2. Preliminaries

We shall now present the main tools we use in this work.

Let M be a $(n+p)$ -dimensional manifold equipped with a Riemannian metric $\langle \, , \rangle$ . Denote by $\nabla$ the Levi-Civita connection and by R the curvature tensor of M. Let $\mathfrak{F}$ be a foliation of codimension p on M and $\mathfrak{F}^{\bot}$ be an orthogonal foliation to $\mathfrak{F}$ . Consider an adapted frame $\{e_1,...,e_n,e_{n+1},...,e_{n+p}\}$ in a neighborhood of a point $x\in M$ . We shall make use of the following convention on the range of indices

\[1\leq A,B,C \leq n+p,\]

\[\ 1\leq i,j,k\leq n,\]

\[n+1\leq \alpha, \beta, \gamma \leq n+p.\]

Let L be a submanifold of M and let $A_{\nu}$ be the shape operator of L defined by an arbitrary normal vector $\nu$ . Note that $A_{\nu}$ and II are related by

\[\langle II(v, w),\nu \rangle=\langle A_{\nu}(v),w\rangle .\]

The submanifold L is said to be totally umbilical if the second fundamental form is $II=\vec{h}\langle \cdot,\cdot\rangle$ which is equivalent to say that the shape operators are always multiple of the identity I. In this case, for each normal vector $\nu$ , $A_{\nu}=\xi_{\nu}I$ , where $\xi_{\nu}=\langle \vec{h} , \nu\rangle$ .

We define the second fundamental form of $\mathfrak{F}$ in the direction $e_{\alpha}$ by

\begin{equation*} H_{\mathfrak{F}}^{\alpha}(e_{i},e_{j})=\langle-\nabla_{e_{i}}e_{\alpha},e_j\rangle.\end{equation*}

Analogously, we define the second fundamental form of $\mathfrak{F}^{\bot}$ in the direction of $e_i$ by

\begin{align*} H_{\mathfrak{F}^{\bot}}^{i}(e_{\alpha},e_{\beta})=\langle -\nabla_{e_{\alpha}}e_{i},e_{\beta}\rangle. \end{align*}

The Weingarten operators of $H_{\mathfrak{F}}^{\alpha}$ and $H_{\mathfrak{F}^{\bot}}^i$ are given, respectively, by

\[A_{e_{\alpha}}(e_i)=-\left( \nabla_{e_i}e_{\alpha}\right)^{\top} \ \ \ \text{and} \ \ \ A_ {e_{\alpha}}(e_i)=-\left( \nabla_{e_{\alpha}}e_{i}\right)^{\bot}.\]

We define the mean curvature vector of $\mathfrak{F}$ and $\mathfrak{F}^{\bot}$ , respectively, by

\[\vec{h}=\sum_{i}{\!\left(\nabla_{e_i}e_{i} \right)^{\bot}} \ \ \ \text{and} \ \ \ \vec{h}^{\bot}=\sum_{\alpha}{\left(\nabla_{e_{\alpha}}e_{\alpha} \right)^{\top}},\]

and finally, the mean curvature function in direction $e_{\alpha}$ is defined by

\[\mathcal{H}_{\mathfrak{F}}^{\alpha}=\frac{1}{n}\langle \vec{h},e_{\alpha} \rangle.\]

We define the norms of the second fundamental form $H_{\mathfrak{F}}^{\alpha}$ and $H_{\mathfrak{F}^{\bot}}^{i}$ of $\mathfrak{F}$ and $\mathfrak{F}^{\bot}$ , respectively, by

\[\|H_ {\mathfrak{F}}^{\alpha}\|=\left(\sum_{i,j}{\langle -\nabla_{e_i}e_{\alpha}, e_j\rangle^2 }\right)^{1/2}\]

and

\[\|H_ {\mathfrak{F}^{\bot}}^{i}\|=\left(\sum_{\alpha,\beta}{\langle -\nabla_{e_{\alpha}}e_{i}, e_{\beta}\rangle^2 }\right)^{1/2}.\]

We say that $\mathfrak{F}^{\bot}$ is totally umbilical if the mean vector field $\vec{h}^{\bot}$ is

\[\vec{h}^{\bot}=\sum_{i}{\lambda(x,e_i)e_i},\]

where $\lambda \,:\,T\mathfrak{F}^{\bot} \rightarrow \mathbb{R}$ is the eigenvalue of the Weingarten operator

\[A^v \,:\, T\mathfrak{F}^{\bot}\rightarrow T\mathfrak{F}^{\bot}\]

defined in the smooth field $T\mathfrak{F}^{\bot}$ of p-planes tangent to the leaves of $T\mathfrak{F}^{\bot}$ . Note that $\lambda(x,v)=\langle \vec{h}^{\bot},v\rangle$ .

For each fixed $\alpha$ , we denote by $(K_{ij}^{\alpha})$ the $n\times n$ matrix with entries given by $R(e_{\alpha}, e_i, e_j, e_{\alpha})$ . The trace of the matrix $(K_{ij}^{\alpha})$ is then given by

\[\text{tr}(K_{ij}^{\alpha})=\sum_{i}{R(e_{\alpha}, e_i, e_i, e_{\alpha})}.\]

The following theorem was proved in [Reference Gomes and Silva11] where the authors find an differential equation that relates the foliations with the ambient manifold.

Proposition 2.1. Let $\mathfrak{F}$ and $\mathfrak{F}^{\bot}$ be complementary foliations of dimensions n and p, respectively, on a $(n+p)$ -dimensional Riemannian manifold. Then, for all $\alpha$

\begin{align*}e_{\alpha} \left\langle \vec{h},e_{\alpha}\right\rangle-\left\|H_{\mathfrak{F}}^{\alpha}\right\|^2-{tr}\!\left(K_{ij}^{\alpha}\right)=\sum\limits_{i=1}^{n}H_{\mathfrak{F}^{\bot}}^{i}\!\left(e_{\alpha},\nabla^{\bot}_{e_{i}}e_{\alpha}-[e_{\alpha},e_{i}]^{\bot}\right)-\mathrm{div}_{\mathfrak{F}}\!\left(\nabla_{e_{\alpha}}e_{\alpha}\right)\end{align*}

where

1. (i) $\vec{h}$ represents the mean curvature vector of $\mathfrak{F}$ ;

2. (ii) $\|H_{\mathfrak{F}}^{\alpha}\|$ is the norm of the second fundamental form of $\mathfrak{F}$ ;

3. (iii) $\mathrm{div}_{\mathfrak{F}}(\nabla_{e_{\alpha}}e_{\alpha})=\sum_{i=1}^{n}\langle e_i,\nabla_{e_i}\nabla_{e_{\alpha}}e_{\alpha}\rangle$ .

Corollary 2.1. Let $\mathfrak{F}$ and $\mathfrak{F}^{\bot}$ be two orthogonal foliations of complementary dimensions on a Riemannian manifold M. Suppose that $\mathfrak{F}^{\bot}$ is totally umbilical. Then,

\begin{align*} n\left\langle \nabla_{e_{\alpha}} \vec{h},e_{\alpha}\right\rangle-\left\|H_{\mathfrak{F}}^{\alpha}\right\|^2-{tr}\!\left(K_{ij}^{\alpha}\right)-\left\|\vec{h}^{\bot}\right\|^{2}+\mathrm{div}_{\mathfrak{F}}\vec{h}^{\bot}=0. \end{align*}

Using the Cauchy-Schwarz inequality, it follows that:

Proposition 2.2. Let $\mathfrak{F}$ and $\mathfrak{F}^{\bot}$ be complementary foliations of dimensions n and p, respectively, on a $(n+p)$ -dimensional Riemannian manifold. Then,

\begin{eqnarray*}e_{\alpha}\!\left(\mathcal{H}_{\mathfrak{F}}^{\alpha}\right) & \geq & \frac{1}{n}\sum_{i=1}^n{H_{\mathfrak{F}^{\bot}}^{i}\!\left(e_{\alpha},\nabla^{\bot}_{e_{i}}e_{\alpha}-[e_{\alpha},e_{i}]^{\bot}\right)}+\frac{1}{n}{tr}\!\left(K_{ij}^{\alpha}\right)\\&&- \frac{1}{n}\mathrm{div}_{\mathfrak{F}}\!\left(\nabla_{e_{\alpha}}e_{\alpha}\right)+\left(\mathcal{H}_{\mathfrak{F}}^{\alpha}\right)^2,\end{eqnarray*}

where $n+1 \leq \alpha \leq n+p$ .

3. Characterization of totally geodesic foliations

Let us analyze orthogonal foliations of complementary dimensions, $\mathfrak{F}$ and $\mathfrak{F}^{\bot}$ defined on a complete Riemannian manifold M. We say that $\mathfrak{F}$ is transversely parallelizable (or $\mathfrak{F}^{\bot}$ is parallelizable) if its normal bundle is trivial, that is, there is a global frame $\{e_{n+1},\cdots,e_{n+p}\}$ on M (for more details see [Reference Moerdijk and Mrcun13]). Now define the number $\mathfrak{G}_{\mathfrak{F}}^{\alpha}$ for all $\alpha\in \{n+1,\cdots,n+p\}$ by

\begin{equation*} \mathfrak{G}_{\mathfrak{F}}^{\alpha} = \inf_{M} \left\{ \frac{1}{n}\sum_{i=1}^n{H_{\mathfrak{F}^{\bot}}^{i}}\!\left(e_{\alpha},\nabla^{\bot}_{e_{i}}e_{\alpha}-[e_{\alpha},e_{i}]^{\bot}\right)+\frac{1}{n}\text{tr}\!\left(K_{ij}^{\alpha}\right)-\frac{1}{n} \mathrm{div}_{\mathfrak{F}}\!\left(\nabla_{e_{\alpha}}e_{\alpha}\right)\right\},\end{equation*}

where $\{e_1,\cdots,e_n,e_{n+1}, \cdots,e_{n+p}\}$ is an adapted frame in a neighborhood of a point. With this number, we will estimate the mean curvature function and give a characterization of the totally geodesic foliations of complementary dimensions on a complete Riemannian manifold M. In fact, we generalize for arbitrary codimension, Theorem 4.2 in [Reference Chaves and da Silva7] and Theorem 2.4 in [Reference Gomes10], following the steps of the proof of the latter and using $\mathfrak{G}_{\mathfrak{F}}^{\alpha}$ .

Theorem 3.1. Let $\mathfrak{F}$ and $\mathfrak{F}^{\bot}$ be orthogonal foliations of complementary dimensions on a complete Riemannian manifolds M. If $\mathfrak{F}^{\bot}$ is parallelizable then, for all $\alpha$

(3.1) \begin{equation}\mathfrak{G}_{\mathfrak{F}}^{\alpha} \leq 0,\end{equation}

(3.2) \begin{equation} \left(\mathcal{H}_{\mathfrak{F}}^{\alpha}\right)^2 \leq -\mathfrak{G}_{\mathfrak{F}}^{\alpha}, \end{equation}

and

\[\mathfrak{F} \ {is\ totally\ geodesic} \Leftrightarrow \mathfrak{G}_{\mathfrak{F}}^{\alpha}=0.\]

Proof. We will use a proof by contradiction. So we assume that

\[\frac{1}{n}\sum_{i=1}^n{H_{\mathfrak{F}^{\bot}}^{i}}\!\left(e_{\alpha},\nabla^{\bot}_{e_{i}}e_{\alpha}-[e_{\alpha},e_{i}]^{\bot}\right)+\frac{1}{n}\sum_{i=1}^n{R\!\left(e_{\alpha},e_i,e_i,e_{\alpha}\right)}-\frac{1}{n} \mathrm{div}_{\mathfrak{F}}\!\left(\nabla_{e_{\alpha}}e_{\alpha}\right)>0,\]

on M. By Proposition 2.2, we have that $d\mathcal{H}_{\mathfrak{F}}^{\alpha}(e_{\alpha})>\left(\mathcal{H}_{\mathfrak{F}}^{\alpha}\right)^2$ holds on M. Let $\gamma(s)$ be an integral curve of the unit vector field $e_{\alpha}$ . Since M is complete, $\gamma$ may be extended to all $\mathbb{R}$ . Thus, along $\gamma$ , the above inequality has the form

\[\left(\mathcal{H}_{\gamma(s)}^{\alpha}\right)^{\prime}>\left(\mathcal{H}_{\gamma(s)}^{\alpha}\right)^{2} \ \ \ \forall s\in \mathbb{R}.\]

We can choose the field $e_{\alpha}$ of such a way that $\mathcal{H}_{\gamma(0)}\geq 0$ . Note that such choose does not change the expression

\[\frac{1}{n}\sum_{i=1}^n{H_{\mathfrak{F}^{\bot}}^{i}}\!\left(e_{\alpha},\nabla^{\bot}_{e_{i}}e_{\alpha}-[e_{\alpha},e_{i}]^{\bot}\right)+\frac{1}{n}\sum_{i=1}^n{R\!\left(e_{\alpha},e_i,e_i,e_{\alpha}\right)}-\frac{1}{n} \mathrm{div}_{\mathfrak{F}}\!\left(\nabla_{e_{\alpha}}e_{\alpha}\right).\]

Therefore, the following inequalities are valid for every $s>0$ ,

(3.3) \begin{equation}\left(\mathcal{H}_{\gamma(s)}^{\alpha}\right)^{\prime}>\left(\mathcal{H}_{\gamma(s)}^{\alpha}\right)^2>0 \ \ \text{ and } \ \ \frac{\left(\mathcal{H}_{\gamma(s)}^{\alpha}\right)^{\prime}}{\left(\mathcal{H}_{\gamma(s)}^{\alpha}\right)^2}>1.\end{equation}

Now consider the real function G given by

\[G(s)=-\frac{1}{\mathcal{H}_{\gamma(s)}^{\alpha}}, \ \ s>0.\]

Take a fixed $b>0$ and apply the Mean Value Theorem for the function G on the interval [b, s], we obtain

\[ -\frac{1}{\mathcal{H}_{\gamma(s)}^{\alpha}}+\frac{1}{\mathcal{H}_{\gamma(b)}^{\alpha}}=\frac{\left(\mathcal{H}_{\gamma(\xi)}^{\alpha}\right)^{\prime}}{\left(\mathcal{H}_{\gamma(\xi)}^{\alpha}\right)^2}(s-b),\]

where $\xi\in (b,s)$ . Consequently, for all $s>b$ , we have

\[ -\frac{1}{\mathcal{H}_{\gamma(s)}^{\alpha}}+\frac{1}{\mathcal{H}_{\gamma(b)}^{\alpha}}>s-b.\]

As s tends to infinity, the right side of this inequality is unlimited, while the left side is limited, which is a contradiction. This Proves (3.1).

To prove (3.2), suppose by contradiction that there exists a point $p\in M$ such that,

\[\left(\mathcal{H}_{p}^{\alpha}\right)^2 > - \mathfrak{G}_{\mathfrak{F}}^{\alpha}.\]

If $\mathfrak{G}_{\mathfrak{F}}^{\alpha}=-\infty,$ then there is nothing to be proved. By (3.1), for some $a\geq 0$ we have

\[\mathfrak{G}_{\mathfrak{F}}^{\alpha}=-a^2,\]

where $a=0$ or $a>0$ . If $a>0$ , we have by hypothesis that, $\left(\mathcal{H}_p^{\alpha}\right)^{2}-a^2>0.$ Let $\gamma$ be an integral curve of $e_{\alpha}$ such that $\gamma(0)=p$ . As previously we can choose a direction, $e_{\alpha}$ such that $\mathcal{H}_p^{\alpha}=\mathcal{H}_{\gamma(0)}^{\alpha}\geq 0$ and then,

\[\mathcal{H}_p^{\alpha}=\mathcal{H}_{\gamma(0)}^{\alpha}>a.\]

By continuity, there exists a maximal interval [0, b) where

\[\left(\mathcal{H}_p^{\alpha}\right)^{2}-a^2>0 \ \ \ \forall s\in [0,b).\]

We claim that $b= +\infty$ .

In fact, if $b<+\infty$ , by continuity we should have $(\mathcal{H}_{\gamma(b)}^{\alpha})^{2}=a^2$ . But, from proposition (2.2), we have:

\[\left(\mathcal{H}_{\gamma(s)}^{\alpha}\right)^{\prime} \geq \left(\mathcal{H}_{\gamma(s)}^{\alpha}\right)^{2}-a^2 > 0, \ \ \ \forall s\in [0,b).\]

Thus, we conclude that $\mathcal{H}_{\gamma(s)}^{\alpha}$ is a strictly increasing function in [0, b], contradiction. Therefore, the following inequality are valid for every $s>0$ ,

\[\mathcal{H}_{\gamma(s)}^{\alpha}>0, \ \ \left(\mathcal{H}_{\gamma(s)}^{\alpha}\right)^{\prime}\geq \left(\mathcal{H}_{\gamma(s)}^{\alpha}\right)^2-a^2>0 \ \text{ and } \ \frac{\left(\mathcal{H}_{\gamma(s)}^{\alpha}\right)^{\prime}}{\left(\mathcal{H}_{\gamma(s)}^{\alpha}\right)^2-a^2}\geq 1.\]

Considering the function G defined by

\[G(s)=\frac{1}{2a}\ln\left(\frac{\mathcal{H}_{\gamma(s)}^{\alpha}-a}{\mathcal{H}_{\gamma(s)}^{\alpha}+a}\right), \ \ s>0.\]

For $b>0$ fixed and by the Mean Value Theorem, we have that there exists $c\in [b,s]$ such that

\[\frac{1}{2a}\ln\left(\frac{\mathcal{H}_{\gamma(s)}^{\alpha}-a}{\mathcal{H}_{\gamma(s)}^{\alpha}+a}\right)-\frac{1}{2a}\ln\left(\frac{\mathcal{H}_{\gamma(b)}^{\alpha}-a}{\mathcal{H}_{\gamma(b)}^{\alpha}+a}\right)=\frac{\left(\mathcal{H}_{\gamma(c)}^{\alpha}\right)^{\prime}}{\left(\mathcal{H}_{\gamma(c)}^{\alpha}\right)^2-a^2}(s-b).\]

Consequently, for all $s>b$ , we have

\[\frac{1}{2a}\ln\left(\frac{\mathcal{H}_{\gamma(s)}^{\alpha}-a}{\mathcal{H}_{\gamma(s)}^{\alpha}+a}\right)-\frac{1}{2a}\ln\left(\frac{\mathcal{H}_{\gamma(b)}^{\alpha}-a}{\mathcal{H}_{\gamma(b)}^{\alpha}+a}\right)\geq s-b.\]

Letting $s\longrightarrow +\infty$ we have a contradiction, because the left side is limited while the right side is unlimited. The case $a=0$ is similar.

Finally, suppose

\[\mathfrak{G}_{\mathfrak{F}}^{\alpha}=0.\]

It follows from Theorem 3.1 that

\[\left(\mathcal{H}_{\mathfrak{F}}^{\alpha}\right)^2\leq -\mathfrak{G}_{\mathfrak{F}}^{\alpha}=0.\]

Therefore, $\mathcal{H}^{\alpha}_{\mathfrak{F}}\equiv 0.$ By Proposition 2.2,

\[0\geq \frac{1}{n}\sum_{i=1}^n{H_{\mathfrak{F}^{\bot}}^{i}}\!\left(e_{\alpha},\nabla^{\bot}_{e_{i}}e_{\alpha}-[e_{\alpha},e_{i}]^{\bot}\right)+\frac{1}{n}\sum_{i=1}^n{R\!\left(e_{\alpha},e_i,e_i,e_{\alpha}\right)}-\frac{1}{n} \mathrm{div}_{\mathfrak{F}}\!\left(\nabla_{e_{\alpha}}e_{\alpha}\right).\]

Thus,

\begin{eqnarray*} 0 & = & \frac{1}{n}\sum_{i=1}^n{H_{\mathfrak{F}^{\bot}}^{i}}\!\left(e_{\alpha},\nabla^{\bot}_{e_{i}}e_{\alpha}-[e_{\alpha},e_{i}]^{\bot}\right)+\frac{1}{n}\sum_{i=1}^n{R\!\left(e_{\alpha},e_i,e_i,e_{\alpha}\right)}-\frac{1}{n} \mathrm{div}_{\mathfrak{F}}\!\left(\nabla_{e_{\alpha}}e_{\alpha}\right),\end{eqnarray*}

on M. By using the Proposition 2.1, we conclude that $\left\|H_{\mathfrak{F}}^{\alpha}\right\|=0$ , i.e., $\mathfrak{F}$ is a totally geodesic foliation. The converse follows from Proposition 2.1.

Proof. Since M has constant sectional curvature, it follows that $\text{tr}(K_{ij}^{\alpha})=n c$ for all $n+1 \leq \alpha \leq n+p$ . Then,

\begin{align*} k \leq \frac{1}{n}\sum_{i=1}^n\left\{{\left(\lambda(x,e_{i})\right)^2}+\frac{1}{n}\text{tr}\!\left(K_{ij}^{\alpha}\right)-\frac{1}{n} \mathrm{div}_{\mathfrak{F}}\!\left(\nabla_{e_{\alpha}}e_{\alpha}\right)\right\},\end{align*}

where in the last identity we use the fact that $\mathfrak{F}^{\bot}$ is umbilical and the vector field $\nabla_{e_{\alpha}}e_{\alpha}$ is free divergent. Consequently $k\leq \mathfrak{G}_{\mathfrak{F}}^{\alpha}$ .

4. Some Applications of the maximum principle on umbilical and minimal foliations

Suppose that one of the foliation is totally umbilical, as in [Reference Gomes and Silva11] and [Reference Almeida, Brito and Colares2] on a complete Riemannian manifold M.

From now on, we assume that $\mathfrak{G}_{\mathfrak{F}}^{\alpha}$ is finite. Then, we use some well-known results of the geometric analysis to get further results for umbilical foliations with integrable normal bundle. First, let us apply the maximum principle due to Yau [Reference Yau22].

1. (1) $\displaystyle{\lim_{k\to \infty}\mathcal{H}_{\mathfrak{F}}^{\alpha}(p_k)}=\sup_{M} \mathcal{H}_{\mathfrak{F}}^{\alpha}$ ;

2. (2) $\displaystyle{\lim_{k\to \infty}\|\nabla \mathcal{H}_{\mathfrak{F}}^{\alpha}(p_k)\|}=0$ ;

3. (3) $\displaystyle{\lim_{k\to \infty}\Delta \mathcal{H}_{\mathfrak{F}}^{\alpha}(p_k)}\leq 0$ .

Proof. From Theorem 3.1, it follows that $\left( \mathcal{H}_{\mathfrak{F}}^{\alpha} \right)^2$ is bounded, since $\mathfrak{G}_{\mathfrak{F}}$ is finite. By Yau [Reference Yau22], we finish the proof.

Denote by $\mathcal{L}^1(N)$ the space of Lebesgue integrable functions on a manifold N. In [Reference Chaves and da Silva7], the authors give the definition of Lebesgue integrable vector fields on foliations. Precisely, a vector field X is Lebesque integrable on a foliation $\mathfrak{F}$ , if and only if $\|X^{\top}\|\in \mathcal{L}^1(L)$ for each leaf L of $\mathfrak{F}$ , where $X^{\top}$ means the tangent projection of X, on L. In this case, we will denote $\|X\|\in \mathcal{L}^1(\mathfrak{F})$ .

The following result gives a sufficient condition for an umbilical foliation with integrable normal bundle on an orientable Riemannian manifold to be totally geodesic.

Theorem 4.1. Let $\mathfrak{F}$ and $\mathfrak{F}^{\bot}$ be two orthogonal foliations of complementary dimensions on a complete Riemannian manifold M. Suppose that $\mathfrak{F}^{\bot}$ is parallelizable and totally umbilical. If $\mathfrak{F}$ has a complete leaf L such that ${tr}(K_{ij}^{\alpha})\geq 0$ on L and the mean curvature of L is constant and satisfies

\[\mathcal{H}_{L}^{\alpha}=\left(-\mathfrak{G}_{\mathfrak{F}}^{\alpha}\right)^{\frac{1}{2}},\]

then $\mathfrak{F}$ is totally geodesic.

Proof. We will do the proof into two steps: the compact case and the complete non-compact case.

Compact case: Let $p\in M$ and $\{e_{1},...,e_{n},e_{n+1},...,e_{n+p}\}$ be a orthonormal adapted frame, in a neighborhood of p. If $\vec{h}(p)\neq 0$ , choose $e_{\alpha}$ as the mean curvature vector normalized, that is, $e_{\alpha}=\dfrac{\vec{h}}{\|\vec{h}\|}$ .

By Theorem 3.1 the mean curvature function $\mathcal{H}_{\mathfrak{F}}^{\alpha}$ attains a maximum on L. Therefore, $d\mathcal{H}_{L}^{\alpha}(e_{\alpha})=0$ on L. Therefore by Corollary 2.1 we have

\begin{equation*}\mathrm{div}_{L}\vec{h}^{\bot}=\left\|H_{L}^{\alpha}\right\|^2+\text{tr}\!\left(K_{ij}^{\alpha}\right)+\big\|\vec{h}^{\bot}\big\|^{2}.\end{equation*}

on L. By the Stokes Theorem and from the fact that $\text{tr}(K_{ij}^{\alpha})\geq 0$ on L, we conclude $H_{L}^{\alpha}=0$ on L. Therefore, L is totally geodesic and

\[\mathfrak{G}_{\mathfrak{F}}^{\alpha}=0.\]

By Theorem 3.1 we conclude the result.

Now if p is a singularity of the field $\vec{h}^{\bot}$ , using the same equation (2.1), we get the equation for the leaf that passes through p say $L_p$ and the same argument is valid.

Complete and non-compact case: Let $p\in M$ and $\{e_{1},\dots,e_{n},e_{n+1},\dots,e_{n+p}\}$ be a orthonormal adapted frame, in a neighborhood of p. If $\vec{h}(p)\neq 0$ choose $e_{\alpha}$ as the mean curvature vector normalized, that is, $e_{\alpha}=\dfrac{\vec{h}}{\|\vec{h}\|}$ .

By Theorem 3.1, the mean curvature function of L attains its maximum on L. Thus, $d\mathcal{H}_{L}^{\alpha}(e_{\alpha})=0$ on L. By Corollary 2.1 we have the equation

\begin{equation*}\mathrm{div}_{L}\vec{h}^{\bot}=\left\|H_{L}^{\alpha}\right\|^2+\text{tr}(K_{ij}^{\alpha})+\big\|\vec{h}^{\bot}\big\|^{2},\end{equation*}

on L. Since L is a complete non-compact and oriented leaf of $\mathfrak{F}$ and $\mathrm{div}_{L}\vec{h}^{\bot}$ does not change signal on L, we can apply Proposition 1 in [Reference Camargo, Caminha and Sousa6] to obtain

\begin{equation*}0=\mathrm{div}_{L}\vec{h}^{\bot}= \left\|H_{L}^{\alpha}\right\|^2+\text{tr}(K_{ij}^{\alpha})+\big\|\vec{h}^{\bot}\big\|^{2}.\end{equation*}

By the same argument above, we conclude that $H_{L}^{^\alpha}=0$ on L. Therefore, L is totally geodesic and

\[\mathfrak{G}_{\mathfrak{F}}^{\alpha}=0.\]

Again by Theorem 3.1, the result follows.

Proof. The proof follows directly from Hopf-Calabi’s maximum principle in [Reference Calabi5] and the fact that $H_{\mathfrak{F}}^{\alpha}(p)=\left(-\mathfrak{G}_{\mathfrak{F}}^{\alpha}\right)^{\frac{1}{2}}$ .

5. Umbilical foliations with integrable normal bundle

Proof. Since $f\,:\,M\longrightarrow {\mathbb{R}}$ is continuous defined on a compact Riemannian manifold M, then, there is a maximum and a minimum point. Let $x\in M$ be a point of maximum of f. Denote by $L_x$ the leaf through x. As f is constant along $L_x$ , we have that $f(y)=\max_{M}\{f(x)\}$ for any $y\in L_{x}$ . Therefore, $L_{x}\subset A$ .

Proof. Let $p\in M$ and $\{e_{1},...,e_{n},e_{n+1},...,e_{n+p}\}$ an orthonormal adapted frame in a neighborhood of p. If $\vec{h}(p)\neq 0$ choose $e_{\alpha}$ as the normalized mean curvature vector, that is, $e_{\alpha}=\frac{\vec{h}}{\|\vec{h}\|}$ . The mean curvature function $\mathcal{H}_{\mathfrak{F}}^{\alpha}\,:\,M\rightarrow \mathbb{R}$ is constant on M or there exists a leaf $L\in \mathfrak{F}$ having the property that

(5.1) \begin{equation} \mathcal{H}_{L}^{\alpha}=\max_{M}{\mathcal{H}_{\mathfrak{F}}^{\alpha}(x)}. \end{equation}

First, assume that $\mathcal{H}_{\mathfrak{F}}^{\alpha}$ is not constant on M. This implies that $e_{\alpha}(\mathcal{H}_{L}^{\alpha})=0$ along L. Then, it follows from Corollary 2.1 that

(5.2) \begin{equation} ne_{\alpha}(\mathcal{H}_{L}^{\alpha})=\left\|H_{L}^{\alpha}\right\|^2+\text{tr}(K_{ij}^{\alpha})+\big\|\vec{h}^{\bot}\big\|^{2}-\mathrm{div}_{L}\vec{h}^{\bot}. \end{equation}

Since $tr(K_{ij}^{\alpha})-\mathrm{div}_{\mathcal{F}}\vec{h}^{\bot}\geq 0$ , we have:

\[\left\|H_{L}^{\alpha}\right\|^2 \leq 0.\]

Therefore, L is totally geodesic, in particular, $\mathcal{H}_{L}^{\alpha}=0$ . From (5.1), we conclude that $\mathcal{H}_{\mathfrak{F}}^{\alpha}\leq 0$ .

Similarly, considering the function $(-\mathcal{H}_{\mathfrak{F}}^{\alpha})$ we conclude $\mathcal{H}_{\mathfrak{F}}^{\alpha}\geq 0$ . Therefore, $\mathcal{H}_{\mathfrak{F}}^{\alpha}=0$ . Contradiction. Hence, $\mathcal{H}_{\mathfrak{F}}^{\alpha}$ is constant along M. Since $e_{\alpha}(\mathcal{H}_{\mathfrak{F}}^{\alpha})=0$ , it follows from (5.2) that

(5.3) \begin{equation} \mathrm{div}_{\mathfrak{F}}\vec{h}^{\bot}= \left\|H_{\mathfrak{F}}^{\alpha}\right\|^2+\text{tr}(K_{ij}^{\alpha})+\big\|\vec{h}^{\bot}\big\|^{2}. \end{equation}

If L is compact, applying Stokes’ theorem we obtain

\[H_{\mathfrak{F}}^{\alpha}\equiv 0, \text{tr}(K_{ij}^{\alpha})\equiv 0 \ \ \text{and} \ \ \vec{h}^{\bot}\equiv 0.\]

If L is complete and non-compact, note that $\mathrm{div}_{L}\vec{h}^{\bot}$ does not change sign and since $\big\|\vec{h}^{\bot}\big\|\in \mathcal{L}^{1}(\mathfrak{F})$ then $\mathrm{div}_{L}{\vec{h}^{\bot}}=0$ . Therefore by equation (5.3), we have

\[H_{\mathfrak{F}}^{\alpha}\equiv 0.\]

Finally if p is a singularity of the field $\vec{h}^{\bot}$ , using (2.1) we get the equation for the leaf that passes through p and the same argument is valid.

As an immediate consequence of our results, we obtain a geometric version of Novikov’s theorem [Reference Novikov16].

Since this holds for compact manifolds with no restriction on dimension, the above is a generalization of the result for odd-dimensional unit spheres, proved in [Reference Almeida, Brito and Colares2].