1 Introduction

It is well known that on a Kähler manifold, the \(J\)-invariant submanifolds (\(J\) being the complex structure of the Kähler manifold) are minimal. On the other hand, for a Sasakian manifold, and more generally for a contact metric manifold, a \(\phi \)-invariant submanifold is also minimal, where \(\phi \) is the structure tensor of the contact metric structure. Similar results are known for a special class of Hermitian manifolds, that is, the class of locally conformally Kähler (lcK) manifolds and in particular for the subclass of Vaisman manifolds. Dragomir and Ornea [12, Theorem 12.1] have shown that a \(J\)-invariant submanifold of an lcK manifold is minimal if and only if the submanifold is tangent to the Lee vector field (and therefore tangent to the anti-Lee vector field). In fact, this result is a slight generalization of the following result of Vaisman [15]: a \(J\)-invariant submanifold of a generalized Hopf manifold (nowadays called Vaisman manifold) inherits a generalized Hopf manifold structure if and only if it is minimal (or, equivalently, if and only if the submanifold is tangent to the Lee vector field). In [9], it was shown that the notion of non-Kähler Vaisman manifold, after constant rescaling of the metric, is equivalent to the notion of normal metric contact pair [7] of type \((h,0)\) and the Lee and anti-Lee vector fields correspond to the Reeb vectors fields of the pair. Moreover, this equivalence enlightened the fact that on a Vaisman manifold, there is another complex structure \(T\) with opposite orientation with respect to \(J\). In terms of normal metric contact pairs, the generalization of Vaisman’s result can be stated as follows: a \(J\)-invariant submanifold of a normal metric contact pair manifold of type \((h,0)\) is minimal if and only if the submanifold is tangent to the Reeb vector fields or, equivalently, if it is also \(T\)-invariant. These observations lead to the study of the invariant submanifolds of normal metric contact pairs of type \((h,k)\) [7] also called Hermitian bicontact structures [11].

More precisely, recall that a metric contact pair [6] of type \((h,k)\) on a manifold \(M\) is \(4\)-tuple \((\alpha _1, \alpha _2, \phi , g)\) such that \((\alpha _1, \alpha _2)\) is a contact pair [2, 5] of type \((h,k)\), \(\phi \) is an endomorphism field of \(M\) such that

$$\begin{aligned} \phi ^2=-Id + \alpha _1 \otimes Z_1 + \alpha _2 \otimes Z_2 , \quad \phi Z_1=\phi Z_2 =0 , \end{aligned}$$

where \(Z_1\) and \(Z_2\) are the Reeb vector fields of \((\alpha _1 , \alpha _2)\), and \(g\) is a Riemannian metric such that \(g(X, \phi Y)= (\mathrm{d}\alpha _1 + \mathrm{d}\alpha _2) (X,Y)\) and \(g(X, Z_i)=\alpha _i(X)\), for \(i=1,2\). The metric contact pair is said to be normal [7] if the two almost complex structures of opposite orientations \(J=\phi - \alpha _2 \otimes Z_1 + \alpha _1 \otimes Z_2 \) and \(T=\phi + \alpha _2 \otimes Z_1 - \alpha _1 \otimes Z_2 \) are integrable. A quite important notion is the one of decomposability of \(\phi \), which means that the tangent spaces of the leaves of the characteristic foliations of the pair are preserved by \(\phi \). The decomposability of \(\phi \) is equivalent to the orthogonality of the two characteristic foliations and implies that the their leaves are \(\phi \)-invariant submanifolds and moreover minimal [8].

In this paper, after giving some characterizations of normal metric contact pairs with decomposable \(\phi \), we address the problem of the minimality of the invariant submanifolds. Observe that on a metric contact pair manifold, we have several notions of invariant submanifold: with respect to \(\phi \), to \(J\), or to \(T\). We first give some general results concerning the invariant submanifolds of a metric contact pair manifold with decomposable \(\phi \), then we specialize to the normal case, and we prove the following:

Theorem 3

Let \((M, \alpha _1, \alpha _2, \phi , g)\) be a normal metric contact pair manifold with decomposable \(\phi \) and Reeb vector fields \(Z_1\) and \(Z_2\). If \(N\) is a \(\phi \)-invariant submanifold of \(M\) such that \(Z_1\) is tangent and \(Z_2\) orthogonal to \(N\), then \(N\) is minimal. Moreover, if \(N\) is connected, then it is a Sasakian submanifold of one of the Sasakian leaves of the characteristic foliation of \(\alpha _2\).

Theorem 4

Let \((M, \alpha _1, \alpha _2, \phi , g)\) be a normal metric contact pair manifold with decomposable \(\phi \) and Reeb vector fields \(Z_1\) and \(Z_2\). Let \(N\) be a \(\phi \)-invariant submanifold of \(M\) nowhere tangent and nowhere orthogonal to \(Z_1\) and \(Z_2\). Then, \(N\) is minimal if and only if the angle between \(Z_1^T\) (\(Z_1^T\) being the tangential component of \(Z_1\) along \(N\)) and \(Z_1\) (or equivalently \(Z_2\)) is constant along the integral curves of \(Z_1^T\).

Theorem 5

Let \((M, \alpha _1, \alpha _2, \phi , g)\) be a metric contact pair manifold with decomposable \(\phi \) and Reeb vector fields \(Z_1\) and \(Z_2\). Suppose that the almost complex structure \(J=\phi - \alpha _2 \otimes Z_1 + \alpha _1 \otimes Z_2 \) is integrable. Then, a \(J\)-invariant submanifold \(N\) of \(M\) is minimal if and only if it is tangent to the Reeb distribution.

The last result, applied to the case of a normal metric contact pair, gives the desired generalization of the result of Vaisman to normal metric contact pairs of type \((h,k)\). Nevertheless, it should be remarked that the full generalization of the original Vaisman result concerning the Vaisman manifolds is not true. In fact, we give an example where the submanifold is both \(J\) and \(T\)-invariant, then tangent to the Reeb distribution, and therefore minimal, but it does not inherit the contact pair structure of the ambient manifold.

In what follows we denote by \(\varGamma (B)\) the space of sections of a vector bundle \(B\). For a given foliation \(\mathcal {F}\) on a manifold \(M\), we denote by \(T\mathcal {F}\) the subbundle of \(TM\) whose fibers are given by the distribution tangent to the leaves. All the differential objects considered are assumed to be smooth.

2 Preliminaries

A contact pair (or bicontact structure) [2, 5, 11] of type \((h,k)\) on a manifold \(M\) is a pair \((\alpha _1, \alpha _2)\) of 1-forms such that

$$\begin{aligned}&\alpha _1\wedge (\mathrm{d}\alpha _1)^{h}\wedge \alpha _2\wedge (\mathrm{d}\alpha _2)^{k} \hbox { is a volume form},\\&\quad (\mathrm{d}\alpha _1)^{h+1}=0 \hbox { and } (\mathrm{d}\alpha _2)^{k+1}=0. \end{aligned}$$

The Élie Cartan characteristic classes of \(\alpha _1\) and \(\alpha _2\) are constant and equal to \(2h+1\) and \(2k+1\), respectively. The distribution \(\mathrm{Ker }\alpha _1 \cap \mathrm{Ker }\mathrm{d}\alpha _1\) (respectively, \(\mathrm{Ker }\alpha _2 \cap \mathrm{Ker }\mathrm{d}\alpha _2\)) is completely integrable [2, 5], and then it determines the characteristic foliation \(\mathcal {F}_1\) of \(\alpha _1\) (respectively, \(\mathcal {F}_2\) of \(\alpha _2\)) whose leaves are endowed with a contact form induced by \(\alpha _2\) (respectively, \(\alpha _1\)). The equations

$$\begin{aligned} \alpha _1 (Z_1)&= \alpha _2 (Z_2)=1 , \quad \alpha _1 (Z_2)=\alpha _2 (Z_1)=0 , \\ i_{Z_1} \mathrm{d}\alpha _1&= i_{Z_1} \mathrm{d}\alpha _2 =i_{Z_2}\mathrm{d}\alpha _1=i_{Z_2} \mathrm{d}\alpha _2=0 , \end{aligned}$$

where \(i_X\) is the contraction with the vector field \(X\), determine uniquely the two vector fields \(Z_1\) and \(Z_2\), called Reeb vector fields. Since they commute [2, 5], they give rise to a locally free \(\mathbb {R}^2\)-action, an integrable distribution called Reeb distribution, and then a foliation \(\mathcal {R}\) of \(M\) by surfaces. The tangent bundle of \(M\) can be split as:

$$\begin{aligned} TM=T\mathcal F _1 \oplus T\mathcal F _2 =\mathcal {H}_1 \oplus \mathcal {H}_2 \oplus \mathcal {V} , \end{aligned}$$

where \(T\mathcal F _i\) is the subbundle determined by the characteristic foliation \(\mathcal F _i, \mathcal {H}_i\) the subbundle whose fibers are given by \(\ker \mathrm{d}\alpha _i \cap \ker \alpha _1 \cap \ker \alpha _2\), \(\mathcal {V} =\mathbb {R} Z_1 \oplus \mathbb {R} Z_2\) and \(\mathbb {R} Z_1, \mathbb {R} Z_2\) the line bundles determined by the Reeb vector fields. Moreover, we have \(T\mathcal F _1=\mathcal {H}_1 \oplus \mathbb {R} Z_2 \) and \(T\mathcal F _2=\mathcal {H}_2 \oplus \mathbb {R} Z_1 \). The fibers of the subbundle \(\mathcal {H}_1 \oplus \mathcal {H}_2\) are given by the distribution \(\ker \alpha _1 \cap \ker \alpha _2\).

Definition 1

We say that a vector field is vertical if it is a section of \(\mathcal {V}\) and horizontal if it is a section of \(\mathcal {H}_1 \oplus \mathcal {H}_2\). A tangent vector will be said vertical if it lies in \(\mathcal {V}\) and horizontal if it lies in \(\mathcal {H}_1 \oplus \mathcal {H}_2\). The subbundles \(\mathcal {V}\) and \(\mathcal {H}_1 \oplus \mathcal {H}_2\) will be called vertical and horizontal, respectively.

The two distributions \(\ker \mathrm{d}\alpha _1\) and \( \ker \mathrm{d}\alpha _2\) are also completely integrable and give rise to the characteristic foliations \(\mathcal {G}_i\) of \(\mathrm{d}\alpha _i\), respectively. We have \( T\mathcal {G} _i =\mathcal {H}_i \oplus \mathcal {V} , \) for \(i=1,2\). The contact pair on \(M\) induces on each leaf of \(\mathcal {G}_1\) (respectively, of \(\mathcal {G}_2\)) a contact pair of type \((0,k)\) (respectively, \((h,0)\)). Each of them is foliated by leaves of \(\mathcal F _1\) (respectively, of \(\mathcal F _2\)) and also by leaves of \(\mathcal {R}\).

A contact pair structure [6] on a manifold \(M\) is a triple \((\alpha _1 , \alpha _2 , \phi )\), where \((\alpha _1 , \alpha _2)\) is a contact pair and \(\phi \) a tensor field of type \((1,1)\) such that:

$$\begin{aligned} \phi ^2=-Id + \alpha _1 \otimes Z_1 + \alpha _2 \otimes Z_2 , \quad \phi Z_1=\phi Z_2 =0 , \end{aligned}$$

where \(Z_1\) and \(Z_2\) are the Reeb vector fields of \((\alpha _1 , \alpha _2)\).

One can see that \(\alpha _i \circ \phi =0\), for \(i=1,2\) and that the rank of \(\phi \) is equal to \(\dim M -2\). The endomorphism \(\phi \) is said to be decomposable if \(\phi (T\mathcal {F}_i) \subset T\mathcal {F}_i\), for \(i=1,2\).

In [7], we defined the notion of normality for a contact pair structure as the integrability of the two natural almost complex structures of opposite orientations \(J=\phi - \alpha _2 \otimes Z_1 + \alpha _1 \otimes Z_2 \) and \(T=\phi + \alpha _2 \otimes Z_1 - \alpha _1 \otimes Z_2 \) on \(M\). This is equivalent to the vanishing of the tensor field

$$\begin{aligned} N^1 (X,Y)= [\phi , \phi ](X, Y) +2 \mathrm{d}\alpha _1 (X,Y) Z_1 +2 \mathrm{d}\alpha _2 (X,Y) Z_2 , \end{aligned}$$

where \([\phi , \phi ]\) is the Nijenhuis tensor of \(\phi \).

A compatible metric [6] with respect to a contact pair structure \((\alpha _1 , \alpha _2 ,\phi )\) on a manifold \(M\), with Reeb vector fields \(Z_1\) and \(Z_2\) is a Riemannian metric \(g\) on \(M\) such that \(g(\phi X,\phi Y)=g(X,Y)-\alpha _1 (X) \alpha _1 (Y)-\alpha _2 (X) \alpha _2 (Y)\) for all \(X,Y \in \varGamma (TM)\). A Riemannian metric \(g\) is said to be an associated metric [6] if \(g(X, \phi Y)= (\mathrm{d}\alpha _1 + \mathrm{d}\alpha _2) (X,Y)\) and \(g(X, Z_i)=\alpha _i(X)\), for \(i=1,2\) and for all \(X,Y \in \varGamma (TM)\).

It is clear that an associated metric is compatible, but the converse is not true. However, a compatible metric always satisfies the second equation \(g(X, Z_i)=\alpha _i(X)\), for \(i=1,2\), and then the subbundles \(\mathcal {H}_1 \oplus \mathcal {H}_2\), \(\mathbb {R} Z_1\), \(\mathbb {R} Z_2\) are pairwise orthogonal.

A metric contact pair (MCP) on a manifold \(M\) is a \(4\)-tuple \((\alpha _1, \alpha _2, \phi , g)\) where \((\alpha _1, \alpha _2, \phi )\) is a contact pair structure and \(g\) an associated metric with respect to it. The manifold \(M\) will be called an MCP manifold or an MCP for short.

For an MCP \((\alpha _1, \alpha _2, \phi , g)\), the endomorphism field \(\phi \) is decomposable if and only if the characteristic foliations \(\mathcal {F}_1\), \(\mathcal {F}_2\) are orthogonal [6]. In this case, \((\alpha _i, \phi , g)\) induces a metric contact structure on the leaves of \(\mathcal {F}_j\) , for \(j\ne i\). Also, the MCP induces MCP’s on the leaves of \(\mathcal {G}_i\).

It has been shown in [9] that a normal MCP structure of type \((h,0)\) is nothing but a non-Kähler Vaisman structure on the manifold.

If the MCP on \(M\) is normal with decomposable endomorphism, then the leaves of \(\mathcal {F}_i\) are Sasakian. Also, those of \(\mathcal {G}_i\) are non-Kähler Vaisman manifolds foliated by leaves of \(\mathcal {F}_i\) (which are Sasakian) and by leaves of \(\mathcal {R}\) (which are complex curves).

Interesting examples and properties of such structures were given in [38].

Example 1

If \((M_1, \alpha _1, \phi _1, g_1)\) and \((M_2, \alpha _2, \phi _2, g_2)\) are two Sasakian manifolds, then the structure \((\alpha _1,\alpha _2, \phi , g)\) with \(\phi =\phi _1\oplus \phi _2\) and \(g=g_1\oplus g_2\) is a normal MCP on the product \(M_1\times M_2\) with decomposable endomorphism. So we have such a structure on \(\mathbb {R}^{2h+2k+2}\) using the standard Sasakian structures on \(\mathbb {R}^{2h+1}\) and \(\mathbb {R}^{2k+1}\) given by

$$\begin{aligned}&\alpha _1=\frac{1}{2}\left( \mathrm{d}z-\sum \limits _{i=1}^h y_{i}\mathrm{d}x_{i}\right) , \quad g_1=\alpha _1\otimes \alpha _1+\frac{1}{4}\sum \limits _{i=1}^h \left( (\mathrm{d}x_{i})^2+(\mathrm{d}y_{i})^2 \right) \\&\alpha _2=\frac{1}{2}\left( \mathrm{d}z^{\prime }-\sum \limits _{j=1}^k y^{\prime }_{j}\mathrm{d}x^{\prime }_{j}\right) , \quad g_2=\alpha _2\otimes \alpha _2+\frac{1}{4}\sum \limits _{j=1}^k \left( (\mathrm{d}x^{\prime }_{j})^2+(\mathrm{d}y^{\prime }_{j})^2 \right) . \end{aligned}$$

The Reeb vector fields are \(Z_1=2\frac{\partial }{\partial z}\) and \(Z_2=2\frac{\partial }{\partial z^{\prime }}\). The endomorphism \(\phi \) sends the vector fields \(X_i=\frac{\partial }{\partial y_i}\) to \(X_{h+i}=\frac{\partial }{\partial x_i}+y_{i}\frac{\partial }{\partial z}\) and \(X^{\prime }_j=\frac{\partial }{\partial y^{\prime }_j}\) to \(X^{\prime }_{k+j}=\frac{\partial }{\partial x^{\prime }_j}+y^{\prime }_{j}\frac{\partial }{\partial z^{\prime }}\).

Remark 1

As already explained in [7, Section 3.4], normal MCP manifolds were already studied in [11] under the name of bicontact Hermitian manifolds and can be regarded as a generalization of the Calabi–Eckmann manifolds. An MCP is a special case of a metric \(f\)-structure with complemented frames in the sense of Yano [16]. The normality condition for such structures is well known and is in fact the same condition we have asked for an MCP to be normal. What is completely new in our context is the fact that the normality condition is equivalent to the integrability of the two almost complex structures \(J\) and \(T\) defined above. Even in the special case of the Vaisman manifolds, this was not known as it was observed in [9] (see the short discussion before Proposition 2.10) where it was used for classification purposes. It should also be observed that \(\mathcal {P}\)-manifolds introduced in [14] by Vaisman are MCP manifolds of type \((h,0)\) with Killing Reeb vector fields.

3 Normal metric contact pairs

We are now interested on metric contact pairs that are at the same time normal. The following proposition is an immediate corollary of [4, Corollary 3.2 and Theorem 3.4].

Proposition 1

Let \((\alpha _1,\alpha _2 , \phi , g)\) be a normal MCP on a manifold, with decomposable \(\phi \), Reeb vector fields \(Z_1\), \(Z_2\), and \(Z=Z_1+Z_2\). Let \(\nabla \) be the Levi–Civita connection of the associated metric \(g\). Then, we have

$$\begin{aligned} g((\nabla _X \phi )Y, W)&= \sum \limits _{i=1} ^2 \left[ \mathrm{d}\alpha _i (\phi Y , X) \alpha _i (W)- \mathrm{d}\alpha _i (\phi W,X) \alpha _i (Y)\right] ;\end{aligned}$$
(1)
$$\begin{aligned} \nabla _X Z&= -\phi X. \end{aligned}$$
(2)

Now, we want to characterize the normal MCP manifolds between the MCP’s as Sasakian manifolds are between the almost contact manifolds.

Theorem 1

Let \((\alpha _1,\alpha _2 , \phi , g)\) be a contact pair structure on a manifold \(M\) with compatible metric \(g\), decomposable \(\phi \) and Reeb vector fields \(Z_1\), \(Z_2\). Then, \((\alpha _1,\alpha _2 , \phi , g)\) is a normal MCP if and only if, for all \(X,Y \in \varGamma (TM)\),

$$\begin{aligned} (\nabla _X \phi ) Y= \sum \limits _{i=1} ^2 [g (X_i, Y_i) Z_i-\alpha _i (Y_i)X_i ], \end{aligned}$$
(3)

where \(X_i\) and \(Y_i\) , \(i=1,2\), are the orthogonal projections of \(X\) and \(Y\), respectively, on the foliation \(\mathcal {F}_j\), with \(j\ne i\).

Proof

Suppose that \((\alpha _1,\alpha _2 , \phi , g)\) is a normal MCP. By (1), for all \(W\in \varGamma (TM)\), we have

$$\begin{aligned} g\left( \left( \nabla _X \phi \right) Y,W\right)&= g((\nabla _{X_1} \phi ) Y,W)+g((\nabla _{X_2} \phi ) Y, W)\\&= \sum \limits _{i=1} ^2 [\mathrm{d}\alpha _i (\phi Y , X_i) \alpha _i (W)- \mathrm{d}\alpha _i (\phi W,X_i) \alpha _i (Y)]\\&= \sum \limits _{i=1} ^2 [(\mathrm{d}\alpha _1+\mathrm{d}\alpha _2) (\phi Y_i , X_i) \alpha _i (W)- (\mathrm{d}\alpha _1+\mathrm{d}\alpha _2) (\phi W,X_i) \alpha _i (Y_i)]\\&= \sum \limits _{i=1} ^2 [g (X_i,Y_i) g (W, Z_i)- \alpha _i (Y_i) g (W,X_i) ]\\&= g \left( W,\sum \limits _{i=1} ^2 [g (X_i,Y_i)Z_i-\alpha _i (Y_i) X_i]\right) , \end{aligned}$$

which is equivalent to (3).

Conversely, suppose that (3) is true. Putting \(Y=Z_j\) in (3), we obtain

$$\begin{aligned} -\phi \nabla _X Z = (\nabla _X \phi ) Z = \alpha _1 (X) Z_1 + \alpha _2 (X) Z_2 -X= \phi ^{2} X , \end{aligned}$$

where \(Z=Z_1+Z_2\). This gives \( \nabla _X Z=- \phi X \) since \(\nabla _X Z\) is horizontal (see [4, Lemma 3.5]). Then, we have

$$\begin{aligned} \mathrm{d}\alpha _1 (X,Y)+\mathrm{d}\alpha _2 (X,Y)&= \frac{1}{2} \sum \limits _{i=1}^2 [X \alpha _i(Y)-Y \alpha _i(X) - \alpha _i ([X,Y])]\\&= \frac{1}{2} \sum \limits _{i=1}^2 [X g(Z_i ,Y)-Y g(Z_i,X) - g (Z_i, \nabla _X Y - \nabla _Y X)]\\&= \frac{1}{2} [g(\nabla _X Z, Y) - g (\nabla _Y Z,X )]\\&= \frac{1}{2}[g(-\phi X,Y)+g(\phi Y,X)]\\&= g (X, \phi Y) , \end{aligned}$$

which means that the compatible metric \(g\) is even associated.

To prove the vanishing of the tensor field \(N^1\), let us compute \([\phi , \phi ]\). Taking \(X \in \varGamma (T\mathcal {F}_1)\) and \(Y \in \varGamma (T\mathcal {F}_2)\) in (3), we obtain \( 0=(\nabla _X \phi ) Y=\nabla _X (\phi Y)-\phi \nabla _X Y , \) which implies \(\nabla _X (\phi Y)=\phi \nabla _X Y\). Then, we obtain

$$\begin{aligned}{}[\phi , \phi ](X,Y)&= \phi ^2 [X,Y]-\phi [\phi X,Y]-\phi [X,\phi Y]+ [\phi X,\phi Y]\\&= \phi ^2 (\nabla _X Y - \nabla _Y X)-\phi (\nabla _{\phi X} Y - \phi \nabla _Y X)\\&\quad -\,\phi (\phi \nabla _X Y - \nabla _{\phi Y} X)+ \phi \nabla _{\phi X} Y - \phi \nabla _{\phi Y } X\\&= 0 \end{aligned}$$

and \(N^1(X,Y)=[\phi , \phi ](X,Y)=0\). Now by (3) with \(X, Y\in \varGamma (T\mathcal {F}_1)\), we have \( (\nabla _X \phi ) Y= g (X, Y) Z_2-\alpha _2 (Y)X. \) Then, we get

$$\begin{aligned}{}[\phi , \phi ](X,Y)&= \phi ^2 (\nabla _X Y - \nabla _Y X)-\phi (\nabla _{\phi X} Y - \phi \nabla _Y X -(\nabla _Y \phi )X)\\&\quad -\,\phi (\phi \nabla _X Y +(\nabla _X \phi )Y - \nabla _{\phi Y} X)+ \phi \nabla _{\phi X} Y - \phi \nabla _{\phi Y } X\\&= g(\phi X,Y)Z_2 - g(X,\phi Y)Z_2\\&= -2 \mathrm{d}\alpha _2 (X, Y) Z_2 . \end{aligned}$$

Hence, \(N^1(X,Y)=[\phi , \phi ](X,Y)+2 \mathrm{d}\alpha _2 (X, Y) Z_2 =0\). In the same way, we obtain \(N^1(X,Y)=[\phi , \phi ](X,Y)+2 \mathrm{d}\alpha _1 (X, Y) Z_1=0\) for all \(X, Y\in \varGamma (T\mathcal {F}_2)\). This shows the normality and completes the proof. \(\square \)

Theorem 2

Let \((M, \alpha _1,\alpha _2 , \phi , g)\) be an MCP manifold with decomposable \(\phi \), \(Z_1\), \(Z_2\) the Reeb vector fields and \(Z=Z_1+Z_2\). Let \(\mathrm{R }\) be the curvature operator of \(g\). Then, the MCP \((\alpha _1,\alpha _2 , \phi , g)\) is normal if and only if

$$\begin{aligned} \mathrm{R }_{X Y} Z=\sum \limits _{i=1} ^2 [\alpha _i (Y_i) X_i-\alpha _i (X_i)Y_i]. \end{aligned}$$
(4)

Proof

Suppose that the MCP is normal. By (2) and (3), for all \(X,Y \in \varGamma (TM)\), we have

$$\begin{aligned} \mathrm{R }_{XY}Z&= -\nabla _X (\phi Y)+\nabla _Y (\phi X)+\phi [X,Y] \\&= -\big (\nabla _{X}\phi \big )Y+\big (\nabla _{Y}\phi \big )X \\&= \sum \limits _{i=1} ^2 [\alpha _i (Y_i) X_i-\alpha _i (X_i)Y_i ]. \end{aligned}$$

Conversely, suppose that (4) is true. Then, for \(Y\) horizontal, we have

$$\begin{aligned} \mathrm{R }_{Z Y}Z=-Y_1-Y_2=-Y. \end{aligned}$$

Using this in the following equation

$$\begin{aligned} \frac{1}{2} \bigl (\mathrm{R }_{Z X}Z - \phi (\mathrm{R }_{Z \phi X}Z) \bigr )= \phi ^2 X + \mathrm{h }^2 X , \end{aligned}$$

which holds for MCP manifolds (see [4, Proposition 4.1]), where \(\mathrm{h }= \frac{1}{2} \mathcal {L}_Z \phi \) and \(\mathcal {L}_Z\) is the Lie derivative along \(Z\), we get

$$\begin{aligned} \frac{1}{2} (-Y-\phi (-\phi Y))= \phi ^2 Y+ \mathrm{h }^2 Y \end{aligned}$$

which implies \(\mathrm{h }=0\). In particular, from the equation \(\nabla _X Z = -\phi X -\phi \mathrm{h }X \) (see [4, Theorem 3.4]), we have \(\nabla _X Z=-\phi X\) . Since \(\mathrm{h }=0\), the vector field \(Z\) is Killing [4, 6], then it is affine, and we have

$$\begin{aligned} \mathrm{R }_{ Z X} Y=-\nabla _X \nabla _Y Z+ \nabla _{\nabla _X Y} Z= \nabla _X {\phi Y}- \phi \nabla _X Y= (\nabla _X \phi )Y . \end{aligned}$$

Then, for every \(X,Y, W \in \varGamma (TM)\), recalling that for an MCP with decomposable \(\phi \) the characteristic foliations are orthogonal, we obtain

$$\begin{aligned} g ((\nabla _X \phi )Y, W)&= g(\mathrm{R }_{Z X}Y , W)= g(\mathrm{R }_{Y W }Z , X)= g \left( \sum \limits _{i=1} ^2 [\alpha _i (W_i) Y_i-\alpha _i (Y_i)W_i ], X\right) \\&= \sum \limits _{i=1} ^2 [g(\alpha _i (W) Y_i, X_i) - g(W , \alpha _i (Y_i) X_i)]\\&= \sum \limits _{i=1} ^2 [g (W, Z_i)g( Y_i, Xi)-g(W , \alpha _i (Y_i) X_i)]\\&= g \left( W , \sum \limits _{i=1} ^2 [g( Y_i, X_i) Z_i- \alpha _i (Y_i) X_i]\right) , \end{aligned}$$

which implies that \((\nabla _X \phi ) Y= \sum _{i=1} ^2 [g (X_i, Y_i) Z_i-\alpha _i (Y_i)X_i ]\) and then the pair is normal by Theorem 1. \(\square \)

4 \(\phi \)-invariant submanifolds

In this section, we study the \(\phi \)-invariant submanifolds of MCP manifolds. We first give some general results and then we specialize to several cases concerning the submanifold position relative to the Reeb distribution.

Let \((\alpha _1, \alpha _2, \phi )\) be a contact pair structure of type \((h,k)\) on a manifold \(M\).

Definition 2

A submanifold \(N\) of \(M\) is said to be invariant with respect to \(\phi \) (or \(\phi \)-invariant) if its tangent space at every point is preserved by \(\phi \), that is if \(\phi _p T_pN \subset T_pN\) for all \(p\in N\).

In the same way, one can define \(J\)-invariant submanifolds and \(T\)-invariant submanifolds for the two almost complex structures defined in Sect. 2.

The simplest examples of \(\phi \)-invariant surfaces are given by the leaves of the foliation \(\mathcal {R}\) tangent to the Reeb distribution. When we suppose the endomorphism field \(\phi \) decomposable, by definition the leaves of the two characteristic foliations \(\mathcal {F}_i\) of the \(1\)-forms \(\alpha _i\) are \(\phi \)-invariant. The same is true for the leaves of the two characteristic foliations \(\mathcal {G}_i\) of the 2-forms \(\mathrm{d}\alpha _i\).

Observe that in the second case, only one of the two Reeb vector fields is tangent to the submanifolds. In the first and third cases, both the Reeb vector fields are tangent and such submanifolds are invariant with respect to \(J\) and \(T\).

Despite the case of a metric contact manifold, where the Reeb vector field is always tangent to a \(\phi \)-invariant submanifold, in our case, the situation can be quite different as we have just seen. We will show several nontrivial examples in the sequel.

In what follows \((M, \alpha _1, \alpha _2, \phi , g)\) will be a given MCP manifold with Reeb vector fields \(Z_1\), \(Z_2\) and \(N\) a \(\phi \)-invariant submanifold of \(M\). We will denote by \(Z_i^T\) (respectively, \(Z_i^\bot \)) the tangential (respectively, normal) component of the two vector fields \(Z_1\) and \(Z_2\) along \(N\).

Proposition 2

Along the \(\phi \)-invariant submanifold \(N\), the tangent vector fields \(Z_1^T\), \(Z_2^T\) and the normal vector fields \(Z_1^\bot \), \(Z_2^\bot \) are vertical.

Proof

For every \(X\in \varGamma (TN)\), we have \(g(\phi Z_i^\bot , X )=-g(Z_i^\bot ,\phi X )=0\) because \(\phi X\in \varGamma (TN)\). Then, the vector fields \(\phi Z_i^\bot \) are also orthogonal to \(N\). As \(0=(\phi Z_i) _{\vert N}= \phi Z_i^T + \phi Z_i^\bot \), we get \(\phi Z_i^T = \phi Z_i^\bot = 0\) because one is tangent and the other is orthogonal to \(N\). We conclude by recalling that the distribution \(\ker \phi \) is spanned by \(Z_1\) and \(Z_2\). \(\square \)

Proposition 3

There is no point \(p\) of the \(\phi \)-invariant submanifold \(N\) such that the tangent vectors \((Z_1)_p\) and \((Z_2)_p\) are both orthogonal to the tangent space \(T_p N\).

Proof

If, at a point \(p\in N\), the two vectors \((Z_i)_p\) (for \(i=1,2\)) are orthogonal to the tangent space \(T_p N\), we have \((Z_i^\bot )_p = (Z_i)_p\) and they are linearly independent. Take an open neighborhood \(U\) of \(p\) in \(M\) such that on \(U\cap N\), the two vector fields \(Z_1^\bot \), \(Z_2^\bot \) still remain linearly independent. By Proposition 2, they span \(\mathbb {R}Z_1 \oplus \mathbb {R}Z_2\) along \(U\) and then the Reeb vector fields \(Z_1\), \(Z_2\) are both orthogonal to \(T_q N\) at each point \(q\in U\cap N\).

Let \(X\) be a vector field defined on \(U\), tangent to \(N\) and such that \(X_p\ne 0\). Then, \(\phi X\) and \([X,\phi X]\) are also tangent to \(N\). Since for every point \(q\in U\cap N\), the tangent space \(T_q N\) is in the kernels of \(\alpha _1\) and \(\alpha _2\) (because it is orthogonal to \((Z_i)_q\)), along \(N\), we have

$$\begin{aligned} 0=\left( \alpha _1+\alpha _2 \right) ([X,\phi X])=-2 \left( \mathrm{d}\alpha _1+\mathrm{d}\alpha _2 \right) (X, \phi X)=2 g(X,X) \end{aligned}$$

contradicting the fact that \(X_p\ne 0\). \(\square \)

4.1 The case \(N\) tangent to only one Reeb vector field

Proposition 4

If the \(\phi \)-invariant submanifold \(N\) is tangent to one of the two Reeb vector fields, say \(Z_1\), and transverse to the other one \(Z_2\), then \(N\) is everywhere orthogonal to \(Z_2\). Moreover, the dimension of \(N\) is odd.

Such submanifolds were called semi-invariant by Blair, Ludden, and Yano [11] in the context of Hermitian manifolds. The semi-invariance is understood with respect to the almost complex structure \(J=\phi - \alpha _2 \otimes Z_1 + \alpha _1 \otimes Z_2 \).

Proof

Since \(Z_1\) is tangent to \(N\), we have \(Z_1^T=(Z_1)_{\vert N}\ne 0\). Now, \(0=g\left( Z_1, Z_2 \right) _{\vert N}=g( Z_1^T, Z_2 ^T)\), which implies \(Z_2^T=0\). Indeed, if at a point \(p\in N\), \(Z_2^T\ne 0\), the two vectors \((Z_i^T)_p\) would be linearly independent. By Proposition 2, they will span the tangent subspace \(\mathbb {R}(Z_1)_p \oplus \mathbb {R}(Z_2)_p\) and then \(Z_2\) will be tangent to \(N\) at \(p\), but \(Z_2\) is supposed to be transverse to N. Now, it is clear that \(\phi \) is almost complex on the orthogonal complement of \(\mathbb {R}Z_1\) in \(TN\). Hence, the dimension of \(N\) is odd. \(\square \)

The following result is a restatement of [11, Propositions 4.2 and 4.3]:

Proposition 5

(Blair et al. [11]) If the \(\phi \)-invariant submanifold \(N\) is tangent to the vector field \(Z_1\) and orthogonal to \(Z_2\), then \((\alpha _1, \phi , g)\) induces a metric contact structure on \(N\). If in addition the MCP on \(M\) is normal, then \(N\) is Sasakian.

Proof

Let \(\tilde{\alpha _1}\) and \(\tilde{\alpha _2}\) denote the forms induced on \(N\) by \(\alpha _1\) and \(\alpha _2\). To prove that \(\tilde{\alpha _1}\) is a contact form, one has just to show that \(\tilde{\mathrm{d}\alpha _1}\) is symplectic on \(\ker \tilde{\alpha _1}\). First, observe that since \(Z_2\) is orthogonal to \(N\), we have \(\tilde{\alpha _2}=0\), then \(\mathrm{d}\tilde{\alpha _1}=\mathrm{d}\tilde{\alpha _1} + \mathrm{d}\tilde{\alpha _2}\). Now, for \(p\in N\), let \(X\in T_pN\) such that \(\tilde{\alpha _1}(X)=0\) and \(\mathrm{d}\tilde{\alpha _1}(X,Y)=0\) for every \(Y\in T_pN\). Then, \((\mathrm{d}\tilde{\alpha _1} + \mathrm{d}\tilde{\alpha _2})(X,Y)=0\) and we get \(g(X,\phi Y)=0\). As we also have \(0=\alpha _1(X)=g(X,Z_1)\), one can say that \(g(X,\cdot )=0\) on \(N\) and then \(X=0\). Hence, \(\tilde{\alpha _1}\) is contact on \(N\). The normality of the induced structure on \(N\) follows from the vanishing of the tensor \(N^1\) and the fact that \(\mathrm{d}\tilde{\alpha _2}=0\) on \(N\). \(\square \)

4.2 The case \(N\) nowhere orthogonal and nowhere tangent to \(Z_1\) and \(Z_2\)

Proposition 6

If the \(\phi \)-invariant submanifold \(N\) is nowhere orthogonal and nowhere tangent to \(Z_1\) and \(Z_2\), then it has odd dimension. Moreover, its tangent bundle \(TN\) intersects the vertical subbundle \(\mathcal {V}\) along a line bundle spanned by the vector field \(Z_1^T\) (or equivalently by \(Z_2^T\)).

Proof

In this case, by Proposition 2, we have necessarily that \(Z_1^T\) and \(Z_2^T\) are vertical, linearly dependent and nonzero. The same holds for \(Z_1^\bot \) and \(Z_2^\bot \). So \(Z_1^T\) spans the intersection of \(TN\) with the vertical subbundle \(\mathcal {V}\). Now, every vector field tangent to \(N\) and orthogonal to \(Z_1^T\) is necessary orthogonal to the Reeb distribution. By the \(\phi \)-invariance of \(TN\), \(\phi \) is almost complex on the orthogonal complement of \(\mathbb {R}Z_1^T\) in \(TN\) and then \(N\) has odd dimension. \(\square \)

Example 2

As a manifold, consider the product \(H^6=\mathbb {H}_3 \times \mathbb {H}_3\) where \(\mathbb {H}_3\) is the \(3\)-dimensional Heisenberg group. Let \(\left\{ \alpha _1, \alpha _2, \alpha _3 \right\} \) (respectively, \(\left\{ \beta _1, \beta _2, \beta _3 \right\} \)) be a basis of the cotangent space at the identity for the first (respectively, second) factor \(\mathbb {H}_3\) satisfying

$$\begin{aligned}&\mathrm{d}\alpha _3=\alpha _1\wedge \alpha _2,\quad \mathrm{d}\alpha _1=\mathrm{d}\alpha _2=0,\\&\mathrm{d}\beta _3=\beta _1\wedge \beta _2 ,\quad \mathrm{d}\beta _1=\mathrm{d}\beta _2=0. \end{aligned}$$

The pair \((\alpha _3 , \beta _3 )\) determines a contact pair of type \((1,1)\) on \(H^6\) with Reeb vector fields \((X_3 , Y_3 )\), the \(X_i\)’s (respectively, the \(Y_i\)’s) being dual to the \(\alpha _i\)’s (respectively, the \(\beta _i\)’s). The left invariant metric

$$\begin{aligned} g=\alpha _3^2 + \beta _3^2 +\frac{1}{2}\left( \alpha _1^2 + \beta _1^2 +\alpha _2^2 + \beta _2^2 \right) \end{aligned}$$

is associated to the pair with decomposable tensor structure \(\phi \) given by \(\phi (X_2)=X_1\) and \(\phi (Y_2)=Y_1\). The MCP manifold \((H^6,\alpha _3, \beta _3, \phi , g)\) is normal because it is the product of two Sasakian manifolds. Let \(\mathfrak {h}_3\) denotes the Lie algebra of \(\mathbb {H}_3\). The three vectors \(Z=X_3+Y_3\), \(X_1 +Y_1\), and \(X_2 +Y_2\) span a \(\phi \)-invariant subalgebra of the Lie algebra \(\mathfrak {h}_3\oplus \mathfrak {h}_3\) of \(H^6\), which determines a foliation in \(H^6\). Each leaf is \(\phi \)-invariant, nowhere tangent and nowhere orthogonal to the Reeb vector fields.

4.3 The case \(N\) tangent to the Reeb distribution

Proposition 7

If the \(\phi \)-invariant submanifold \(N\) is tangent to both \(Z_1\) and \(Z_2\), then it has even dimension.

Proof

If the Reeb distribution is tangent to \(N\), then on its orthogonal complement in \(TN\) the endomorphism \(\phi \) is almost complex and this completes the proof. \(\square \)

Example 3

Take the same MCP on \(H^6=\mathbb {H}_3 \times \mathbb {H}_3\) as in Example 2. The four vectors \(X_3\), \(Y_3\), \(X_1 +Y_1\), and \(X_2 +Y_2\) span a \(\phi \)-invariant Lie subalgebra \(\mathfrak {n}_4\) of \(\mathfrak {h}_3\oplus \mathfrak {h}_3\), which determines a foliation on \(H^6\). Each leaf of this foliation is \(\phi \)-invariant and tangent to the Reeb distribution.

Remark 2

When the \(\phi \)-invariant submanifold \(N\) is tangent to both the Reeb vector fields \(Z_1\) and \(Z_2\), the contact pair \((\alpha _1, \alpha _2)\) on \(M\) does not induce necessarily a contact pair on \(N\). Indeed, from Example 3, take any leaf \(L^4\) of the foliation determined by the subalgebra \(\mathfrak {n}_4\). Then, \(L^4\) is a \(\phi \)-invariant submanifolds of the MCP manifold \(H^6\). However, the contact pair \((\alpha _3 , \beta _3 )\) induces a pair of \(1\)-forms on the \(4\)-dimensional manifold \(L^4\) whose Élie Cartan classes are both equal to 3. Then, the induced pair on \(L^4\) is not a contact one.

From this construction, one can also have a most interesting example where, in addition, the submanifold is closed without being a contact pair submanifold.

Example 4

Consider once again the normal MCP on the nilpotent Lie group \(H^6=\mathbb {H}_3 \times \mathbb {H}_3\) defined in Example 2, and the foliation on \(H^6\) defined by the Lie algebra \(\mathfrak {n}_4\) described in Example 3. Let \(L_e\) be the leaf passing through the identity element of the Lie group \(H^6\). One can see that the Lie subgroup \(L_e\) is isomorphic to \(\mathbb {H}_3 \times \mathbb {R}\). In fact, by using the change of basis of its Lie algebra \(\mathfrak {n}_4\), \(U_i=X_i+Y_i\) for \(i=1,2,3\) and \(U_4=X_3\), we get \([U_1,U_2]=U_3\) and the other brackets are zero. Since the structure constants of the nipotent Lie algebra \(\mathfrak {n}_4\) are rational, there exists cocompact lattices \(\varGamma \) of \(L_e\). For example, since \(\mathbb H_3\) can be considered as the group of the real matrices

$$\begin{aligned} \gamma (x,y,z)= \begin{pmatrix} 1 &{} y &{} z \\ 0 &{} 1 &{} x \\ 0 &{} 0 &{} 1 \end{pmatrix}, \end{aligned}$$

take \(\varGamma \simeq \varGamma _r \times \mathbb {Z}\) where \(\mathbb {Z}\) acts on the factor \(\mathbb {R}\) and \(\varGamma _r=\{\gamma (x,y,z) \vert x\in \mathbb Z,y\in r\mathbb Z, z\in \mathbb Z\}\), with \(r\) a positive integer, acts on the first factor \(\mathbb {H}_3\) by left multiplication (see, e.g., [13]). Because \(L_e\) is a subgoup of \(H^6\), it is a lattice of \(H^6\) too. Now, the closed nilmanifold \(N^4=L_e/\varGamma \) is a submanifold of the nilmanifold \(M^6= H^6 /\varGamma \). Since the MCP on \(H^6\) is left invariant, it descends to the quotient \(M^6\) as a normal MCP \((\tilde{\alpha _3}, \tilde{\beta _3}, \tilde{\phi }, \tilde{g})\) of type \((1,1)\) with decomposable endomorphism \(\tilde{\phi }\). Moreover, the closed submanifold \(N^4\) is \(\tilde{\phi }\)-invariant and tangent to the Reeb distribution. Note that the contact pair \((\tilde{\alpha _3}, \tilde{\beta _3})\) on \(M^6\) does not induce a contact pair on the submanifold \(N^4\), because the Élie Cartan classes of the induced \(1\)-forms are equal to \(3\) and the dimension of \(N^4\) is \(4\).

We know (see [7]) that a normal MCP with decomposable endomorphism is nothing but a Hermitian bicontact manifold of bidegree \((1,1)\) [11]. As we will see later in Paragraph 4.4, in a normal MCP, a \(\phi \)-invariant submanifold tangent to the Reeb distribution is a complex submanifold. So according to Example 4, we can state what follows:

Proposition 8

There exists a Hermitian bicontact manifold of bidegree \((1,1)\) carrying a closed complex submanifold, which does not inherit a bicontact structure.

Remark 3

This contradicts a statement of Abe (see [1, Theorem 2.2]). The construction of the MCP manifold \(M^6\) and its submanifold \(N^4\) in Example 4 gives clearly a counterexample.

4.4 Relationship with \(T\) and \(J\)-invariance

Put \(\rho = \alpha _2 \otimes Z_1 - \alpha _1 \otimes Z_2 \). One can easily see that a connected submanifold of \(M\) is \(\rho \)-invariant if and only if it is tangent or orthogonal to the Reeb distribution. The following holds:

Proposition 9

Let \(M'\) be a submanifold of the MCP manifold \(M\). If \(M'\) is orthogonal to the Reeb distribution, then none of the endomorphisms \(\phi \), \(J\) and \(T\) leaves \(M'\) invariant.

Proof

Let \(M'\) be a submanifold of \(M\) orthogonal to the Reeb distribution. By Proposition 3, it cannot be \(\phi \)-invariant. Suppose that \(M'\) is invariant with respect to \(J\) or \(T\). Since it is orthogonal to the Reeb vector fields, it is also \(\rho \)-invariant. Now, by the relations \(\phi =J+\rho =T-\rho \), we obtain that \(M'\) is \(\phi \)-invariant, and this is not possible. \(\square \)

Proposition 10

Let \(M'\) be a submanifold of the MCP manifold \(M\). Then, any two of the following properties imply the others:

  1. (a)

    \(M'\) is \(\phi \)-invariant,

  2. (b)

    \(M'\) is \(J\)-invariant,

  3. (c)

    \(M'\) is \(T\)-invariant,

  4. (d)

    \(M'\) is tangent to the Reeb distribution.

Proof

From the relations \(J=\phi -\rho \) and \(T=\phi +\rho \), one can remark that any two of the four endomorphisms fields \(\{\phi , J, T, \rho \}\) are linear combinations of the remaining two. So if we replace the property (d) with “\(M'\) is \(\rho \)-invariant,” then the conclusion is obvious. Suppose without the loss of generality that \(M'\) is connected. We have seen that the property “\(M'\) is \(\rho \)-invariant” is equivalent to “\(M'\) is tangent or orthogonal to the Reeb vector fields.” But by Proposition 9, the property “\(M'\) is orthogonal to the Reeb distribution” is not compatible with Properties (a), (b), and (c), and this completes the proof. \(\square \)

Example 5

Consider \(\mathbb {R}^{2h+2k+2}\) together with the normal MCP described in Example 1 with \(h>0\). For any pair of integers \(n_1, n_2\) such that \(0< n_1\le h\) and \(0\le n_2\le k\), the \(2(n_1+n_2)\)-dimensional distribution spanned by the vector fields \(Y_i=X_i+\frac{1}{2}x_iZ_1\), \(JY_i\) for \(i=1, \ldots ,n_1\) and (in the case \(n_2 > 0\)) by \(Y^{\prime }_j=X^{\prime }_j+\frac{1}{2}x^{\prime }_jZ_2\), \(JY^{\prime }_j\) for \(j=1, \ldots ,n_2\) is completely integrable. On the open set \(\{ x_i\ne 0, x^\prime _j\ne 0\}\), this distribution is invariant with respect to the complex structure \(J\) but it is not invariant with respect to \(\phi \). So it gives rise to a foliation by \(2(n_1+n_2)\)-dimensional \(J\)-invariant submanifolds, which are not \(\phi \)-invariant.

5 Minimal \(\phi \)-invariant submanifolds

In Sect. 4, we observed that the leaves of the two characteristic foliations of an MCP with decomposable endomorphism \(\phi \) are \(\phi \)-invariant submanifolds. Moreover, in [8], we have seen that these submanifolds are minimal. In this section, we extend this result to further \(\phi \)-invariant submanifolds of normal or complex MCP manifolds (the latter terminology meaning that just one of the two natural almost complex structure is supposed to be integrable).

Theorem 3

Let \((M, \alpha _1, \alpha _2, \phi , g)\) be a normal MCP manifold with decomposable \(\phi \) and Reeb vector fields \(Z_1\) and \(Z_2\). If \(N\) is a \(\phi \)-invariant submanifold of \(M\) such that \(Z_1\) is tangent and \(Z_2\) orthogonal to \(N\), then \(N\) is minimal. Moreover, if \(N\) is connected, then it is a Sasakian submanifold of one of the Sasakian leaves of the characteristic foliation of \(\alpha _2\).

Proof

Denote by \(\mathrm{B }\) the second fundamental form of the submanifold \(N\), by \(\nabla \) the Levi–Civita connection of the metric \(g\) on \(M\), and by \(\tilde{\nabla }\) its induced connection on \(N\). By Proposition 5, \((\alpha _1,Z_1,\phi ,g)\) induces a Sasakian structure on \(N\), say \((\tilde{\alpha _1},Z_1,\tilde{\phi },\tilde{g})\). Then, for every \(X\), \(Y\in \varGamma (TN)\), we have \(\left( \tilde{\nabla }_X \tilde{\phi } \right) Y=\tilde{g}(X,Y)Z_1-\tilde{\alpha _1}(Y)X\) (see, e.g., [10]). Using this and (3) for all \(X\), \(Y\in \varGamma (TN)\) orthogonal to \(Z_1\), we obtain

$$\begin{aligned} \mathrm{B }(X,\phi Y)-\phi \mathrm{B }(X,Y)=\left( \nabla _X \phi \right) Y-\left( \tilde{\nabla }_X \tilde{\phi } \right) Y=g(X_2,Y_2)(Z_2-Z_1) \end{aligned}$$
(5)

since \(X\), \(Y\) are horizontal, \(X_2\) and \(Y_2\) being, respectively, the orthogonal projections of \(X\) and \(Y\) on \(T\mathcal {F}_1\). But the vector field \(\mathrm{B }(X,\phi Y)-\phi \mathrm{B }(X,Y)\) must be orthogonal to \(N\) by the \(\phi \)-invariance of \(N\). Then, \(g(X_2,Y_2)=0\) for every \(X\), \(Y\in \varGamma (TN)\) orthogonal to \(Z_1\), which gives \(X_2=0\). This implies that \(N\) is tangent to the characteristic distribution of \(\alpha _2\).

Equation (5) becomes

$$\begin{aligned} \mathrm{B }(X,\phi Y)-\phi \mathrm{B }(X,Y)=0. \end{aligned}$$

If we interchange the roles of \(X\) and \(Y\) and take the difference, we get \(\mathrm{B }(X,\phi Y)=\mathrm{B }(Y,\phi X)\), which implies \(\mathrm{B }(X,Y)=-\mathrm{B }(\phi X,\phi Y)\). Now, locally, take an orthonormal \(\phi \)-basis of the metric contact structure on \(N\)

$$\begin{aligned} Z_1,e_1, \phi e_1, \ldots , e_s , \phi e_s. \end{aligned}$$

We have \(\mathrm{B }(Z_1,Z_1)=0\) since \(\nabla _{Z_1}Z_1=0\) (see [6]). As \(e_j\) are orthogonal to \(Z_1\), we obtain

$$\begin{aligned} \mathrm{trace }(\mathrm{B })=\mathrm{B }(Z_1 ,Z_1)+\sum \limits _{j=1} ^s\left( \mathrm{B }(e_j,e_j)+ \mathrm{B }(\phi e_j,\phi e_j)\right) =0, \end{aligned}$$

which means that \(N\) is minimal. \(\square \)

Consider a \(\phi \)-invariant submanifold \(N\) of an MCP, which is nowhere tangent and nowhere orthogonal to the Reeb vector fields. In Proposition 6, we have seen that at every point, its tangent space intersects the Reeb distribution giving rise to the distribution on \(N\) spanned by \(Z_1^T\) (or equivalently by \(Z_2^T\)). For such a submanifold, we have

Theorem 4

Let \((M, \alpha _1, \alpha _2, \phi , g)\) be a normal metric contact pair manifold with decomposable \(\phi \) and Reeb vector fields \(Z_1\) and \(Z_2\). Let \(N\) be a \(\phi \)-invariant submanifold of \(M\) nowhere tangent and nowhere orthogonal to \(Z_1\) and \(Z_2\). Then, \(N\) is minimal if and only if the angle between \(Z_1^T\) and \(Z_1\) (or equivalently \(Z_2\)) is constant along the integral curves of \(Z_1^T\).

Proof

Put \(\zeta =\frac{1}{\Vert Z_1^T\Vert }Z_1^T=\pm \frac{1}{\Vert Z_2^T\Vert }Z_2^T\). Using (3) for all \(X\), \(Y\in \varGamma (TN)\) orthogonal to \(\zeta \), we obtain

$$\begin{aligned} \mathrm{B }(X,\phi Y)-\phi \mathrm{B }(X,Y)=\left( \left( \nabla _X \phi \right) Y\right) ^\bot =g(X_1,Y_1)Z_1^\bot +g(X_2,Y_2)Z_2^\bot \end{aligned}$$

since \(X\) are \(Y\) are horizontal because they are necessarily orthogonal to \(Z_1\) and \(Z_2\). The term on the right is symmetric on \((X,Y)\), then we get

$$\begin{aligned} \mathrm{B }(X,\phi Y)-\phi \mathrm{B }(X,Y)=0. \end{aligned}$$

As previously, this yields \(\mathrm{B }(X,Y)=-\mathrm{B }(\phi X,\phi Y)\). Now, take a local orthonormal basis on \(N\) in this manner

$$\begin{aligned} \zeta ,e_1, \phi e_1, \ldots , e_s , \phi e_s. \end{aligned}$$

We obtain

$$\begin{aligned} \mathrm{trace }(\mathrm{B })=\mathrm{B }(\zeta ,\zeta )+\sum \limits _{j=1} ^s\left( \mathrm{B }(e_j,e_j)+ \mathrm{B }(\phi e_j,\phi e_j)\right) =\mathrm{B }(\zeta ,\zeta ). \end{aligned}$$

In order to compute \(\mathrm{B }(\zeta ,\zeta )\), observe that there exists a smooth function \(\theta \) on \(N\) taking nonzero values in \(] -\pi /2,\pi /2[\) and for which \(\zeta =(\cos \theta ) Z_1+(\sin \theta ) Z_2\). This function is well defined on \(N\) since \(\zeta \) lies in \(\mathcal {V}=\mathbb {R} Z_1 \oplus \mathbb {R} Z_2\) and \(g(\zeta , Z_1)>0\). It represents the oriented angle \((Z_1,\zeta )\) in the oriented orthonormal basis \((Z_1,Z_2)\) of \(\mathcal {V}\) along \(N\). One can easily show that \(Z_1^T=(\cos \theta ) \zeta \), \(Z_2^T=(\sin \theta ) \zeta \) and then since \(J\zeta =-(\sin \theta ) Z_1+ (\cos \theta )Z_2\), we have \(Z_1^\bot =-(\sin \theta ) J\zeta \) and \(Z_2^\bot =(\cos \theta ) J\zeta \). Hence, \(J\zeta \) is a nonvanishing section of the normal bundle \(TN^\bot \) of \(N\) in \(M\).

Using the equations \(\nabla _{Z_i}Z_j=0\), for \(i,j=1,2\), concerning MCP’s [6], we obtain

$$\begin{aligned} \nabla _\zeta \zeta =\zeta (\theta )J\zeta . \end{aligned}$$

This yields

$$\begin{aligned} \mathrm{trace }(\mathrm{B })=\mathrm{B }(\zeta ,\zeta )=\zeta (\theta )J\zeta \end{aligned}$$

which is zero if and only if \(\zeta (\theta )=0\). \(\square \)

Example 6

For the \(\phi \)-invariant submanifolds described in Example 2, the Reeb vector fields \(X_3\) and \(Y_3\) make a constant angle with their orthogonal projection \(\frac{1}{2}(X_3+Y_3)\). Hence, they are minimal.

A theorem of Vaisman [15] states that on a Vaisman manifold, a complex submanifold inherits the structure of Vaisman manifold if and only if it is minimal or equivalently if and only if it is tangent to the Lee vector field (and therefore tangent to the anti-Lee one). This result has been generalized to the lcK manifolds as follows (see [12, Theorem 12.1]): a complex submanifold of an lcK manifold is minimal if and only if it is tangent to the Lee vector field. Non-Kähler Vaisman manifolds are special lcK manifolds. According to [9], they are, up to a constant rescaling of the metric, exactly normal MCP manifolds of type \((h,0)\), the Reeb vector fields being the Lee and the anti-Lee vector field. What follows is a generalization of the theorem of Vaisman to complex MCP manifolds of any type \((h,k)\).

Theorem 5

Let \((M, \alpha _1, \alpha _2, \phi , g)\) be an MCP manifold with decomposable \(\phi \) and Reeb vector fields \(Z_1\) and \(Z_2\). Suppose that the almost complex structure \(J=\phi - \alpha _2 \otimes Z_1 + \alpha _1 \otimes Z_2 \) is integrable. Then, a \(J\)-invariant submanifold \(N\) of \(M\) is minimal if and only if it is tangent to the Reeb distribution.

We have the same conclusion if we replace \(J\) with the almost complex structure \(T=\phi + \alpha _2 \otimes Z_1 - \alpha _1 \otimes Z_2 \). Recall that for a submanifold tangent to the Reeb distribution, we have equivalence between invariance with respect to \(J\), \(T\), and \(\phi \) (see Proposition 10). Hence, minimal \(J\)-invariant submanifolds of an MCP are necessarily \(\phi \)-invariant and \(T\)-invariant. We can restate Theorem 5 for a normal MCP as follows:

Corollary 1

Let \((M, \alpha _1, \alpha _2, \phi , g)\) be a normal MCP manifold with decomposable \(\phi \). Then, a \(J\)-invariant submanifold \(N\) of \(M\) is minimal if and only if it is \(T\)-invariant.

The \(J\)-invariant submanifolds described in Example 5 are not minimal. Of course, they are not tangent to the Reeb distribution.

We have seen that the MCP on \(\mathbb {H}_3 \times \mathbb {H}_3\), given in Example 2, is normal because each factor is a Sasakian manifold. The submanifolds described in Example 3 are tangent to the Reeb distribution and then they are minimal. The following statement gives further interesting examples.

Corollary 2

Consider an MCP \(( \alpha _1, \alpha _2, \phi , g)\) with decomposable \(\phi \) on a manifold. Suppose that \(J\) (or \(T\)) is integrable. Then, the leaves of the characteristic foliations \(\mathcal {G} _1\) and \(\mathcal {G} _2\) of \(\mathrm{d}\alpha _1\) and \(\mathrm{d}\alpha _2\) are minimal.

Proof (of Theorem 5)

In order to compute the normal mean curvature \(H\) of the \(J\)-invariant submanifold \(N\), one needs the expression of the tensor field \(F(X,Y)=(\nabla _X J)Y\) where \(\nabla \) is the Levi–Civita connection of \(g\). Since \(J\) is integrable, \(g\) is Hermitian with fundamental \(2\)-form

$$\begin{aligned} \varPhi =\mathrm{d}\alpha _1+\mathrm{d}\alpha _2-2\alpha _1\wedge \alpha _2. \end{aligned}$$

First, observe that \(\alpha _2 \circ J=\alpha _1\) and \(\mathrm{d}\alpha _i(JX,JY)=\mathrm{d}\alpha _i(X,Y)\). Moreover, by the decomposability of \(\phi \), we have \(\pi _i \circ J=J\circ \pi _i\) where \(\pi _i:TM\rightarrow T\mathcal {G}_j\) (for \(j\ne i\) with \(i,j=1,2\)) denote the orthogonal projections. Next, using this and the classical equation for a Hermitian structure \(4g((\nabla _X J)Y,W)=6\mathrm{d}\Phi (X,JY,JW)-6\mathrm{d}\Phi (X,Y,W)\), after a straightforward calculation, we get

$$\begin{aligned} F(X,Y)&= [-\mathrm{d}\alpha _2(X,Y)-\mathrm{d}\alpha _1(X,JY)]Z_1 +[\mathrm{d}\alpha _1(X,Y)-\mathrm{d}\alpha _2 (X, JY)]Z_2 \\&\quad +\,\alpha _2(Y)\pi _1 JX-\alpha _1(Y) \pi _2 JX-\alpha _1(Y) \pi _1 X -\alpha _2 (Y) \pi _2 X. \end{aligned}$$

Any vector \(v\) tangent to \(M\) at a point of \(N\) decomposes as \(v=v^T+v^\bot \) where \(v^T\) and \(v^\bot \) are tangent and orthogonal to \(N\), respectively. The \(J\)-invariance implies that \(J(v^T)=(Jv)^T\) and \(J(v^\bot )=(Jv)^\bot \). Denote by \(\mathrm{B }\) the second fundamental form of the submanifold \(N\). Then, \( \mathrm{B }(X,JY)=J\mathrm{B }(X,Y)+F(X,Y)^\bot \) and \( \mathrm{B }(JX,JY)+\mathrm{B }(X,Y)=JF(Y,X)^\bot +F(JX,Y)^\bot \). Hence, we obtain

$$\begin{aligned} \mathrm{B }(X,X)+\mathrm{B }(JX,JX)&= -2||\pi _2 X ||^2 Z_1^\bot +2||\pi _1 X ||^2Z_2^\bot \\&\quad +\,2[-\alpha _1(X)\pi _1 JX-\alpha _2(X) \pi _2 JX-\alpha _2(X) \pi _1 X +\alpha _1 (X) \pi _2 X]^\bot \end{aligned}$$

Let \(N^\prime \) be the open set of \(N\) consisting of all points where \(Z_1^T\ne 0\). It is also a \(J\)-invariant submanifold of \(M\). Take an orthonormal (local) \(J\)-basis on \(N^\prime \)

$$\begin{aligned} e_1, J e_1, \ldots , e_n , J e_n. \end{aligned}$$

One can choose it in such a way that \(e_1=\frac{1}{||Z_1^T||}Z_1^T\) and then \(Je_1=\frac{1}{||Z_1^T||}Z_2^T\). Since the \(e_l\) and \(Je_l\), for \(l=2, \ldots ,n\) are orthogonal to \(Z_1^T\), \(Z_2^T\), \(Z_1^\bot \) and \(Z_2^\bot \), they are orthogonal to \(Z_1\) and \(Z_2\) too. So they are horizontal. Now, the normal mean curvature along \(N^\prime \) is \(H_{\vert N^\prime }=\frac{1}{2n}\mathrm{trace }(\mathrm{B })=\frac{1}{2n}\sum \limits _{l=1}^n\left( \mathrm{B }(e_l,e_l)+\mathrm{B }(Je_l,Je_l)\right) \), and becomes

$$\begin{aligned} H_{\vert N^\prime }=\frac{1}{n}\left( -\sum \limits _{l=1}^n ||\pi _2 e_l||^2 Z_1^\bot +\sum \limits _{l=1}^n ||\pi _1 e_l||^2 Z_2 ^\bot +(\pi _2Z_1^T -\pi _1 Z_2^T)^\bot \right) . \end{aligned}$$
(6)

If we suppose \(N\) minimal, then \(H_{\vert N^\prime }=0\) and the scalar products with \(Z_i^\bot \) yield

$$\begin{aligned} 0&= ng(H_{\vert N^\prime },Z_1^\bot )= -||\pi _2 Z_1^T||^2 - ||Z_1^\perp ||^2 \sum \limits _{l=1}^n ||\pi _2 e_l||^2 \\ 0&= ng(H_{\vert N^\prime },Z_2^\bot )= ||\pi _1 Z_2^T||^2 + ||Z_2^\perp ||^2 \sum \limits _{l=1}^n ||\pi _1 e_l||^2. \end{aligned}$$

Then, we get \(\pi _2 Z_1^T=0\) and \(\pi _1 Z_1^T=J\pi _1 Z_2^T=0\) along \(N^\prime \), which means that \(Z_i^T=Z_i\) at these points. Hence, \(Z_1\) and \(Z_2\) are tangent to \(N^\prime \). Now, we have to prove that \(N=N^\prime \). Every point \(p\) of \(N\) is in the closure of \(N^\prime \) in \(N\). For otherwise, there exists an open neighborhood \(U_p\) of \(p\) in \(N\), which does not intersect \(N^\prime \), i.e., \(Z_1\) and \(Z_2\) are orthogonal to the \(J\)-invariant (and also \(\phi \)-invariant) submanifold \(U_p\). But this contradicts Proposition 3. Now, since \(Z_1^T=Z_1\) on \(N^\prime \), by continuity of \(Z_1^T\), we have \((Z_1^T)_p=(Z_1)_p\) and then \(p\in N^\prime \). Hence, \(N=N^\prime \) so that \(Z_1\) and \(Z_2=JZ_1\) are tangent to \(N\).

Conversely, suppose that \(Z_1\) and \(Z_2\) are tangent to \(N\). Then, \(Z_i^T=(Z_i)_{\vert N}\), which implies that \(N=N^\prime \), and replacing in Eq. 6, we get \(H=0\). This completes the proof. \(\square \)

Remark 4

One could hope on a full generalization of the original Vaisman result, which could be stated as follows: a \(J\) and \(T\)-invariant submanifold of a normal MCP inherits the structure of normal MCP if and only if it is minimal. In fact, this kind of generalization is not possible, because the submanifold in Example 4 is both \(J\) and \(T\)-invariant, therefore it is minimal, but it does not inherit the normal MCP of the ambient manifold by Proposition 8.