1 Introduction

Recently, with some new insight, we obtained several optimal pinching results on compact minimal submanifolds of the round sphere in terms of either the scalar curvature or the Ricci curvature (cf. [9, 10]). The purpose of this note is to apply this new idea to the Ricci curvature pinching problem for two further interesting situations. Historically, towards that problem, Ejiri [6] for the first time proved an optimal pinching theorem for compact minimal submanifolds in \({\mathbb {S}}^{n+p}(1)\).

Theorem A

Let \(M^n\) be an n-dimensional simply connected compact orientable minimal submanifold immersed in \({\mathbb {S}}^{n+p}(1)\) such that the immersion is full. If \(n\ge 4\) and the Ricci curvature of \(M^n\) satisfies \(\mathrm{Ric}\,(M^n)\ge n-2\), then \(M^n\) is either \({\mathbb {S}}^n(1)\) (totally geodesic), or \(n=2m\) and \(M^n={\mathbb {S}}^m(\sqrt{1/2})\times {\mathbb {S}}^m(\sqrt{1/2})\), or \(n=4\) and \(M^4={\mathbb {C}}P^2(4/3)\rightarrow {\mathbb {S}}^7(1)\). Here, \({\mathbb {C}}P^2(4/3)\) denotes the two-dimensional complex projective space with holomorphic sectional curvature 4/3.

However, we should see that even though in the above theorem the pinching constant for Ricci curvature is optimal if n is even, it is not if n is odd (cf. [12, Theorem 3.2] for \(n\ge 5\)). In particular, we observe that Theorem A is valid also for the case \(n=3\). This latter assertion follows actually from the combinations of Shen [18, Theorem 1], Li [12, Theorem 3.3], the result of Chern–do Carmo–Kobayashi [3] for \(p\le 2\), and Li–Li [15, Theorem 3] for \(p\ge 3\). Moreover, this observation further motivates us to consider the following interesting question:

Question

For 3-dimensional compact minimal submanifolds of the unit sphere \({\mathbb {S}}^{3+p}(1)\), what is the best possible Ricci curvature condition so that submanifolds next to the totally geodesic one can be characterized?

Regarding a special case about Lagrangian submanifolds in the nearly Kähler 6-sphere \({\mathbb {S}}^6(1)\), very recently in [10], we adopted a strategy that has been used successfully in [9]. As the result, we can improve the previous results of Li [14] and Antić–Djorić–Vrancken [1] by giving a satisfactory answer to the above question (cf. [10, Main Theorem]).

In this paper, with continuing the concern about the above Question, we shall study two further interesting situations: the hypersurfaces of \({\mathbb {S}}^4(1)\) and the Legendrian submanifolds of \({\mathbb {S}}^7(1)\). Before stating the main results, we look at two examples.

Example 1.1

The Clifford torus \(\mathbf {Cl}_{1,2}:={\mathbb {S}}^1(\sqrt{1/3})\times {\mathbb {S}}^2(\sqrt{2/3})\) in \({\mathbb {S}}^4(1)\) is a compact minimal hypersurface with two distinct principal curvatures \(\kappa _1=\pm \sqrt{2}\) and \(\kappa _2=\kappa _3=\mp \sqrt{1/2}\). Its Ricci curvature satisfies \(0\le \text {Ric}(\mathbf {Cl}_{1,2})\le 3/2\).

Remark 1.1

Conversely, making use of the integral formula technique due to Peng–Terng [17], Li [13] proved that if a compact minimal hypersurface M in \({\mathbb {S}}^4(1)\) satisfies the Ricci curvature condition \(0\le \,\mathrm{Ric}(M)\le 3/2\), then it must be the case \(M=\mathbf {Cl}_{1,2}\).

Example 1.2

The Calabi torus \(\mathbf {Ca}_{1,2}\) in \({\mathbb {S}}^7(1)\).

Let \(\tau :{\mathbb {S}}^2(1)\hookrightarrow {\mathbb {R}}^3\) be the inclusion mapping, and \(\gamma :\,{\mathbb {S}}^1(1)\rightarrow {\mathbb {S}}^3(1)\) be the standard embedding with a parametrization

$$\begin{aligned} \gamma (t)=\left( \tfrac{\sqrt{3}}{2}\exp (\tfrac{it}{\sqrt{3}}),\tfrac{1}{2}\exp (-i\sqrt{3}t)\right) =(\gamma _1(t),\gamma _2(t))\in {\mathbb {C}}^2,\ \ i=\sqrt{-1}. \end{aligned}$$

Putting \(F:\,{\mathbb {S}}^1(1)\times {\mathbb {S}}^2(1)\rightarrow {\mathbb {S}}^7(1)\) such that

$$\begin{aligned} F(t,x)=(\gamma _1(t)\tau (x),\gamma _2(t))\in {\mathbb {C}}^4\simeq {\mathbb {R}}^8, \end{aligned}$$
(1.1)

then following [16], we call \(\mathbf {Ca}_{1,2}:=F({\mathbb {S}}^1(1)\times {\mathbb {S}}^2(1))\) in \({\mathbb {S}}^7(1)\) the Calabi torus.

Considering the unit sphere \({\mathbb {S}}^7(1)\) as a Sasakian space form with standard contact metric structure \((\varphi ,\xi ,\eta ,g)\) (cf. [8]), the Calabi torus \(\mathbf {Ca}_{1,2}\) in \({\mathbb {S}}^7(1)\) then becomes a minimal Legendrian submanifold. Exactly, one can choose a local orthonormal frame field \(\{E_1,E_2,E_3\}\) on \(\mathbf {Ca}_{1,2}\) such that the second fundamental form h takes the form (cf. [5] and [16]):

$$\begin{aligned} \left\{ \begin{aligned}&h(E_1,E_1)=\tfrac{2}{\sqrt{3}}\varphi {E_1},\ \ h(E_2,E_2)=h(E_3,E_3)=-\tfrac{1}{\sqrt{3}}\varphi {E_1},\\&h(E_1,E_2)=-\tfrac{1}{\sqrt{3}}\varphi E_2,\ \ h(E_1,E_3)=-\tfrac{1}{\sqrt{3}}\varphi E_3,\ \ h(E_2,E_3)=0. \end{aligned}\right. \end{aligned}$$
(1.2)

From the Gauss equation and (1.2), the curvature tensor of \(\mathbf {Ca}_{1,2}\) satisfies

$$\begin{aligned} \begin{aligned}&R(E_1,E_2)E_3=R(E_2,E_3)E_1=R(E_3,E_1)E_2=0,\\&R(E_1,E_2)E_2=R(E_1,E_3)E_3=0,\\&R(E_2,E_3)E_3=(4/3)E_2. \end{aligned} \end{aligned}$$
(1.3)

Thus its Ricci curvature satisfies \(0\le \text {Ric}(\mathbf {Ca}_{1,2})\le 4/3\). Moreover, we can say more about the sectional curvatures. Indeed, for any plane section \(\sigma \) of \(T_p(\mathbf {Ca}_{1,2})\), we can find an orthonormal basis \(\{X,Y\}\) of \(\sigma \) such that \(X=\cos \theta E_2(p)+\sin \theta E_3(p)\) and \(Y=\sin \rho E_1(p)-\cos \rho \sin \theta E_2(p)+\cos \rho \cos \theta E_3(p)\) with \(\theta ,\rho \in {\mathbb {R}}\). Then direct calculations show that the sectional curvature of \(\sigma \) satisfies

$$\begin{aligned} K(\sigma )=R(X,Y,Y,X)=(4/3)\cos ^2\rho . \end{aligned}$$
(1.4)

Remark 1.2

The above immersion \(F:\,{\mathbb {S}}^1(1)\times {\mathbb {S}}^2(1)\rightarrow {\mathbb {S}}^7(1)\) is called the Calabi torus because it is one of the generalized Calabi product Legendrian immersions that have been intensively studied by Castro–Li–Urbano [4] (cf. [4, Theorem 3.1]). Obviously, \(\mathbf {Ca}_{1,2}\) with the induced metric is isometric with the Riemannian product \({\mathbb {S}}^1(\sqrt{3})\times {\mathbb {S}}^2(\sqrt{3}/2)\).

To introduce our first result, let \(M^3\) be a compact minimal hypersurface in the unit sphere \({\mathbb {S}}^4(1)\) with h the second fundamental form and N the unit normal vector field. Denote by g the metric on \({\mathbb {S}}^4(1)\) as well as the induced metric on \(M^3\). Let \(UM^3\) be the unit tangent bundle over \(M^3\), i.e., \(U_qM^3=\{v\in T_qM^3\,|\,g(v,v)=1\}\) for \(q\in M^3\). Then we can define a function \(f_q(v)=g(A_Nv,v)\) on \(U_qM^3\), where \(A_N\) denotes the shape operator of \(M^3\) in \({\mathbb {S}}^4(1)\). Since \(U_qM^3\) is compact, there exists an element \(e\in U_qM^3\) such that \(f_q(e)=\max _{v\in U_qM^3}f_q(v)\). Put

$$\begin{aligned} {\mathcal {U}}_q:=\{u\in U_qM^3~|~f_q(u)=f_q(e)\}. \end{aligned}$$

Then, for any fixed \(e\in {\mathcal {U}}_q\), we have a well-defined function \(\Phi _e:\ U_qM^3\rightarrow {\mathbb {R}}\) defined by

$$\begin{aligned} \Phi _e(v)=[g(A_Ne,v)]^2,\ \ v\in U_qM^3. \end{aligned}$$

Now, our first result can be stated as:

Theorem 1.1

Let \(M^3\) be a compact minimal hypersurface in the unit sphere \({\mathbb {S}}^4(1)\). If the Ricci curvature \(\mathrm{Ric}(v):=\mathrm{trace} \{X\mapsto R(X,v)v\}/\Vert v\Vert ^2\) of \(M^3\) satisfies, at any \(q\in M^3\), \(\mathrm{Ric}(v)\ge \tfrac{3}{2}-\tfrac{3}{4}\Phi _e(v)\) for all \(v\in U_qM^3\) and a fixed \(e\in {\mathcal {U}}_q\), then either

  1. (a)

    \(M^3={\mathbb {S}}^3(1)\) is totally geodesic with \(\mathrm{Ric}(v)\equiv 2\), or

  2. (b)

    \(M^3=\mathbf {Cl}_{1,2}\), which satisfies \(\mathrm{Ric}(v)\equiv \tfrac{3}{2} -\tfrac{3}{4}\Phi _e(v)\) and \(0\le \mathrm{Ric}(v)\le \tfrac{3}{2}\).

To introduce the second result of this paper, let \(M^3\) be a compact minimal Legendrian submanifold in the unit sphere \({\mathbb {S}}^7(1)\), being equipped with the standard contact metric structure \(\{\varphi ,\xi ,\eta ,g\}\). Then, letting h be the second fundamental form, we can follow [2] and consider the function \(F_q(v)=g(h(v,v),\varphi v)\), defined on \(U_qM^3\). In view of the compactness of \(U_qM^3\), there is an element \(e\in U_qM^3\) such that \(F_q(e)=\mathrm{max}_{v\in U_qM^3}F_q(v)\). Similarly, we put

$$\begin{aligned} {\mathcal {V}}_q:=\{u\in U_qM^3~|~F_q(u)=F_q(e)\}. \end{aligned}$$

Then, for any fixed \(e\in {\mathcal {V}}_q\), we have a well-defined function \(\Psi _e:\ U_qM^3\rightarrow {\mathbb {R}}\) defined by

$$\begin{aligned} \Psi _e(v)=[g(h(e,e),\varphi v)]^2,\ \ v\in U_qM^3. \end{aligned}$$

Our second result can be stated as:

Theorem 1.2

Let \(M^3\) be a compact minimal Legendrian submanifold in the unit sphere \({\mathbb {S}}^7(1)\). If the Ricci curvature \(\mathrm{Ric}(v)=\mathrm{trace} \{X\mapsto R(X,v)v\}/\Vert v\Vert ^2\) of \(M^3\) satisfies, at any \(q\in M^3\), \(\mathrm{Ric}(v)\ge \tfrac{4}{3}-\Psi _e(v)\) for all \(v\in U_qM^3\) and a fixed \(e\in {\mathcal {V}}_q\), then either

  1. (a)

    \(M^3={\mathbb {S}}^3(1)\) is totally geodesic with \(\mathrm{Ric}(v)\equiv 2\), or

  2. (b)

    \(M^3=\mathbf {Ca}_{1,2}\), which satisfies \(\mathrm{Ric}(v)\equiv \tfrac{4}{3} -\Psi _e(v)\) and \(0\le \mathrm{Ric}(v)\le \tfrac{4}{3}\).

Remark 1.3

The statement of our Ricci curvature condition in Theorems 1.1 and 1.2 is a little different from that of Ejiri’s Theorem A. Actually, in Theorem A, the Ricci curvature condition \(\mathrm{Ric}\,(M^n)\ge n-2\) just means that \(\mathrm{Ric}\,(v):=\text {trace}\,\{X\mapsto R(X,v)v\}/\Vert v\Vert ^2\ge n-2\) for all nonzero \(v\in T_qM^n\) and any \(q\in M^n\).

Remark 1.4

In [5], Dillen and Vrancken classified all 3-dimensional compact minimal C-totally real (this is equivalent to Legendrian) submanifolds in \({\mathbb {S}}^7(1)\) with nonnegative sectional curvatures. Their classification consists of three examples amongst which the two cases in Theorem 1.2 appeared.

2 Basic lemmas

In this section, we briefly review several facts and lemmas on 3-dimensional submanifolds \(M^3\) in the unit sphere \({\mathbb {S}}^{3+p}(1)\). This includes the hypersurface case for \(p=1\) (cf. [17]) and the Legendrian submanifold case for \(p=4\) (cf. [8]).

2.1 Hypersurfaces in \({\mathbb {S}}^4(1)\)

In this subsection, we assume that \(M^3\) is a minimal hypersurface in the unit sphere \({\mathbb {S}}^4(1)\). For the sake of simplicity, we adopt the notations of Peng and Terng [17]. We begin with the following well-known result.

Lemma 2.1

Let \(M^3\) be a minimal hypersurface in the unit sphere \({\mathbb {S}}^4(1)\). Then it holds that

$$\begin{aligned} \tfrac{1}{2}\Delta \Vert h\Vert ^2=\Vert \nabla h\Vert ^2+\Vert h\Vert ^2(3-\Vert h\Vert ^2). \end{aligned}$$
(2.1)

Next, noting that \(g(A_NX,Y)=g(h(X,Y),N)\), for later purposes, we state the following easy fact:

Lemma 2.2

Let \(M^3\) be a minimal hypersurface in the unit sphere \({\mathbb {S}}^{4}(1)\) with unit normal vector field N. Then, for each \(q\in M^3\), there exists an orthonormal basis \(\{e_1, e_2,e_3\}\) of \(T_qM^3\) and numbers \(\kappa _1,\kappa _2\) such that the second fundamental form h of \(M^3\) satisfies

$$\begin{aligned} \begin{aligned} h(e_1,e_1)=(\kappa _1+\kappa _2)N,\ \ h(e_2,e_2)=-\kappa _1N,\\ h(e_3,e_3)=-\kappa _2N,\ \ h(e_i,e_j)=0,~\mathrm{for}~i\ne j, \end{aligned} \end{aligned}$$
(2.2)

where \(\kappa _1+\kappa _2=\max \{\kappa _1+\kappa _2,-\kappa _1,-\kappa _2\}=\max _{v\in U_qM^3}g(A_Nv,v)\).

2.2 Legendrian submanifolds in \({\mathbb {S}}^7(1)\)

In this subsection, we assume that \(M^3\) is a minimal Legendrian submanifold in the unit sphere \({\mathbb {S}}^7(1)\) which is regarded as a Sasakian space form with contact metric structure \(\{\varphi ,\xi ,\eta ,g\}\). According to Chern–do Carmo–Kobayashi [3], and by using the notations introduced in [8] for integral (in the present case, this is also equivalent to the Legendrian condition) submanifolds, we have (cf. also [16, Lemma 2.1])

Lemma 2.3

Let \(M^3\) be a minimal Legendrian submanifold in the unit sphere \({\mathbb {S}}^7(1)\). Then, in terms of \(H_{i}=(h^{i^*}_{jk})\), we have the Laplacian of \(\Vert h\Vert ^2\) as below:

$$\begin{aligned} \tfrac{1}{2}\Delta \Vert h\Vert ^2=\Vert {\bar{\nabla }}^\xi h\Vert ^2+4\Vert h\Vert ^2 -\sum _{i,j}N(H_{i}H_{j}-H_{j}H_{i})-\sum _{i,j}(S_{ij})^2, \end{aligned}$$
(2.3)

where \(\Vert {\bar{\nabla }}^\xi h\Vert ^2=\sum _{i,j,k,l}(h^{l^*}_{ij,k})^2\), \(S_{ij}=\mathrm{trace}(H_iH_j)\), and \(N(A)=\sum _{i,j}(a_{ij})^2\) for \(A=(a_{ij})\).

Next, following an idea due to Ejiri [7] and based on [8, Lemma 3.6], we can easily get the following result.

Lemma 2.4

Let \(M^3\) be a minimal Legendrian submanifold in \({\mathbb {S}}^7(1)\). Then, for each \(q\in M^3\), there exists an orthonormal basis \(\{e_1,e_2,e_3\}\) of \(T_qM^3\) and numbers \(\{\lambda _1,\lambda _2,\mu _1,\mu _2\}\) such that the second fundamental form h of \(M^3\) has the following form:

$$\begin{aligned} \left\{ \begin{aligned}&h(e_1,e_1)=(\lambda _1+\lambda _2)\varphi {e_1},\ \ h(e_1,e_2)=-\lambda _1\varphi {e_2},\ \ h(e_1,e_3)=-\lambda _2\varphi {e_3},\\&h(e_2,e_2)=-\lambda _1\varphi {e_1}+\mu _1\varphi {e_2}+\mu _2\varphi {e_3},\ \ h(e_2,e_3)=\mu _2\varphi {e_2}-\mu _1\varphi {e_3},\\&h(e_3,e_3)=-\lambda _2\varphi {e_1}-\mu _1\varphi {e_2}-\mu _2\varphi {e_3}, \end{aligned} \right. \end{aligned}$$
(2.4)

where for \(F_q(v)=g(h(v,v),\varphi v)\) defined on \(U_qM^3\), it holds

$$\begin{aligned} \left\{ \begin{aligned}&\lambda _1+\lambda _2=\max \nolimits _{v\in U_qM^3}F_q(v)\ge 0,\\&\lambda _1+\lambda _2\ge -2\lambda _1,\ \ \lambda _2+\lambda _1\ge -2\lambda _2,\\&-(\lambda _1+\lambda _2)\le \mu _i\le \lambda _1+\lambda _2,\ \ i=1,2. \end{aligned} \right. \end{aligned}$$
(2.5)

Then, with the Legendrian condition, straightforward calculations give

Lemma 2.5

If (2.4) holds, then, by using the notations of Lemma 2.3, we have

$$\begin{aligned}&\Vert h\Vert ^2=\sum _{i,j,\alpha }({h^\alpha _{ij}})^2=4\lambda ^2_1+4\lambda ^2_2 +2\lambda _1\lambda _2+4\mu ^2_1+4\mu ^2_2, \end{aligned}$$
(2.6)
$$\begin{aligned}&\sum _{i,j}N(H_{i}H_{j}-H_{j}H_{i})+\sum _{i,j}(S_{ij})^2\nonumber \\&\quad =24(\lambda ^4_1+\lambda ^3_1\lambda _2+\lambda ^2_1\lambda ^2_2 +\lambda _1\lambda ^3_2+\lambda ^4_2)-36\lambda _1\lambda _2(\mu ^2_1+\mu ^2_2)\nonumber \\&\qquad +18(\lambda ^2_1+\lambda ^2_2)(\mu ^2_1+\mu ^2_2)+24(\mu ^2_1+\mu ^2_2)^2. \end{aligned}$$
(2.7)

3 Proof of Theorem 1.1

Let \(M^3\) be a compact minimal hypersurface in \({\mathbb {S}}^4(1)\). For any fixed \(q\in M^3\), by choosing the orthonormal basis \(\{e_1,e_2,e_3\}\) of \(T_qM^3\) as stated in Lemma 2.2, the Gauss equation implies that the Ricci curvature \(R_{ij}=\sum _kg(R(e_i,e_k)e_k,e_j)\) is given by

$$\begin{aligned} (R_{ij})= \begin{pmatrix} 2-(\kappa _1+\kappa _2)^2 &{}\quad 0 &{} \quad 0 \\ 0 &{} \quad 2-\kappa ^2_1 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 2-\kappa ^2_2 \end{pmatrix}. \end{aligned}$$

According to the assumption of Theorem 1.1 and the statement of Lemma 2.2, the Ricci curvature of \(M^3\) satisfies \(\mathrm{Ric}(v)\ge \tfrac{3}{2}-\tfrac{3}{4}\Phi _{e_1}(v)\) for all \(v\in U_qM^3\), where \(\Phi _{e_1}(e_1)=(\kappa _1+\kappa _2)^2\) and \(\Phi _{e_1}(e_i)=[g(A_Ne_1,e_i)]^2=0\) for \(i=2,3\). It follows that

$$\begin{aligned} \left\{ \begin{aligned}&2-(\kappa _1+\kappa _2)^2\ge \tfrac{3}{2}-\tfrac{3}{4}(\kappa _1+\kappa _2)^2,\\&2-\kappa ^2_1\ge \tfrac{3}{2},\ \ 2-\kappa ^2_2\ge \tfrac{3}{2}. \end{aligned} \right. \end{aligned}$$
(3.1)

From (3.1), interchanging \(e_2\) and \(e_3\) if necessary, we can assume that

$$\begin{aligned} 0\le \kappa ^2_1\le \tfrac{1}{2},\ \ 0\le \kappa ^2_2\le \tfrac{1}{2},\ \ 0\le (\kappa _1+\kappa _2)^2\le 2,\ \ \kappa _2\le \kappa _1. \end{aligned}$$
(3.2)

Now, by using (2.2), we can rewrite

$$\begin{aligned} \begin{aligned}&\Vert h\Vert ^2(3-\Vert h\Vert ^2)\\&=-2(\kappa ^2_1+\kappa ^2_2+\kappa _1\kappa _2)(2\kappa ^2_1+2\kappa ^2_2+2\kappa _1\kappa _2-3)\\&=-(\kappa ^2_1+\kappa ^2_2+\kappa _1\kappa _2)\big [3(2\kappa ^2_1-1) +3(2\kappa ^2_2-1)-2(\kappa _1-\kappa _2)^2\big ]. \end{aligned} \end{aligned}$$
(3.3)

It follows from (3.2) and (3.3) that

$$\begin{aligned} \Vert h\Vert ^2(3-\Vert h\Vert ^2)\ge 0. \end{aligned}$$
(3.4)

Moreover, from the fact \(\kappa _1+\kappa _2=\max \{\kappa _1+\kappa _2,-\kappa _1,-\kappa _2\}\), we see that the equality sign holds in (3.4) if and only if either \(\kappa _1=\kappa _2=0\), or \(\kappa _1=\kappa _2=\sqrt{2}/2\) so that \(\Vert h\Vert ^2=3\).

On the other hand, since \(M^3\) is compact, Lemma 2.1 implies that

$$\begin{aligned} \mathop \int \limits _{M^3}\Big \{\Vert \nabla h\Vert ^2+\Vert h\Vert ^2(3-\Vert h\Vert ^2)\Big \}dV_M=0. \end{aligned}$$
(3.5)

Therefore, from (3.4) and the arbitrariness of \(q\in M^3\), we have

$$\begin{aligned} \Vert \nabla h\Vert ^2=\Vert h\Vert ^2(3-\Vert h\Vert ^2)\equiv 0\ \ \text {on}\ \ M^3. \end{aligned}$$

This immediately shows that \(M^3\) in \({\mathbb {S}}^4(1)\) is either totally geodesic, or the Clifford torus (see Lawson [11] or Chern–do Carmo–Kobayashi [3]). \(\square \)

4 Proof of Theorem 1.2

Let \(M^3\) be a compact minimal Legendrian submanifold in the unit sphere \({\mathbb {S}}^7(1)\). For any fixed point \(q\in M^3\), let \(\{e_1,e_2,e_3\}\) be the orthonormal basis of \(T_qM^3\) as stated in Lemma 2.4 such that \(e_1=e\). Then, by using the Gauss equation and the fact that \(R_{ij}=\sum _kg(R(e_i,e_k)e_k,e_j)\), we can express \((R_{ij})\) by

$$\begin{aligned} \begin{pmatrix} 2-2\lambda ^2_1-2\lambda ^2_2-2\lambda _1\lambda _2 &{}\quad (\lambda _1-\lambda _2)\mu _1 &{}\quad (\lambda _1-\lambda _2)\mu _2 \\ (\lambda _1-\lambda _2)\mu _1 &{}\quad 2-2\lambda ^2_1-2\mu ^2_1-2\mu ^2_2 &{} \quad 0 \\ (\lambda _1-\lambda _2)\mu _2 &{}\quad 0 &{} \quad 2-2\lambda ^2_2-2\mu ^2_1-2\mu ^2_2 \end{pmatrix}. \end{aligned}$$

According to the assumption of Theorem 1.2 and the statement of Lemma 2.4, the Ricci curvature of \(M^3\) satisfies \(\mathrm{Ric}(v)\ge \tfrac{4}{3}-\Psi _{e_1}(v)\) for all \(v\in U_qM^3\), where \(\Psi _{e_1}(e_1)=(\lambda _1+\lambda _2)^2\) and \(\Psi _{e_1}(e_i)=[g(h(e_1,e_1),\varphi e_i)]^2=0\) for \(i=2,3\). It follows, by taking \(v=e_1,e_2,e_3\) with \(\mathrm{Ric}(e_i)=R_{ii}\), that

$$\begin{aligned} \left\{ \begin{aligned}&2-2\lambda ^2_1-2\lambda ^2_2-2\lambda _1\lambda _2 \ge \tfrac{4}{3}-(\lambda _1+\lambda _2)^2,\\&2-2\lambda ^2_1-2\mu ^2_1-2\mu ^2_2 \ge \tfrac{4}{3},\\&2-2\lambda ^2_2-2\mu ^2_1-2\mu ^2_2 \ge \tfrac{4}{3}. \end{aligned} \right. \end{aligned}$$
(4.1)

From (4.1), interchanging \(e_2\) and \(e_3\) if necessary, we can assume that \(\lambda ^2_2\le \lambda ^2_1\). Then the inequalities in (4.1) together with (2.5) imply the following relations:

$$\begin{aligned} 0\le \lambda _1\le \tfrac{\sqrt{3}}{3},\ \ \lambda _2\le \lambda _1,\ \ -\tfrac{1}{3}\lambda _1\le \lambda _2,\ \ \mu ^2_1+\mu ^2_2\le \tfrac{1}{3}-\lambda ^2_1. \end{aligned}$$
(4.2)

On the other hand, as \(M^3\) is compact, by using Lemmas 2.3 and 2.5, we have the integral identity

$$\begin{aligned} \begin{aligned} 0&=\mathop \int \limits _{M^3}\Big \{\Vert {\bar{\nabla }}^\xi h\Vert ^2 -\big \{24[\lambda ^4_1+\lambda ^4_2+\lambda ^2_1\lambda ^2_2 +\lambda _1\lambda _2(\lambda ^2_1+\lambda ^2_2)]\\&\quad +18(\mu ^2_1+\mu ^2_2)(\lambda _1-\lambda _2)^2+24(\mu ^2_1+\mu ^2_2)^2\\&\quad -16(\mu ^2_1+\mu ^2_2)-16(\lambda ^2_1+\lambda ^2_2+\tfrac{1}{2}\lambda _1\lambda _2)\big \}\Big \}~dV_M. \end{aligned} \end{aligned}$$
(4.3)

Now we put

$$\begin{aligned} \begin{aligned} \Omega&:=24[\lambda ^4_1+\lambda ^4_2+\lambda ^2_1\lambda ^2_2 +\lambda _1\lambda _2(\lambda ^2_1+\lambda ^2_2)] +18(\mu ^2_1+\mu ^2_2)(\lambda _1-\lambda _2)^2 \\&\quad +\,24(\mu ^2_1+\mu ^2_2)^2-16(\mu ^2_1+\mu ^2_2) -16(\lambda ^2_1+\lambda ^2_2+\tfrac{1}{2}\lambda _1\lambda _2). \end{aligned} \end{aligned}$$
(4.4)

To achieve the desired goal, we first make the following claim.

Claim

Under the restrictions of (4.2), it holds that \(\Omega \le 0\), and \(\Omega =0\) if and only if either \(\lambda _1=\lambda _2=\mu _1=\mu _2=0\), or \(\lambda _1=\lambda _2=\sqrt{3}/{3}\) and \(\mu _1=\mu _2=0\).

To verify the Claim, it suffices to consider the point \(q\in M^3\) for which \(h\not =0\). In that case, we have \(\lambda _1>0\), and by (4.2), we further get

$$\begin{aligned} 3\lambda _1+5\lambda _2=\tfrac{4}{3}\lambda _1+5(\tfrac{1}{3}\lambda _1+\lambda _2)>0. \end{aligned}$$

Thus, the Claim follows immediately from (4.2) and the following expression

$$\begin{aligned} \begin{aligned} \Omega&=24\big [2(\mu ^2_1+\mu ^2_2)+2\lambda ^2_1+\lambda _1\lambda _2+2\lambda ^2_2\big ] \big (\mu ^2_1+\mu ^2_2+\lambda ^2_1-\tfrac{1}{3}\big )\\&\quad -\,3\big [2(\mu ^2_1+\mu ^2_2)+2\lambda ^2_1+\lambda _1\lambda _2+2\lambda ^2_2\big ] (3\lambda _1+5\lambda _2)(\lambda _1-\lambda _2)\\&\quad -\,24(\mu ^2_1+\mu ^2_2)^2-3(\lambda _1-\lambda _2)^2 (2\lambda ^2_1+2\lambda ^2_2-3\lambda _1\lambda _2)\\&\quad -\,12(5\lambda ^2_1+5\lambda ^2_2+4\lambda _1\lambda _2)(\mu ^2_1+\mu ^2_2). \end{aligned} \end{aligned}$$
(4.5)

Finally, from the Claim and the arbitrariness of \(q\in M^3\), by using (4.3), we conclude that \(M^3\) is a Legendrian submanifold with C-parallel second fundamental form (i.e. \({\bar{\nabla }}^\xi h=0\)). Moreover, it is either totally geodesic and \(M^3={\mathbb {S}}^3(1)\), or by continuity, it is such that \(\lambda _1=\lambda _2=\sqrt{3}/{3}\), \(\mu _1=\mu _2=0\) and \(\Vert h\Vert ^2=10/3\) hold identically on \(M^3\).

In the latter case, it is easily seen from (2.4) and the Gauss equation that the compact Legendrian submanifold \(M^3\) in \({\mathbb {S}}^7(1)\) is of nonnegative sectional curvatures, and the sectional curvatures have the same expression as we have described for the Calabi torus \(\mathbf {Ca}_{1,2}\) in the introduction. Then, according to [5, Main theorem] and its proof ([5, Example 5.3] in particular), we conclude that \(M^3\) is congruent to \(\mathbf {Ca}_{1,2}\) with the immersion given by (1.1).

This completes the proof of Theorem 1.2. \(\square \)