We begin this section with the following definition.
Definition 4
Let \((M,g,S(TM))\) be an \((n+r)\)-dimensional r-lightlike submanifold of a semi-Riemannian manifold and \(S(TM)\) be an integrable distribution of index q. The bounded screen Ricci tensor, denoted by \(\operatorname{Ric}_{S(TM)}\), is defined by
$$ \operatorname{Ric}_{S(TM)}(X,Y)=\operatorname{tr} \bigl\{ Z \rightarrow R(X,Z)Y \bigr\} $$
(34)
for any \(X,Y\in\Gamma(S(TM))\).
Suppose \(\{e_{1},\ldots,e_{n}\}\) be an orthonormal basis of \(\Gamma(S(TM))\). The bounded screen Ricci curvature at a unit vector \(e_{i}\in\Gamma(S(TM))\), denoted by \(\operatorname{Ric}_{S(TM)}(e_{i})\), is given by
$$ \operatorname{Ric}_{S(TM)}(e_{i}) = \sum _{j\neq i =1}^{n}R(e_{i},e_{j},e_{j},e_{i})= \sum_{j\neq i =1}^{n}K_{ij}. $$
(35)
We note that:
-
(a)
If \(n=1\), then the bounded screen Ricci curvature vanishes identically.
-
(b)
If \(n=2\), then the bounded screen Ricci curvature becomes the bounded sectional curvature.
Remark 1
We note that the screen Ricci curvature is bounded when the screen distribution of a lightlike submanifold is Riemannian. This map was first of all introduced by Duggal in [55] and named by the authors in [56, 57] in the case of a lightlike hypersurface of a Lorentzian manifold in which we know that \(S(TM)\) is Riemannian.
Theorem 4
Let
\((M,g,S(TM))\)
be an
\((r+3)\)-dimensional
r-lightlike submanifold of a semi-Riemannian manifold and
\(S(TM)\)
be an integrable distribution. The bounded screen Ricci curvature is constant at every unit vector on
\(\Gamma(S(TM))\)
if and only if the bounded sectional curvature is constant.
Proof
Let \(\{e_{1},e_{2},e_{3}\}\) be an orthonormal basis of \(\Gamma(S(TM))\). If \(\operatorname{Ric}_{S(TM)}\) is constant, then we can write
$$\begin{aligned}& \operatorname{Ric}_{S(TM)}(e_{1})=K_{12}+K_{13}= \lambda, \\& \operatorname{Ric}_{S(TM)}(e_{2})=K_{21}+K_{23}= \lambda, \\& \operatorname{Ric}_{S(TM)}(e_{3})=K_{31}+K_{32}= \lambda, \end{aligned}$$
where λ is a constant. Thus, we have
$$ K_{12}=\frac {1}{2} \bigl[\operatorname{Ric}_{S(TM)}(e_{1})+ \operatorname{Ric}_{S(TM)}(e_{2})-\operatorname{Ric}_{S(TM)}(e_{3}) \bigr]=\frac{1}{2}\lambda, $$
which shows that \(K_{12}\) is constant. The converse part of this theorem is straightforward. □
Taking the trace in (18) with respect to \(S(TM)\) and putting (35) in it, we have the following result.
Lemma 1
Let
\((M,g,S(TM))\)
be an
\((n+r)\)-dimensional
r-lightlike submanifold of an
m̃-dimensional semi-Riemannian manifold of index
\((q+\tilde{q})\)
and
\(S(TM)\)
be an integrable distribution. Suppose
\(\{e_{1},\ldots,e_{n}\}\)
is an orthonormal basis of
\(\Gamma(S(TM))\). For any unit vector
\(X\in\Gamma(S(TM))\), we have
$$ \operatorname{Ric}_{S(TM)}(X)=\widetilde{\operatorname{Ric}}_{S(TM)}(X)+S(X), $$
(36)
where
$$ \widetilde{\operatorname{Ric}}_{S(TM)}(X)=\varepsilon\sum _{j=1}^{n}\varepsilon _{j} \widetilde{R}(X,e_{j},e_{j},X),\qquad g(X,X)=\varepsilon= \mp1 $$
(37)
and
$$\begin{aligned} S(X) =&\varepsilon \Biggl[\sum_{j=1}^{n} \varepsilon_{j} \Biggl[\sum_{l=1}^{r}B^{l}(e_{j},e_{j})C^{l}(X,X) -\sum_{l=1}^{r}B^{\ell}(X,e_{j})C^{\ell}(e_{j},X) \Biggr] \\ &{}-\sum_{j=1}^{n}\varepsilon_{j} \Biggl[ \sum_{\alpha=1}^{m} \varepsilon _{\alpha} D^{\alpha}(X,e_{j})D^{\alpha}(e_{j},X) -D^{\alpha}(e_{j},e_{j})D^{\alpha}(X,X) \Biggr] \Biggr]. \end{aligned}$$
(38)
Here, \(\widetilde{\operatorname{Ric}}_{S(TM)}\)
is the Ricci curvature of
n-plane section (screen distribution) of
M̃
given in [21].
Theorem 5
Let
\((M,g,S(TM))\)
be an
\((n+r)\)-dimensional minimal
r-lightlike submanifold of an
m̃-dimensional semi-Riemannian space form
\(\widetilde{M}(c)\)
and
\(S(TM)\)
be an integrable distribution. For any spacelike unit vector
\(X\in\Gamma(S(TM))\), we have:
-
(a)
$$\begin{aligned} \operatorname{Ric}_{S(TM)}(X) \leq& (n-1)c+ \bigl\vert h^{\ell} \vert _{{\mathcal{H}}_{1}\times{\mathcal{V}}} \bigr\vert \bigl\vert h^{*} \vert_{{\mathcal{H}}_{1}\times{\mathcal{V}}} \bigr\vert - \bigl\vert h^{\ell} \vert_{{\mathcal{H}}_{1}\times {\mathcal{H}}} \bigr\vert \bigl\vert h^{*} \vert_{{\mathcal{H}}_{1}\times{\mathcal{H}}} \bigr\vert \\ &{}+ \bigl\vert h^{s}\vert_{{\mathcal{H}}_{1}\times{\mathcal{V}}}^{\widetilde{{\mathcal{H}}}} \bigr\vert ^{2} + \bigl\vert h^{s}\vert_{{\mathcal{H}}_{1}\times{\mathcal{H}}}^{\widetilde{{\mathcal{V}}}} \bigr\vert ^{2} \end{aligned}$$
(39)
and
$$\begin{aligned} \operatorname{Ric}_{S(TM)}(X) \geq& (n-1)c+ \bigl\vert h^{\ell} \vert _{{\mathcal{H}}_{1}\times{\mathcal{V}}} \bigr\vert \bigl\vert h^{*} \vert_{{\mathcal{H}}_{1}\times{\mathcal{V}}} \bigr\vert - \bigl\vert h^{\ell} \vert_{{\mathcal{H}}_{1}\times {\mathcal{H}}} \bigr\vert \bigl\vert h^{*} \vert_{{\mathcal{H}}_{1}\times{\mathcal{H}}} \bigr\vert \\ &{}- \bigl\vert h^{s}\vert_{{\mathcal{H}}_{1}\times{\mathcal{V}}}^{\widetilde{{\mathcal{V}}}} \bigr\vert ^{2} - \bigl\vert h^{s}\vert_{{\mathcal{H}}_{1}\times{\mathcal{H}}}^{\widetilde{{\mathcal{H}}}} \bigr\vert ^{2}, \end{aligned}$$
(40)
where
\({\mathcal{H}}_{1}=\operatorname{Span}\{X\}\).
-
(b)
The equality cases of both the inequalities (39) and (40) are true simultaneously for all spacelike vector
\(X\in\Gamma(S(TM))\)
if and only if
D
vanishes on
\(S(TM)\).
Proof
(a) From (36) and (38) we get
$$\begin{aligned} \operatorname{Ric}_{S(TM)}(e_{i}) =& \sum _{j=1}^{n}\varepsilon_{i} \varepsilon_{j} \Biggl[\sum_{l=1}^{r}B^{l}(e_{j},e_{j})C^{l}(e_{i},e_{i}) -\sum_{l=1}^{r}B^{\ell}(e_{i},e_{j})C^{\ell}(e_{j},e_{i}) \\ &{}+\sum_{\alpha=1}^{m} \varepsilon_{\alpha} D^{\alpha }(e_{j},e_{j})D^{\alpha}(e_{i},e_{i})-D^{\alpha}(e_{i},e_{j})D^{\alpha }(e_{j},e_{i}) \Biggr] \\ &{}+(n-1)c. \end{aligned}$$
(41)
Since M is minimal we obtain
$$\begin{aligned} \operatorname{Ric}_{S(TM)}(X) =& (n-1)c+ \bigl\vert h^{\ell} \vert _{{\mathcal{H}}_{1}\times{\mathcal{V}}} \bigr\vert \bigl\vert h^{*} \vert_{{\mathcal{H}}_{1}\times{\mathcal{V}}} \bigr\vert - \bigl\vert h^{\ell} \vert_{{\mathcal{H}}_{1}\times {\mathcal{H}}} \bigr\vert \bigl\vert h^{*} \vert_{{\mathcal{H}}_{1}\times{\mathcal{H}}} \bigr\vert \\ &{}+ \bigl\vert h^{s}\vert_{{\mathcal{H}}_{1}\times{\mathcal{V}}}^{\widetilde{{\mathcal{H}}}} \bigr\vert ^{2} + \bigl\vert h^{s}\vert_{{\mathcal{H}}_{1}\times{\mathcal{H}}}^{\widetilde{{\mathcal{V}}}} \bigr\vert ^{2} - \bigl\vert h^{s}\vert_{{\mathcal{H}}_{1}\times{\mathcal{V}}}^{\widetilde{{\mathcal{V}}}} \bigr\vert ^{2} - \bigl\vert h^{s}\vert_{{\mathcal{H}}_{1}\times{\mathcal{H}}}^{\widetilde{{\mathcal{H}}}} \bigr\vert ^{2}. \end{aligned}$$
(42)
Taking into consideration (42), we have both the inequalities (39) and (40).
(b) The equality cases of both (39) and (40) inequalities are true simultaneously for all spacelike vector \(X\in\Gamma(S(TM))\) if and only if
$$ \bigl\vert h^{s}\vert_{{\mathcal{H}}_{1}\times{\mathcal{V}}}^{\widetilde{{\mathcal{H}}}} \bigr\vert = \bigl\vert h^{s}\vert_{{\mathcal{H}}_{1}\times{\mathcal{H}}}^{\widetilde{{\mathcal{V}}}} \bigr\vert = \bigl\vert h^{s}\vert_{{\mathcal{H}}_{1}\times{\mathcal{V}}}^{\widetilde{{\mathcal{V}}}} \bigr\vert = \bigl\vert h^{s}\vert_{{\mathcal{H}}_{1}\times{\mathcal{H}}}^{\widetilde{{\mathcal{H}}}} \bigr\vert =0, $$
(43)
which implies that D vanishes on \(S(TM)\). □
With similar arguments as in the proof of Theorem 5, we obtain the following theorem.
Theorem 6
Let
\((M,g,S(TM))\)
be an
\((n+r)\)-dimensional minimal
r-lightlike submanifold of an
m̃-dimensional semi-Riemannian space form
\(\widetilde {M}(c)\)
and
\(S(TM)\)
be an integrable distribution. For any timelike unit vector
\(Y\in\Gamma(S(TM))\), we have:
-
(a)
$$\begin{aligned} \operatorname{Ric}_{S(TM)}(Y) \leq& (n-1)c- \bigl\vert h^{\ell} \vert _{{\mathcal{V}}_{1}\times{\mathcal{V}}} \bigr\vert \bigl\vert h^{*} \vert_{{\mathcal{V}}_{1}\times{\mathcal{V}}} \bigr\vert + \bigl\vert h^{\ell} \vert_{{\mathcal{V}}_{1}\times {\mathcal{H}}} \bigr\vert \bigl\vert h^{*} \vert_{{\mathcal{V}}_{1}\times{\mathcal{H}}} \bigr\vert \\ &{}+ \bigl\vert h^{s}\vert_{{\mathcal{V}}_{1}\times{\mathcal{V}}}^{\widetilde{{\mathcal{V}}}} \bigr\vert ^{2} + \bigl\vert h^{s}\vert_{{\mathcal{V}}_{1}\times{\mathcal{H}}}^{\widetilde{{\mathcal{H}}}} \bigr\vert ^{2} \end{aligned}$$
(44)
and
$$\begin{aligned} \operatorname{Ric}_{S(TM)}(Y) \geq& (n-1)c- \bigl\vert h^{\ell} \vert _{{\mathcal{V}}_{1}\times{\mathcal{V}}} \bigr\vert \bigl\vert h^{*} \vert_{{\mathcal{V}}_{1}\times{\mathcal{V}}} \bigr\vert + \bigl\vert h^{\ell} \vert_{{\mathcal{V}}_{1}\times {\mathcal{H}}} \bigr\vert \bigl\vert h^{*} \vert_{{\mathcal{V}}_{1}\times{\mathcal{H}}} \bigr\vert \\ &{}- \bigl\vert h^{s}\vert_{{\mathcal{V}}_{1}\times{\mathcal{V}}}^{\widetilde{{\mathcal{H}}}} \bigr\vert ^{2} - \bigl\vert h^{s}\vert_{{\mathcal{V}}_{1}\times{\mathcal{H}}}^{ \widetilde{{\mathcal{V}}}} \bigr\vert ^{2}, \end{aligned}$$
(45)
where
\({\mathcal{V}}_{1}=\operatorname{Span}\{Y\}\).
-
(b)
The equality cases of both the inequalities (44) and (45) are true simultaneously for all timelike vector
\(X\in\Gamma(S(TM))\)
if and only if
D
vanishes on
\(S(TM)\).
Now, we give the following definition.
Definition 5
Let \((M,g,S(TM))\) be an \((n+r)\)-dimensional r-lightlike submanifold of semi-Riemannian manifold and \(S(TM)\) be an integrable distribution of index q. Suppose \(\{e_{1},\ldots,e_{n}\}\) is an orthonormal basis of \(\Gamma(S(TM))\). The bounded screen scalar curvature at a point \(p\in M\), denoted by \(r_{S(TM)}(p)\), is given by
$$ r_{S(TM)}(p)=\frac{1}{2}\sum_{i,j=1}^{n}K_{ij}. $$
(46)
With similar arguments to the proof of Theorem 4.7 in [56], we have the following proposition immediately.
Proposition 4
Let
\((M,g,S(TM))\)
be a
\((2n+r)\)-dimensional
r-lightlike submanifold and
\(S(TM)\)
be an integrable distribution. Then the bounded screen Ricci curvature is constant if and only if
$$ r_{S(TM)}(\pi_{n})=r_{S(TM)} \bigl( \pi_{n}^{\bot} \bigr), $$
(47)
where
\(\pi_{n}\)
is an
n-dimensional non-degenerate sub-plane section of
\(\Gamma(S(TM))\)
and
\(\pi_{n}^{\bot}\)
is complementary vector bundle of
\(\pi_{n}\)
in
\(\Gamma(S(TM))\).
Taking the trace in equation (36), we have the following result.
Lemma 2
Let
\((M,g,S(TM))\)
be an
\((n+r)\)-dimensional
r-lightlike submanifold and
\(S(TM)\)
be an integrable distribution. Then we have
$$\begin{aligned} \begin{aligned}[b] 2r_{S(TM)}(p)={}&2\tilde{r}_{S(TM)}(p)+n \mu_{1} \sum _{\ell =1}^{r}(\operatorname{trace} A_{N_{\ell}}) +n\mu_{2} \sum_{\alpha=1}^{m}( \operatorname{trace} A_{u_{\alpha}}) \\ &{}+2 \bigl\vert h^{\ell}\vert_{{\mathcal{V}}\times{\mathcal{H}}} \bigr\vert \bigl\vert h^{*}\vert_{{\mathcal{V}}\times{\mathcal{H}}} \bigr\vert - \bigl\vert h^{\ell} \vert_{{\mathcal{V}}\times{\mathcal{V}}} \bigr\vert \bigl\vert h^{*} \vert_{{\mathcal{V}}\times{\mathcal{V}}} \bigr\vert \\ &{}- \bigl\vert h^{\ell}\vert_{{\mathcal{H}}\times{\mathcal{H}}} \bigr\vert \bigl\vert h^{*}\vert_{{\mathcal{H}}\times{\mathcal{H}}} \bigr\vert + \bigl\vert h^{s} \vert_{{\mathcal{V}}\times{\mathcal{V}}}^{\widetilde{{\mathcal{V}}}} \bigr\vert ^{2} + \bigl\vert h^{s}\vert_{{\mathcal{H}}\times{\mathcal{H}}}^{\widetilde{{\mathcal{V}}}} \bigr\vert ^{2} \\ &{}- \bigl\vert h^{s}\vert_{{\mathcal{V}}\times{\mathcal{V}}}^{\widetilde{{\mathcal{H}}}} \bigr\vert ^{2} - \bigl\vert h^{s}\vert_{{\mathcal{H}}\times{\mathcal{H}}}^{\widetilde{{\mathcal{H}}}} \bigr\vert ^{2} +2 \bigl\vert h^{s} \vert_{{\mathcal{V}}\times{\mathcal{H}}}^{\widetilde{{\mathcal{H}}}} \bigr\vert ^{2} -2 \bigl\vert h^{s}\vert_{{\mathcal{V}}\times{\mathcal{H}}}^{\widetilde{{\mathcal{V}}}} \bigr\vert ^{2}, \end{aligned} \end{aligned}$$
(48)
where
$$ \tilde{r}_{S(TM)}(p)=\frac{1}{2}\sum _{i,j=1}^{n}\widetilde{K}_{ij}. $$
(49)
Here, \(\tilde{r}_{S(TM)}(e_{i})\)
is the scalar curvature of
n-plane section (screen distribution) of
M̃
given in [21].
Theorem 7
Let
\((M,g,S(TM))\)
be an
\((n+r)\)-dimensional
r-lightlike submanifold of a semi-Riemannian space form
\(\widetilde{M}(c)\)
and
\(S(TM)\)
be an integrable distribution. Then we have:
-
(a)
$$\begin{aligned} 2r_{S(TM)}(p) \leq&n(n-1)c+n \mu_{1} \sum _{\ell=1}^{r}(\operatorname{trace} A_{N_{\ell}}) +n \mu_{2} \sum_{\alpha=1}^{m}( \operatorname{trace} A_{u_{\alpha}}) \\ &{}+2 \bigl\vert h^{\ell}\vert_{{\mathcal{V}}\times{\mathcal{H}}} \bigr\vert \bigl\vert h^{*}\vert_{{\mathcal{V}}\times{\mathcal{H}}} \bigr\vert - \bigl\vert h^{\ell} \vert_{{\mathcal{V}}\times{\mathcal{V}}} \bigr\vert \bigl\vert h^{*} \vert_{{\mathcal{V}}\times{\mathcal{V}}} \bigr\vert \\ &{}- \bigl\vert h^{\ell}\vert_{{\mathcal{H}}\times{\mathcal{H}}} \bigr\vert \bigl\vert h^{*}\vert_{{\mathcal{H}}\times{\mathcal{H}}} \bigr\vert + \bigl\vert h^{s} \vert_{{\mathcal{V}}\times{\mathcal{V}}}^{\widetilde{{\mathcal{V}}}} \bigr\vert ^{2} + \bigl\vert h^{s}\vert_{{\mathcal{H}}\times{\mathcal{H}}}^{\widetilde{{\mathcal{V}}}} \bigr\vert ^{2} \\ &{}+2 \bigl\vert h^{s}\vert_{{\mathcal{V}}\times{\mathcal{H}}}^{\widetilde{{\mathcal{H}}}} \bigr\vert ^{2}. \end{aligned}$$
(50)
The equality case of (50) is true for all
\(p\in M\)
if and only if
M
is spacelike
\({\mathcal{V}}\)-geodesic, spacelike
\({\mathcal{H}}\)-geodesic and timelike mixed geodesic.
-
(b)
$$\begin{aligned} 2r_{S(TM)}(p) \geq&n(n-1)c+n \mu_{1} \sum _{\ell=1}^{r}(\operatorname{trace} A_{N_{\ell}}) +n \mu_{2} \sum_{\alpha=1}^{m}( \operatorname{trace} A_{u_{\alpha}}) \\ &{}+2 \bigl\vert h^{\ell}\vert_{{\mathcal{V}}\times{\mathcal{H}}} \bigr\vert \bigl\vert h^{*}\vert_{{\mathcal{V}}\times{\mathcal{H}}} \bigr\vert - \bigl\vert h^{\ell} \vert_{{\mathcal{V}}\times{\mathcal{V}}} \bigr\vert \bigl\vert h^{*} \vert_{{\mathcal{V}}\times{\mathcal{V}}} \bigr\vert \\ &{}- \bigl\vert h^{\ell}\vert_{{\mathcal{H}}\times{\mathcal{H}}} \bigr\vert \bigl\vert h^{*}\vert_{{\mathcal{H}}\times{\mathcal{H}}} \bigr\vert - \bigl\vert h^{s} \vert_{{\mathcal{V}}\times{\mathcal{V}}}^{\widetilde{{\mathcal{H}}}} \bigr\vert ^{2} - \bigl\vert h^{s}\vert_{{\mathcal{H}}\times{\mathcal{H}}}^{\widetilde{{\mathcal{H}}}} \bigr\vert ^{2} \\ &{}-2 \bigl\vert h^{s}\vert_{{\mathcal{V}}\times{\mathcal{H}}}^{\widetilde{{\mathcal{V}}}} \bigr\vert ^{2}. \end{aligned}$$
(51)
The equality case of (51) is true for all
\(p\in M\)
if and only if
M
is timelike
\({\mathcal{V}}\)-geodesic, timelike
\({\mathcal{H}}\)-geodesic and spacelike mixed geodesic.
Now, we recall a class of r-lightlike submanifolds of a semi-Riemannian manifold of an arbitrary signature which admits an integrable unique screen distribution as follows.
Definition 6
[47]
An r-lightlike submanifold is called a screen locally conformal if
$$ C^{\ell}(X,Y)=\varphi_{\ell}B^{\ell}(X,Y), \quad \forall X,Y\in \Gamma(TM|_{{\mathcal{U}}}), \ell\in\{1,\ldots,r\}, $$
(52)
where each \(\varphi_{\ell}\) is a conformal smooth function on a neighborhood \({\mathcal{U}}\) in M. If each \(\varphi_{\ell}\) is a non-zero constant, then the submanifold is called screen homothetic.
Lemma 3
[39]
If
\(a_{1},\ldots,a_{n}\)
are
n-real numbers (\(n>1\)), then
$$ \frac{1}{n} \Biggl(\sum_{i=1}^{n}a_{i} \Biggr)^{2}\leq\sum_{i=1}^{n}a_{i}^{2}, $$
(53)
with equality if and only if
\(a_{1}=\cdots=a_{n}\).
Theorem 8
Let
\((M,g,S(TM))\)
be an
\((n+r)\)-dimensional screen conformal (\(\varphi_{\ell}>0\)) r-lightlike submanifold of an
m̃-dimensional semi-Riemannian space form
\(\widetilde{M}(c)\), \(S(TM)\)
be an integrable distribution of index
q
and
\(S(TM^{\bot})\)
be Riemannian. Then we have
$$\begin{aligned} 2r_{S(TM)}(p) \leq& n(n-1)c+\sum_{\ell=1}^{r}n^{2} \varphi_{\ell} \mu^{2}_{1} +n\mu_{2} \sum_{\alpha=1}^{m}(\operatorname{trace} A_{u_{\alpha}})-q\mu _{1}^{2}|_{{\mathcal{V}}} \\ &{}- (n-q)\mu_{1}^{2}|_{{\mathcal{H}}} +2 \varphi_{\ell} \bigl\vert h^{\ell}\vert_{{\mathcal{V}}\times{\mathcal{H}}} \bigr\vert ^{2} +2 \bigl\vert h^{s}\vert_{{\mathcal{V}}\times{\mathcal{H}}}^{\widetilde{{\mathcal{H}}}} \bigr\vert ^{2}. \end{aligned}$$
(54)
The equality case of (54) is true for all
\(p\in M\)
if and only if
\(h^{\ell}(X,X)=h^{\ell}(Y,Y)\)
and
\(h^{s}(X,Y)=0\)
for all two timelike or spacelike vectors
\(X,Y\in\Gamma(S(TM))\).
Proof
Let \(\{e_{1},\ldots,e_{n}\}\) be an orthonormal basis of \(\Gamma(S(TM))\). If \(S(TM^{\bot})\) is a Riemannian distribution, then we have \(\widetilde{{\mathcal{V}}}=0\). From equation (52), it follows that
$$ h^{*}(X,Y)=\varphi_{\ell}h^{\ell}(X,Y), \quad \forall X,Y\in\Gamma(TM). $$
(55)
Taking into account Lemma 3 and equation (55), we get
$$\begin{aligned}& \mu_{1}\sum_{\ell=1}^{n} \operatorname{trace} A_{N_{\ell}}=\sum_{\ell =1}^{n}n^{2} \varphi_{\ell}\mu^{2}_{1}, \end{aligned}$$
(56)
$$\begin{aligned}& q\mu_{1}^{2}|_{{\mathcal{V}}}\leq \bigl\vert h^{\ell}\vert_{{\mathcal{V}}\times{\mathcal{V}}} \bigr\vert \bigl\vert h^{*} \vert_{{\mathcal{V}}\times{\mathcal{V}}} \bigr\vert , \end{aligned}$$
(57)
$$\begin{aligned}& (n-q)\mu_{1}^{2}|_{{\mathcal{H}}}\leq \bigl\vert h^{\ell}\vert_{{\mathcal{H}}\times{\mathcal{H}}} \bigr\vert \bigl\vert h^{*} \vert_{{\mathcal{H}}\times{\mathcal{H}}} \bigr\vert , \end{aligned}$$
(58)
where
$$ \mu_{1}|_{{\mathcal{V}}}=-\frac{1}{q} \bigl(B(e_{1},e_{1})+ \cdots +B(e_{q},e_{q}) \bigr) $$
and
$$ \mu_{1}|_{{\mathcal{H}}}=\frac{1}{n-q} \bigl(B(e_{q+1},e_{q+1})+ \cdots +B(e_{n},e_{n}) \bigr). $$
If we put (56), (57), and (58) in (48), we obtain the inequality (54).
Assuming the equality case of (54), in view of Lemma 3 in (57) and (58), for each \(\ell\in\{1,\ldots,r\}\), we have
$$ B^{\ell}(e_{1},e_{1})=\cdots=B^{\ell}(e_{q},e_{q}), B^{\ell }(e_{q+1},e_{q+1})=\cdots=B^{\ell}(e_{n},e_{n}), $$
and for each \(i,j\in\{1,\ldots, q\}\), \(a,b\in\{q+1,\ldots,n\}\), \(\alpha\in\{1,\ldots,m\}\), we have
$$ D^{\ell}(e_{i},e_{j})=D^{\ell}(e_{a},e_{b})=0. $$
This completes the proof of the theorem. □