1 Introduction

In 1979, Kulkarni [1] proved that the sectional curvature of a semi-Riemannian manifold M is unbounded from above and below at each point unless the manifold has constant sectional curvature. Later, Nomizu [2] showed that if there exists a real number d such that at any point \(p\in M\), the sectional curvature \(K(\Pi)\) of a 2-plane section Π satisfies

$$ K(\Pi)\leq d, $$
(1)

then M is of constant sectional curvature.

In [3], Dajczer and Nomizu, and in [4], Harris remarked that if the absolute value of the sectional curvature \(|K(\Pi)|\) is bounded for all timelike 2-planes Π (or for all spacelike 2-planes Π) at \(p\in M\), then M is of constant sectional curvature.

In Riemannian geometry, there are various relations between the intrinsic and extrinsic curvature invariants of a submanifold, known as Chen inequalities, in the literature [5–44]. But different from the Riemannian context, from the Kulkarni result, it is too restrictive to relate the intrinsic invariant of a submanifold with the extrinsic ones for a submanifold of a semi-Riemannian manifold. This reveals the necessity to re-investigate or modify the domain of sectional curvature map in semi-Riemannian geometry and lightlike geometry. For this purpose, the authors showed in [45, 46] that the domain of the sectional curvature map in a Lorentzian manifold is not a linear subspace as it was used in the literature but it is a polynomial subspace of a projective vector space which makes it possible for the sectional curvature map on any Lorentzian manifold to be bounded. This is revolutionary information which might lead one to require a revision for many studies related to the sectional curvature map in semi-Riemannian geometry and lightlike geometry.

In this paper, we extend this modified sectional curvature from Lorentzian manifolds to semi-Riemannian manifolds under the name of ’bounded sectional curvature’. We introduce some special r-lightlike submanifolds and establish some relationships involving intrinsic curvatures and extrinsic curvatures for r-lightlike submanifolds of a semi-Riemannian manifold.

2 Preliminaries

Let \((\widetilde{M},\tilde{g})\) be a real \((\widetilde{m}+\tilde {n})\)-dimensional semi-Riemannian manifold, where \(\tilde{n}\geq1\), \(\widetilde{m}> 1\) with g̃ a semi-Riemannian metric on M̃ of constant index \(\tilde{q}\in\{1,\ldots,\widetilde{m}+\tilde{n}-1\}\). Suppose \((M,g)\) to be an ñ-dimensional lightlike submanifold of \((\widetilde{M},\tilde{g})\) where g denotes the restriction of g̃ to M which we assume be degenerate. Then there exists a smooth distribution, called radical space of the tangent space \(T_{p}M\) at \(p\in M\), defined by

$$ \operatorname{Rad}T_{p}M=T_{p}M\cap T_{p}M^{\bot}\neq\{0\}, $$
(2)

where

$$ T_{p}M^{\bot}= \bigl\{ v_{p}\in T_{p}\widetilde{M}: \tilde{g}_{p}(v_{p},w_{p})=0, \forall w_{p}\in T_{p}M \bigr\} . $$
(3)

Let us consider the rank of \(\operatorname{Rad} T_{p}M\) to be r (\(r>0\)), \(\tilde{n}=n+r\) (\(n\geq0\)), and \(\widetilde{m}=n+m+2r\) (\(m\geq 0\)). Then there exist the following four possible cases:

Case 1.:

M is called a r-lightlike submanifold if \(1\leq r< \min\{n+r,m+r\}\).

Case 2.:

M is called a coisotropic submanifold if \(m=0\).

Case 3.:

M is called a isotropic submanifold if \(n=0\).

Case 4.:

M is called a totally lightlike submanifold if \(m=n=0\).

Let \((M,g)\) be an \((n+r)\)-dimensional lightlike submanifold of \((\widetilde{M},\tilde{g})\). Let \(S(TM)\) and \(S(TM^{\bot})\) be a complementary non-degenerate vector bundle of \(\operatorname{Rad} TM\) in TM and \(TM^{\bot}\), respectively, \(\operatorname{tr} (TM)\) be a complementary vector bundle to TM in \(T\widetilde{M}|_{M}\). Then we have

$$ T\widetilde{M}|_{M}= \bigl(\operatorname{Rad} TM\oplus \operatorname{ltr} (TM) \bigr)\oplus _{\mathrm{orth}}S(TM)\oplus_{\mathrm{orth}}S \bigl(TM^{\bot} \bigr), $$
(4)

where \(\oplus_{\mathrm{orth}}\) denotes the orthogonal direct sum and ⊕ denotes the direct sum, but it is not orthogonal.

For any r-lightlike submanifold, there exists a local quasi-orthonormal frame field \(\{\xi_{1},\ldots,\xi_{r},e_{1},\ldots,e_{n},N_{1},\ldots,N_{r},u_{1},\ldots,u_{m}\}\) on a local coordinate neighborhood of \({\mathcal{U}}\) of M such that this basis satisfies the following relation:

$$ \tilde{g}(N_{i},\xi_{j})= \delta_{ij}, \qquad \tilde {g}(N_{i},N_{j})= \tilde{g}(N_{i},u_{j})=\tilde{g}( \xi_{i},u_{j})=0,\quad \forall i,j\in\{1,\ldots,r\}, $$
(5)

where \(\delta_{ij}\) is the Kronecker delta function and

$$\begin{aligned}& \Gamma(\operatorname{Rad} TM|_{{\mathcal{U}}})=\operatorname{Span}\{ \xi_{1},\ldots ,\xi_{r}\} ,\qquad \Gamma\bigl( \operatorname{ltr} (TM)|_{{\mathcal{U}}}\bigr)=\operatorname{Span} \{N_{1},\ldots ,N_{r}\} , \\& \Gamma\bigl(S(TM)|_{{\mathcal{U}}}\bigr)=\operatorname{Span}\{e_{1}, \ldots,e_{n}\} ,\qquad \Gamma\bigl(S\bigl(TM^{\bot} \bigr)|_{{\mathcal{U}}}\bigr)=\operatorname{Span}\{u_{1},\ldots ,u_{m}\} . \end{aligned}$$

Let ∇̃ be the Levi-Civita connection of M̃ and P be the projection morphism of \(\Gamma(TM)\) to \(\Gamma(S(TM))\). The Gauss and Weingarten formulas are given by

$$\begin{aligned}& \widetilde{\nabla}_{X}Y=\nabla _{X}Y+\sum _{l=1}^{r}B^{l}(X,Y)N_{l}+\sum _{\alpha=1}^{m} D^{\alpha}(X,Y)u_{\alpha}, \end{aligned}$$
(6)
$$\begin{aligned}& \widetilde{\nabla}_{X}N_{k}=-A_{N_{k}}X+\sum _{l=1}^{r}\rho _{kl}(X)N_{l}+ \sum_{\alpha=1}^{m}\rho_{\alpha k}(X)u_{\alpha}, \end{aligned}$$
(7)
$$\begin{aligned}& \widetilde{\nabla}_{X}u_{\beta} =-A_{u_{\beta}}X+\sum _{l=1}^{r}\varepsilon _{\beta l}(X)N_{l}+ \sum_{\alpha=1}^{m}\varepsilon_{\beta \alpha}(X)u_{\alpha}, \end{aligned}$$
(8)
$$\begin{aligned}& \nabla_{X}PY=\nabla_{X}^{\ast}PY+\sum _{l=1}^{r}C^{l}(X,PY)\xi_{l} , \end{aligned}$$
(9)
$$\begin{aligned}& \nabla_{X}\xi_{k} =-A_{\xi_{k} }^{\ast}X-\sum _{l=1}^{r}\rho _{kl}(X) \xi_{l} \end{aligned}$$
(10)

for any \(X,Y\in\Gamma(TM)\), where ∇ and \(\nabla^{\ast}\) are the induced linear connection on TM and \(S(TM)\), respectively; \(B^{l}\) and \(D^{\alpha}\) are coefficients of the lightlike second fundamental form and coefficients of the screen second fundamental form of TM, respectively, \(C^{l}\) are the coefficients of the local second fundamental form on \(S(TM)\), \(A_{N_{l}}\), \(A_{u_{\alpha}}\) are the shape operators on M, \(A_{\xi_{k} }^{\ast}\) is the shape operator on \(S(TM)\) and \(\varepsilon_{l}\), \(\varepsilon_{\alpha}\), \(\rho_{l}\), \(\rho_{\alpha}\) are 1-forms on M [47].

The second fundamental form h and the local second fundamental form \(h^{\ast}\) are given by

$$ h(X,Y)=\sum_{l=1}^{r}B^{l}(X,Y)N_{l} +\sum_{\alpha=1}^{m} D^{\alpha}(X,Y)u_{\alpha} $$
(11)

and

$$ h^{\ast}(X,PY) = \sum_{l=1}^{r}C^{l}(X,PY) \xi_{l}, $$
(12)

respectively. The submanifold \((M,g,S(TM))\) is called totally geodesic if

$$ h(X,Y)=0 $$
(13)

for all \(X,Y\in\Gamma(TM)\) and it is called totally umbilical [48] if there exists a smooth transversal vector field \(H\in\Gamma(\operatorname{tr}(TM))\) such that

$$ h(X,Y)=\tilde{g}(X,Y) H $$
(14)

for all \(X,Y\in\Gamma(TM)\).

Let \(\{ e_{1},\ldots,e_{n} \} \) be an orthonormal basis of \(\Gamma(S(TM))\). Consider

$$ \mu_{1}=\frac{1}{n}\sum _{i=1}^{n}\sum_{l=1}^{r} \varepsilon _{i}B^{l}(e_{i},e_{i}) \quad \text{and} \quad \mu_{2}=\frac{1}{n}\sum _{i=1}^{n}\sum_{\alpha=1}^{r} \varepsilon_{i}\varepsilon_{\alpha}D^{\alpha}(e_{i},e_{i}), $$
(15)

where \(\varepsilon_{i}=g(e_{i},e_{i})\), \(\varepsilon_{\alpha }=g(e_{\alpha},e_{\alpha})\) for any \(i\in\{1,\ldots, n\}\) and \(\alpha\in\{1,\ldots,m\}\). The mean curvature vectors on TM and on \(\Gamma(S(TM))\) at \(p\in M\), denoted by \(H(p)\) and \(H^{*}(p)\), are given by

$$\begin{aligned}& H(p)=\frac{1}{n}\operatorname{trace}|_{S(TM)}h=\sum _{l=1}^{r}\mu _{1}N_{l}+\sum _{\alpha=1}^{m} \mu_{2}u_{\alpha}, \end{aligned}$$
(16)
$$\begin{aligned}& H^{*}(p)=\frac{1}{n}\sum_{i=1}^{n} \varepsilon_{i}\sum_{\ell =1}^{r}C^{\ell}(e_{i},e_{i}) \xi_{\ell} +\frac{1}{n}\sum_{l=1}^{r} \mu_{1}N_{l}+\frac{1}{n}\sum _{\alpha=1}^{m} \mu_{2}u_{\alpha}, \end{aligned}$$
(17)

respectively. From equation (16), we can see that the submanifold is minimal if and only if \(H(p)\) vanishes identically and \(D^{\alpha}=0\) on \(\operatorname{Rad} (TM)\) [49, 50].

Let us denote curvature tensors of the ambient manifold and the submanifold by R̃ and R, respectively. Then the following relation between these tensors holds:

$$\begin{aligned} \tilde{g} \bigl(\widetilde{R}(X,Y)PZ,PW \bigr) =&g \bigl(R(X,Y)PZ,PW \bigr)+ \sum_{l=1}^{r}B^{\ell}(X,PZ)C^{\ell}(Y,PW) \\ &{}-\sum_{l=1}^{r}B^{l}(Y,PZ)C^{l}(X,PW) \\ &{}+ \sum_{\alpha=1}^{m} \varepsilon _{\alpha} \bigl[ D^{\alpha}(X,PZ)D^{\alpha}(Y,PW)-D^{\alpha}(Y,PZ)D^{\alpha}(X,PW) \bigr] \end{aligned}$$
(18)

for all \(X,Y,Z,U\in\Gamma(TM)\) [47].

Let \(\Pi=\operatorname{Span}\{X,Y\}\) be a 2-dimensional non-degenerate plane in \(T_{p}M\). Then the sectional curvature at p is expressed by

$$ K(\Pi)=\frac{g(R_{p}(X,Y)Y,X)}{g_{p}(X,X)g_{p}(Y,Y) - g_{p}(X,Y)^{2}}. $$
(19)

We note that since \(C^{l}\) is not symmetric, the sectional curvature map does not need to be symmetric on any lightlike submanifold of a semi-Riemannian manifold [47].

Now, we recall the following result [51].

Theorem 1

Let \((M,g,S(TM))\) be an r-lightlike submanifold of a semi-Riemannian manifold \((\widetilde{M},\tilde{g})\). Then the following assertions are equivalent:

  1. (i)

    \(S(TM)\) is integrable.

  2. (ii)

    \(h^{*}\) is symmetric on \(\Gamma(S(TM))\).

  3. (iii)

    \(A_{N}\) is self-adjoint on \(\Gamma(S(TM))\) with respect to g.

As a consequence of Theorem 1, we obtain the following theorem.

Theorem 2

Let \((M,g,S(TM))\) be an r-lightlike submanifold of a semi-Riemannian manifold \((\widetilde{M},\tilde{g})\). The sectional curvature map is symmetric if and only if \(S(TM)\) is integrable.

3 Bounded sectional curvature

We start by taking into consideration a quotient space given by

$$ S(TM)\oplus_{\mathrm{orth}} S(TM)/\operatorname{SL}(2,\mathbb{R}), $$
(20)

where \(\operatorname{SL}(2,R)\) denotes the special linear transformation. For any given two vector pairs \((X,Y)\) and \((A,B)\) in this space, \((X,Y)\sim(A,B)\) if \(A=aX+bY\) and \(B=cX+dY\) with \(ad-bc=1\). It is clear that the ∼ relation is an equivalence relation. Furthermore, for this relation in this space, we can write \((X,Y)\sim(A,B)\) if and only if \(A\wedge B=X\wedge Y\), where ∧ is the wedge product. Since \(X\wedge Y\) is an element of the vector space of anti-symmetric contravariant two tensors \(\wedge^{2}S(TM)\) which is also known as the second exterior power of \(S(TM)\) [52], any element of \(S(TM)\oplus_{\mathrm{orth}} S(TM)/\operatorname{SL}(2,\mathbb{R})\) can be considered as an element of \(\wedge^{2}S(TM)\) satisfying

$$ \Pi\wedge\Pi=0 $$
(21)

for all \(\Pi\in\wedge^{2}S(TM)\). We note that equation (21) holds because of the property of anti-symmetry of the wedge product. Thus, we have

$$ \bigl(S(TM) \wedge S(TM) \bigr)/\operatorname{SL}(2,\mathbb{R})\cong \bigl\{ \Pi \in \wedge ^{2}S(TM):\Pi\wedge\Pi=0 \bigr\} . $$
(22)

Now, we consider the space of planes in \(S(TM)\). It is well known that any vector pair spanned a plane section in \(S(TM)\) are related by a general linear group \(\operatorname{GL}(2,\mathbb{R})\). Therefore, the space of planes in \(S(TM)\), denoted by the Grassmanian \(G_{r_{2}}(S(TM))\), is given by

$$ G_{r_{2}} \bigl(S(TM) \bigr)\equiv \bigl( S(TM) \oplus_{\mathrm{orth}} S(TM) \bigr)/\operatorname{GL}(2,\mathbb{R}). $$
(23)

Since the Grassmanian can be embedded into the real projective space \(\mathbb{P}(\wedge^{2}S(TM))\) but is not embedded into the space \(\wedge^{2}S(TM)\) (this embedding is also known as the Plücker embedding [52]) it can be written

$$ G_{r_{2}} \bigl(S(TM) \bigr)= \bigl\{ \Pi=X\wedge Y \in \mathbb{P} \bigl(\wedge^{2} S(TM) \bigr): \Pi \wedge\Pi=0 \bigr\} . $$
(24)

Eventually, if \(S(TM)\) is semi-Riemannian, then the sectional curvature map is defined by

$$ K:G_{r_{2}} \bigl(S(TM) \bigr)\cap \bigl\{ \Pi=X\wedge Y:G( \Pi, \Pi)\neq0 \bigr\} \rightarrow \mathbb{R}, $$
(25)

where

$$ G(\Pi,\Pi)=g(X,X)g(Y,Y)-g(X,Y)^{2}. $$
(26)

In the case of \(S(TM)\) is Riemannian, then \(G(\Pi,\Pi)\neq0\) for all \(\Pi\in \mathbb{P}(\wedge^{2}S(TM))\) and thereby the sectional curvature map in the Riemannian context is given by

$$ K:G_{r_{2}} \bigl(S(TM) \bigr)\rightarrow\mathbb{R}. $$
(27)

As a consequence of the above information, we give the following definition.

Definition 1

Let \((M,g,S(TM))\) be an \((n+r)\)-dimensional r-lightlike submanifold of an m̃-dimensional semi-Riemannian manifold \((\widetilde{M},\tilde{g})\) and \(S(TM)\) be integrable. The map

$$ K:G_{r_{2}} \bigl(S(TM) \bigr)\cap \bigl\{ \Pi:G(\Pi,\Pi) \neq0 \bigr\} \rightarrow\mathbb{R}, $$
(28)

which is defined by

$$ K(\Pi)=\frac{R(\Pi,\Pi)}{G(\Pi,\Pi)}, $$
(29)

is called bounded sectional curvature map.

Proposition 1

Let \((M,g)\) be an \((n+r)\)-dimensional r-lightlike submanifold of an m̃-dimensional semi-Riemannian manifold \((\widetilde{M},\tilde{g})\) and \(S(TM)\) be integrable. Then the bounded sectional curvature map is well defined, bounded, and independent of the choice of basis on Π.

Proof

Let \(\{e_{a_{1}}\wedge e_{a_{2}}:a_{1}< a_{2}\}\) be a basis \(\wedge^{2}S(TM)\). Suppose that \(\Pi=e_{a_{1}}\wedge e_{a_{2}}=e_{a'_{1}}\wedge e_{a'_{2}}\). Then one can write

$$\begin{aligned}& e_{a'_{1}} = ae_{a_{1}}+be_{a_{2}}, \\& e_{a'_{2}} = ce_{a_{1}}+de_{a_{2}}, \end{aligned}$$

with \(ad-bc\neq0\). Here, it is clear that the area obeys

$$ G (e_{a'_{1}}\wedge e_{a'_{2}},e_{a'_{1}}\wedge e_{a'_{2}} )= (ad-bc)^{2}G (e_{a_{1}}\wedge e_{a_{2}},e_{a_{1}}\wedge e_{a_{2}} ). $$

Since \(S(TM)\) is integrable and R is symmetric, we have

$$ R(e_{a'_{1}},e_{a'_{2}},e_{a'_{2}},e_{a'_{1}})=(ad-bc)^{2}R(e_{a_{1}},e_{a_{2}},e_{a_{2}},e_{a_{1}}), $$

which implies that \(K(\Pi)\) is independent of the choice of basis on Π, it is well defined and both bounded from above or bounded from below. □

4 Special lightlike submanifolds

We begin this section with the following definition of [53, 54].

Definition 2

Let \((\widetilde{M},\overline{g})\) be an m̃-dimensional semi-Riemannian manifold of index q̃. A distribution on M̃ is called maximally timelike if it is timelike and has rank q̃. A distribution on M̃ is called maximally spacelike if it is spacelike and has rank \((\widetilde{m}-\tilde{q})\).

Now, we recall the following theorem and proposition of Baum in [53].

Theorem 3

(Existence of maximally timelike-spacelike distributions)

Let \((\widetilde{M},\tilde{g})\) be a semi-Riemannian manifold. Then there is a g̃-orthogonal decomposition such that \(T\widetilde{M}={\widetilde{\mathcal{V}}}\oplus_{\mathrm{orth}} {\widetilde{\mathcal{H}}}\), where \({\widetilde{\mathcal{V}}}\) is a maximally timelike and \({\widetilde{\mathcal{H}}}\) is a maximally spacelike distribution on M̃.

Proposition 2

(Maximally timelike-spacelike distributions are isomorphic)

Let \((\widetilde{M},\tilde{g})\) be a semi-Riemannian manifold. Every maximally timelike (or spacelike) distributions on M̃ are isomorphic as smooth vector bundles over M̃.

Let \((M,g,S(TM))\) be an \((n+r)\)-dimensional r-lightlike submanifold and \(S(TM)\) be an integrable distribution of index q. Consider \(\{e_{1},\ldots,e_{q},e_{q+1},\ldots,e_{n}\}\) to be an orthonormal basis of \(S(TM)\). Then there exists a g-orthogonal decomposition given by

$$ S(TM)={\mathcal{V}}\oplus_{\mathrm{orth}} {\mathcal{H}}, $$
(30)

where \({\mathcal{V}} = \operatorname{Span}\{e_{1},\ldots,e_{q}\}\) is the maximally timelike distribution and \({\mathcal{H}}=\operatorname{Span}\{e_{q+1},\ldots,e_{n}\}\) is the maximally spacelike distribution.

The aforementioned concepts can be constructed on the coscreen distribution \(S(TM^{\bot})\). Let \(\widetilde{{\mathcal{V}}}\) be a maximally timelike and \(\widetilde{{\mathcal{V}}}\) be a maximally spacelike distributions on \(S(TM^{\bot})\). Then there exists also a g̃-orthogonal decomposition of \(S(TM^{\bot})\) given by

$$ S \bigl(TM^{\bot} \bigr)=\widetilde{{\mathcal{V}}} \oplus_{\mathrm{orth}} \widetilde {{\mathcal{H}}}, $$
(31)

where \(\widetilde{{\mathcal{V}}}=\operatorname{Span}\{\tilde{e}_{1},\ldots ,\tilde{e}_{\tilde{q}}\}\), \(\widetilde{{\mathcal{H}}}=\operatorname{Span}\{\tilde {e}_{\tilde{q}+1},\ldots,\tilde{e}_{m}\}\).

From (11), we can write

$$ h ( X,Y ) = \sum_{l=1}^{r}B^{l}(X,Y)N_{l}+h^{\widetilde {{\mathcal{V}}}} ( X,Y ) +h^{\widetilde{{\mathcal{H}}}} ( X,Y ), $$
(32)

where

$$ h ^{\widetilde{{\mathcal{V}}}} ( X,Y )=\sum_{\alpha =1}^{\tilde{q}}D^{\alpha}(X,Y) \tilde{e}_{\alpha} \quad \text{and} \quad h^{\widetilde{{\mathcal{H}}}} ( X,Y ) \sum _{\alpha=\tilde{q}+1}^{m}D^{\alpha}(X,Y) \tilde{e}_{\alpha} $$
(33)

for all \(X,Y\in TM\).

Now, we shall state some special r-lightlike submanifolds definitions.

Definition 3

Let \((M,g,S(TM))\) be an r-lightlike submanifold of a semi-Riemannian manifold \((\widetilde {M},\tilde{g})\) of index \(( q+\tilde{q} ) \) and \(S(TM)\) be an integrable distribution of index q. The submanifold will be called:

  1. 1.

    Timelike \({\mathcal{V}}\)-geodesic if \(h^{s} \vert^{\widetilde{{\mathcal{V}}}} _{{\mathcal{V}}\times{\mathcal{V}}}=0\), i.e., \(D^{\alpha}(X,Y)=0\) for all \(X,Y\in\mathcal{V}\) and \(\alpha\in\{1,\ldots,\tilde{q}\}\).

  2. 2.

    Timelike \({\mathcal{H}}\)-geodesic if \(h^{s} \vert^{\widetilde{{\mathcal{V}}}} _{{\mathcal{H}}\times{\mathcal{H}}}=0\), i.e., \(D^{\alpha}(X,Y)=0\) for all \(X,Y\in\mathcal{H}\) and \(\alpha\in\{1,\ldots,\tilde{q}\}\).

  3. 3.

    Timelike mixed geodesic if \(h^{s}\vert ^{\widetilde{{\mathcal{V}}}}_{{\mathcal{V}}\times{\mathcal{H}}}=0\), i.e., \(D^{\alpha}(X,Y)=0\) for all \(X\in\mathcal{V}\), \(Y\in\mathcal{H}\) and \(\alpha\in\{ 1,\ldots,\tilde{q}\}\).

  4. 4.

    Timelike geodesic if \(h ^{\widetilde{{\mathcal{V}}}}=0\), i.e., \(D^{\alpha}(X,Y)=0\) for all \(X,Y\in TM\) and \(\alpha\in\{1,\ldots,\tilde{q}\}\).

  5. 5.

    Timelike screen geodesic if \(h^{*}\vert _{{\mathcal{V}}}=0\), i.e., \(C^{l}(X,Y)=0\) for all \(X,Y\in\mathcal{V}\) and \(l \in\{1,\ldots,r\}\).

  6. 6.

    Spacelike \({\mathcal{V}}\)-geodesic if \(h^{s} \vert^{\widetilde{{\mathcal{H}}}}_{{\mathcal{V}}\times{\mathcal{V}}}=0\), i.e., \(D^{\alpha}(X,Y)=0\) for all \(X,Y\in\mathcal{V}\) and \(\alpha\in\{\tilde{q}+1,\ldots,m\}\).

  7. 7.

    Spacelike \({\mathcal{H}}\)-geodesic if \(h^{s} \vert^{\widetilde{{\mathcal{H}}}}_{{\mathcal{H}}\times{\mathcal{H}}}=0\), i.e., \(D^{\alpha}(X,Y)=0\) for all \(X,Y\in\mathcal{H}\) and \(\alpha\in\{\tilde{q}+1,\ldots ,m\}\).

  8. 8.

    Spacelike mixed geodesic if \(h^{s}\vert ^{\widetilde{{\mathcal{H}}}} _{{\mathcal{V}}\times{\mathcal{H}}}=0\), i.e., \(D^{\alpha}(X,Y)=0\) for all \(X\in\mathcal{V}\), \(Y\in\mathcal{H}\), and \(\alpha\in\{ \tilde{q}+1,\ldots,m\}\).

  9. 9.

    Spacelike geodesic if \(h^{\widetilde{{\mathcal{H}}}}=0\), i.e., \(D^{\alpha}(X,Y)=0\) for all \(X,Y\in TM\) and \(\alpha\in\{\tilde{q}+1,\ldots,m\}\).

  10. 10.

    Spacelike screen geodesic if \(h^{*}\vert _{{\mathcal{H}}}=0\), i.e., \(C^{l}(X,Y)=0\) for all \(X,Y\in\mathcal{H}\) and \(l\in\{1,\ldots,r\}\).

  11. 11.

    Mixed geodesic if \(h ^{s}\vert_{{\mathcal{V}}\times {\mathcal{H}}}=0\), i.e., \(D^{\alpha}(X,Y)=0\) for all \(X\in\mathcal{V}\), \(Y\in\mathcal{H}\) and \(\alpha\in\{ 1,\ldots,m\}\).

  12. 12.

    Mixed screen geodesic if \(h^{*}\vert _{{\mathcal{V}}\times {\mathcal{H}}}=0\), i.e., \(C^{l}(X,Y)=0\) for all \(X,Y\in TM\) and \(l\in\{1,\ldots,r\}\).

We also note that the submanifold is:

  1. 1.

    timelike geodesic if and only if \(h^{s}\vert ^{\widetilde{{\mathcal{V}}}}_{{\mathcal{V}}\times{\mathcal{V}}}=h^{s}\vert^{\widetilde {{\mathcal{V}}}} _{{\mathcal{H}}\times{\mathcal{H}}}=h^{s}\vert^{\widetilde{{\mathcal{V}}}}_{{\mathcal{V}}\times{\mathcal{H}}}=0\),

  2. 2.

    spacelike geodesic if and only if \(h^{s}\vert ^{\widetilde{{\mathcal{H}}}}_{{\mathcal{V}}\times{\mathcal{V}}}=h^{s}\vert^{\widetilde {{\mathcal{H}}}} _{{\mathcal{H}}\times{\mathcal{H}}}= h^{s}\vert^{\widetilde{{\mathcal{H}}}}_{{\mathcal{V}}\times{\mathcal{H}}}=0\),

  3. 3.

    mixed geodesic if and only if \(h^{s}\vert^{\widetilde {{\mathcal{V}}}} _{{\mathcal{V}}\times{\mathcal{H}}}= h^{s}\vert^{\widetilde {{\mathcal{H}}}} _{{\mathcal{V}}\times{\mathcal{H}}}=0\).

In view of Definition 3, we give the following proposition.

Proposition 3

Let \((M,g,S(TM))\) be an r-lightlike submanifold of a semi-Riemannian manifold \((\widetilde {M},\tilde{g})\) of index \(( q+\tilde{q} ) \) and \(S(TM)\) be an integrable distribution of index q. Then the following statements are true:

  1. (a)

    The submanifold is timelike \({\mathcal{V}}\)-geodesic and timelike \({\mathcal{H}}\)-geodesic, then the mean curvature vector on \(\Gamma(S(TM))\) is spacelike.

  2. (b)

    The submanifold is spacelike \({\mathcal{V}}\)-geodesic and spacelike \({\mathcal{H}}\)-geodesic, then the mean curvature vector on \(\Gamma(S(TM))\) is timelike.

Example 1

Let us consider the submanifold M of the semi-Euclidean space \(\mathbb {R}_{4}^{8}\) with the signature \((-,-,-,-,+,+,+,+)\) given by

$$\phi ( x_{1},x_{2},x_{3},x_{4} ) = \biggl( \frac{1}{\sqrt {2}}x_{1},\cos x_{2},\sin x_{2},\sinh x_{3},\cosh x_{3}, \frac{1}{\sqrt{2}}x_{1},\frac {1}{\sqrt{2}}x_{4}, \frac{1}{\sqrt{2}}x_{4} \biggr) $$

for all \(( x_{1},x_{2},x_{3},x_{4} ) \in\mathbb{R}^{4}\). Then we have

$$\begin{aligned}& \xi_{1} = \frac{1}{\sqrt{2}}\frac{\partial }{\partial x_{1}}+\frac {1}{\sqrt{2}} \frac{\partial }{\partial x_{6}},\qquad e_{1}=-\sin x_{2}\frac{\partial }{\partial x_{2}}+\cos x_{2}\frac{\partial }{\partial x_{3}}, \\& e_{2} = \cosh x_{3}\frac{\partial }{\partial x_{4}}+\sinh x_{3}\frac{\partial }{\partial x_{5}},\qquad e_{3}=\frac{1}{\sqrt{2}} \frac{\partial }{\partial x_{7}}+\frac{1}{\sqrt{2}}\frac{\partial }{\partial x_{8}}, \\& N_{1} = -\frac{1}{\sqrt{2}}\frac{\partial }{\partial x_{1}}+\frac {1}{\sqrt{2}} \frac{\partial }{\partial x_{6}},\qquad e_{4}=\cos x_{2}\frac{\partial }{\partial x_{2}}+ \sin x_{2}\frac{\partial }{\partial x_{3}}, \\& e_{5} = \sinh x_{3}\frac{\partial }{\partial x_{4}}+\cosh x_{3}\frac{\partial }{\partial x_{5}},\qquad e_{6}=\frac{1}{\sqrt{2}} \frac{\partial }{\partial x_{7}}-\frac{1}{\sqrt{2}}\frac{\partial }{\partial x_{8}}. \end{aligned}$$

It is easy to see that \(e_{1}\), \(e_{2}\), \(e_{4}\) are timelike unit vectors, \(e_{3}\), \(e_{5}\), \(e_{6}\) are spacelike unit vectors and M is a 1-lightlike submanifold with screen distribution \(S(TM)=\operatorname{Span}\{e_{1},e_{2},e_{3}\}\), \(\operatorname{Rad}(TM)=\operatorname{Span}\{\xi _{1}\}\), \(\operatorname{ltr}(TM)=\operatorname{Span}\{N_{1}\}\), and \(S(TM^{\bot})=\operatorname{Span}\{e_{4},e_{5},e_{6}\}\).

Here, we have also

$$S(TM)={\mathcal{V}}\oplus_{\mathrm{orth}} {\mathcal{H}}, $$

where \({\mathcal{V}}=\operatorname{Span} \{ e_{1},e_{2} \} \) and \({\mathcal{H}}=\operatorname{Span} \{ e_{3} \} \) and

$$S \bigl(TM^{\bot} \bigr)=\widetilde{{\mathcal{V}}}\oplus_{\mathrm{orth}} \widetilde {{\mathcal{H}}}, $$

where \(\widetilde{{\mathcal{V}}}=\operatorname{Span} \{ e_{4} \} \) and \(\widetilde{{\mathcal{H}}}=\operatorname{Span} \{ e_{5},e_{6} \} \). By a straightforward computation, we have \(B=C=0\) and

$$\begin{aligned}& D_{11}^{4} =1, \qquad D_{11}^{5}=D_{11}^{6}=0, \\& D_{12}^{4} =D_{12}^{5}=D_{12}^{6}=D_{13}^{4}=D_{13}^{5}=D_{13}^{6}=0, \\& D_{22}^{5} =1,\qquad D_{22}^{4}=D_{22}^{6}=D_{23}^{4}=D_{23}^{5}=D_{23}^{6}=0, \\& D_{33}^{4}=D_{33}^{5}=D_{33}^{6}=0. \end{aligned}$$

Thus, we have

$$\begin{aligned}& \bigl\vert h\vert^{\widetilde{{\mathcal{V}}}} _{{\mathcal{V}}\times{\mathcal{V}}} \bigr\vert ^{2} \neq 0, \qquad \bigl\vert h\vert^{\widetilde {{\mathcal{V}}}} _{{\mathcal{H}}\times{\mathcal{H}}} \bigr\vert ^{2}=0,\qquad \bigl\vert h\vert^{\widetilde{{\mathcal{V}}}}_{{\mathcal{V}}\times{\mathcal{H}}} \bigr\vert ^{2}=0, \\& \bigl\vert h\vert^{\widetilde{{\mathcal{H}}}} _{{\mathcal{V}}\times{\mathcal{V}}} \bigr\vert ^{2} \neq 0,\qquad \bigl\vert h\vert^{\widetilde {{\mathcal{H}}}}_{{\mathcal{H}}\times{\mathcal{H}}} \bigr\vert ^{2}=0,\qquad \bigl\vert h\vert^{\widetilde{{\mathcal{H}}}}_{{\mathcal{V}}\times{\mathcal{H}}} \bigr\vert ^{2}=0, \end{aligned}$$

which shows that the submanifold is not timelike \({\mathcal{V}}\)-geodesic and spacelike \({\mathcal{V}}\)-geodesic but it is timelike \({\mathcal{H}}\)-geodesic, spacelike \({\mathcal{H}}\)-geodesic, and mixed geodesic.

Similarly, examples for the other cases can be given.

5 Some relations for r-lightlike submanifolds

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:

  1. (a)

    If \(n=1\), then the bounded screen Ricci curvature vanishes identically.

  2. (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:

  1. (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 \({\mathcal{H}}_{1}=\operatorname{Span}\{X\}\).

  2. (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:

  1. (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 \({\mathcal{V}}_{1}=\operatorname{Span}\{Y\}\).

  2. (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:

  1. (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.

  2. (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. □