On the sectional curvature of lightlike submanifolds

Abstract

The main purpose of this paper is to show how to obtain rigidity theorems with the help of curvature invariants in submanifolds of a semi-Riemannian manifold. For this purpose, the bounded sectional curvature is introduced and some special submanifolds of r-lightlike submanifolds of a semi-Riemannian manifold are investigated.

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:

• (i) $S(TM)$ is integrable.

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

• (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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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:

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

• (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:

• (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. □