\ & +g(AX,N)BY-g(AY,N)BX+\eta (AX)\varphi BY\\ & -\eta (AX)\rho (Y)\xi -\eta (AY)\varphi BX+\eta (AY)\rho (X)\xi ,\end{array}$$

$$\begin{array}{rl}\mathrm{Ric}X& =(2m-1)X-3\eta (X)\xi +g(A\xi ,\xi )BX-g(AX,N)\varphi A\xi \\ & +g(AX,\xi )A\xi +hSX-S^{2}X,\end{array}$$

$$N=\cos (t)Z_1+\sin (t)J{Z}_2$$

$$\{\begin{array}{l}\xi =\sin (t)Z_2-\cos (t)J{Z}_1,\\ AN=\cos (t)Z_1-\sin (t)J{Z}_2,\\ A\xi =\sin (t)Z_2+\cos (t)J{Z}_1.\end{array}$$

$${\mathcal{Q}}_{[z]}=\{X\in T_{[z]}M\mid AX\in T_{[z]}M\mathrm{for}\mathrm{all}A\in {\mathfrak{A}}_{[z]}\}.$$

$$X\alpha =(\xi \alpha )\eta (X)+2g(A\xi ,\xi )g(X,AN)$$

$$\begin{array}{rl}& 2g(S\varphi SX,Y)-\alpha g((\varphi S+S\varphi )X,Y)-2g(\varphi X,Y)\\ & +g(X,AN)g(Y,A\xi )-g(Y,AN)g(X,A\xi )\\ & -g(X,A\xi )g(JY,A\xi )+g(Y,A\xi )g(JX,A\xi )\\ & -2g(X,AN)g(\xi ,A\xi )\eta (Y)+2g(Y,AN)g(\xi ,A\xi )\eta (X)=0\end{array}$$

$$AN=\sum _{i=1}^{2m}g(AN,e_i)e_i+g(AN,N)N=g(AN,N)N,$$

$$g(A\xi ,N)=g(A\xi ,\xi )=g(AN,N)=0$$

$$\begin{split}0=X\big{(}g(AN,N)\big{)}&=g\big{(}{\bar{\nabla}}_{X}(AN),N\big{)}+g\big{(}AN,{\bar{\nabla}}_{X}N\big{)}\\ &=g\big{(}(\bar{\nabla}_{X}A)N+A(\bar{\nabla}_{X}N),N\big{)}+g\big{(}AN,\bar{\nabla}_{X}N\big{)}\\ &=g\big{(}q(X)JAN-ASX,N\big{)}-g\big{(}AN,SX\big{)}\\ &=-2g\big{(}ASX,N\big{)},\end{split}$$

$$\begin{split}0&=g\big{(}{\bar{\nabla}}_{X}(A{\xi}),N\big{)}+g\big{(}A{\xi},{\bar{\nabla}}_{X}N\big{)}\\ &=g\big{(}({\bar{\nabla}}_{X}A){\xi}+A({\bar{\nabla}}_{X}\xi),N\big{)}+g\big{(}A{\xi},{\bar{\nabla}}_{X}N\big{)}\\ &=g\big{(}({\bar{\nabla}}_{X}A){\xi},N\big{)}+g\big{(}\nabla_{X}\xi+\sigma(X,\xi),AN\big{)}+g\big{(}A{\xi},{\bar{\nabla}}_{X}N\big{)}\\ &=g\big{(}q(X)JA{\xi},N\big{)}+g\big{(}{\phi}SX+g(SX,{\xi})N,AN\big{)}-g\big{(}A{\xi},SX\big{)}\\ &=-2g(A{\xi},SX),\end{split}$$

$$T_{[z]}M=[\xi ]\oplus [A\xi ,AN]\oplus \mathcal{Q},$$

$$(2\lambda -\alpha )S\varphi X=(\alpha \lambda +2)\varphi X$$

$$\begin{array}{rl}{\nabla }_X(A\xi )& ={\bar{\nabla }}_X(A\xi )-\sigma (X,A\xi )\\ & =q(X)JA\xi +A({\nabla }_X\xi )+g(SX,\xi )AN-g(SX,A\xi )N\end{array}$$

$$\left\{\begin{array}[]{l}\mathrm{Tangent\ Part:}\ \ \nabla_{X}(A\xi)=q(X)\phi A\xi+B\phi SX-g(SX,\xi)\phi A\xi\\ \\ \mathrm{Normal\ Part:}\ \ q(X)g(A\xi,\xi)=-g(AN,\nabla_{X}\xi)+g(SX,\xi)g(A\xi,\xi)+g(SX,A\xi)\end{array}\right.$$

$$q(\xi )g(A\xi ,\xi )=2\alpha g(A\xi ,\xi ).$$

$$({\nabla }_{\xi }{R}_{\xi })Y=0$$

$$\begin{array}{rl}g(R_{\xi }Y,Z)& =g(R(Y,\xi )\xi ,Z)\\ & =g(Y,Z)-\eta (Y)\eta (Z)+g(A\xi ,\xi )g(AY,Z)-g(Y,A\xi )g(A\xi ,Z)\\ & -g(AY,N)g(AN,Z)+\alpha g(SY,Z)-{\alpha }^2\eta (Y)\eta (Z),\end{array}$$

$$\begin{array}{rl}{R}_{\xi }Y& =Y-\eta (Y)\xi +g(A\xi ,\xi )BY-g(A\xi ,Y)A\xi \\ & -g(\varphi A\xi ,Y)\varphi A\xi +\alpha SY-{\alpha }^2\eta (Y)\xi .\end{array}$$

$$\begin{split}&(\nabla_{X}{R}_{\xi})Y\\ &\quad=-g(Y,\nabla_{X}\xi)\xi-\eta(Y)\nabla_{X}\xi+g\big{(}\nabla_{X}(A\xi),\xi\big{)}BY+g(A\xi,\nabla_{X}\xi)BY\\ &\quad\ \ \ +g(A\xi,\xi)(\nabla_{X}B)Y-g\big{(}\nabla_{X}(A\xi),Y\big{)}A\xi-g(A\xi,Y)\nabla_{X}(A\xi)\\ &\quad\ \ \ -g\big{(}(\nabla_{X}\phi)A\xi,Y\big{)}\phi A\xi+g\big{(}\nabla_{X}(A\xi),\phi Y\big{)}\phi A\xi\\ &\quad\ \ \ -g(\phi A\xi,Y)(\nabla_{X}\phi)A\xi-g(\phi A\xi,Y)\phi\big{(}\nabla_{X}(A\xi)\big{)}\\ &\quad\ \ \ +(X\alpha)SY+\alpha(\nabla_{X}S)Y-2\alpha(X\alpha)\eta(Y)\xi-\alpha^{2}g(Y,\nabla_{X}\xi)\xi-\alpha^{2}\eta(Y)\nabla_{X}\xi\\ &\quad=-g(Y,\phi SX)\xi-\eta(Y)\phi SX+g(B\phi SX,\xi)BY+g(A\xi,\phi SX)BY\\ &\quad\ \ \ +g(A\xi,\xi)\Big{\{}q(X)JAY+g(SX,Y)AN-q(X)g(AY,\xi)N\Big{\}}\\ &\quad\ \ \ +g(A\xi,\xi)\Big{\{}g(SX,Y)g(A\xi,\xi)N+g(AN,Y)SX\Big{\}}\\ &\quad\ \ \ -\Big{\{}\big{(}q(X)-\alpha\eta(X)\big{)}g(\phi A\xi,Y)+g(B\phi SX,Y)\Big{\}}A\xi\\ &\quad\ \ \ -g(A\xi,Y)\Big{\{}\big{(}q(X)-\alpha\eta(X)\big{)}\phi A\xi+B\phi SX\Big{\}}\\ &\quad\ \ \ -\Big{\{}g(A\xi,\xi)g(SX,Y)-g(SX,A\xi)\eta(Y)\Big{\}}\phi A\xi+\big{(}q(X)-\alpha\eta(X)\big{)}g(A\xi,Y)\phi A\xi\\ &\quad\ \ \ -\Big{\{}\big{(}q(X)-\alpha\eta(X)\big{)}g(A\xi,\xi)\eta(Y)-g(B\phi SX,\phi Y)\Big{\}}\phi A\xi\\ &\quad\ \ \ -g(\phi A\xi,Y)\Big{\{}\eta(A\xi)SX-g(SX,A\xi)\xi\Big{\}}\\ &\quad\ \ \ +g(\phi A\xi,Y)\Big{\{}\big{(}q(X)-\alpha\eta(X)\big{)}A\xi-g(A\xi,\xi)\big{(}q(X)-\alpha\eta(X)\big{)}\xi-\phi B\phi SX\Big{\}}\\ &\quad\ \ \ +(X\alpha)SY+\alpha(\nabla_{X}S)Y-2\alpha(X\alpha)\eta(Y)\xi-\alpha^{2}g(Y,\phi SX)\xi-\alpha^{2}\eta(Y)\phi SX,\end{split}$$

$$\begin{split}(\nabla_{X}B)Y&=\nabla_{X}(BY)-B(\nabla_{X}Y)\\ &={\bar{\nabla}}_{X}(BY)-\sigma(X,BY)-B(\nabla_{X}Y)\\ &={\bar{\nabla}}_{X}\big{(}AY-g(AY,N)N\big{)}-g(SX,BY)N-B(\nabla_{X}Y)\\ &=({\bar{\nabla}}_{X}A)Y+A\big{(}{\bar{\nabla}}_{X}Y\big{)}-g\big{(}{\bar{\nabla}}_{X}(AY),N\big{)}N-g(AY,{\bar{\nabla}}_{X}N)N\\ &\quad\ \ -g(AY,N){\bar{\nabla}}_{X}N-g(SX,BY)N-B(\nabla_{X}Y)\\ &=q(X)JAY+A(\nabla_{X}Y)+g(SX,Y)AN-q(X)g(JAY,N)N\\ &\quad\ \ -g(\nabla_{X}Y,AN)N-g(SX,Y)g(AN,N)N\\ &\quad\ \ +g(AY,SX)N+g(AY,N)SX-g(SX,BY)N-B(\nabla_{X}Y)\\ &=q(X)JAY+g(SX,Y)AN-q(X)g(AY,\xi)N\\ &\quad\ \ +g(SX,Y)g(A\xi,\xi)N+g(AY,N)SX.\end{split}$$

Full text

• This work was supported by grant Proj. Nos. NRF-2018-R1D1A1B-05040381 and NRF-2019-R1I1A1A 01050300 from National Research Foundation of Korea.

1. Introduction

For Hermitian symmetric space of compact type different from the above ones, we can give the example of complex quadric ${Q^{m}}=SO_{m+2}/SO_{m}SO_{2}$, which is a complex hypersurface in the complex projective space ${\mathbb{C}}P^{m+1}$ (see Romero [23], [24], Smyth [25], Suh [28], [29]). The complex quadric can also be regarded as a kind of real Grassmann manifolds of compact type with rank 2 (see Besse [1], Helgason [6], and Knap [12] ). Accordingly, the complex quadric ${Q^{m}}$ admits two important geometric structures, a complex conjugation structure $A$ and a Kähler structure $J$, which anti-commute with each other, that is, $AJ=-JA$. Then for $m{\geq}2$ the triple $({Q^{m}},J,g)$ is a Hermitian symmetric space of compact type with rank $2$ and its maximal sectional curvature is equal to $4$ (see Kobayashi and Nomizu [14], Reckziegel [22]).

In addition to the complex structure $J$ there is another distinguished geometric structure on $Q^{m}$, namely a parallel rank two vector bundle ${\mathfrak{A}}$ which contains an $S^{1}$-bundle of real structures, that is, complex conjugations $A$ on the tangent spaces of $Q^{m}$. The set is denoted by ${\mathfrak{A}}_{[z]}=\{A_{{\lambda}\bar{z}}{|}\,{\lambda}\in S^{1}{\subset}{\mathbb{C}}\}$, $[z]\in{Q^{m}}$, and it is the set of all complex conjugations defined on $Q^{m}$. Then ${\mathfrak{A}}_{[z]}$ becomes a parallel rank $2$-subbundle of $\text{End}\ T_{[z]}{Q^{m}}$, $[z]\in{Q^{m}}$. This geometric structure determines a maximal ${\mathfrak{A}}$-invariant subbundle ${\mathcal{Q}}$ of the tangent bundle $TM$ of a real hypersurface $M$ in $Q^{m}$. Here the notion of parallel vector bundle ${\mathfrak{A}}$ means that $({\bar{\nabla}}_{X}A)Y=q(X)JAY$ for any vector fields $X$ and $Y$ on $Q^{m}$, where $\bar{\nabla}$ and  $q$ denote a connection and a certain $1$-form defined on $T_{[z]}{Q^{m}}$, $[z]\in{Q^{m}}$ respectively (see Smyth [25]).

Recall that a nonzero tangent vector $W\in T_{[z]}{Q^{m}}$ is called singular if it is tangent to more than one maximal flat in $Q^{m}$. There are two types of singular tangent vectors for the complex hyperbolic quadric $Q^{m}$:

• •

If there exists a conjugation $A\in{\mathfrak{A}}$ such that $W\in V(A)=\{X\in T_{[z]}{Q^{m}}{|}\,AX=X\}$, then $W$ is singular. Such a singular tangent vector is called ${\mathfrak{A}}$-principal.

• •

If there exist a conjugation $A\in{\mathfrak{A}}$ and orthonormal vectors $Z_{1}$, $Z_{2}\in V(A)$ such that $W/||W||=(Z_{1}+JZ_{2})/\sqrt{2}$, then $W$ is singular. Such a singular tangent vector is called ${\mathfrak{A}}$-isotropic, where $V(A)=\{X\in T_{[z]}{Q^{m}}{|}\,AX=X\}$ and $JV(A)=\{X\in T_{[z]}{Q^{m}}{|}\,AX=-X\}$ are the $(+1)$-eigenspace and $(-1)$-eigenspace for the involution $A$ on $T_{[z]}{Q^{m}}$, $[z]\in{Q^{m}}$.

On the other hand, Okumura [17] proved that the Reeb flow on a real hypersurface in ${\mathbb{C}}P^{m}=SU_{m+1}/S(U_{1}U_{m})$ is isometric if and only if $M$ is an open part of a tube around a totally geodesic ${\mathbb{C}}P^{k}$ in ${\mathbb{C}}P^{m}$ for some $k\in\{0,\ldots,m-1\}$. For the complex $2$-plane Grassmannian $G_{2}({\mathbb{C}}^{m+2})=SU_{m+2}/S(U_{2}U_{m})$ a classification was obtained by Berndt and Suh [2]. The Reeb flow on a real hypersurface in $G_{2}({\mathbb{C}}^{m+2})$ is isometric if and only if $M$ is an open part of a tube around a totally geodesic $G_{2}({\mathbb{C}}^{m+1})$ in $G_{2}({\mathbb{C}}^{m+2})$. For the complex quadric $Q^{m}=SO_{m+2}/SO_{2}SO_{m}$, Berndt and Suh [3] have obtained the following result:

Theorem A**.**

Let $M$ be a real hypersurface in the complex quadric $Q^{m}$, $m\geq 3$. Then the Reeb flow on $M$ is isometric if and only if $m$ is even, say $m=2k$, and $M$ is an open part of a tube around a totally geodesic ${\mathbb{C}}P^{k}$ in $Q^{2k}$.

For the complex hyperbolic space ${\mathbb{C}}H^{m}$ a classification was obtained by Montiel and Romero [16]. They proved that the Reeb flow on a real hypersurface in ${\mathbb{C}}H^{m}$ is isometric if and only if $M$ is an open part of a tube around a totally geodesic ${\mathbb{C}}H^{k}$ in ${\mathbb{C}}H^{m}$ for some $k\in\{0,\ldots,m-1\}$. The classification problems related to the Reeb parallel shape operator, parallel Ricci tensor, and harmonic curvature for real hypersurfaces in the complex quadric $Q^{m}$ were recently given in Suh [26], [28] and [29] respectively.

The notion of isometric Reeb flow was introduced by Hutching and Taubes [7] and the geometric construction of horospheres in a non-compact manifold of negative curvature was mainly discussed in the book due to Eberlein [5].

On the other hand, Jacobi fields along geodesics of a given Riemannian manifold $(\bar{M},g)$ satisfy a well known differential equation. This equation naturally inspires the so-called Jacobi operator. That is, if $R$ denotes the curvature operator of $\bar{M}$, and $X$ is a tangent vector field to $\bar{M}$, then the Jacobi operator $R_{X}\in\text{End}(T_{z}{\bar{M}})$ with respect to $X$ at $z\in\bar{M}$, defined by $(R_{X}Y)(z)=(R(Y,X)X)(z)$ for any $Y\in T_{z}{\bar{M}}$, becomes a self adjoint endomorphism of the tangent bundle $T{\bar{M}}$ of $\bar{M}$. Thus, each tangent vector field $X$ to $\bar{M}$ provides a Jacobi operator $R_{X}$ with respect to $X$. In particular, for the Reeb vector field $\xi$, the Jacobi operator $R_{\xi}$ is said to be the structure Jacobi operator.

Actually, many geometers have considered the fact that a real hypersurface $M$ in Kähler manifolds has parallel structure Jacobi operator (or Reeb parallel structure Jacobi operator, respectively), that is, $\nabla_{X}R_{\xi}=0$ (or $\nabla_{\xi}R_{\xi}=0$, respectively) for any tangent vector field $X$ on $M$. Recently Ki, Pérez, Santos and Suh [10] have investigated the Reeb parallel structure Jacobi operator in the complex space form $M^{m}(c)$, $c\neq 0$, and have used it to study some principal curvatures for a tube over a totally geodesic submanifold. In particular, Pérez, Jeong and Suh [19] have investigated real hypersurfaces $M$ in $G_{2}({\mathbb{C}}^{m+2})$ with parallel structure Jacobi operator, that is, ${\nabla}_{X}{R}_{\xi}=0$ for any tangent vector field $X$ on $M$. Jeong, Suh and Woo [9] and Pérez and Santos [20] have generalized such a notion to the recurrent structure Jacobi operator, that is, $({\nabla}_{X}{R}_{\xi})Y={\beta}(X){R}_{\xi}Y$ for a certain $1$-form $\beta$ and any vector fields $X,Y$ on $M$ in $G_{2}({\mathbb{C}}^{m+2})$ or ${\mathbb{C}}P^{m}$. In [8], Jeong, Lee, and Suh have considered a Hopf real hypersurface with Codazzi type of structure Jacobi operator, $(\nabla_{X}R_{\xi})Y=(\nabla_{Y}R_{\xi})X$, in $G_{2}(\mathbb{C}^{m+2})$. Moreover, Pérez, Santos and Suh [21] have further investigated the property of the Lie $\xi$-parallel structure Jacobi operator in complex projective space ${\mathbb{C}}P^{m}$, that is, ${\mathcal{L}}_{\xi}R_{\xi}=0$.

Motivated by these results, in this paper we want to give a classification of Hopf real hypersurfaces in $Q^{m}$ with non-vanishing geodesic Reeb flow and Reeb parallel structure Jacobi operator, that is, ${\nabla}_{\xi}{R}_{\xi}=0$. Here a real hypersurface $M$ is said to be Hopf if the Reeb vector field $\xi$ of $M$ is principal by the shape operator $S$, that is, $S\xi=g(S\xi,\xi)\xi=\alpha\xi$. In particular, if the Reeb curvature function $\alpha=g(S\xi,\xi)$ identically vanishes, we say that $M$ has a vanishing geodesic Reeb flow. Otherwise, $M$ has a non-vanishing geodesic Reeb flow.

Under these background and motivation, first we prove the following:

Theorem 1**.**

There does not exist any Hopf real hypersurface in the complex quadric $Q^{m}$, $m\geq 3$, with Reeb parallel structure Jacobi operator and $\mathfrak{A}$-principal singular normal vector field, provided with non-vanishing geodesic Reeb flow.

Now let us consider a Hopf real hypersurface with $\mathfrak{A}$-isotropic singular normal vector field $N$ in $Q^{m}$. Then by virtue of Theorem $\rm A$ we can give a complete classification of Hopf real hypersurfaces in $Q^{m}$ with Reeb parallel structure Jacobi operator as follows:

Theorem 2**.**

Let $M$ be a Hopf real hypersurface in the complex quadric $Q^{m}$, $m\geq 3$, with Reeb parallel structure Jacobi operator and non-vanishing geodesic Reeb flow. If $M$ has the $\mathfrak{A}$-isotropic singular normal vector field in $Q^{m}$, then $M$ is locally congruent to a tube around the totally geodesic ${\mathbb{C}}P^{k}$ in ${Q^{2k}}$, where $m=2k$, and $r\in(0,\frac{\pi}{4})\cup(\frac{\pi}{4},\frac{\pi}{2})$.

2. The complex quadric

For more background to this section we refer to [11], [14], [22], [26], [27], [29] and [32]. The complex quadric $Q^{m}$ is the complex hypersurface in ${\mathbb{C}}P^{m+1}$ which is defined by the equation $z_{1}^{2}+\cdots+z_{m+2}^{2}=0$, where $z_{1},\cdots,z_{m+2}$ are homogeneous coordinates on ${\mathbb{C}}P^{m+1}$. We equip $Q^{m}$ with the Riemannian metric which is induced from the Fubini Study metric on ${\mathbb{C}}P^{m+1}$ with constant holomorphic sectional curvature $4$. The Kähler structure on ${\mathbb{C}}P^{m+1}$ induces canonically a Kähler structure $(J,g)$ on the complex quadric. For a nonzero vector $z\in\mathbb{C}^{m+2}$ we denote by $[z]$ the complex span of $z$, that is, $[z]=\mathbb{C}z=\{\lambda z\,|\,\lambda\in S^{1}\subset\mathbb{C}\}$. Note that by definition $[z]$ is a point in $\mathbb{C}P^{m+1}$. For each $[z]\in Q^{m}\subset\mathbb{C}P^{m+1}$ we identify $T_{[z]}{\mathbb{C}}P^{m+1}$ with the orthogonal complement ${\mathbb{C}}^{m+2}\ominus{\mathbb{C}}z$ of ${\mathbb{C}}z$ in ${\mathbb{C}}^{m+2}$ (see Kobayashi and Nomizu [14]). The tangent space $T_{[z]}Q^{m}$ can then be identified canonically with the orthogonal complement ${\mathbb{C}}^{m+2}\ominus({\mathbb{C}}z\oplus{\mathbb{C}}\rho)$ of ${\mathbb{C}}z\oplus{\mathbb{C}}\rho$ in ${\mathbb{C}}^{m+2}$, where $\rho\in\nu_{[z]}Q^{m}$ is a normal vector of $Q^{m}$ in ${\mathbb{C}}P^{m+1}$ at the point $[z]$.

The complex projective space ${\mathbb{C}}P^{m+1}$ is a Hermitian symmetric space of the special unitary group $SU_{m+2}$, namely ${\mathbb{C}}P^{m+1}=SU_{m+2}/S(U_{m+1}U_{1})$. We denote by $o=[0,\ldots,0,1]\in{\mathbb{C}}P^{m+1}$ the fixed point of the action of the stabilizer $S(U_{m+1}U_{1})$. The special orthogonal group $SO_{m+2}\subset SU_{m+2}$ acts on ${\mathbb{C}}P^{m+1}$ with cohomogeneity one. The orbit containing $o$ is a totally geodesic real projective space ${\mathbb{R}}P^{m+1}\subset{\mathbb{C}}P^{m+1}$. The second singular orbit of this action is the complex quadric $Q^{m}=SO_{m+2}/SO_{m}SO_{2}$. This homogeneous space model leads to the geometric interpretation of the complex quadric $Q^{m}$ as the Grassmann manifold $G_{2}^{+}({\mathbb{R}}^{m+2})$ of oriented $2$-planes in ${\mathbb{R}}^{m+2}$. It also gives a model of $Q^{m}$ as a Hermitian symmetric space of rank $2$. The complex quadric $Q^{1}$ is isometric to a sphere $S^{2}$ with constant curvature, and $Q^{2}$ is isometric to the Riemannian product of two $2$-spheres with constant curvature. For this reason we will assume $m\geq 3$ from now on.

For a unit normal vector $\rho$ of $Q^{m}$ at a point $[z]\in Q^{m}$ we denote by $A=A_{\rho}$ the shape operator of $Q^{m}$ in ${\mathbb{C}}P^{m+1}$ with respect to $\rho$. The shape operator is an involution on the tangent space $T_{[z]}Q^{m}$ and

[TABLE]

where $V(A_{\rho})$ is the $(+1)$-eigenspace and $JV(A_{\rho})$ is the $(-1)$-eigenspace of $A_{\rho}$. Geometrically this means that the shape operator $A_{\rho}$ defines a real structure on the complex vector space $T_{[z]}Q^{m}$, or equivalently, is a complex conjugation on $T_{[z]}Q^{m}$. Since the real codimension of $Q^{m}$ in ${\mathbb{C}}P^{m+1}$ is $2$, this induces an $S^{1}$-subbundle ${\mathfrak{A}}$ of the endomorphism bundle ${\rm End}(TQ^{m})$ consisting of complex conjugations. There is a geometric interpretation of these conjugations. The complex quadric $Q^{m}$ can be viewed as the complexification of the $m$-dimensional sphere $S^{m}$. Through each point $[z]\in Q^{m}$ there exists a one-parameter family of real forms of $Q^{m}$ which are isometric to the sphere $S^{m}$. These real forms are congruent to each other under action of the center $SO_{2}$ of the isotropy subgroup of $SO_{m+2}$ at $[z]$. The isometric reflection of $Q^{m}$ in such a real form $S^{m}$ is an isometry, and the differential at $[z]$ of such a reflection is a conjugation on $T_{[z]}Q^{m}$. In this way the family ${\mathfrak{A}}$ of conjugations on $T_{[z]}Q^{m}$ corresponds to the family of real forms $S^{m}$ of $Q^{m}$ containing $[z]$, and the subspaces $V(A)$ in $T_{[z]}Q^{m}$ correspond to the tangent spaces $T_{[z]}S^{m}$ of the real forms $S^{m}$ of $Q^{m}$.

The Gauss equation for $Q^{m}\subset{\mathbb{C}}P^{m+1}$ implies that the Riemannian curvature tensor $\bar{R}$ of $Q^{m}$ can be described in terms of the complex structure $J$ and the complex conjugations $A\in{\mathfrak{A}}$:

[TABLE]

It is well known that for every unit tangent vector $U\in T_{[z]}Q^{m}$ there exist a conjugation $A\in{\mathfrak{A}}$ and orthonormal vectors $Z_{1}$, $Z_{2}\in V(A)$ such that

[TABLE]

for some $t\in[0,\pi/4]$ (see [22]). The singular tangent vectors correspond to the values $t=0$ and $t=\pi/4$. If $0<t<\pi/4$ then the unique maximal flat containing $U$ is ${\mathbb{R}}Z_{1}\oplus{\mathbb{R}}JZ_{2}$. Later we will need the eigenvalues and eigenspaces of the Jacobi operator $\bar{R}_{U}=\bar{R}(\,\cdot\,,U)U$ for a singular unit tangent vector $U$.

1. (1)

If $U$ is an ${\mathfrak{A}}$-principal singular unit tangent vector with respect to $A\in{\mathfrak{A}}$, then the eigenvalues of $\bar{R}_{U}$ are [math] and $2$ and the corresponding eigenspaces are ${\mathbb{R}}U\oplus J(V(A)\ominus{\mathbb{R}}U)$ and $(V(A)\ominus{\mathbb{R}}U)\oplus{\mathbb{R}}JU$, respectively. 2. (2)

If $U$ is an ${\mathfrak{A}}$-isotropic singular unit tangent vector with respect to $A\in{\mathfrak{A}}$ and $X$, $Y\in V(A)$, then the eigenvalues of $\bar{R}_{U}$ are [math], $1$ and $4$ and the corresponding eigenspaces are ${\mathbb{R}}U\oplus{\mathbb{C}}(JZ_{1}+Z_{2})$, $T_{[z]}Q^{m}\ominus({\mathbb{C}}Z_{1}\oplus{\mathbb{C}}Z_{2})$ and ${\mathbb{R}}JU$, respectively.

3. Real hypersurfaces in $Q^{m}$

Let $M$ be a real hypersurface in $Q^{m}$ and denote by $(\phi,\xi,\eta,g)$ the induced almost contact metric structure. By using the Gauss and Wingarten formulas the left-hand side of (2.1) becomes

[TABLE]

where $R$ and $S$ denote the Riemannian curvature tensor and the shape operator of $M$ in $Q^{m}$, respectively. Taking tangent and normal components of (2.1) respectively, we obtain

[TABLE]

and

[TABLE]

where $X$, $Y$, $Z$ and $W$ are tangent vector fields of $M$.

Note that $JX=\phi X+\eta(X)N$ and $JN=-\xi$, where $\phi X$ is the tangential component of $JX$ and $N$ is a (local) unit normal vector field of $M$. The tangent bundle $TM$ of $M$ splits orthogonally into $TM={\mathcal{C}}\oplus{\mathbb{R}}\xi$, where ${\mathcal{C}}=\mathrm{ker}\,\eta$ is the maximal complex subbundle of $TM$. The structure tensor field $\phi$ restricted to ${\mathcal{C}}$ coincides with the complex structure $J$ restricted to ${\mathcal{C}}$, and $\phi\xi=0$. Moreover, since the complex quadric $Q^{m}$ has also a real structure $A$, we decompose $AX$ into its tangential and normal components for a fixed $A\in\mathfrak{A}_{[z]}$ and $X\in T_{[z]}M$:

[TABLE]

where $BX$ is the tangential component of $AX$ and

[TABLE]

From these notations, the equations (3.1) and (3.2) can be written as

[TABLE]

and

[TABLE]

which are called the equations of Gauss and Codazzi, respectively. Moreover, from (3.1) the Ricci tensor $\mathrm{Ric}$ of $M$ is given by

[TABLE]

where $h=\mathrm{Tr}S$.

As mentioned in section 2, since the normal vector field $N$ belongs to $T_{[z]}Q^{m}$, $[z]\in M$, we can choose $A\in{\mathfrak{A}}_{[z]}$ such that

[TABLE]

for some orthonormal vectors $Z_{1}$, $Z_{2}\in V(A)$ and $0\leq t\leq\frac{\pi}{4}$ (see Proposition 3 in [22]). Note that $t$ is a function on $M$. If $t=0$, then $N=Z_{1}\in V(A)$, therefore we see that $N$ becomes the $\mathfrak{A}$-principal singular tangent vector field. On the other hand, if $t=\frac{\pi}{4}$, then $N=\frac{1}{\sqrt{2}}(Z_{1}+JZ_{2})$. That is, $N$ is to be the $\mathfrak{A}$-isotropic singular tangent vector field. In addition, since $\xi=-JN$, we have

[TABLE]

This implies $g(\xi,AN)=0$ and $g(A\xi,\xi)=-g(AN,N)=-\cos(2t)$ on $M$. At each point $[z]\in M$ we define the maximal ${\mathfrak{A}}$-invariant subspace of $T_{[z]}M$, $[z]\in M$ as follows:

[TABLE]

It is known if $N_{[z]}$ is ${\mathfrak{A}}$-principal, then ${\mathcal{Q}}_{[z]}={\mathcal{C}}_{[z]}$ (see [26]).

We now assume that $M$ is a Hopf hypersurface in the complex quadric $Q^{m}$. Then the shape operator $S$ of $M$ in $Q^{m}$ satisfies $S\xi=\alpha\xi$ with the Reeb function $\alpha=g(S\xi,\xi)$ on $M$. By virtue of the Codazzi equation, we obtain the following lemma.

Lemma 3.1** ([31]).**

Let $M$ be a Hopf hypersurface in $Q^{m}$, $m\geq 3$. Then we obtain

[TABLE]

and

[TABLE]

for any tangent vector fields $X$ and $Y$ on $M$.

Remark 3.2**.**

By virtue of (3.6) we know if $M$ has vanishing geodesic Reeb flow (or constant Reeb curvature, respectively), then the normal vector $N$ is singular. In fact, under this assumption (3.6) becomes $g(A\xi,\xi)g(X,AN)=0$ for any tangent vector field $X$ on $M$. Since $g(A\xi,\xi)=-\cos(2t)$, the case of $g(A\xi,\xi)=0$ implies that $N$ is $\mathfrak{A}$-isotropic. Besides, if $g(A\xi,\xi)\neq 0$, that is, g(AN, X)=0 for all $X\in TM$, then

[TABLE]

which implies that $N=A^{2}N=g(AN,N)AN$. Taking an inner product with $N$, it follows $g(AN,N)=\pm 1$. Since $g(AN,N)=\cos(2t)$ where $t\in[0,\frac{\pi}{4})$, we obtain $AN=N$. Hence $N$ should be $\mathfrak{A}$-principal.**

Lemma 3.3** ([26]).**

Let $M$ be a Hopf hypersurface in $Q^{m}$ such that the normal vector field $N$ is ${\mathfrak{A}}$-principal everywhere. Then $\alpha$ is constant. Moreover, if $X\in{\mathcal{C}}$ is a principal curvature vector of $M$ with principal curvature $\lambda$, then $2\lambda\neq\alpha$ and its corresponding vector $\phi X$ is a principal curvature vector of $M$ with principal curvature $\frac{\alpha\lambda+2}{2{\lambda}-{\alpha}}$.

Lemma 3.4** ([26]).**

Let $M$ be a Hopf hypersurface in $Q^{m}$, $m\geq 3$, such that the normal vector field $N$ is ${\mathfrak{A}}$-isotropic everywhere. Then $\alpha$ is constant.

If the normal vector $N$ is $\mathfrak{A}$-isotropic, then we obtain

[TABLE]

from (3.5) and the notation of $N$. Taking the covariant derivative of $g(AN,N)=0$ along the direction of any $X\in T_{[z]}M$, $[z]\in M$, it becomes

[TABLE]

where we have used the covariant derivative of the complex structure $A$, that is, $(\bar{\nabla}_{X}A)Y=q(X)JAY$ and the formula of Weingarten. Then the above formula gives $SAN=0$, because $AN$ becomes a tangent vector field on $M$ for $\mathfrak{A}$-isotropic unit normal vector field $N$.

On the other hand, by differentiating $g(A{\xi},N)=0$ and using the formula of Gauss, we have:

[TABLE]

where $\sigma$ is the second fundamental form of $M$ and ${\phi}AN=JAN=-AJN=A{\xi}$. By $g(A\xi,N)=0$, the vector field $A\xi$ becomes a tangent vector field on $M$ with $\mathfrak{A}$-isotropic unit normal vector field $N$. Then the above formula gives $SA{\xi}=0$.

Moreover, when the normal vector $N$ is $\mathfrak{A}$-isotropic, the tangent vector space $T_{[z]}M$, $[z]\in M$, is decomposed

[TABLE]

where ${\mathcal{C}}\ominus{\mathcal{Q}}={\mathcal{Q}}^{\bot}=\text{Span}[A{\xi},AN]$. From the equation (3.7), we obtain

[TABLE]

for some principal curvature vector $X\in\mathcal{Q}\subset T_{[z]}M$ such that $SX=\lambda X$. If $2\lambda-\alpha=0$ (i.e. $\lambda=\frac{\alpha}{2}$), then $\alpha\lambda+2=\frac{\alpha^{2}+4}{2}=0$, which makes a contradiction. Hence we obtain:

Lemma 3.5**.**

Let $M$ be a Hopf hypersurface in $Q^{m}$ such that the normal vector field $N$ is ${\mathfrak{A}}$-isotropic. Then $SA\xi=0$ and $SAN=0$. Moreover, if $X\in{\mathcal{Q}}$ is a principal curvature vector of $M$ with principal curvature $\lambda$, then $2\lambda\neq\alpha$ and its corresponding vector $\phi X$ is a principal curvature vector of $M$ with principal curvature $\frac{\alpha\lambda+2}{2{\lambda}-{\alpha}}$.

On the other hand, from the property of $g(A\xi,N)=0$ on a real hypersurface $M$ in $Q^{m}$ we see that the non-zero vector field $A\xi$ is tangent to $M$. Hence by Gauss formula it induces

[TABLE]

for any $X\in TM$. From $AN=AJ\xi=-JA\xi$ and $JA\xi=\phi A\xi+\eta(A\xi)N$, it gives us

[TABLE]

In particular, if $M$ is Hopf, then the second equation in (3.8) becomes

[TABLE]

4. Proof of Theorem 1

• Reeb parallel structure Jacobi operator with $\mathfrak{A}$-principal normal -

Let us $M$ be a real hypersurface in the complex quadric $Q^{m}$, $m\geq 3$, with Reeb parallel structure Jacobi operator, that is,

[TABLE]

for all tangent vector fields $Y$ of $M$.

As mentioned in section 1 the structure Jacobi operator ${R}_{\xi}\in\mathrm{End}(TM)$ with respect to the unit tangent vector field $\xi\in TM$ is induced from the curvature tensor ${R}$ of $M$ given in section 3 as follows: for any tangent vector fields $Y$, $Z\in TM$

[TABLE]

where we have used $J\xi=N$, $JA=-AJ$, and $g(A\xi,N)=0$.

Remark**.**

For any tangent vector field $X$ of $M$ the vector field $AX$ belongs to $TQ^{m}$, that is, $AX=BX+\rho(X)N\in TM\oplus(TM)^{\bot}=TQ^{m}$. Therefore, from (4) the structure Jacobi operator of $M$ is given by

[TABLE]

Here we have used that $A\xi=B\xi\in TM$ (i.e. $\rho(\xi)=g(AN,\xi)=0$) and $AN=AJ\xi=-JA\xi=-\phi A\xi-\eta(A\xi)N$.

Taking the covariant derivative of (4.1) along the direction of $X\in TM$, then we have

[TABLE]

where we have used (3.8) and

[TABLE]

Since $M$ is a Hopf real hypersurface in $Q^{m}$ with Reeb parallel structure Jacobi operator, it yields that

[TABLE]

[TABLE]

From now on, we assume that $M$ is a Hopf real hypersurface with non-vanishing geodesic Reeb flow and with Reeb parallel structure Jacobi operator in the complex quadric $Q^{m}$, $m\geq 3$. In addition, we suppose that the normal vector field $N$ of $M$ is $\mathfrak{A}$-principal. Then this assumption gives us

[TABLE]

from (3.5). So it follows that $AY\in TM$ for all $Y\in TM$, that is, $g(AY,N)=g(Y,AN)=0$. Moreover, taking the derivative to $AN=N$ with respect to the Levi-Civita connection $\bar{\nabla}$ of $Q^{m}$ and using (3.8), we get

[TABLE]

together with $({\bar{\nabla}}_{Y}A)X=q(Y)JAX$ and ${\bar{\nabla}}_{Y}N=-SY$.

From these properties, the equation (4.2) can be rearranged as follows.

[TABLE]

In addition, from (3.9) we know $q({\xi})=2{\alpha}$. By Lemma 3.3 and our assumption, the Reeb curvature function $\alpha$ is non-zero constant on $M$. So (4.4) reduces to the following

[TABLE]

together with $JAY=\phi AY+\eta(AY)N=\phi AY-\eta(Y)N$.

On the other hand, by using the equation of Codazzi in section (3.6), we have

[TABLE]

Since $M$ is Hopf and Lemma 3.3, it leads to

[TABLE]

From this, together with (4.5), it follows that

[TABLE]

By virtue of Lemma 3.1, for the $\mathfrak{A}$-principal unit normal vector field, we obtain

[TABLE]

Therefore, (4.6) can be written as

[TABLE]

Inserting $Y=SX$ for $X\in\mathcal{C}$ into (4.8) and taking the structure tensor $\phi$ leads to

[TABLE]

where $\mathcal{C}=\mathrm{ker}\,\eta$ denotes the maximal complex subbundle of $TM$, which is defined by a distribution $\mathcal{C}=\{X\in T_{[z]}M\,|\,\eta(X)=0\}$ in $T_{[z]}M$, $[z]\in M$. By using (4.3) and (4.7) this equation gives us

[TABLE]

for all $X\in\mathcal{C}$.

On the other hand, in this subsection we have assumed that the normal vector field $N$ of $M$ is $\mathfrak{A}$-principal. It follows that $AY\in TM$ for all $Y\in TM$. From this, the anti-commuting property with respect to $J$ and $A$ implies $\phi AX=-A\phi X$. Hence (4.8) can be expressed as

[TABLE]

Putting $Y=\phi X$ into (4.10), it gives

[TABLE]

for all $X\in\mathcal{C}$. Inserting this into (4.9) gives

[TABLE]

Taking the complex conjugate $A$ to (4.11) and using (4.3) again, we get

[TABLE]

for all $X\in\mathcal{C}$. Summing up (4.11) and (4.12), gives $AX=X$ for all $X\in\mathcal{C}$. This gives a contradiction. In fact, it is well known that the trace of the real structure $A$ is zero, that is, $\mathrm{Tr}A=0$ (see Lemma 1 in [25]). For an orthonormal basis $\{\,e_{1},e_{2}\cdots,e_{2m-2},e_{2m-1}=\xi,e_{2m}=N\,\}$ for $T{Q^{m}}$, where $e_{j}\in\mathcal{C}$ $(j=1,2,\cdots,2m-2)$, the trace of $A$ is given by

[TABLE]

It implies that $m=1$. But we now consider for the case $m\geq 3$.

Consequently, this completes the proof that there does not exists a Hopf real hypersurface $(\alpha\neq 0)$ in complex quadrics ${Q^{m}}$, $m\geq 3$, with Reeb parallel structure Jacobi operator and $\mathfrak{A}$-principal normal vector field.

5. Proof of Theorem 2

• Reeb parallel structure Jacobi operator with $\mathfrak{A}$-isotropic normal -

In this section, we assume that the unit normal vector field $N$ is $\mathfrak{A}$-isotropic and $M$ is a real hypersurface in complex quadric $Q^{m}$ with non-vanishing geodesic Reeb flow and with Reeb parallel structure Jacobi operator. Then the normal vector field $N$ can be written as

[TABLE]

for some orthonormal vectors $Z_{1}$, $Z_{2}\in V(A)$, where $V(A)$ denotes a $(+1)$-eigenspace of the complex conjugation $A\in~{}{\mathfrak{A}}$. Then it follows that

[TABLE]

Then it gives that

[TABLE]

which means that both vector fields $AN$ and $A\xi$ are tangent to $M$. From this and Lemma 3.4, we see that the shape operator $S$ of $M$ becomes to be Reeb parallel, that is, $(\nabla_{\xi}S)Y=0$ for all tangent vector field $Y$ on $M$.

On the other hand, from the Codazzi equation (3.2) we obtain

[TABLE]

where the third equality holds from Lemmas 3.1 and 3.4. From this and $M$ has non-vanishing geodesic Reeb flow, we see that $M$ has isometric Reeb flow, that is, $S\phi=\phi S$.

Consequently, we obtain:

Proposition 5.1**.**

Let $M$ be a real hypersurface with non-vanishing geodesic Reeb flow in the complex quadrics ${Q^{m}}$, $m\geq 3$. If the unit normal vector field $N$ of $M$ is $\mathfrak{A}$-isotropic and the structure Jacobi operator $R_{\xi}$ of $M$ is Reeb parallel, then the shape operator $S$ of $M$ satisfies the property of Reeb parallelism. Moreover, it means that the Reeb flow of $M$ is isometric.

Theorem B**.**

Let $M$ be a real hypersurface of the complex quadric $Q^{m}$, $m\geq 3$. The Reeb flow on $M$ is isometric if and only if $m$ is even, say $m=2k$, and $M$ is an open part of a tube around a totally geodesic $\mathbb{C}P^{k}$ in $Q^{2k}$.

Then by virtue of Theorem $\rm B$, we assert: if $M$ is a real hypersurface in $Q^{m}$, $m\geq 3$, with the assumptions given in Proposition 5.1, then $M$ is an open part of $(\mathcal{T}_{A})$. Here the model space $(\mathcal{T}_{A})$ is a tube over a totally geodesic complex projective space $\mathbb{C}P^{k}$ in $Q^{2k}$, $m=2k$.

From now on, let us check the converse problem, that is, the model space $(\mathcal{T}_{A})$ satisfies the all assumptions stated in Proposition 5.1. In order to do this, we first introduce one proposition given in [26].

Proposition A**.**

Let $(\mathcal{T}_{A})$ be the tube of radius $0<r<\frac{\pi}{2}$ around the totally geodesic $\mathbb{C}P^{k}$ in $Q^{2k}$. Then the following statements hold:

1. (i)

$(\mathcal{T}_{A})$* is a Hopf hypersurface.* 2. (ii)

Every unit normal vector $N$ of $(\mathcal{T}_{A})$ is $\mathfrak{A}$-isotropic and therefore can be written in the form $N=(Z_{1}+JZ_{2})/\sqrt{2}$ with some orthonormal vectors $Z_{1}$, $Z_{2}\in V(A)$ and $A\in\mathfrak{A}$. 3. (iii)

$(\mathcal{T}_{A})$* has four distinct constant principal curvatures and the property that the shape operator leaves invariant the maximal complex subbundle $\mathcal{C}$ of $T(\mathcal{T}_{A})$ are $J$-invariant. The principal curvatures and corresponding principal curvature spaces of $(\mathcal{T}_{A})$ are as follows.*

[TABLE] 4. (iv)

Each of the two focal sets of $(\mathcal{T}_{A})$ is a totally geodesic $\mathbb{C}P^{k}\subset Q^{2k}$. 5. (v)

$S\phi=\phi S$* (isometric Reeb flow).* 6. (vi)

$(\mathcal{T}_{A})$* is a homogeneous hypersurface of $Q^{2k}$. More precisely, it is an orbit of the $U_{k+1}$-action on $Q^{2k}$ isomorphic to $U_{k+1}/U_{k-1}U_{1}$, an $S^{2k-1}$-bundle over $\mathbb{C}P^{k}$.*

By virtue of $\rm(i)$ and $\rm(ii)$ in Proposition $\rm A$, $(\mathcal{T}_{A})$ is a Hopf real hypersurface with $\mathfrak{A}$-isotropic normal vector $N$ in $Q^{m}$. Moreover, the structure Jacobi operator $R_{\xi}$ of $(\mathcal{T}_{A})$ should be Reeb parallel, because of $\alpha=2\cot(2r)\neq 0$, $0<r<\frac{\pi}{2}$.