Screen Cauchy–Riemann (SCR)-lightlike submanifolds of metallic semi-Riemannian manifolds

Shanker, Gauree; Yadav, Ankit; Kaur, Ramandeep

doi:10.1108/AJMS-11-2021-0286

Purpose

The screen Cauchy–Riemann (SCR)-lightlike submanifold is an important class of submanifolds of semi-Riemannian manifolds. It contains various other classes of submanifolds as its sub-cases. It has been studied under various ambient space. The purpose of this research is to study the geometry of SCR-lightlike submanifolds of metallic semi-Riemannian manifolds.

Design/methodology/approach

The article is divided into five sections. The first section is introductory section which represents brief overview of the conducted research of this article. The second section outlines the key results that are utilized throughout the paper. In section three, the definition of SCR-lightlike submanifold is constructed with one non-trivial example. In section four and five, the important results on integrability, totally geodesic foliations and warped product are given.

Findings

The SCR-lightlike submanifold is introduced. One non-trivial example is constructed which helps to understand the given structure. The necessary and sufficient conditions for the integrability and to be totally geodesic for various distributions are obtained. The necessary and sufficient conditions for induced connection on totally umbilical SCR-lightlike submanifolds to be a metric connection are discussed. Various results are found on totally umbilical SCR-lightlike submanifolds. Finally, the existence of the warped product lightlike submanifold of the type N^⊥×_λN_T is studied.

Originality/value

SCR-lightlike submanifolds have been explored within ambient manifolds possessing various structures, such as Kaehler, Sasakian and Kenmotsu structures. In this article, we investigate this structure on submanifolds of metallic semi-Riemannian manifolds. This original and authentic research will aid researchers in advancing the study of semi-Riemannian manifolds.

1. Introduction

The study of lightlike submanifolds in semi-Riemannian manifolds presents a significantly greater level of complexity compared to study of submanifolds of Riemannian manifolds. In case of semi-Riemannian manifolds, the induced metric on submanifold has two cases, either degenerate or non-degenerate. If metric is non-degenerate, various structures of the ambient space can be induced uniquely and with same characteristic to ambient space. But this is not true for the case of degenerate metric. Also, if the submanifold has a degenerate metric then the intersection of the normal bundle and the tangent bundle is non-trivial. Due to this, we cannot study the geometry of degenerate or lightlike submanifolds by using tools of non-degenerate submanifold. The whole study of how to do calculus on lightlike submanifolds is known as lightlike geometry. The lightlike geometry is known as mathematical language of general theory of relativity. Due to this reason, it becomes a developing research topic in mathematics and physics.

In 1996, Duggal and Bejancu accumulated research on lightlike submanifolds in Ref. [1]. Later, various category of lightlike submanifolds, on the basis of action of (1,1) tensor on tangent bundle, of semi-Riemannian manifolds are studied.

The root $x = \frac{1 + \sqrt{5}}{2}$ of the equation x² − x − 1 = 0 is known as a golden ratio [4]. This ratio has many applications in paintings, pictures, temples, fractals etc. Crasmeranue and Hretcanu [2] introduced golden manifold by defining the tensor ϕ on Riemannian manifold, such that ϕ² = ϕ + 1. The metallic means, the generalization of golden means, was introduced by Spinadel [3–5]. Later, on the basis of metallic structure, geometry of various lightlike submanifolds of golden and metallic semi-Riemannian manifold are introduced in [6–23] and [32–34]. In Ref. [24], the author introduced Cauchy–Riemann (CR)- submanifolds of Kaehler manifolds which contains invariant and totally-real submanifolds as its subcases. But it was observed that CR-lightlike submanifolds does not contain invariant and screen real lightlike submanifolds as its sub-cases. Therefore, a new class SCR-lighlike submanifolds has been introduced by Duggal and Sahin [25] which contains invariant lightlike submanifolds and screen-lightlike submanifolds as its sub-cases. Later, the concept of SCR-lightlike submanifolds has been studied in indefinite Sasakian manifolds with the name of contact SCR-lightlike submanifolds [26]. In Ref. [17], author introduced SCR-lightlike submanifolds and investigated the existence of warped product SCR-lightlike submanifolds of indefinite nearly Kahler. The metallic semi-Riemannian manifolds is an important class of semi-Riemannian manifolds, therefore, we introduce SCR-lightlike submanifolds of metallic semi-Riemannian manifolds which contains invariant lightlike submanifolds [18] and screen real lightlike submanifolds [27] of metallic semi-Riemannian manifolds as its special cases. The sections of this article are as follows:

In section 2, some basics of lightlike geometry and definition of metallic semi-Riemannian manifolds are recalled. In section 3, we introduce SCR-lightlike submanifold of metallic semi-Riemannian manifold. One example is also constructed. In section 4, we establish the necessary and sufficient conditions for the integrability of the distributions Rad(TN), D′ and D^⊥, and the foliations determined by the distribution Rad(TN), D′ and D^⊥ to be totally geodesic, where Rad(TN), D′ and D^⊥ represents radical distribution, invariant distribution and anti-invariant distribution of the tangent bundle TN. In section 5, the necessary and sufficient conditions for induced connection on totally umbilical SCR-lightlike submanifolds to be a metric connection are derived. Also, it is studied that the totally umbilical SCR-lightlike submanifolds with smooth vector fields H^s ∈ Γ(μ) ⊆Γ(S(TM^⊥)) cannot be the warped product lightlike submanifold of the type N^⊥×_λN_T.

2. Preliminaries

Suppose $({\tilde{N}}^{m + n}, \tilde{g})$ be a semi-Riemannian manifold with constant index q (1 ≤ q ≤ m + n − 1, m, n ≥ 1), then the submanifold (N^m, g) is known as degenerate(lightlike) submanifold if the induced metric g is degenerate [28].

If there exists a smooth distribution $N$ (known as null distribution) such that $N_{p} = T N_{p} \cap T N_{p}^{⊥}$ for every p ∈ N and with rank r then N is called the r-lightlike submanifold. The null distribution $N$ is also denoted by Rad(TN). Also, on the basis of non-degenerate complementary subbundles S(TN)(screen distribution) and S(TN^⊥)(screen transversal distribution) of Rad(TN) in TN and TN^⊥ respectively,

we have the following decomposition (for detail see Ref. [28])

T \tilde{N} |_{N} = T N \oplus t r (T N) = [R a d (T N) \oplus l t r (T N)] ⊥ S (T N) ⊥ S (T N^{⊥})

(2.1)

where

t r (T N) = l t r (T N) ⊥ S (T N^{⊥})

(2.2)

Theorem 2.1.

[28] Let $(\tilde{N}, \tilde{g})$ be a semi-Riemannian manifold and (N, g, S(TN), S(TN^⊥)) be its r-lightlike submanifold. Then, for a coordinate neighborhood u of N, there exists a vector bundle ltr(TN) and a basis of Γ(ltr(TN)|u) contains a smooth section {N_i} of $S {(T N^{⊥})}^{⊥} |_{u}$ such that, for any i, j ∈ {1, 2, …, r}, where {ξ_i}(1 ≤ i ≤ r,)

is a lightlike basis of Γ(Rad(TN)).

{\tilde{g}}_{i j} (N_{i}, ξ_{j}) = δ_{i j}, {\tilde{g}}_{i j} (N_{i}, N_{j}) = 0,

(2.3)

For any X₁, X₂ ∈ Γ(TN) and W ∈ Γ(tr(TN)), the Gauss and Weingarten formula are

{\tilde{\nabla}}_{X_{1}} X_{2} = \nabla_{X_{1}} X_{2} + h (X_{1}, X_{2}),

(2.4)

{\tilde{\nabla}}_{X_{1}} W = - A_{W} X_{1} + \nabla_{X_{1}}^{⊥} W,

(2.5)

where ${\nabla_{X_{1}} X_{2}, A_{U} X_{1}}$ and ${h (X_{1}, X_{2}), \nabla_{X_{1}}^{⊥} W}$ belong to Γ(TN) and Γ(tr(TN)) respectively, and ∇ is induced connection on N. Further (2.4) and (2.5) reduce to

{\tilde{\nabla}}_{X_{1}} X_{2} = \nabla_{X_{1}} X_{2} + h^{l} (X_{1}, X_{2}) + h^{s} (X_{1}, X_{2}),

(2.6)

{\tilde{\nabla}}_{X_{1}} N = - A_{N} X_{1} + \nabla_{X_{1}}^{l} (N) + D^{s} (X_{1}, N), N \in Γ (l t r (T N)),

(2.7)

{\tilde{\nabla}}_{X_{1}} W = - A_{W} X_{1} + \nabla_{X_{1}}^{s} (W) + D^{l} (X_{1}, W), W \in Γ (S (T N^{⊥})) .

(2.8)

For the lightlike submanifold N of $\tilde{N}$ ⁠, (2.4), (2.6) and (2.5), (2.7), (2.8) are known as Gauss equations and Weingarten equations respectively.

Also, we have the following equations [28]:

\tilde{g} (h^{s} (X_{1}, X_{2}), W) + \tilde{g} (X_{1}, D^{l} (X_{2}, W)) = g (A_{W} X_{1}, X_{2}),

(2.9)

\tilde{g} (h^{l} (X_{1}, X_{2}), ξ) + \tilde{g} (X_{1}, h^{l} (X_{2}, ξ)) = - g (X_{1}, \nabla_{X_{2}} ξ),

(2.10)

for any ξ ∈ Γ(Rad(TN)), X₁, X₂ ∈ Γ(TN) and W ∈ Γ(S(TN^⊥)).

Since the induced connection from the ambient connection is not always a Levi-Civita connection. Then, for any X₁, X₂, Z ∈ Γ(TN) and U, V ∈ Γ(tr(TN)), we have following formulas

(\nabla_{X_{1}} g) (X_{2}, Z) = \tilde{g} (h^{l} (X_{1}, X_{2}), Z) + \tilde{g} (h^{l} (X_{1}, Z), X_{2})

(2.11)

Let us consider S a projection map on S(TN) from TN. Then, we have the following equations, for any X₁, X₂ ∈ Γ(TN) and ξ ∈ Γ(Rad(TN)):

\nabla_{X_{1}} S Y = \nabla_{X_{1}}^{*} S X_{2} + h^{*} (X_{1}, S X_{2})

(2.12)

\nabla_{X_{1}} ξ = - A_{ξ}^{*} X_{1} + \nabla_{X_{1}}^{* t} (ξ),

(2.13)

where ${h^{*} (X_{1}, P Y), \nabla_{X_{1}}^{* t} (ξ)}$ and ${\nabla_{X_{1}}^{*} S Y, A_{ξ}^{*} X_{1}}$ belong to Γ(Rad(TN)) and Γ(S(TN))), respectively.

See pg. 196–198 [28], for detail explanation of equations (2.4)-(2.13).

Definition 2.1.

[18] A smooth semi-Riemannian manifold $(\tilde{N}, \tilde{g})$ with (1,1) tensor field $\tilde{P}$ such that

and $\tilde{g}$ is $\tilde{P}$ -compatible, i.e.,

is said to be a metallic semi-Riemannian manifold.

{\tilde{P}}^{2} = p \tilde{P} + q I,

(2.14)

\tilde{g} (\tilde{P} X_{1}, X_{2}) = \tilde{g} (X_{1}, \tilde{P} X_{2})

(2.15)

From (2.14) in (2.15), we get

\tilde{g} (\tilde{P} X_{1}, \tilde{P} X_{2}) = p \tilde{g} (\tilde{P} X_{1}, X_{2}) + q \tilde{g} (X_{1}, X_{2}),

(2.16)

for any X₁, X₂ ∈ Γ(TN) [2,8].

Also, if $({\tilde{\nabla}}_{X_{1}} \tilde{P}) X_{2} = 0$ then $\tilde{P}$ is called locally metallic structure. Throughout the paper, we assume that $({\tilde{\nabla}}_{X_{1}} \tilde{P}) X_{2} = 0$ ⁠.

3. SCR-lightlike submanifolds

Definition 3.1.

A screen Cauchy–Riemann (SCR)-lightlike submanifold of a metallic semi-Riemannian manifold is a lightlike submanifold (N, g, S(TN)) which satisfies the following conditions:

(i)
$\tilde{P} (R a d (T N)) = R a d (T N)$ ,
i.e., Rad(TN) is invariant with respect to $\tilde{P}$ ⁠.
(ii)
There exist non-null distribution D and D^⊥ of S(TN) such that
$\tilde{P} (D) = D, \tilde{P} (D^{⊥}) \subset S (T N^{⊥})$ and D ∩ D^⊥ = {0}, where D^⊥ is orthogonal complementary to D in S(TN).

Here, N is proper if, D ≠ {0} and D^⊥ ≠ {0}.

From above definition, ltr(TN) is invariant w.r.t $\tilde{P}$ and

T N = D^{'} ⊥ D,

where D′ = Rad(TN) ^⊥ D.

Suppose μ be the orthogonal complementary part to $\tilde{P} D^{⊥}$ in S(TN^⊥), then

t r (T N) = l t r (T N) ⊥ μ ⊥ \tilde{P} (D^{⊥})

Example 3.1.

Suppose $(\tilde{N} = R_{1}^{7}, \tilde{g}, \tilde{P})$ be a 7-dimensional metallic semi-Euclidean space with semi-Euclidean metric $\tilde{g}$ and sign (−+ + + + + −).

Let us define, for the standard coordinate system (y₁, y₂, y₃, y₄, y₅, y₆, y₇) of $R_{1}^{7}$ ,

Then, $\tilde{P}$ is a metallic structure.

Suppose N is a submanifold of $\tilde{N}$ such that

from above system, we have following tangent vectors:

thus,

where $\tilde{P} ξ = σ ξ$ , $\tilde{P} (N) = σ N$ , $\tilde{P} (Z_{2}) \in W$ and $\tilde{P} (Z_{3}) = σ Z_{3}$ ⁠. Here, D′ = D ^⊥ Rad(TN) = span{Z₁, Z₃} and D^⊥ = span{Z₂}. Hence, N is SCR-lightlike submanifold.

\tilde{P} (y_{1}, y_{2}, y_{3}, y_{4}, y_{5}, y_{6}, y_{7}) = (σ y_{1}, σ y_{2}, (p - σ) y_{3}, σ y_{4}, σ y_{5}, σ y_{6}, (p - σ) y_{7}) .

y_{1} = y_{2} = u_{1}, y_{3} = \frac{σ}{\sqrt{q}} u_{2},

y_{4} = u_{2}, y_{5} = - \sin σ u_{3},

y_{6} = \cos σ u_{3} .

Z_{1} = \frac{\partial}{\partial y_{1}} + \frac{\partial}{\partial y_{2}}, Z_{2} = \frac{σ}{\sqrt{q}} \frac{\partial}{\partial y_{3}} + \frac{\partial}{\partial y_{4}},

Z_{3} = - \sin σ \frac{\partial}{\partial y_{5}} + \cos σ \frac{\partial}{\partial y_{6}} .

R a d (T N) = s p a n {Z_{1} = ξ}, S (T N) = s p a n {Z_{2}, Z_{3}},

l t r (T N) = s p a n \{N = - \frac{1}{2} (\frac{\partial}{\partial y_{1}} - \frac{\partial}{\partial y_{2}})\},

W = - \sqrt{q} \frac{\partial}{\partial y_{3}} + σ \frac{\partial}{\partial y_{4}} \in S (T N^{⊥}),

Let Q₁, Q₂ and R be the projection maps on Rad(TN), D and D^⊥, respectively.

For any X ∈ Γ(TN), we have

X = Q_{1} X + Q_{2} X + R X

(3.1)

or

X = Q X + R X,

(3.2)

where Q is projection map on D′ such that QX = Q₁X + Q₂X.

Applying $\tilde{P}$ on (3.2), we obtain

\tilde{P} X = \tilde{P} Q_{1} X + \tilde{P} Q_{2} X + \tilde{P} R X .

(3.3)

We write above equation as

\tilde{P} X = T X + w X,

(3.4)

where TX and wX are the tangential and transversal components of $\tilde{P} X$ ⁠, respectively.

For any U ∈ tr(TN), we assume

\tilde{P} U = B U + C U,

(3.5)

where BU and CU are tangential and transversal components of $\tilde{P} U$ ⁠, respectively.

For any X, Y ∈ Γ(TN), expanding the expression $({\tilde{\nabla}}_{X} \tilde{P}) Y = 0$ ⁠, by using (3.3)-(3.5), we obtain

\begin{align} - A_{T Q_{1} Y}^{*} X + \nabla_{X}^{*} T Q_{1} Y + h^{l} (X, T Q_{1} Y) + h^{l} (X, T Q_{1} Y) + \nabla_{X} T Q_{2} Y + h^{s} (X, T Q_{2} Y) \\ + h^{s} (X, T Q_{2} Y) - A_{wRY} X + D^{l} (X, w R Y) + \nabla_{X}^{s} w R Y = \tilde{P} Q_{1} \nabla_{X} Y + \tilde{P} Q_{2} \nabla_{X} Y + \tilde{P} R \nabla_{X} Y \\ + C h^{l} (X, Y) + B h^{s} (X, Y) + C h^{s} (X, Y) . \end{align}

(3.6)

By comparing components of Rad(TN), D, D^⊥, ltr(TN), and S(TN^⊥), we get the following equations

\nabla_{X}^{*} T Q_{1} Y + Q_{1} \nabla_{X} T Q_{2} Y = Q_{1} A_{wRY} X + T Q_{1} \nabla_{X} Y,

(3.7)

Q_{2} \nabla_{X} T Q_{2} Y = Q_{2} A_{T Q_{1} Y}^{*} X + Q_{2} A_{wRY} X + T Q_{2} \nabla_{X} Y,

(3.8)

R \nabla_{X} T Q_{2} Y = R A_{T Q_{1} Y}^{*} X + R A_{wRY} X + B h^{s} (X, Y),

(3.9)

h^{l} (X, T Q_{1} Y) + h^{l} (X, T Q_{2} Y) + D^{l} (X, w R Y) = C h^{l} (X, Y),

(3.10)

and

h^{s} (X, T Q_{1} Y) + h^{s} (X, T Q_{2} Y) + \nabla_{X}^{s} w R Y = w R \nabla_{X} Y + C h^{s} (X, Y) .

(3.11)

Lemma 3.1.

Let (N, g, S(TN)) be a SCR-lightlike submanifold of a metallic semi-Riemannian manifold $(\tilde{N}, \tilde{P})$ ⁠. Then we have

and

where (∇_XT)Y = ∇_XTY − T∇_XY and $(\nabla_{X}^{t} w) Y = \nabla_{X}^{t} Y - w \nabla_{X} Y$ ⁠.

Proof. For any X, Y ∈ Γ(TN), we have

Expanding above equation by using (3.4), (3.5) and Gauss and Weingarten formula, we obtain

this reduces to

By comparing tangential and transversal components in above equation, we obtain

and

□

(\nabla_{X} T) Y = A_{w Y} X + B h^{s} (X, Y)

(3.12)

(\nabla_{X}^{t} w) Y = - h (X, T Y) + D^{l} (X, w Y) + C h^{l} (X, Y) + C h^{s} (X, Y),

(3.13)

{\tilde{\nabla}}_{X} \tilde{P} Y = \tilde{P} {\tilde{\nabla}}_{X} Y .

{\tilde{\nabla}}_{X} (T Y + w Y) = \tilde{P} (\nabla_{X} Y + h^{l} (X, Y) + h^{s} (X, Y)),

\begin{align} \nabla_{X} T Y + h (X, T Y) - A_{w Y} X + \nabla_{X}^{s} Y + D^{l} (X, w Y) = T \nabla_{X} Y + w \nabla_{X} Y + \\ C h^{l} (X, Y) + B h^{s} (X, Y) + C h^{s} (X, Y) \end{align}

(\nabla_{X} T) Y = A_{w Y} X + B h^{s} (X, Y)

(3.15)

(\nabla_{X}^{t} w) Y = - h (X, T Y) + D^{l} (X, w Y) + C h^{l} (X, Y) + C h^{s} (X, Y) .

(3.16)

4. Integrability and totally geodesic foliations

In this section, we find necessary and sufficient conditions for the distributions Rad(TN), D and D′ to be integrable and to define totally geodesic foliations.

Theorem 4.1.

Let (N, g, S(TN)) be a SCR-lightlike submanifold of a metallic semi-Riemannian manifold $(\tilde{N}, \tilde{P})$ ⁠. Then,

Proof. (i) Let ξ₁, ξ₂ ∈ Rad(TN), from (3.7), we obtain

Interchanging role of ξ₁ and ξ₂ in (4.1), we obtain

Subtracting (4.2) from (4.1), we obtain

Similarly, from (3.11), we obtain

Interchanging role of ξ₁ and ξ₂ in (4.4) and subtract resulting equation from (4.4), we obtain

From (4.3) and (4.5) we obtain, Rad(TN) is integrable if and only if,

$Q_{2} A_{T Q_{1} ξ_{2}}^{*} ξ_{1} = Q_{2} A_{T Q_{1} ξ_{1}}^{*} ξ_{2}$ and h^s(ξ₁, TQ₁ξ₂) = h^s(ξ₂, TQ₁ξ₁).

(ii) For any ξ₁, ξ₂ ∈ Rad(TN) and Y ∈ Γ(S(TN)), from (2.4), we have

Since $\tilde{\nabla}$ is metric connection, using (2.16) in above equation, we obtain

Using $\tilde{P} {\tilde{\nabla}}_{ξ_{1}} Y = {\tilde{\nabla}}_{ξ_{1}} \tilde{P} Y$ in (4.7), (4.7) reduces to

Using (2.6) and (2.8), we get □

Since Rad(TN) defines totally geodesic foliation if and only if $\nabla_{ξ_{1}} ξ_{2} \in R a d (T N)$ , from (4.9) and (4.6), we obtain $\nabla_{ξ_{1}} ξ_{2} \in R a d (T N)$ if and only if

(i)
the distribution Rad(TN) is integrable if and only if,
$Q_{2} A_{T Q_{1} ξ_{2}}^{*} ξ_{1} = Q_{2} A_{T Q_{1} ξ_{1}}^{*} ξ_{2}$ and h^s(ξ₁, TQ₁ξ₂) = h^s(ξ₂, TQ₁ξ₁).
(ii)
the distribution Rad(TN) defines totally geodesic foliations if and only if,

p h^{l} (ξ_{1}, Y) = h^{l} (ξ_{1}, T Y) + D^{l} (ξ_{1}, w Y) .

Q_{2} A_{T Q_{1} ξ_{1}}^{*} ξ_{2} = T Q_{2} \nabla_{ξ_{1}} ξ_{2}

(4.1)

Q_{2} A_{T Q_{1} ξ_{2}}^{*} ξ_{1} = T Q_{2} \nabla_{ξ_{2}} ξ_{1}

(4.2)

Q_{2} A_{T Q_{1} ξ_{2}}^{*} ξ_{1} - Q_{2} A_{T Q_{1} ξ_{1}}^{*} ξ_{2} = T Q_{2} [ξ_{1}, ξ_{2}] .

(4.3)

h^{s} (ξ_{1}, T Q_{1} ξ_{2}) = w R \nabla_{ξ_{1}} ξ_{2} + C h^{s} (ξ_{1}, ξ_{2})

(4.4)

h^{s} (ξ_{1}, T Q_{1} ξ_{2}) - h^{s} (ξ_{2}, T Q_{1} ξ_{1}) = w R [ξ_{1}, ξ_{2}]

(4.5)

\tilde{g} (\nabla_{ξ_{1}} ξ_{2}, Y) = \tilde{g} ({\tilde{\nabla}}_{ξ_{1}} ξ_{2}, Y)

(4.6)

\tilde{g} (ξ_{2}, {\tilde{\nabla}}_{ξ_{1}} Y) = - \frac{1}{q} \tilde{g} (\tilde{P} ξ_{2}, \tilde{P} {\tilde{\nabla}}_{ξ_{1}} Y) + \frac{p}{q} \tilde{g} (\tilde{P} ξ_{2}, {\tilde{\nabla}}_{ξ_{1}} Y) .

(4.7)

\tilde{g} (ξ_{2}, {\tilde{\nabla}}_{ξ_{1}} Y) = - \frac{1}{q} \tilde{g} (\tilde{P} ξ_{2}, {\tilde{\nabla}}_{ξ_{1}} (T Y + w Y)) + \frac{p}{q} \tilde{g} (\tilde{P} ξ_{2}, {\tilde{\nabla}}_{ξ_{1}} Y) .

(4.8)

- q \tilde{g} (ξ_{2}, {\tilde{\nabla}}_{ξ_{1}} Y) = \tilde{g} (\tilde{P} ξ_{2}, h^{l} (ξ_{1}, T Y) + D^{l} (ξ_{1}, w Y) - p h^{l} (ξ_{1}, Y)) .

(4.9)

p h^{l} (ξ_{1}, Y) = h^{l} (ξ_{1}, T Y) + D^{l} (ξ_{1}, w Y) .

Theorem 4.2.

Let (N, g, S(TN)) be a SCR-lightlike submanifold of metallic semi-Riemannian manifold $(\tilde{N}, \tilde{P})$ ⁠. Then,

Proof.

Interchanging role of X and Y in (4.10), we get

Subtracting (4.10) from (4.11), we obtain

Above equation can be written as

From (4.12), D′ is integrable if and only if

For any Z ∈ Γ(D^⊥), from (2.6), we have

Using (2.16) in above equation, we get

Using (2.6) in (4.14), we get

Above equation reduces to

From (4.16), we obtain ∇_XY ∈ Γ(TN) if and only if $h^{s} (X, \tilde{P} Y) - p h^{s} (X, Y) \in Γ (μ)$ ⁠.

□

(i)
the distribution D′ is integrable if and only if,

h^{s} (X, \tilde{P} Y) = h^{s} (Y, \tilde{P} X)

(ii)
the distribution D′ defines totally geodesic foliations if and only if,

h^{s} (X, \tilde{P} Y) - p h^{s} (X, Y) \in Γ (μ) .

(i)
For any X, Y ∈ Γ(D′), from (3.11), we obtain

h^{s} (X, T Q_{1} Y) + h^{s} (X, T Q_{2} Y) = w R \nabla_{X} Y + C h^{s} (X, Y) .

(4.10)

h^{s} (Y, T Q_{1} X) + h^{s} (Y, T Q_{2} X) = w R \nabla_{Y} X + C h^{s} (Y, X) .

(4.11)

h^{s} (X, T Q_{1} Y) + h^{s} (X, T Q_{2} Y) - h^{s} (Y, T Q_{1} X) - h^{s} (Y, T Q_{2} X) = w R [X, Y]

(4.12)

h^{s} (X, \tilde{P} Y) - h^{s} (Y, \tilde{P} X) = w R [X, Y] .

(4.13)

h^{s} (X, \tilde{P} Y) = h^{s} (Y, \tilde{P} X)

(ii)
For any X, Y ∈ Γ(D′), the distribution D′ defines totally geodesic foliation if and only if ∇_XY ∈ Γ(D′).

g (\nabla_{X} Y, Z) = g ({\tilde{\nabla}}_{X} Y, Z) .

q g (\nabla_{X} Y, Z) = q \tilde{g} ({\tilde{\nabla}}_{X} Y, Z) = \tilde{g} (\tilde{P} {\tilde{\nabla}}_{X} Y, \tilde{P} Z) - p \tilde{g} ({\tilde{\nabla}}_{X} Y, \tilde{P} Z) .

(4.14)

q g (\nabla_{X} Y, Z) = \tilde{g} (h^{s} (X, \tilde{P} Y), \tilde{P} Z) - p \tilde{g} (h^{s} (X, Y), \tilde{P} Z) .

(4.15)

q g (\nabla_{X} Y, Z) = \tilde{g} (h^{s} (X, \tilde{P} Y) - p h^{s} (X, Y), \tilde{P} Z)

(4.16)

Theorem 4.3.

Let (N, g, S(TN)) be a SCR-lightlike submanifold of metallic semi-Riemannian manifold $\tilde{N}$ ⁠. Then,

Proof.

Interchanging Z₁ and Z₂, we obtain

Subtracting (4.18) from (4.17), we obtain

From (4.19), we obtain D^⊥ is integrable if and only if

Using (2.16) in (4.20), we obtain

Since ${\tilde{\nabla}}_{X} \tilde{P} Y = \tilde{P} {\tilde{\nabla}}_{X} Y$ ⁠, above equation reduces to

Expanding above equation using (2.6) and (2.8) in (4.22), we obtain

On the other hand, for any N ∈ ltr(TN), we have

Using (2.16) in (4.24), we obtain

Further, expanding (4.25) by using (2.8), we obtain

Since the distribution D^⊥ defines totally geodesic foliation if and only if $\nabla_{Z_{1}} Z_{2} \in Γ (D^{⊥})$ ⁠, from (4.23) and (4.26), we get, $\nabla_{Z_{1}} Z_{2} \in Γ (D^{⊥})$ if and only if $B \nabla_{Z_{1}}^{s} \tilde{P} Z_{2} = - A_{\tilde{P} Z_{2}} Z_{1} .$ and $T Q_{1} A_{\tilde{P} Z_{2}} Z_{1} = p Q_{1} A_{\tilde{P} Z_{2}} Z_{1}$ ⁠.

□

(i)
the distribution D^⊥ is integrable if and only if,

T Q A_{w R Z_{1}} Z_{2} = T Q A_{w R Z_{2}} Z_{1} .

(ii)
the distribution D^⊥ defines totally geodesic foliation if and only if, $B \nabla_{Z_{1}}^{s} \tilde{P} Z_{2} = - A_{\tilde{P} Z_{2}} Z_{1} .$ and $T Q_{1} A_{\tilde{P} Z_{2}} Z_{1} = p Q_{1} A_{\tilde{P} Z_{2}} Z_{1}$ ⁠.

(i)
For any Z₁, Z₂ ∈ Γ(D^⊥), from (3.7) and (3.8), we obtain

- Q A_{wRY} Z_{1} = T Q \nabla_{Z_{1}} Z_{2},

(4.17)

- Q A_{wRX} Z_{2} = T Q \nabla_{Z_{2}} Z_{1},

(4.18)

- Q A_{wRX} Z_{2} + Q A_{wRY} Z_{1} = T Q [Z_{1}, Z_{2}]

(4.19)

Q A_{w R Z_{1}} Z_{2} = Q A_{w R Z_{2}} Z_{1} .

(ii)
For any Z₁, Z₂ ∈ Γ(D^⊥) and X ∈ Γ(D), from (2.6), we have

g (\nabla_{Z_{1}} Z_{2}, X) = \tilde{g} ({\tilde{\nabla}}_{Z_{1}} Z_{2}, X) .

(4.20)

q g (\nabla_{Z_{1}} Z_{2}, X) = \tilde{g} (\tilde{P} {\tilde{\nabla}}_{Z_{1}} Z_{2}, \tilde{P} X) - p \tilde{g} ({\tilde{\nabla}}_{Z_{1}} Z_{2}, \tilde{P} X) .

(4.21)

q g (\nabla_{Z_{1}} Z_{2}, X) = \tilde{g} ({\tilde{\nabla}}_{Z_{1}} \tilde{P} Z_{2}, \tilde{P} X) - p \tilde{g} ({\tilde{\nabla}}_{Z_{1}} \tilde{P} Z_{2}, X) .

(4.22)

q g (\nabla_{Z_{1}} Z_{2}, X) = \tilde{g} (B \nabla_{Z_{1}}^{s} \tilde{P} Z_{2}, X) + p \tilde{g} (A_{\tilde{P} Z_{2}} Z_{1}, X) .

(4.23)

\tilde{g} (\nabla_{Z_{1}} Z_{2}, N) = \tilde{g} ({\tilde{\nabla}}_{Z_{1}} Z_{2}, N) = \frac{1}{q} \tilde{g} (\tilde{P} {\tilde{\nabla}}_{Z_{1}} Z_{2}, \tilde{P} N) - \frac{p}{q} \tilde{g} (\tilde{P} {\tilde{\nabla}}_{Z_{1}} Z_{2}, N) .

(4.24)

\tilde{g} (\nabla_{Z_{1}} Z_{2}, N) = \frac{1}{q} \tilde{g} (\tilde{P} {\tilde{\nabla}}_{Z_{1}} Z_{2}, \tilde{P} N) - \frac{p}{q} \tilde{g} (\tilde{P} {\tilde{\nabla}}_{Z_{1}} Z_{2}, N) .

(4.25)

\tilde{g} (\nabla_{Z_{1}} Z_{2}, N) = - \frac{1}{q} \tilde{g} (A_{\tilde{P} Z_{2}} Z_{1}, \tilde{P} N) + \frac{p}{q} \tilde{g} (A_{\tilde{P} Z_{2}} Z_{1}, N) .

(4.26)

5. Totally umbilical SCR-lightlike submanifolds

Definition 5.1.

[1] A lightlike submanifolds (N, g, S(TN)) of a semi-Riemannian manifold $(\tilde{N}, \tilde{g})$ is said to be totally umbilical in $\tilde{N}$ if there is smooth transversal vector field $H \in Γ (t r (T \tilde{N}))$ on $\tilde{N}$ called the transversal curvature vector field of $\tilde{N}$ , such that, for all X₁, X₂ ∈ Γ(TN)

h (X_{1}, X_{2}) = H \tilde{g} (X_{1}, X_{2}) .

(5.1)

Alternatively, N is totally umbilical, if and only if on each coordinate neighborhood $U$ there exist a smooth vector fields $H^{l} \in Γ (l t r (T N))$ and $H^{s} \in Γ (S (T N^{⊥}))$ such that

h^{l} (X_{1}, X_{2}) = H^{l} \tilde{g} (X_{1}, X_{2}),

(5.2)

h^{s} (X_{1}, X_{2}) = H^{s} \tilde{g} (X_{1}, X_{2}),

(5.3)

D^{l} (X_{1}, W) = 0,

(5.4)

for any X₁, X₂ ∈ Γ(TN) and W ∈ S(TN^⊥).

Theorem 5.1.

Let (N, g, S(TN)) be a totally umbilical r-lightlike SCR submanifold of metallic semi-Riemannian manifold $\tilde{N}$ , then the induced connection ∇ is metric connection if and only if, $\tilde{g} (X, T Q_{2} Y) = p \tilde{g} (X, Y)$ ⁠.

Proof. The induced connection ∇ is metric connection if and only if, for any X ∈ Γ(TN) and ξ ∈ Γ(Rad(TN)), ∇_Xξ ∈ Rad(TN).

Suppose N is totally umbilical lightlike submanifold and Y ∈ S(TN), from (2.16), we get

Since $\tilde{\nabla}$ is metric connection, (5.5) reduces to

Using (5.2) and (5.3) in (5.6), we get

Since $D^{l} (X \tilde{P} Q_{2} Y) = 0$ and $\tilde{g} (H^{l}, \tilde{P} ξ)$ is not identically zero, from (5.7) g(∇_Xξ, Y) = 0 if and only if, $\tilde{g} (X, T Q_{2} Y) = p \tilde{g} (X, Y)$ ⁠.

This implies, the induced connection ∇ is metric connection if and only if, $\tilde{g} (X, T Q_{2} Y) = p \tilde{g} (X, Y)$ ⁠. □

g (\nabla_{X} ξ, Y) = \frac{1}{q} g (\tilde{P} {\tilde{\nabla}}_{X} ξ, \tilde{P} Y) - \frac{p}{q} g (\tilde{P} {\tilde{\nabla}}_{X} ξ, Y) .

(5.5)

g (\nabla_{X} ξ, Y) = - \frac{1}{q} \tilde{g} (\tilde{P} ξ, {\tilde{\nabla}}_{X} (T Y + w Y)) + \frac{p}{q} \tilde{g} (\tilde{P} ξ, {\tilde{\nabla}}_{X} Y) .

(5.6)

g (\nabla_{X} ξ, Y) = \tilde{g} (H^{l}, \tilde{P} ξ) (- \frac{1}{q} \tilde{g} (X, T Q_{2} Y) + \frac{p}{q} \tilde{g} (X, Y)) - \frac{1}{q} \tilde{g} (\tilde{ξ}, D^{l} (X, w Y)) .

(5.7)

Theorem 5.2.

Let N be a totally umbilical lightlike submanifold of metallic semi-Riemannian manifold. Then

Proof. (i) Let ξ₁, ξ₂ ∈ Rad(TN) and X ∈ Γ(S(TN)), from (2.16), we get

Above equation reduces to

Using (2.6) and (2.8) in above equation, we get

Since N is totally umbilical, D^l(ξ, wX) = 0 and h^l(ξ, wX) = 0. From (5.10), we obtain

Therefore, [ξ₁, ξ₂] ∈ Rad(TN) implies Rad(TN) is integrable.

(ii) Let ξ₁, ξ₂ ∈ Rad(TN) and X ∈ Γ(S(TN)), from (2.16), we obtain

this implies

Since N is totally umbilical, $h (X, \tilde{P} ξ_{1}) = H \tilde{g} (X, \tilde{P} ξ_{1}) = 0$ ⁠, $h (X, \tilde{P} ξ_{1}) = H \tilde{g} (X, ξ_{1}) = 0$ and D^l(ξ, wX) = 0. From (5.13), we get

Therefore $\nabla_{ξ_{1}} ξ_{2} \in R a d (T N)$ ⁠. □

(i)
Rad(TN) is always integrable.
(ii)
Rad(TN) always defines totally geodesic foliation.

\tilde{g} ([ξ_{1}, ξ_{2}], X) = \frac{1}{q} \tilde{g} ({\tilde{\nabla}}_{ξ_{1}} \tilde{P} ξ_{2}, \tilde{P} X) - \frac{p}{q} \tilde{g} ({\tilde{\nabla}}_{ξ_{1}} ξ_{2}, \tilde{P} X) - \frac{1}{q} \tilde{g} ({\tilde{\nabla}}_{ξ_{2}} \tilde{P} ξ_{1}, \tilde{P} X) + \frac{p}{q} \tilde{g} ({\tilde{\nabla}}_{ξ_{2}} ξ_{1}, \tilde{P} X) .

(5.8)

\tilde{g} ([ξ_{1}, ξ_{2}], X) = - \frac{1}{q} \tilde{g} (\tilde{P} ξ_{2}, {\tilde{\nabla}}_{ξ_{1}} \tilde{P} X) + \frac{p}{q} \tilde{g} (ξ_{2}, {\tilde{\nabla}}_{ξ_{1}} \tilde{P} X) + \frac{1}{q} \tilde{g} (\tilde{P} ξ_{2}, {\tilde{\nabla}}_{ξ_{1}} \tilde{P} X) - \frac{p}{q} \tilde{g} (ξ_{2}, {\tilde{\nabla}}_{ξ_{1}} \tilde{P} X) .

(5.9)

\begin{align} \tilde{g} ([ξ_{1}, ξ_{2}], X) & = - \frac{1}{q} \tilde{g} (\tilde{P} ξ_{2}, D^{l} (ξ_{1}, w X) + \frac{p}{q} \tilde{g} (ξ_{2}, D^{l} (ξ_{1}, w X)) + \frac{1}{q} \tilde{g} (\tilde{P} ξ_{2}, D^{l} (ξ_{1}, w X) \\ - \frac{p}{q} \tilde{g} (ξ_{2}, D^{l} (ξ_{1}, w X)) - \frac{1}{q} \tilde{g} (\tilde{P} ξ_{2}, h^{l} (ξ_{1}, T X) + \frac{p}{q} \tilde{g} (ξ_{2}, h^{l} (ξ_{1}, T X)) \\ + \frac{1}{q} \tilde{g} (\tilde{P} ξ_{2}, h^{l} (ξ_{1}, T X) - \frac{p}{q} \tilde{g} (ξ_{2}, h^{l} (ξ_{1}, T X)) . \end{align}

(5.10)

\tilde{g} ([ξ_{1}, ξ_{2}], X) = 0 .

(5.11)

\tilde{g} (\nabla_{ξ_{1}} ξ_{2}, X) = \frac{1}{q} \tilde{g} ({\tilde{\nabla}}_{ξ_{1}} \tilde{P} ξ_{2}, T X) + \frac{1}{q} \tilde{g} ({\tilde{\nabla}}_{ξ_{1}} \tilde{P} ξ_{2}, w X) - \frac{p}{q} \tilde{g} ({\tilde{\nabla}}_{ξ_{1}} ξ_{2}, T X) - \frac{p}{q} \tilde{g} ({\tilde{\nabla}}_{ξ_{1}} ξ_{2}, T X),

(5.12)

\tilde{g} (\nabla_{ξ_{1}} ξ_{2}, X) = \frac{1}{q} \tilde{g} (h^{l} (ξ_{1}, T X), \tilde{P} ξ_{2}) + \frac{1}{q} \tilde{g} (D^{l} (ξ_{1}, w X), \tilde{P} ξ_{2}) - \frac{p}{q} \tilde{g} (h^{l} (ξ_{1}, T X), ξ_{2}) - \frac{p}{q} \tilde{g} (D^{l} (ξ_{1}, w X), ξ_{2}) .

(5.13)

\tilde{g} (\nabla_{ξ_{1}} ξ_{2}, X) = 0 .

Theorem 5.3.

Let N be a totally umbilical lightlike submanifold of metallic semi-Riemannian manifold. Then

Proof. (i) From Theorem 4.2 (i), D′ is integrable if and only if,

Since N is totally umbilical,

and

Since $\tilde{g} (X, \tilde{P} Y) = \tilde{g} (\tilde{P} X, Y)$ ⁠, from (5.14) and (5.15), we obtain

Therefore, D′ is always integrable.

(ii) From Theorem 4.2 (ii), D′ defines totally geodesic foliation if and only if, $h^{s} (X, \tilde{P} Y) - p h^{s} (X, Y) \in Γ (μ)$ ⁠.

Since N is totally umbilical, D′ defines totally geodesic foliation if and only if,

This implies, D′ defines totally geodesic foliation if and only if, H^s ∈ Γ(μ)). □

(i)
D′ is integrable.
(ii)
D′ defines totally geodesic foliation if and only if, H^s ∈ Γ(μ)).

h^{s} (X, \tilde{P} Y) = h^{s} (\tilde{P} X, Y) .

h^{s} (X, \tilde{P} Y) = H^{s} \tilde{g} (X, \tilde{P} Y)

(5.14)

h^{s} (\tilde{P} X, Y) = H^{s} \tilde{g} (\tilde{P} X, Y) .

(5.15)

h^{s} (X, \tilde{P} Y) = h^{s} (\tilde{P} X, Y) = H^{s} \tilde{g} (\tilde{P} X, Y) = h^{s} (\tilde{P} X, Y) .

(5.16)

H^{s} (\tilde{g} (X, \tilde{P} Y) - p \tilde{g} (X, Y) \in Γ (μ)) .

Theorem 5.4.

Let N be a totally umbilical lightlike submanifold of metallic semi-Riemannian manifold. Then, for any Z₁ and Z₂ ∈ D^⊥

Proof. The Proof is same as Theorem 4.3. □

(i)
D^⊥ is integrable if and only if,

T Q A_{w R Z_{1}} Z_{2} = T Q A_{w R Z_{2}} Z_{1} .

(ii)
D^⊥ defines totally geodesic foliation if and only, if, H^l = 0.

The notion of warped product was defined by Bishop and O′ Niell in Ref. [29] as a generalization of the notion of the Cartesian product. Further, various types of warped product submanifolds have been studied in Riemannian manifolds with certain metric structures. Later, the warped product lightlike submanifolds of a semi-Riemannian manifold have been studied by Sahin [22]. In the Theorem 5.6, we investigate the existence of warped product lightlike submanifolds of the type N^⊥×_λN_T in metallic semi-Riemannian manifold $\tilde{N}$ when H^s ∈ Γ(μ).

Definition 5.2.

[29] Let B × F be a product manifold of Riemannain manifolds B and F with Riemannian metric g_B and g_F, respectively, and π: B × F → B and η: F × B → F be projection maps. The manifold B × F equipped with Riemannian metric g such that

that is, for any X ∈ TN at (p, q),

is known as warped product N = B ×_λF.

g = g_{B} + λ^{2} g_{F}

‖ X ‖ = ‖ π_{*} (X) ‖ + λ^{2} ‖ η_{*} (X) ‖^{2},

If λ (Known as warping function) is constant then the warped product manifold is trivial. For differentiable function λ on N, the gradient ∇λ is defined by

g (\nabla λ, Z) = Z λ, \forall Z \in Γ (T N) .

Theorem 5.5.

[29] Let N = B ×_λF be a warped product manifolds. If X₁, X₂ ∈ T(B) and Y₁, Y₂ ∈ T(F), then

\nabla_{X_{1}} X_{2} \in T (B),

(5.17)

\nabla_{X_{1}} Y_{1} = \nabla_{Y_{1}} X_{1} = (\frac{X_{1} λ}{λ}) Y_{1},

(5.18)

\nabla_{Y_{1}} Y_{2} = - \frac{g (Y_{1}, Y_{2})}{λ} \nabla λ .

(5.19)

Theorem 5.6.

Let (N, g, S(TN)) be totally umbilical SCR-lightlike submanifold of metallic semi-Riemannian manifold $\tilde{N}$ with H^s ∈ Γ(μ) Then, there does not exist any non-trivial lightlike warped product of the type N^⊥×_λN_T

Proof. Let X ∈ Γ(D_T) and Z ∈ Γ(D^⊥). Then, from (5.18), we obtain

\nabla_{X} Z = \nabla_{Z} X = (Z \ln λ) X

(5.20)

this implies

g (\nabla_{X} Z, X) = Z l n λ g (X, X) .

(5.21)

Using (2.16) in (5.21), we get

Z \ln λ g (X, X) = Z \ln λ {\frac{1}{q} g (P X, P X) - \frac{p}{q} g (P X, X)} .

(5.22)

Above equation can be re-written as

Z \ln λ g (X, X) = \frac{1}{q} g (Z \ln λ P X, P X) - \frac{p}{q} g (Z \ln λ P X, X) .

(5.23)

Again, using (5.18) in (5.23), we get

Z \ln λ g (X, X) = \frac{1}{q} g (\nabla_{P X} Z, P X) - \frac{p}{q} g (\nabla_{P X} Z, X) .

(5.24)

Using (2.6) in equation in (5.24), we obtain

Z \ln λ g (X, X) = \frac{1}{q} \tilde{g} ({\tilde{\nabla}}_{P X} Z - h^{l} (P X, Z) - h^{s} (P X, Z), P X)

(5.25)

- \frac{p}{q} \tilde{g} ({\tilde{\nabla}}_{P X} Z - h^{l} (P X, Z) - h^{s} (P X, Z), X)

(5.26)

Since N is totally umbilical, h^lg(PX, Z) = H^lg(PX, Z) and h^s(PX, Z) = H^sg(PX, Z). Therefore, equation (5.25) reduces to

Z \ln λ g (X, X) = \frac{1}{q} \tilde{g} ({\tilde{\nabla}}_{P X} Z, P X) - \frac{p}{q} \tilde{g} ({\tilde{\nabla}}_{P X} Z, X) .

(5.27)

Since the $\tilde{\nabla}$ is Levi-Civita connection, (5.27) can be re-written as

Z \ln λ g (X, X) = - \frac{1}{q} \tilde{g} ({\tilde{\nabla}}_{P X} P X, Z) + \frac{p}{q} \tilde{g} ({\tilde{\nabla}}_{P X} X, Z) .

(5.28)

Above equation reduces to

Z \ln λ g (X, X) = - \frac{1}{q} g (\nabla_{P X} P X, Z) + \frac{p}{q} g (\nabla_{P X} X, Z) .

(5.29)

From theorem 5.3 if H^s ∈ Γ(μ) then ∇_PXX and ∇_PXX belong to D′. Therefore, g(∇_PXPX, Z) = 0 and g(∇_PXX, Z) = 0. From (5.29), we get (Z ln λ)g(X, X) = 0.

For Y ∈ D ⊂ D′, we have

(Z \ln λ) g (Y, Y) = 0 .

(5.30)

Since D is non-degenerate distribution, we get Z ln λ = 0 this leads to λ = 0 which is a contradiction.

Hence, there does not exist any non-trivial lightlike warped product of type N^⊥×_λN_T.□

The first author is thankful to DST Government of India for providing financial support in terms of DST-FST label-I grant vide sanction number SR/FST/MS-I/2021/104(C).

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Screen Cauchy–Riemann (SCR)-lightlike submanifolds of metallic semi-Riemannian manifolds

1. Introduction

2. Preliminaries

3. SCR-lightlike submanifolds

4. Integrability and totally geodesic foliations

5. Totally umbilical SCR-lightlike submanifolds

References

Further reading

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Screen Cauchy–Riemann (SCR)-lightlike submanifolds of metallic semi-Riemannian manifolds Open Access

1. Introduction

2. Preliminaries

3. SCR-lightlike submanifolds

4. Integrability and totally geodesic foliations

5. Totally umbilical SCR-lightlike submanifolds

References

Further reading

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Screen Cauchy–Riemann (SCR)-lightlike submanifolds of metallic semi-Riemannian manifolds