2 Preliminaries

In this section, we shall introduce the basic materials only about the complex quadric $Q^m$ and its real hypersurfaces. For those about the complex hyperbolic quadric $Q^{m*}$ that are completely similar, we refer to [Reference Klein and Suh13, Reference Suh25, Reference Suh, Pérez and Woo27, Reference Van der Veken and Wijffels28] for details.

2.1 The complex quadric

As described in the introduction section, the complex quadric $Q^m$ is equipped with a canonical Kähler structure, denoted by $\{J,g\}$ , induced from that of $\mathbb {C}P^{m+1}$ . For the purpose of this paper, we shall assume $m\ge 3$ in the sequel.

For a nonzero vector $z\in \mathbb {C}^{m+2}$ , we denote by $[z]$ the complex span of z, that is, $[z]=\{\lambda z|\lambda \in \mathbb {C}^\ast \}$ . For $[z]\in Q^m$ , the tangent space $T_{[z]}Q^m$ can be identified canonically with $\mathbb {C}^{m+2}\ominus ([z]\oplus [\bar {z}])$ in $\mathbb {C}^{m+2}$ . Note that $\zeta =-\bar {z}$ is a unit normal vector of $Q^m$ in $\mathbb {C}P^{m+1}$ at the point $[z]$ , and the shape operator $A_{\zeta }$ of $Q^m$ in $\mathbb {C}P^{m+1}$ with respect to $\zeta $ is given by $A_{\zeta }w=\bar {w}$ for all $w\in T_{[z]}Q^m$ . Thus, restricted to $T_{[z]}Q^m$ , $A_{\zeta }$ is the complex conjugation. Moreover, acting on the complex vector space $T_{[z]}Q^m$ , the shape operator $A_{\zeta }$ is an anticommuting involution such that $A_{\zeta }^2=Id$ and $AJ=-JA$ . Hence, we have $T_{[z]}Q^m=V(A_{\zeta })\oplus JV(A_{\zeta })$ , where $V(A_{\zeta })=\mathbb {R}^{m+2}\cap T_{[z]}Q^m$ is the $(+1)$ -eigenspace and $JV(A_{\zeta })=\mathbf {i}\mathbb {R}^{m+2}\cap T_{[z]}Q^m$ is the $(-1)$ -eigenspace of $A_{\zeta }$ , respectively. Because the normal space $\nu _{[z]}Q^m$ of $Q^m$ in $\mathbb {C}P^{m+1}$ at $[z]$ is a complex subspace of $T_{[z]}\mathbb {C}P^{m+1}$ of complex dimension one, every unit normal vector in $\nu _{[z]}Q^m$ can be written as $\lambda \bar {z}$ with some $\lambda \in S^1\subset \mathbb {C}$ , and it holds that $V(A_{\lambda \bar {z}})=\lambda V(A_{\bar {z}})$ . We denote by $\mathfrak {A}$ the set of all shape operators of $Q^m\hookrightarrow \mathbb {C}P^{m+1}$ associated with unit normal vector fields. Then, $\mathfrak {A}$ is an $S^1$ -sub-bundle of the endomorphism bundle End( $TQ^m$ ), consisting of complex conjugations on the tangent spaces of $Q^m$ . In summary, we have

Lemma 2.1 (cf. [Reference Reckziegel22, Reference Smyth23])

For each $A\in \mathfrak {A}$ , it holds that

(2.1) $$ \begin{align} A^2=Id, \ \ g(AX,Y)=g(X,AY), \ \ AJ=-JA, \ \ \forall X,Y\in TQ^m. \end{align} $$

The sub-bundle $\mathfrak {A}$ is commonly called consisting of complex conjugations (cf. [Reference Reckziegel22]). On the other hand, Lemma 2.1 implies, in particular, that $\mathfrak {A}$ is a family of almost product structures on $Q^m$ (cf. [Reference Li, Ma, Van der Veken, Vrancken and Wang16]), and we will use the latter term in this paper. Furthermore, because the second fundamental form of the embedding $Q^m\hookrightarrow \mathbb {C}P^{m+1}$ is parallel, $\mathfrak {A}$ is contained in a parallel sub-bundle of End( $TQ^m$ ). It follows that, for each almost product structure $A\in \mathfrak {A}$ , there exists a one-form q on $Q^m$ such that it holds the relation (see [Reference Smyth23])

(2.2) $$ \begin{align} (\bar{\nabla}_XA)Y=q(X)JAY,\ \ \forall\, X,Y\in TQ^m, \end{align} $$

where $\bar {\nabla }$ denotes the Levi–Civita connection of $Q^m$ .

The Gauss equation for the complex hypersurface $Q^m\hookrightarrow \mathbb {C}P^{m+1}$ implies that the Riemannian curvature tensor $\bar {R}$ of $Q^m$ can be expressed in terms of the Riemannian metric g, the complex structure J, and a generic $A\in \mathfrak {A}$ as follows (cf. [Reference Reckziegel22]):

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

where it should be noted that $\bar {R}$ is independent of the special choice of $A\in \mathfrak {A}$ .

According to Reckziegel [Reference Reckziegel22], a real $2$ -dimensional linear subspace $\sigma $ of $T_{[z]}Q^m$ is called a $2$ -flat if the curvature tensor $\bar {R}$ of $Q^m$ vanishes identically on $\sigma $ . A nonzero tangent vector $W\in T_{[z]}Q^m$ is called $\mathfrak {A}$ -singular if it is contained in more than one $2$ -flat. There are two types of $\mathfrak {A}$ -singular tangent vectors for $Q^m$ , namely, $\mathfrak {A}$ -principal and $\mathfrak {A}$ -isotropic tangent vectors, described as follows (cf. [Reference Berndt and Suh3, Reference Reckziegel22]):

1. 1. If there exists an almost product structure $A\in \mathfrak {A}_{[z]}$ such that $W\in V(A)$ , then W is $\mathfrak {A}$ -singular. Such $W\in T_{[z]}Q^m$ is called an $\mathfrak {A}$ -principal tangent vector.

2. 2. If there exists an almost product structure $A\in \mathfrak {A}_{[z]}$ and two orthonormal vectors $X,Y\in V(A)$ such that $\frac {W}{\|W\|}=\frac {X+JY}{\sqrt {2}}$ , then W is $\mathfrak {A}$ -singular. In this case, we have $g(AW,W)=0$ and $W\in T_{[z]}Q^m$ is called an $\mathfrak {A}$ -isotropic tangent vector.

Further results about the complex quadric are referred to [Reference Berndt and Suh3, Reference Klein12, Reference Reckziegel22].

2.2 Hypersurfaces of the complex quadric

We begin with introducing the basic formulas. Let M be a hypersurface of $Q^m$ and N its unit normal vector field. For a generic almost product structure $A\in \mathfrak {A}$ and an arbitrary tangent vector field X of M, we have the decomposition

(2.4) $$ \begin{align} JX=\phi X+ \eta(X)N,\ \ AX=TX+ \mu(X)N, \end{align} $$

where $\phi X$ (resp. $TX$ ) and $\eta (X)N$ (resp. $\mu (X)N$ ) are the tangent and normal parts of $JX$ (resp. $AX$ ), respectively. Here, $\phi $ and T are tensor fields of type $(1,1)$ , and $\eta $ and $\mu $ are $1$ -forms over M. Then, by definition, we have the following relations:

(2.5) $$ \begin{align} \left\{ \begin{aligned} &\eta(X)=g(X,\xi),\ \ \eta(\phi X)=0,\ \ \phi^2X=-X+\eta(X)\xi,\\ &g(\phi X,Y)=-g(X,\phi Y),\ \ g(\phi X,\phi Y)=g(X,Y)-\eta(X)\eta(Y),\\ &g(TX,Y)=g(X,TY),\ \ \mu(X)=g(AX,N), \end{aligned}\right. \end{align} $$

where $\xi :=-JN$ is called the Reeb vector field of M. The equations in (2.5) show that $\{\phi ,\xi ,\eta \}$ determines an almost contact structure on M.

Let $\nabla $ be the induced connection on M with R its Riemannian curvature tensor. The formulas of Gauss and Weingarten state that

(2.6) $$ \begin{align} \bar \nabla_X Y=\nabla_X Y + g(SX,Y)N,\quad \bar \nabla_X N=- S X , \ \ \forall\, X,Y \in TM, \end{align} $$

where S is the shape operator of $M\hookrightarrow Q^m$ .

Next, we calculate the covariant derivatives of the tensors $\phi ,\eta ,T$ , and $\mu $ . These basic equations are necessary for our proof of Theorem 1.1.

Lemma 2.2 The covariant derivatives of the tensors $\phi ,T,\mu $ , and $\eta $ are given by $:$

(2.7) $$ \begin{align} \nabla_{X} \xi=\phi SX, \end{align} $$

(2.8) $$ \begin{align} (\nabla_{X} \phi)Y=-g(SX,Y)\xi+\eta(Y)SX, \end{align} $$

(2.9) $$ \begin{align} (\nabla_X T)Y=q(X)[\phi TY-\mu(Y)\xi]+g(SX,Y)[-\phi T\xi+\mu(\xi)\xi]+\mu(Y)SX, \end{align} $$

(2.10) $$ \begin{align} (\nabla_X \mu)Y=q(X)g(TY,\xi)-g(SX,TY)-g(SX,Y)g(T\xi,\xi), \end{align} $$

(2.11) $$ \begin{align} (\nabla_X \eta)Y=-g(SX,\phi Y). \end{align} $$

Proof These equations are derived by direct calculations. Specifically, we have the following derivation by using the basic equations in (2.2), (2.5), and (2.6):

$$ \begin{align*} \nabla_X \xi&=-\bar{\nabla}_X JN-g(SX,\xi)N=-J\bar{\nabla}_X N-g(SX,\xi)N\\ &=JSX-g(SX,\xi)N=\phi SX, \end{align*} $$

$$ \begin{align*} (\nabla_X \phi)Y&=\nabla_X {\phi Y}-\phi\nabla_X Y=\bar{\nabla}_X {\phi Y} -g(SX,\phi Y)N-\phi\nabla_X Y\\ &=[\bar{\nabla}_X (JY-\eta(Y)N)]^\top-\phi\nabla_X Y\\ &=[(\bar{\nabla}_X J)Y+J\bar{\nabla}_X Y]^\top-\eta(Y)\bar{\nabla}_X N-\phi\nabla_X Y\\ &=-g(SX,Y)\xi+\eta(Y)SX, \end{align*} $$

$$ \begin{align*} (\nabla_X T)Y&=\nabla_X {TY}-T\nabla_X Y=\bar{\nabla}_X {TY}-g(SX,TY)N-T\nabla_X Y\\ &=[\bar{\nabla}_X (AY-\mu(Y)N)]^\top-T\nabla_X Y\\ &=[(\bar{\nabla}_X A)Y+A\bar{\nabla}_X Y]^\top-\mu(Y)\bar{\nabla}_X N-T\nabla_X Y\\ &=[q(X)JAY+g(SX,Y)AN]^\top+\mu(Y)SX\\ &=q(X)[\phi TY-\mu(Y)\xi]+g(SX,Y)[-\phi T\xi+\mu(\xi)\xi]+\mu(Y)SX, \end{align*} $$

$$ \begin{align*} (\nabla_X \mu)Y&=X(\mu(Y))-\mu(\nabla_X Y)\\ &=g(\bar{\nabla}_X {AY},N) +g(AY,\bar{\nabla}_X N)-g(A\bar{\nabla}_X Y-g(SX,Y)AN,N)\\ &=g((\bar{\nabla}_X A)Y,N)-g(SX,TY)+g(SX,Y)g(AN,N)\\ &=g(q(X)JAY,N)-g(SX,TY)+g(SX,Y)g(AN,N)\\ &=q(X)g(TY,\xi)-g(SX,TY)-g(SX,Y)g(T\xi,\xi), \end{align*} $$

$$ \begin{align*} (\nabla_X \eta)Y=X(\eta(Y))-\eta(\nabla_X Y)=g(Y,\nabla_X \xi)=-g(SX,\phi Y), \end{align*} $$

where $\cdot ^\top $ denotes the tangential part.▪

Third, by using the decomposition of A and the expression of the curvature tensor of $Q^m$ , we have the following Gauss and Codazzi equations of M:

(2.12) $$ \begin{align} R(X,Y)Z=&g(Y,Z)X-g(X,Z)Y+g(\phi Y,Z)\phi X-g(\phi X,Z)\phi Y\nonumber\\ &-2g(\phi X,Y)\phi Z+g(AY,Z)(AX)^\top-g(AX,Z)(AY)^\top\nonumber\\ &+g(JAY,Z)(JAX)^\top-g(JAX,Z)(JAY)^\top\nonumber\\ &+g(SZ,Y)SX-g(SZ,X)SY, \end{align} $$

(2.13) $$ \begin{align} g\big((\nabla_X S)Y&-(\nabla_Y S)X,Z\big)=\eta(X)g(\phi Y,Z)-\eta(Y)g(\phi X,Z)\nonumber\\ &-2g(\phi X,Y)\eta(Z)+\mu(X)g(TY,Z)-\mu(Y)g(TX,Z)\nonumber\\ &-g(TX,\xi)g(TY,\phi Z)-g(TX,\xi)\mu(Y)\eta(Z)\nonumber\\ &+g(TY,\xi)g(TX,\phi Z)+g(TY,\xi)\mu(X)\eta(Z). \end{align} $$

Contracting Y and Z in (2.12), we get the Ricci tensor of M (cf. (4.1) of [Reference Suh24]):

(2.14) $$ \begin{align} \text{Ric}\,(X,Y)&=(2m-1)g(X,Y)-3\eta(X)\eta(Y)-g(AN,N)g(AX,Y)\nonumber\\ &\quad +g(AX,N)g(AN,Y)-g(A\xi,N)g(JAX,Y)\nonumber\\ &\quad +g(A\xi,X)g(A\xi,Y)+Hg(SX,Y)-g(S^2X,Y)\nonumber\\ &=(2m-1)g(X,Y)-3\eta(X)\eta(Y)+g(T\xi,\xi)g(TX,Y)\nonumber\\ &\quad +\mu(X)\mu(Y)+\mu(\xi)g(TX,\phi Y)+\mu(\xi)\mu(X)\eta(Y)\nonumber\\ &\quad +g(TX,\xi)g(TY,\xi)+Hg(SX,Y)-g(S^2X,Y), \end{align} $$

where $H=\mathrm {tr}\,S$ denotes the mean curvature of the hypersurface M.

Let $\nabla ^2 S$ denote the second covariant derivative of S, defined by:

$$ \begin{align*}(\nabla^{2}S)(X,Y,Z):=\nabla_X[(\nabla_Y S)Z]-(\nabla_{\nabla_X Y} S)Z-(\nabla_Y S){\nabla_X Z}. \end{align*} $$

Then, we have the following Ricci identity:

(2.15) $$ \begin{align} g\big((\nabla^{2}S)(X,Y,Z),&W\big)-g\big((\nabla^{2}S)(Y,X,Z),W\big)\nonumber\\ &=-g\big(R(X,Y)Z,SW\big)-g\big(R(X,Y)W,SZ\big). \end{align} $$

Notice that the tangent bundle $TM$ of M splits orthogonally into $TM=\mathcal {C}\oplus \mathcal {F}$ , where $\mathcal {C}=\text {ker}(\eta )$ is the maximal complex sub-bundle of $TM$ and $\mathcal {F}=\mathbb {R}\xi $ . When restricted to $\mathcal {C}$ , the structure tensor field $\phi $ coincides with the complex structure J. Moreover, at each point $[z]\in M$ , the set

$$ \begin{align*}\mathcal{Q}_{[z]}=\{X\in T_{[z]}M\mid AX\in T_{[z]}M\ \text{for all}\ A\in \mathfrak{A}_{[z]}\} \end{align*} $$

defines a maximal $\mathfrak {A}_{[z]}$ -invariant subspace of $T_{[z]}M$ .

According to Proposition 3 of [Reference Reckziegel22], at each point $[z]\in M$ , we can choose $A\in \mathfrak {A}_{[z]}$ and two orthonormal vectors $Z_1,Z_2\in V(A)$ such that the unit normal vector field N takes the form

(2.16) $$ \begin{align} N=\cos t\,Z_1+\sin t\,JZ_2, \end{align} $$

where the parameter function t satisfies $0\leq t\leq \frac {\pi }{4}$ . Using $\xi =-JN$ , we further get

$$ \begin{align*} AN=\cos t\,Z_1-\sin t\,JZ_2,\\ \xi=\sin t\,Z_2-\cos t\,JZ_1,\\ A\xi=\sin t\,Z_2+\cos t\,JZ_1. \end{align*} $$

This implies that $g(A\xi ,N)=0$ and $A\xi \in T_{[z]}M$ .

Remark 2.1 From the above facts, we can get the following conclusions:

1. (1) If $N_{[z]}$ is $\mathfrak {A}$ -principal, we have $t=0$ at $[z]$ , $\mathcal {Q}_{[z]}=\mathcal {C}_{[z]}$ , and $\dim \mathcal {Q}_{[z]}=2m-2$ . For each $A\in \mathfrak {A}_{[z]}$ , (2.1) implies that $A|_{\mathcal {Q}_{[z]}}$ has two eigenvalues $1$ and $-1$ . For $\varepsilon \in \{1,-1\}$ , if denoting by $\mathcal {Q}(\varepsilon )$ the eigenspace of $A|_{\mathcal {Q}}$ corresponding to $\varepsilon $ , then it holds that $J\mathcal {Q}(1)=\mathcal {Q}(-1)$ and $\dim \mathcal {Q}(1)=\dim \mathcal {Q}(-1)=m-1$ .

2. (2) If $N_{[z]}$ is not $\mathfrak {A}$ -principal, we get $\mathcal {Q}_{[z]}=\{N_{[z]},\xi _{[z]},AN_{[z]},A\xi _{[z]}\}^\bot $ for any $A\in \mathfrak {A}_{[z]}$ , and $\dim \mathcal {Q}_{[z]}=2(m-2)$ . For each $A\in \mathfrak {A}_{[z]}$ , (2.1) implies that $A|_{\mathcal {Q}_{[z]}}$ has two eigenvalues $1$ and $-1$ . Moreover, it holds that $J\mathcal {Q}(1)=\mathcal {Q}(-1)$ and $\dim \mathcal {Q}(1)=\dim \mathcal {Q}(-1)=m-2$ .

Finally, we recall the following lemma due to Berndt and Suh [Reference Berndt and Suh3] (cf. also [Reference Loo17]). This lemma is very helpful for simplifying our latter calculations. Specifically, it allows us to choose locally, under appropriate conditions, the almost product structure $A\in \mathfrak {A}$ such that $A\xi \in TM$ and $g(AN,\xi )=0$ .

Lemma 2.3 (cf. [Reference Berndt and Suh3] and [Reference Loo17])

Let M be a hypersurface of $Q^m$ with unit normal vector field N and $\xi =-JN$ . Then, we have $:$

1. (a) If N is $\mathfrak {A}$ -principal on an open set $U\subset M$ , then there exists an almost product structure $A\in \mathfrak {A}$ on U such that $AN=N$ .

2. (b) If N is not $\mathfrak {A}$ -principal at $[z]\in M$ , then there exist a neighborhood U of $[z]$ and an almost product structure $A\in \mathfrak {A}$ on U such that $A\xi \in TM$ .

3 Key lemmas related to the Tsinghua principle

We begin with establishing the following general lemma about real hypersurfaces of the complex quadric.

Lemma 3.1 Let M be a real hypersurface in the complex quadric $Q^m\ (m\geq 3)$ . Then, for a generic almost product structure $A\in \mathfrak {A}$ and any tangent vector fields $W,X,Y,Z\in TM$ , we have

(3.1) $$ \begin{align} -\mathop{\mathfrak{S}}\limits_{WXY} \big\{g(R(W,X)Y,SZ) +g(R(W,X)Z,SY)\big\}=\mathop{\mathfrak{S}}\limits_{WXY}\mathbf{I}(W,X,Y,Z), \end{align} $$

where $\mathop {\mathfrak {S}}\limits _{WXY}$ denotes the cyclic summation over $W,X$ , and Y, and

$$ \begin{align*} \mathbf{I}&(W,X,Y,Z):=-g(SW,\phi X)g(\phi Y,Z)+g(SW,\phi Y)g(\phi X,Z)\\ &+2g(SW,\phi Z)g(\phi X,Y)-3g(SW,Y)\eta(X)\eta(Z)+3g(SW,X)\eta(Y)\eta(Z)\\ &-[g(TX,SW)+\eta(T\xi)g(SW,X)]g(TY,Z)+\mu(X)g(SW,Y)\mu(Z)\\ &+[g(TY,SW)+\eta(T\xi)g(SW,Y)]g(TX,Z)-\mu(Y)g(SW,X)\mu(Z)\\ &-[g(SW,X)\mu(\xi)+\mu(X)\eta(SW)]g(TY,\phi Z)-g(TX,\phi SW)g(TY,\phi Z)\\ &+\eta(TX)g(SW,Y)\eta(TZ)-g(SW,X)\mu(\xi)\mu(Y)\eta(Z)-g(TX,\phi SW)\mu(Y)\eta(Z)\\ &+[g(SW,Y)\mu(\xi)+\mu(Y)\eta(SW)]g(TX,\phi Z)+g(TY,\phi SW)g(TX,\phi Z)\\ &-\eta(TY)g(SW,X)\eta(TZ)+g(SW,Y)\mu(\xi)\mu(X)\eta(Z)+g(TY,\phi SW)\mu(X)\eta(Z). \end{align*} $$

Proof We shall calculate the expression of the cyclic summation

(3.2) $$ \begin{align} \mathfrak{B}:=\mathop{\mathfrak{S}}\limits_{WXY} \big\{g((\nabla^2 S)(W,X,Y),Z)- g((\nabla^2 S)(W,Y,X),Z)\big\} \end{align} $$

in two different ways. On the one hand, taking the covariant derivative of the Codazzi equation (2.13), we can get

$$ \begin{align*} &g((\nabla^2 S)(W,X,Y),Z)-g((\nabla^2 S)(W,Y,X),Z)\\ &=(\nabla_W\eta)(X)g(\phi Y,Z)+\eta(X)g((\nabla_W\phi)Y,Z)-(\nabla_W\eta)(Y)g(\phi X,Z)\\ &-\eta(Y)g((\nabla_W\phi)X,Z)-2g((\nabla_W\phi)X,Y)\eta(Z)-2g(\phi X,Y)(\nabla_W\eta)(Z)\\ &+(\nabla_W\mu)(X)g(TY,Z)+\mu(X)g((\nabla_WT)Y,Z)-(\nabla_W\mu)(Y)g(TX,Z)\\ &-\mu(Y)g((\nabla_WT)X,Z)-g((\nabla_WT)X,\xi)g(TY,\phi Z)-g(TX,\nabla_W\xi)g(TY,\phi Z)\\ &-g(TX,\xi)g((\nabla_WT)Y,\phi Z)-g(TX,\xi)g(TY,(\nabla_W\phi)Z)\\ &-g((\nabla_WT)X,\xi)\mu(Y)\eta(Z)-g(TX,\nabla_W\xi)\mu(Y)\eta(Z)-g(TX,\xi)(\nabla_W\mu)(Y)\eta(Z)\\ &-g(TX,\xi)\mu(Y)(\nabla_W\eta)(Z)+g((\nabla_WT)Y,\xi)g(TX,\phi Z)\\ &+g(TY,\nabla_W\xi)g(TX,\phi Z)+g(TY,\xi)g((\nabla_WT)X,\phi Z)\\ &+g(TY,\xi)g(TX,(\nabla_W\phi)Z)+g((\nabla_WT)Y,\xi)\mu(X)\eta(Z)\\&+g(TY,\nabla_W\xi)\mu(X)\eta(Z)+g(TY,\xi)(\nabla_W\mu)(X)\eta(Z)+g(TY,\xi)\mu(X)(\nabla_W\eta)(Z). \end{align*} $$

Then, by straightforward calculations of the RHS, with the use of equations in (2.5) and Lemma 2.2, we can obtain

(3.3) $$ \begin{align} g((\nabla^2 S)(W,X,Y),Z)-g((\nabla^2 S)(W,Y,X),Z)=\mathbf{I}(W,X,Y,Z). \end{align} $$

Here, we note that the terms involving $q(W)$ are cancelled out with each other. Interestingly, when taking the covariant derivative of the Codazzi equation for a Lagrangian submanifold of $Q^m$ , this phenomenon also occurred in [Reference Li, Ma, Van der Veken, Vrancken and Wang16].

Now, from (3.3), we have

(3.4) $$ \begin{align} \mathfrak{B}=\mathop{\mathfrak{S}}\limits_{WXY}\mathbf{I}(W,X,Y,Z). \end{align} $$

On the other hand, the cyclic summation $\mathfrak {B}$ can be rewritten as

(3.5) $$ \begin{align} \mathfrak{B}=\mathop{\mathfrak{S}}\limits_{WXY}\big\{g((\nabla^{2}S)(W,X,Y),Z) -g((\nabla^{2}S)(X,W,Y),Z)\big\}. \end{align} $$

Then, we can apply the Ricci identity (2.15), so that, from (3.5), we obtain

(3.6) $$ \begin{align} \mathfrak{B}=-\mathop{\mathfrak{S}}\limits_{WXY} \big\{g(R(W,X)Y,SZ) +g(R(W,X)Z,SY)\big\}. \end{align} $$

From (3.4) and (3.6), we get the assertion (3.1). ▪

Next, for a conformally flat real hypersurface M of $Q^m$ , Lemma 3.1 reduces to the following lemma, which is crucial for the proof of Theorem 1.1.

Lemma 3.2 Let M be a conformally flat real hypersurface in the complex quadric $Q^m\ (m\geq 3)$ . Then, for a generic almost product structure $A\in \mathfrak {A}$ and any tangent vector fields $W,X,Y,Z\in TM$ , we have

(3.7) $$ \begin{align} \mathop{\mathfrak{S}}\limits_{WXY}\mathbf{II}(W,X,Y,Z) =\mathop{\mathfrak{S}}\limits_{WXY}\mathbf{I}(W,X,Y,Z), \end{align} $$

where $\mathbf {I}(W,X,Y,Z)$ is defined as in Lemma 3.1 , and

$$ \begin{align*} \mathbf{II}&(W,X,Y,Z):=\frac{3}{2m-3}\big[\eta(Y)\eta(SX)-\eta(X)\eta(SY)\big]g(W,Z)\\ &+\frac{1}{2m-3}\Big\{\eta(T\xi)\big[g(TX,SY)-g(TY,SX)\big]+\mu(X)\mu(SY)-\mu(Y)\mu(SX)\\ &-\mu(\xi)\big[g(TY,\phi SX)-g(TX,\phi SY)+\mu(Y)\eta(SX)-\mu(X)\eta(SY)\big]\\ &+\eta(TX)\eta(TSY)-\eta(TY)\eta(TSX)\Big\}g(W,Z). \end{align*} $$

Proof The conformally flatness of the real hypersurface M implies that it has vanishing Weyl curvature tensor or, equivalently, its Riemannian curvature tensor R takes the following form:

(3.8) $$ \begin{align} g(R(X,Y)Z,W)=&\frac{1}{2m-3}\big\{\text{Ric}(Y,Z)g(X,W)-\text{Ric}(X,Z)g(Y,W)\nonumber\\ &+\text{Ric}(X,W)g(Y,Z)-\text{Ric}(Y,W)g(X,Z)\big\}\nonumber\\ &+\frac{r}{(2m-2)(2m-3)}\big\{g(X,Z)g(Y,W)-g(Y,Z)g(X,W)\big\}, \end{align} $$

where r denotes the scalar curvature of M. Substituting (3.8) into (3.6), we obtain

(3.9) $$ \begin{align} \mathfrak{B}=\frac{1}{2m-3}\mathop{\mathfrak{S}}\limits_{WXY} \big\{g(W,Z)[\text{Ric}(X,SY)-\text{Ric}(Y,SX)]\big\}. \end{align} $$

By using (2.14), we finally have that

(3.10) $$ \begin{align} \mathfrak{B}=\mathop{\mathfrak{S}}\limits_{WXY}\mathbf{II}(W,X,Y,Z). \end{align} $$

By (3.4) and (3.10), the assertion (3.7) follows.▪

4 Proof of Theorem 1.1 for $Q^m$

In order to complete the proof of Theorem 1.1 for $Q^m\ (m\ge 3)$ , we suppose on the contrary that $Q^m$ admits a conformally flat real hypersurface M. First of all, with the help of Lemma 3.2, we can prove the following lemma.

Proof We argue by contradiction. Assume that at some point $[z]\in M$ , the unit normal vector $N_{[z]}$ is not $\mathfrak {A}$ -principal. By Lemma 2.3, there exist a neighborhood U around $[z]$ in M and an almost product structure $A\in \mathfrak {A}$ on U such that $A\xi \in TM$ and $\mu (\xi )=g(A\xi ,N)=0$ . It follows that there exist a unit tangent vector field $e_1\in \mathcal {C}$ and functions $a,c$ with $c>0$ such that

(4.1) $$ \begin{align} AN=aN+ce_1,\ \ A\xi=cJe_1-a\xi,\ \ a^2+c^2=1. \end{align} $$

Then, we get $\eta (T\xi )=g(T\xi ,\xi )=-a$ .

Put $e_2=Je_1,e_{2m-1}=\xi $ . From (2.1) and that $\dim {\mathcal {Q}}=2m-4$ on U, we can choose orthogonal unit tangent vector fields $\{e_3,\ldots ,e_{2m-2}\}$ such that

$$ \begin{align*}e_{2p-3}\in\mathcal{Q}(1),\ \ e_{2p-2}=Je_{2p-3}\in\mathcal{Q}(-1),\ \ 3\leq p\leq m. \end{align*} $$

Then, $\{e_i\}_{i=1}^{2m-1}$ forms a local orthonormal frame field of M with the following properties:

(4.2) $$ \begin{align} \left\{ \begin{aligned} &AN=aN+ce_1,\ \ Ae_1=cN-ae_1,\ \ Ae_2=ce_{2m-1}+ae_2,\\[-1mm] &Ae_{2p-3}=e_{2p-3},\ \ Ae_{2p-2}=-e_{2p-2},\ \ 3\le p\le m,\\[-1mm] &Ae_{2m-1}=ce_{2}-ae_{2m-1}. \end{aligned}\right. \end{align} $$

Put $Se_i=\sum _{j=1}^{2m-1}a_{ij}e_j$ , $1\leq i\leq 2m-1$ , where $a_{ij}=a_{ji},\ 1\leq i,j\leq 2m-1$ .

Now, we apply for Lemma 3.2 with $A\in \mathfrak {A}$ being chosen as above.

By choosing appropriate $W=e_i,\, X=e_j,\, Y=e_k,\, Z=e_l$ , $1\leq i,j,k,l\leq 2m-1$ , with the use of (2.4), (4.2), $\mu (\xi )=0$ , and $\eta (T\xi )=-a$ , we can calculate $\mathbf {I}(W,X,Y,Z)$ and $\mathbf {II}(W,X,Y,Z)$ directly. As the result, by (3.7), we shall obtain a system of linear equations of the components $\{a_{ij}\}$ . For instance, for $3\leq p\leq m$ , letting $(W,X,Y,Z)=(e_1,e_2,e_{2m-1},e_{2p-3}),\ (e_1,e_2,e_{2m-1},e_{2p-2})$ , respectively, by direct calculations, we have

$$ \begin{align*} \left\{ \begin{aligned} &\mathbf{I}(e_1,e_2,e_{2m-1},e_{2p-3})=\mathbf{I}(e_2,e_{2m-1},e_1,e_{2p-3})=0,\\ &\mathbf{I}(e_{2m-1},e_1,e_2,e_{2p-3})=2a_{{2p-2},{2m-1}};\\ &\mathbf{I}(e_1,e_2,e_{2m-1},e_{2p-2})=\mathbf{I}(e_2,e_{2m-1},e_1,e_{2p-2})=0,\\ &\mathbf{I}(e_{2m-1},e_1,e_2,e_{2p-2})=-2a_{{2p-3},{2m-1}};\\ &\mathbf{II}(e_1,e_2,e_{2m-1},e_{2p-3})=\mathbf{II}(e_2,e_{2m-1},e_1,e_{2p-3}) =\mathbf{II}(e_{2m-1},e_1,e_2,e_{2p-3})=0;\\ &\mathbf{II}(e_1,e_2,e_{2m-1},e_{2p-2})=\mathbf{II}(e_2,e_{2m-1},e_1,e_{2p-2}) =\mathbf{II}(e_{2m-1},e_1,e_2,e_{2p-2})=0. \end{aligned}\right. \end{align*} $$

It then follows from (3.7) that

(4.3) $$ \begin{align} a_{{2p-3},{2m-1}}=a_{{2p-2},{2m-1}}=0,\ \ 3\leq p\leq m. \end{align} $$

In the following, we shall carry the above procedures repeatedly, but the detailed calculation process will be omitted. To start, for $3\leq p\leq m$ , taking in (3.7),

$$ \begin{align*}(W,X,Y,Z)=(e_1,e_{2p-3},e_{2p-2},e_{2m-1}),\ (e_2,e_{2p-3},e_{2p-2},e_{2m-1}), \end{align*} $$

respectively, and using the fact $c>0$ , we obtain

(4.4) $$ \begin{align} a_{{2p-3},{2p-3}}=a_{{2p-2},{2p-2}},\ \ a_{{2p-3},{2p-2}}=0,\ \ 3\leq p\leq m. \end{align} $$

For $3\leq p\leq m$ , taking in (3.7),

$$ \begin{align*}(W,X,Y,Z)=(e_1,e_{2p-3},e_{2p-2},e_1),\ (e_1,e_{2p-3},e_{2p-2},e_2),\ (e_2,e_{2p-3},e_{2p-2},e_1), \end{align*} $$

respectively, with the use of (4.4), we obtain

(4.5) $$ \begin{align} a_{12}=0,\ \ a_{11}=a_{22}=a_{{2p-3},{2p-3}},\ \ 3\leq p\leq m. \end{align} $$

For $3\leq p\leq m$ , taking in (3.7),

$$ \begin{align*}(W,X,Y,Z)=(e_{2p-3},e_{2p-2},e_{2m-1},e_1),\ (e_{2p-3},e_{2p-2},e_{2m-1},e_2), \end{align*} $$

respectively, with the use of (4.4), we obtain

(4.6) $$ \begin{align} a_{1,{2m-1}}=a_{2,{2m-1}}=0. \end{align} $$

For $3\leq p\leq m$ , taking in (3.7),

$$ \begin{align*}(W,X,Y,Z)=(e_1,e_{2p-3},e_{2m-1},e_{2p-2}), \end{align*} $$

respectively, together with (4.6) and $c>0$ , we obtain

(4.7) $$ \begin{align} a_{11}=a_{{2m-1},{2m-1}}. \end{align} $$

For $3\leq p\leq m$ , taking in (3.7),

$$ \begin{align*}(W,X,Y,Z)=(e_2,e_{2p-3},e_{2m-1},e_2),\ (e_2,e_{2p-2},e_{2m-1},e_2), \end{align*} $$

respectively, together with (4.3) and $c>0$ , we obtain

$$ \begin{align*}\frac{2m-3+2a}{2m-3}a_{2,{2p-3}}=0, \ \ \frac{2m-3-2a}{2m-3}a_{2,{2p-2}}=0,\ \ 3\leq p\leq m. \end{align*} $$

Then, as $m\geq 3$ and $-1 < a < 1$ , we deduce that

(4.8) $$ \begin{align} a_{2,{2p-3}}=a_{2,{2p-2}}=0,\ \ 3\leq p\leq m. \end{align} $$

For $3\leq p\leq m$ , taking in (3.7),

$$ \begin{align*}(W,X,Y,Z)=(e_{2p-3},e_{2p-2},e_{2m-1},e_{2p-3}),\ (e_{2p-3},e_{2p-2},e_{2m-1},e_{2p-2}), \end{align*} $$

respectively, together with (4.3), (4.8), and $c>0$ , we obtain

(4.9) $$ \begin{align} a_{1,{2p-3}}=a_{1,{2p-2}}=0,\ \ 3\leq p\leq m. \end{align} $$

If $m=3$ , then the above calculations show that $Se_i=a_{ii}e_i,\ 1\le i\le 5$ , and $a_{11}=\cdots =a_{55}$ . Hence, M is totally umbilical and $S\phi =\phi S$ holds. This is equivalent to that M has isometric Reeb flow. But, Corollary 1.3 of [Reference Berndt and Suh3] states that there are no real hypersurfaces with isometric Reeb flow in the odd-dimensional complex quadric $Q^{2k+1}$ for each $k\ge 1$ . As desired, we get a contradiction.

Next, we continue with the discussions for $m\geq 4$ .

For $3\leq p,s\leq m$ and $p\neq s$ , taking in (3.7),

$$ \begin{align*} (W,X,Y,Z)=&(e_{2p-3},e_1,e_2,e_{2s-3}),\ (e_{2p-3},e_1,e_2,e_{2s-2}),\\ &(e_{2p-2},e_1,e_2,e_{2s-3}),\ (e_{2p-2},e_1,e_2,e_{2s-2}), \end{align*} $$

respectively, we obtain

(4.10) $$ \begin{align} a_{{2p-3},{2s-3}}=a_{{2p-3},{2s-2}}=a_{{2p-2},{2s-3}}=a_{{2p-2},{2s-2}}=0,\ 3\le p\not=s\leq m. \end{align} $$

From the above calculations, we see that $Se_i=a_{ii}e_i,\ 1\le i\le 2m-1$ , and $a_{11}=\cdots =a_{2m-1,2m-1}$ . Hence, M is totally umbilical and $S\phi =\phi S$ holds, so M has isometric Reeb flow. But, according to Theorem 1.1 and Proposition 4.1 of Berndt and Suh [Reference Berndt and Suh3], there are no totally umbilical real hypersurfaces with isometric Reeb flow in the complex quadric $Q^m\ (m\geq 4)$ . This gives again a contradiction.

From the above contradictions, we have completed the proof of Lemma 4.1. ▪

Now, by Lemma 4.1, the unit normal vector field N of the conformally flat real hypersurface M is $\mathfrak {A}$ -principal everywhere. According to Lemma 2.3, there exists an almost product structure $A\in \mathfrak {A}$ such that $AN=N$ and $A\xi =-\xi $ . Thus, we have $T\xi =-\xi $ and $\mu (X)=g(AX,N)=0$ for any $X\in TM$ .

Put $e_{2m-1}=\xi $ . From (2.1) and that $\dim {\mathcal {Q}}=2m-2$ , we can choose unit tangent vector fields

$$ \begin{align*}e_{2p-3}\in\mathcal{Q}(1),\ e_{2p-2}=Je_{2p-3}\in\mathcal{Q}(-1),\ \ 2\leq p\leq m, \end{align*} $$

so that $\{e_i\}_{i=1}^{2m-1}$ forms a local orthonormal frame field of M with the following properties:

(4.11) $$ \begin{align} Ae_{2p-3}=e_{2p-3},\ \ Ae_{2p-2}=-e_{2p-2},\ \ Ae_{2m-1}=-e_{2m-1},\ 2\leq p\leq m. \end{align} $$

Put again $Se_i=\sum _{j=1}^{2m-1}a_{ij}e_j$ , where $a_{ij}=a_{ji}$ , $1\leq i,j\leq 2m-1$ .

Now, we apply for Lemma 3.2 with $A\in \mathfrak {A}$ being chosen as above.

By choosing appropriate $W=e_i,\, X=e_j,\, Y=e_k,\, Z=e_l$ , $1\leq i,j,k,l\leq 2m-1$ , with the use of (2.4), (4.11), $T\xi =-\xi $ , and that $\mu (X)=0$ , for any $X\in TM$ , we can calculate $\mathbf {I}(W,X,Y,Z)$ and $\mathbf {II}(W,X,Y,Z)$ directly. Then, by (3.7), we will obtain a system of linear equations of the components $\{a_{ij}\}$ . The procedure is essentially totally similar to that in the proof of Lemma 4.1. Thus, for simplicity, most of the detailed calculations below will be omitted.

For $2\leq p,s\leq m$ and $p\neq s$ , taking in (3.7),

$$ \begin{align*}(W,X,Y,Z)=(e_{2p-3},e_{2p-2},e_{2m-1},e_{2s-3}),\ (e_{2p-3},e_{2p-2},e_{2m-1},e_{2s-2}), \end{align*} $$

respectively, we can obtain

(4.12) $$ \begin{align} a_{{2s-3},{2m-1}}=a_{{2s-2},{2m-1}}=0,\ \ 2\leq s\leq m. \end{align} $$

It follows that $S\xi =a_{{2m-1},{2m-1}} \xi $ , and therefore, M is a Hopf hypersurface such that its unit normal vector field is $\mathfrak {A}$ -principal everywhere. Then, we can apply for Theorem 14 of [Reference Loo17] or Theorem 2 of [Reference Lee and Suh15] to obtain that M is an open part of a tube around a totally geodesic $Q^{m-1}\hookrightarrow Q^m$ . Thus, by Proposition 4.1 of [Reference Berndt and Suh2], we have

$$ \begin{align*}Se_{2p-3}=-\frac{2}{\alpha}e_{2p-3},\ \ Se_{2p-2}=0, \ \ S\xi=\alpha \xi,\ \ \text{for all}\ 2\leq p\leq m. \end{align*} $$

Finally, letting $(W,X,Y,Z)=(e_3,e_2,e_4,e_1)$ , by direct calculations, we have

$$ \begin{align*} \left\{ \begin{aligned} &\mathbf{I}(e_3,e_2,e_4,e_1)=-\frac{4}{\alpha},\ \ \mathbf{I}(e_2,e_4,e_3,e_1)= \mathbf{I}(e_4,e_3,e_2,e_1)=0;\\ &\mathbf{II}(e_3,e_2,e_4,e_1)=\mathbf{II}(e_2,e_4,e_3,e_1)= \mathbf{II}(e_4,e_3,e_2,e_1)=0. \end{aligned}\right. \end{align*} $$

It then follows from (3.7) that $\frac {4}{\alpha }=0$ . This is a contradiction, by which we have completed the proof of Theorem 1.1 for the complex quadric $Q^m\ (m\ge 3)$ .

5 Proof of Theorem 1.1 for $Q^{m*}$

To begin, we first note that, like the complex quadric $Q^{m}$ , the set $\mathfrak {A}$ of all almost product structures of $Q^{m\ast }$ is also an $S^1$ -sub-bundle of the endomorphism bundle End( $TQ^{m*}$ ), and there are also two types of singular tangent vectors for the complex hyperbolic quadric $Q^{m*}$ : $\mathfrak {A}$ -principal and $\mathfrak {A}$ -isotropic.

Now, suppose on the contrary that the complex hyperbolic quadric $Q^{m\ast }\ (m\geq 3)$ admits a conformally flat real hypersurface $\bar {M}$ , with N and S a unit normal vector field and the shape operator, respectively, and g and $\nabla $ be the induced metric and the corresponding Levi–Civita connection on $\bar {M}$ , respectively. Then, there is also naturally an almost contact structure $\{\phi ,\xi ,\eta \}$ , a type $(1,1)$ tensor field T, and a $1$ -form $\mu $ induced from a generic almost product structure $A\in \mathfrak {A}$ on $\bar {M}$ .

In the following, we outline the proof in three steps, which are essentially the same as the proof of Theorem 1.1 for $Q^m\ (m\ge 3)$ .

Step 1. The real hypersurface $\bar {M}$ in $Q^{m\ast }\ (m\geq 3)$ satisfies equation (3.7).

As mentioned in the introduction, the almost product structure A and the Kähler structure J of $Q^{m\ast }$ also satisfy (2.1) and (2.2) (see [Reference Klein and Suh13, Reference Suh25, Reference Suh, Pérez and Woo27, Reference Van der Veken and Wijffels28]), and it makes the tensors $\phi ,T,\mu $ , and $\eta $ on $\bar {M}$ also satisfy the equations (2.7)–(2.11) in Lemma 2.2. Because the Riemannian curvature tensor of $Q^{m\ast }$ is exactly opposite to the Riemannian curvature tensor of $Q^m$ , it makes both the Gauss–Codazzi equations and the Ricci tensor of $\bar {M}$ a slightly different from (2.12)–(2.14): some terms change at most by negative signs. However, when we apply for the Tsinghua principle with $\bar {M}$ , following the same discussions as in Section 3, we see that Lemma 3.2 still holds for conformally flat real hypersurfaces in $Q^{m\ast }\ (m\geq 3)$ .

Step 2. The unit normal vector field of $\bar {M}$ must be $\mathfrak {A}$ -principal everywhere.

In fact, if otherwise, the unit normal vector field N is assumed locally not $\mathfrak {A}$ -principal. By Lemma 3.2 of [Reference Suh25], there exists an almost product structure $A\in \mathfrak {A}$ in local such that $A\xi \in T\bar {M}$ and $\mu (\xi )=g(A\xi ,N)=0$ . Similarly, we can also choose a local frame field $\{e_i\}_{i=1}^{2m-1}$ of $\bar {M}$ satisfying equations (4.1) and (4.2) as in the proof of Lemma 4.1. Then, following the same calculations as in the proof of Lemma 4.1, we can show that $\bar {M}$ is totally umbilical and $S\phi =\phi S$ holds. Thus, $\bar {M}$ has isometric Reeb flow. But, according to Theorem 1.1 and Proposition 4.1 of Suh [Reference Suh25], there are no totally umbilical real hypersurfaces with isometric Reeb flow in the complex hyperbolic quadric $Q^{m\ast }\ (m\geq 3)$ . This gives a contradiction.

Step 3. The unit normal vector field of $\bar {M}$ cannot be $\mathfrak {A}$ -principal everywhere.

In fact, if the unit normal vector field N is $\mathfrak {A}$ -principal, there exists an almost product structure $A\in \mathfrak {A}$ such that $AN=N$ and $A\xi =-\xi $ . We can find a local frame field $\{e_i\}_{i=1}^{2m-1}$ ( $e_{2m-1}=\xi $ ) of $\bar {M}$ satisfying the same equations as (4.11). Then, by similar calculations as taking in (3.7),

$$ \begin{align*}(W,X,Y,Z)=(e_{2p-3},e_{2p-2},e_{2m-1},e_{2s-3}),\ (e_{2p-3},e_{2p-2},e_{2m-1},e_{2s-2}), \end{align*} $$

for $2\leq p,s\leq m$ and $p\neq s$ , respectively, we can show that $S\xi =\alpha \xi $ . Therefore, $\bar {M}$ is a Hopf hypersurface such that the unit normal vector field is $\mathfrak {A}$ -principal everywhere. Then, according to Lemma 4.2 (ii) of [Reference Suh, Pérez and Woo27], by choosing the almost product structure $A\in \mathfrak {A}$ such that $AN=N$ , it holds $ASX=SX$ for any $X\in \mathcal {C}$ .

As $\bar {M}$ is a Hopf hypersurface with $\mathfrak {A}$ -principal unit normal vector field, the above fact implies that $SAX=SX$ for all $X\in \mathcal {C}$ . It follows that $S\mathcal {Q}(-1)=0$ , where $\mathcal {Q}(-1)$ is the eigenspace of $A|_{\mathcal {C}}$ corresponding to the eigenvalue $-1$ . Then, for any $X\in \mathcal {Q}(-1)$ , Lemma 3.4 of [Reference Suh, Pérez and Woo27] shows that $\alpha S\phi X=2\phi X$ . It follows that $\alpha \not =0$ and $SX=\frac {2}{\alpha }X$ for all $X\in \mathcal {Q}(1)=J\mathcal {Q}(-1)$ . Thus, we have

$$ \begin{align*}Se_{2p-3}=\frac{2}{\alpha}e_{2p-3},\ \ Se_{2p-2}=0, \ \ S\xi=\alpha \xi,\ \ \text{for all}\ 2\leq p\leq m. \end{align*} $$

Finally, taking $(W,X,Y,Z)=(e_3,e_2,e_4,e_1)$ in (3.7), by direct calculations, we have $\frac {4}{\alpha }=0$ . This is a contradiction.

In summary, we have completed the proof of Theorem 1.1 for $Q^{m*}\ (m\ge 3)$ .