The squared norm of the second fundamental form
ζ is defined by
$$\begin{aligned} \Vert \zeta \Vert ^{2} =& \sum_{i,j = 1}^{m}g \bigl(\zeta(\mathcal{E}_{i}, \mathcal{E}_{j}), \zeta( \mathcal{E}_{i}, \mathcal{E}_{j})\bigr) = \sum _{r = m + 1}^{2n + 1}\sum_{i,j = 1}^{m}g \bigl(\zeta(\mathcal{E}_{i}, \mathcal{E}_{j}), \mathcal{E}_{r}\bigr)^{2}. \end{aligned}$$
From the assumed frames, the above equation can be written as
$$ \begin{aligned}[b] \Vert \zeta \Vert ^{2} ={}& \sum _{r = m + 1}^{2n + 1}\sum_{i,j = 1}^{2a + 1}g \bigl(\zeta(\mathcal{E}_{i}, \mathcal{E}_{j}), \mathcal{E}_{r}\bigr)^{2} + 2\sum _{r = m + 1}^{2n + 1}\sum_{i = 1}^{2a + 1} \sum_{j = 1}^{2b}g\bigl(\zeta\bigl( \mathcal{E}_{i}, \mathcal{E}^{*}_{j}\bigr), \mathcal{E}_{r}\bigr)^{2} \\ &{} + \sum_{r = m + 1}^{2n + 1}\sum _{i,j = 1}^{2b}g\bigl(\zeta\bigl(\mathcal{E}^{*}_{i}, \mathcal{E}^{*}_{j}\bigr), \mathcal{E}_{r} \bigr)^{2}. \end{aligned} $$
(17)
Using the hypothesis and leaving the first term on the right-hand side of (
17) to introduce the inequality, we obtain
$$\begin{aligned} \Vert \zeta \Vert ^{2} \geq 2\sum_{r = m + 1}^{2n + 1} \sum_{i = 1}^{2a + 1}\sum _{j = 1}^{2b}g\bigl(\zeta\bigl(\mathcal{E}_{i}, \mathcal{E}^{*}_{j}\bigr), \mathcal{E}_{r} \bigr)^{2}. \end{aligned}$$
Decomposing the above equation according to (
5), we derive
$$ \begin{aligned}[b] \Vert \zeta \Vert ^{2}\geq{}& 2 \Biggl[\sum_{r = m + 1}^{2a + m}\sum _{i = 1}^{2a + 1}\sum_{j = 1}^{2b}g \bigl(\zeta\bigl(\mathcal{E}_{i}, \mathcal{E}^{*}_{j} \bigr), \tilde{\mathcal{E}}_{r}\bigr)^{2} \\ &{} + \sum_{r = 2a + m + 1}^{2b + 2a + m}\sum _{i = 1}^{2a + 1}\sum_{j = 1}^{2b}g \bigl(\zeta\bigl(\mathcal{E}_{i}, \mathcal{E}^{*}_{j} \bigr), \tilde{\mathcal{E}}_{r}\bigr)^{2} + \sum _{r = 2m}^{2n + 1}\sum_{i = 1}^{2a + 1} \sum_{j = 1}^{2b}g\bigl(\zeta\bigl( \mathcal{E}_{i}, \mathcal{E}^{*}_{j}\bigr), \tilde{\mathcal{E}}_{r}\bigr)^{2} \Biggr]. \end{aligned} $$
(18)
Removing all the terms except for
\(\mathcal{F}\mathcal{D}_{2}\)-components, we arrive at
$$ \begin{aligned} \Vert \zeta \Vert ^{2} &\geq 2 \Biggl(\sum_{r = 2a + m + 1}^{2b + 2a + m}\sum _{i = 1}^{2a + 1}\sum_{j = 1}^{2b}g \bigl(\zeta\bigl(\mathcal{E}_{i}, \mathcal{E}^{*}_{j} \bigr), \tilde{\mathcal{E}}_{r}\bigr)^{2} \Biggr), \\ \mbox{or}\quad \Vert \zeta \Vert ^{2} &\geq 2 \Biggl( \sum_{r = 1}^{2b}\sum _{i = 1}^{2a + 1}\sum_{j = 1}^{2b}g \bigl(\zeta\bigl(\mathcal{E}_{i}, \mathcal{E}^{*}_{j} \bigr), \tilde{\mathcal{E}}_{r}\bigr)^{2} \Biggr). \end{aligned} $$
(19)
Thus, by using the orthonormal frame fields of
\(\mathcal{D}_{1}, \mathcal{D}_{2}\) and
\(\mathcal{F}\mathcal{D}_{2}\), the above inequality reduces to
$$\begin{aligned} \Vert \zeta \Vert ^{2}\geq{}& 2 \Biggl[ \csc^{2} \vartheta_{2} \sum_{i = 1}^{a} \sum_{j = 1}^{b}g\bigl(\zeta\bigl( \mathcal{E}_{i}, \mathcal{E}^{*}_{j}\bigr), \mathcal{F}\mathcal{E}^{*}_{j}\bigr)^{2} \\ &{} + \csc^{2} \vartheta_{2} \sec^{2} \vartheta_{2} \sum_{i = 1}^{a}\sum _{j = 1}^{b}g\bigl(\zeta\bigl( \mathcal{E}_{i}, \mathcal{E}^{*}_{j}\bigr), \mathcal{F}\mathcal{P}\mathcal{E}^{*}_{j}\bigr)^{2} \\ &{} + \csc^{2} \vartheta_{2} \sec^{2} \vartheta_{2} \sum_{i = 1}^{a}\sum _{j = 1}^{b}g\bigl(\zeta\bigl( \mathcal{E}_{i}, \mathcal{P}\mathcal{E}^{*}_{j} \bigr), \mathcal{F}\mathcal{E}^{*}_{j}\bigr)^{2} \\ &{} + \csc^{2} \vartheta_{2} \sec^{2} \vartheta_{1} \sum_{i = 1}^{a}\sum _{j = 1}^{b}g\bigl(\zeta\bigl(P \mathcal{E}_{i}, \mathcal{E}^{*}_{j}\bigr), \mathcal{F}\mathcal{E}^{*}_{j}\bigr)^{2} \\ &{} + \csc^{2} \vartheta_{2} \sec^{2} \vartheta_{1} \sec^{2} \vartheta_{2} \sum _{i = 1}^{a}\sum_{j = 1}^{b}g \bigl(\zeta\bigl(\mathcal{P}\mathcal{E}_{i}, \mathcal{P} \mathcal{E}^{*}_{j}\bigr), \mathcal{F}\mathcal{E}^{*}_{j} \bigr)^{2} \\ &{} + \csc^{2} \vartheta_{2} \sec^{2} \vartheta_{1} \sec^{2} \vartheta_{2} \sum _{i = 1}^{a}\sum_{j = 1}^{b}g \bigl(\zeta\bigl(\mathcal{P}\mathcal{E}_{i}, \mathcal{E}^{*}_{j} \bigr), \mathcal{F}\mathcal{P}\mathcal{E}^{*}_{j} \bigr)^{2} \\ &{} + \csc^{2} \vartheta_{2} \sec^{4} \vartheta_{2} \sum_{i = 1}^{a}\sum _{j = 1}^{b}g\bigl(\zeta\bigl( \mathcal{E}_{i}, \mathcal{P}\mathcal{E}^{*}_{j} \bigr), \mathcal{F}\mathcal{P}\mathcal{E}^{*}_{j} \bigr)^{2} \\ &{} + \csc^{2} \vartheta_{2} \sec^{2} \vartheta_{1} \sec^{4} \vartheta_{2} \sum _{i = 1}^{a}\sum_{j = 1}^{b}g \bigl(\zeta\bigl(\mathcal{P}\mathcal{E}_{i}, \mathcal{P} \mathcal{E}^{*}_{j}\bigr), \mathcal{F}\mathcal{P} \mathcal{E}^{*}_{j}\bigr)^{2} \\ &{} + \csc^{2} \vartheta_{2} \sum _{j = 1}^{b}g\bigl(\zeta\bigl(\xi, \mathcal{E}^{*}_{j}\bigr), \mathcal{F}\mathcal{E}^{*}_{j} \bigr)^{2} \\ &{} + \csc^{2} \vartheta_{2} \sec^{2} \vartheta_{2} \sum_{j = 1}^{b}g \bigl(\zeta\bigl(\xi, \mathcal{E}^{*}_{j}\bigr), \mathcal{F} \mathcal{P}\mathcal{E}^{*}_{j}\bigr)^{2} \\ &{} + \csc^{2} \vartheta_{2} \sec^{2} \vartheta_{2} \sum_{j = 1}^{b}g \bigl(\zeta\bigl(\xi, P\mathcal{E}^{*}_{j}\bigr), \mathcal{F} \mathcal{E}^{*}_{j}\bigr)^{2} \\ &{} + \csc^{2} \vartheta_{2} \sec^{4} \vartheta_{2} \sum_{j = 1}^{b}g \bigl(\zeta\bigl(\xi, P\mathcal{E}^{*}_{j}\bigr), \mathcal{F} \mathcal{P}\mathcal{E}^{*}_{j}\bigr)^{2} \Biggr]. \end{aligned}$$
Using Lemma
2, the hypothesis, and the fact that
$$\begin{aligned} \Vert \nabla \ln h \Vert ^{2} = \sum_{i = 1}^{a} (\mathcal{E}_{i} \ln h)^{2} + \sum _{i = 1}^{a} \sec^{2} \vartheta_{1} (\mathcal{P}\mathcal{E}_{i} \ln h)^{2} + (\xi \ln h), \end{aligned}$$
we derive
$$ \begin{aligned} \Vert \zeta \Vert ^{2}\geq {}&4b \csc^{2} \vartheta_{2}\biggl(\cos^{2} \vartheta_{1} + \frac{1}{9} \cos^{2} \vartheta_{2}\biggr) \sum^{a}_{i = 1} \bigl[ \Vert \nabla \ln h \Vert ^{2} - 2\mu \upsilon( \mathcal{E}_{i}) (\mathcal{E}_{i} \ln h) \bigr] \\ &~{} + 4 b \csc^{2} \vartheta_{2} \lambda\Biggl(\lambda + \sum^{a}_{i = 1}\upsilon(\mathcal{E}_{i}) (\mathcal{P}\mathcal{E}_{i} \ln h)\Biggr). \end{aligned} $$
In view of the assumed orthonormal frame, the 1-form
\(\upsilon(\mathcal{E}_{i})\) is identically zero for all
\(i \in \{1, \dots, 2a\}\), the above expression can be modified as
$$\begin{aligned} \Vert \zeta \Vert ^{2} \geq& 4 b \csc^{2} \vartheta_{2} \biggl[\biggl(\cos^{2} \vartheta_{1} + \frac{1}{9} \cos^{2} \vartheta_{2}\biggr) \bigl( \Vert \nabla \ln h \Vert ^{2} - \mu^{2}\bigr) + \lambda^{2} \biggr]. \end{aligned}$$
This is the required inequality (i). Now, we discuss the following cases:
(a)
For
\(\lambda = 1\) and
\(\mu = 0\), we have
$$\begin{aligned} \Vert \zeta \Vert ^{2} \geq& 4b \csc^{2} \vartheta_{2} \biggl[\biggl(\cos^{2} \vartheta_{1} + \frac{1}{9} \cos^{2} \vartheta_{2}\biggr) \bigl( \Vert \nabla \ln h \Vert ^{2}\bigr) + 1 \biggr]. \end{aligned}$$
(b)
For
\(\lambda = 0\) and
\(\mu = 1\), we have
$$\begin{aligned} \Vert \zeta \Vert ^{2} \geq& 4b \csc^{2} \vartheta_{2} \biggl(\cos^{2} \vartheta_{1} + \frac{1}{9} \cos^{2} \vartheta_{2}\biggr) \bigl( \Vert \nabla \ln h \Vert ^{2} - 1\bigr). \end{aligned}$$
(c)
For
\(\lambda = 0\) and
\(\mu = 0\), we have
$$\begin{aligned} \Vert \zeta \Vert ^{2} \geq& 4b \csc^{2} \vartheta_{2} \biggl(\cos^{2} \vartheta_{1} + \frac{1}{9} \cos^{2} \vartheta_{2}\biggr) \bigl( \Vert \nabla \ln h \Vert ^{2}\bigr). \end{aligned}$$
If the equality holds in (
13), then from (
17) and the hypothesis of the theorem, we find that
$$\begin{aligned} \zeta(\mathcal{D}_{1}, \mathcal{D}_{1}) =& 0 \end{aligned}$$
(20)
and
$$\begin{aligned} \zeta(\mathcal{D}_{2}, \mathcal{D}_{2}) =& 0. \end{aligned}$$
(21)
Similarly, from (
18), we get
\(\zeta(\mathcal{D}_{1}, \mathcal{D}_{2}) \perp \mathcal{F}\mathcal{D}_{1}\) and
\(\zeta(\mathcal{D}_{1}, \mathcal{D}_{2}) \perp \nu\), which further give
$$\begin{aligned} \zeta(\mathcal{D}_{1}, \mathcal{D}_{2}) \subset& \mathcal{F}\mathcal{D}_{2}. \end{aligned}$$
(22)
Let
\(\zeta_{2}\) be the second fundamental form of
\(\mathcal{M}_{2}\) in
\(\mathcal{M}\), then for any
\(\mathcal{X} \in \Gamma(\mathcal{D}_{1})\) and
\(\mathcal{Z}, \mathcal{W} \in \Gamma(\mathcal{D}_{2})\), we have
\(g(\zeta_{2}(\mathcal{Z}, \mathcal{W}), \mathcal{X}) = g(\nabla_{\mathcal{Z}}\mathcal{W}, \mathcal{X}) = - g(\mathcal{W}, \nabla_{\mathcal{Z}}\mathcal{X})\). Thus, we derive from Lemma
1(ii) and (
6) that
\(g(\zeta_{2}(\mathcal{Z}, \mathcal{W}), \mathcal{X}) = - g(\nabla h, \mathcal{X}) g(\mathcal{Z}, \mathcal{W})\), or
$$\begin{aligned} \zeta_{2}(\mathcal{Z}, \mathcal{W}) =& - \nabla h g( \mathcal{Z}, \mathcal{W}). \end{aligned}$$
(23)
By (
20) and Lemma
1(i) (
\(\mathcal{M}_{1}\) is totally geodesic in
\(\mathcal{M}\)), we conclude that
\(\mathcal{M}_{1}\) is totally geodesic in
\(\overline{\mathcal{M}}\). On the other hand, both (
21) and (
23) say that
\(\mathcal{M}_{2}\) is totally umbilical in
\(\overline{\mathcal{M}}\). Furthermore, relations (
20), (
21) and (
22) imply that
\(\mathcal{M}\) is a minimal submanifold of
\(\overline{\mathcal{M}}\). Hence, our assertion (ii) is proved. □