Geodesic Lightlike Submanifolds of Indefinite Kenmotsu Manifolds

Abstract

The aim of the present paper is to study geodesic contact screen Cauchy Riemannian (SCR-) lightlike submanifolds, geodesic screen transversal lightlike, and geodesic transversal lightlike submanifolds of indefinite Kenmotsu manifolds.

1. Introduction

The study of the geometry of submanifolds of a Riemannian or semi-Riemannian manifold is one of the interesting topics of differential geometry. Despite of some similarities between semi-Riemannian manifolds and Riemannian manifolds, the lightlike submanifolds are different since their normal vector bundle intersect with the tangent bundle making it more interesting and difficult to study. These submanifolds were introduced and studied by Duggal and Bejancu [1]. On the other hand, geodesic CR-lightlike submanifolds in Kähler manifolds were studied by Sahin and Gunes [2], and geodesic lightlike submanifolds of indefinite Sasakian manifolds were investigated by Dong and Liu [3]. In 2006, Sahin [4] initiated the study of transversal lightlike submanifolds of an indefinite Kähler manifold which are different from CR-lightlike [1] and screen CR-lightlike submanifolds [5]. Recently, Sahin [6] introduced the notion of screen transversal lightlike submanifolds of indefinite Kähler manifolds and obtained some useful results. In this paper, we study geometric conditions under which some lightlike submanifolds of an indefinite Kenmotsu manifold are totally geodesic.

2. Preliminaries

We follow [1] for the notation and fundamental equations for lightlike submanifolds used in this paper. A submanifold 𝑀^𝑚 immersed in a semi-Riemannian manifold (𝑀^𝑚+𝑛,𝑔) is called a lightlike submanifold if it admits a degenerate metric 𝑔 induced from 𝑔 whose radical distribution Rad𝑇𝑀=𝑇𝑀∩𝑇𝑀^⟂ is of rank 𝑟, where 1≤𝑟≤𝑚 and 𝑇𝑀^⟂=∪𝑈∈𝑇_𝑥𝑀∶𝑔(𝑈,𝑉)=0,∀𝑉∈𝑇_𝑥𝑀. (2.1) Let 𝑆(𝑇𝑀) be a screen distribution which is a semi-Riemannian complementary distribution of Rad𝑇𝑀 in 𝑇𝑀, that is, 𝑇𝑀=Rad𝑇𝑀⟂𝑆(𝑇𝑀). (2.2) Consider a screen transversal vector bundle 𝑆(𝑇𝑀^⟂), which is a semi-Riemannian complementary vector bundle of Rad𝑇𝑀 in 𝑇𝑀^⟂. Since for any local basis {𝜉_𝑖} of Rad𝑇𝑀, there exists a local null frame {𝑁_𝑖} of sections with values in the orthogonal complement of 𝑆(𝑇𝑀^⟂) in [𝑆(𝑇𝑀)]^⟂ such that 𝑔(𝜉_𝑖,𝑁_𝑗)=𝛿_𝑖𝑗 and 𝑔(𝑁_𝑖,𝑁_𝑗)=0, it follows that there exists a lightlike transversal vector bundle ltr(𝑇𝑀) locally spanned by {𝑁_𝑖} [1, page 144]. Let tr(𝑇𝑀) be the complementary (but not orthogonal) vector bundle to 𝑇𝑀 in 𝑇𝑀|_𝑀.

Then, tr(𝑇𝑀)=ltr(𝑇𝑀)⟂𝑆𝑇𝑀^⟂,𝑇𝑀=𝑆(𝑇𝑀)⟂[Rad𝑇𝑀⊕ltr(𝑇𝑀)]⟂𝑆𝑇𝑀^⟂. (2.3) Let ∇ be the Levi-Civita connection on 𝑀. Then, in view of the decomposition (2.3), the Gauss and Weingarten formulas are given by ∇_𝑋𝑌=∇_𝑋𝑌+ℎ(𝑋,𝑌)∀𝑋,𝑌∈Γ(𝑇𝑀),∇_𝑋𝑈=−𝐴_𝑈𝑋+∇^𝑡_𝑋𝑈,∀𝑋∈Γ(𝑇𝑀),𝑈∈Γ(tr(𝑇𝑀)), (2.4)(2.5) where {∇_𝑋𝑌,𝐴_𝑈𝑋} and {ℎ(𝑋,𝑌),∇^𝑡_𝑋𝑈} belong to Γ(𝑇𝑀) and Γ(tr(𝑇𝑀)), respectively, ∇ and ∇^𝑡 are linear connection on 𝑀 and on the vector bundle tr(𝑇𝑀), respectively. Moreover, we have ∇_𝑋𝑌=∇_𝑋𝑌+ℎ^𝑙(𝑋,𝑌)+ℎ^𝑠(𝑋,𝑌),∇_𝑋𝑁=−𝐴_𝑁𝑋+∇^𝑙_𝑋𝑁+𝐷^𝑠(𝑋,𝑁),∇_𝑋𝑊=−𝐴_𝑊𝑋+∇^𝑠_𝑋𝑊+𝐷^𝑙(𝑋,𝑊) (2.6)(2.7)(2.8) for all, 𝑋,𝑌∈Γ(𝑇𝑀),𝑁∈Γ(ltr(𝑇𝑀)), and 𝑊∈Γ(𝑆(𝑇𝑀^⟂)). If we denote the projection of 𝑇𝑀 on 𝑆(𝑇𝑀) by 𝑃, then by using (2.6)–(2.8) and the fact that ∇ is a metric connection, we obtain 𝑔ℎ^𝑠(𝑋,𝑌),𝑊+𝑔𝑌,𝐷^𝑙(𝑋,𝑊)=𝑔𝐴_𝑊𝑋,𝑌,𝑔𝐷^𝑠(𝑋,𝑁),𝑊=𝑔𝑁,𝐴_𝑊𝑋. (2.9)(2.10) From the decomposition of the tangent bundle of a lightlike submanifold, we have ∇_𝑋𝑃𝑌=∇^∗_𝑋𝑃𝑌+ℎ^∗(),∇𝑋,𝑃𝑌_𝑋𝜉=−𝐴^∗_𝜉𝑋+∇_𝑋^∗𝑡𝜉 (2.11)(2.12) for any 𝑋,𝑌∈Γ(𝑇𝑀), and 𝜉∈Γ(Rad𝑇𝑀). By using above equation we obtain 𝑔ℎ^𝑙(𝑋,𝑃𝑌),𝜉=𝑔𝐴^∗_𝜉𝑋,𝑃𝑌,𝑔ℎ^∗(𝑋,𝑃𝑌),𝑁=𝑔𝐴_𝑁𝑋,𝑃𝑌,𝑔ℎ^𝑙(𝑋,𝜉),𝜉=0,𝐴^∗_𝜉𝜉=0. (2.13)(2.14)

An odd dimensional semi-Riemannian manifold (𝑀,𝑔) is said to be an indefinite contact metric manifold [7] if there exists a (1,1) tensor field 𝜙, a vector field 𝑉, called the characteristic vector field, and its 1-form 𝜂 satisfying 𝑔𝜙𝑋,𝜙𝑌=𝑔(𝑋,𝑌)−𝜖𝜂(𝑋)𝜂(𝑌),𝑔()𝜙𝑉,𝑉=𝜖,₂𝑋=−𝑋+𝜂(𝑋)𝑉,𝑔(𝑋,𝑉)=𝜖𝜂(𝑋),∀𝑋,𝑌∈Γ(𝑇𝑀), (2.15)(2.16) where 𝜖=±1. It is not difficult to show that 𝜙𝑉=0,𝜂𝑜𝜙=0,𝜂(𝑉)=𝜖.

An indefinite almost contact metric manifold 𝑀 is said to be an indefinite Kenmotsu manifold [8] if ∇_𝑋𝑉=−𝑋+𝜂(𝑋)𝑉,∇_𝑋𝜙𝑌=−𝑔𝜙𝑋,𝑌𝑉+𝜖(𝑌)𝜙𝑋 (2.17)(2.18) for any 𝑋,𝑌∈Γ(𝑇𝑀).

Without loss of generality, we take 𝜖=1. For any vector field 𝑋 tangent to 𝑀, we put 𝜙𝑋=𝑇𝑋+𝜔𝑋, (2.19) where 𝑇𝑋 and 𝜔𝑋 are the tangential and transversal parts of 𝜙𝑋, respectively.

For any 𝑈∈Γ(tr(𝑇𝑀)), we have 𝜙𝑈=𝐵𝑈+𝐶𝑈, (2.20) where 𝐵𝑈 and 𝐶𝑈 are the tangential and transversal parts of 𝜙𝑈, respectively.

From now on, we denote (𝑀,𝑔,𝑆(𝑇𝑀),𝑆(𝑇𝑀^⟂)) by 𝑀 in this paper.

3. Geodesic Contact 𝑆𝐶𝑅-Lightlike Submanifolds

In this section, we study the geometric conditions under which the distributions involved in the definition of a contact 𝑆𝐶𝑅-lightlike submanifold 𝑀 and the submanifold itself are totally geodesic. We recall the following definition of contact 𝑆𝐶𝑅-lightlike submanifold of an indefinite Kenmotsu manifold given by Duggal and Sahin [5].

Definition 3.1. A lightlike submanifold 𝑀, tangent to structure vector field 𝑉, immersed in an indefinite Kenmotsu manifold (𝑀,𝑔) is said to be contact 𝑆𝐶𝑅-lightlike submanifold of 𝑀 if the following conditions are satisfied.  (a) There exists real nonnull distribution 𝐷 and 𝐷^⟂ such that 𝑆(𝑇𝑀)=𝐷⟂𝐷^⟂⟂{𝑉},𝜙𝐷^⟂⊂𝑆𝑇𝑀^⟂, (3.1)

𝐷∩𝐷^⟂=0, where 𝐷^⟂ is the orthogonal complementary to 𝐷⟂{𝑉} in 𝑆(𝑇𝑀). (b) 𝜙𝐷=𝐷,𝜙Rad𝑇𝑀=Rad𝑇𝑀,𝜙ltr(𝑇𝑀)=ltr(𝑇𝑀).

The tangent bundle of a contact 𝑆𝐶𝑅-lightlike submanifold is decomposed as 𝑇𝑀=𝐷⟂𝐷^⟂⟂{𝑉},𝐷=𝐷⟂Rad𝑇𝑀. (3.2) We will use the symbol 𝜇 to denote the orthogonal complement of 𝜙𝐷^⟂ in 𝑆(𝑇𝑀^⟂).

Definition 3.2. A contact 𝑆𝐶𝑅-lightlike submanifold 𝑀 of an indefinite Kenmostsu manifold 𝑀 is said to be(i)𝐷^⟂-totally geodesic contact 𝑆𝐶𝑅-lightlike submanifold if ℎ(𝑋,𝑌)=0 for any 𝑋,𝑌∈Γ(𝐷^⟂),(ii)mixed totally geodesic contact 𝑆𝐶𝑅-lightlike submanifold if ℎ(𝑋,𝑌)=0 for any 𝑋∈Γ((𝐷)⟂{𝑉}) and 𝑌∈Γ(𝐷^⟂).

Let 𝑀 be a contact 𝑆𝐶𝑅-lightlike submanifold of indefinite Kenmotsu manifold 𝑀 and let 𝑃 and 𝑄 be the projection morphisms on 𝐷 and 𝐷^⟂, respectively. Then for any vector field 𝑋 tangent to 𝑀, we can write 𝑋=𝑃𝑋+𝑄𝑋+𝜂(𝑋)𝑉. (3.3) Applying 𝜙 to (3.3) and using (2.19), we obtain 𝜙𝑋=𝑇𝑃𝑋+𝜔𝑄𝑋. (3.4) If we denote 𝑇𝑃𝑋 by 𝑇𝑋 and 𝜔𝑄𝑋 by 𝜔𝑋, then (3.4) can be rewritten as 𝜙𝑋=𝑇𝑋+𝜔𝑋, (3.5) where 𝑇𝑋∈Γ(𝐷) and 𝜔𝑋∈Γ(𝜙(𝐷^⟂))⊂𝑆(𝑇𝑀^⟂).

For any 𝑊∈Γ(𝑆(𝑇𝑀^⟂)), we have 𝜙𝑊=𝐵𝑊+𝐶𝑊, (3.6) where 𝐵𝑊∈Γ(𝐷^⟂) and 𝐶𝑊∈Γ(𝜇)⊂𝑆(𝑇𝑀^⟂).

In view of the above arguments, we are in a position to prove the following characterization theorem for the existence of a 𝐷^⟂-totally geodesic contact SCR-lightlike submanifold immersed in indefinite Kenmotsu manifolds.

Theorem 3.3. Let 𝑀 be a contact 𝑆𝐶𝑅−lightlike submanifold of an indefinite Kenmotsu manifolds 𝑀. Then 𝑀 is 𝐷^⟂-totally geodesic if and only if(i)ℎ^𝑠(𝑋,𝜙𝜉)∈𝜇.(ii)𝐴_𝜔𝑌𝑋∉𝐷^⟂ and ∇^𝑠_𝑋𝜔𝑌∉𝜇 for any 𝑋,𝑌∈Γ(𝐷^⟂).

Proof. Suppose that the contact 𝑆𝐶𝑅-lightlike submanifold 𝑀 is totally geodesic. Then we see that 𝑔(ℎ^𝑙(𝑋,𝑌),𝜉)=0 and 𝑔(ℎ^𝑠(𝑋,𝑌),𝑊)=0  for all,𝑋,𝑌∈Γ(𝐷^⟂).
Also, from (2.6) and (2.15), we obtain 𝑔ℎ^𝑙(𝑋,𝑌),𝜉=𝑔∇_𝑋𝜙𝑌−∇_𝑋𝜙𝑌,𝜙𝜉, (3.7) from which we derive 𝑔ℎ^𝑙(𝑋,𝑌),𝜉=𝑔𝐷^𝑙(𝑋,𝜔𝑌),𝜙𝜉, (3.8) where we have used (2.8), (2.18), and (3.5). Using (2.9) in the above equation, we get 𝑔ℎ^𝑙(𝑋,𝑌),𝜉=−𝑔ℎ^𝑠𝑋,𝜙𝜉,𝜔𝑌. (3.9) On the other hand, making use of (2.6), (2.8), (2.15), (2.18), (3.5), and (3.6), we arrive at 𝑔ℎ^𝑠(𝑋,𝑌),𝑊=−𝑔𝐴_𝜔𝑌𝑋,𝐵𝑊+𝑔∇^𝑠_𝑋𝜔𝑌,𝐶𝑊. (3.10) Hence, (i) and (ii) follows from (3.9) and (3.10) together with the fact that 𝑔(ℎ^𝑙(𝑋,𝑌),𝜉)=0 and 𝑔(ℎ^𝑠(𝑋,𝑌),𝑊)=0.
Converse part directly follow from (3.9) and (3.10).

The necessary and sufficient conditions for contact SCR-lightlike submanifolds to be mixed totally geodesic is given by the following theorem.

Theorem 3.4. Let 𝑀 be a contact SCR-lightlike submanifold of an indefinite Kenmotsu manifold 𝑀. Then 𝑀 is mixed totally geodesic if and only if(i)ℎ^𝑠(𝑋,𝜙𝜉)∈𝜇, (ii)𝐴_𝜔𝑌𝑋∉𝐷^⟂ and ∇^𝑠_𝑋𝜔𝑌∉𝜇 for any 𝑋∈Γ(𝐷⟂{𝑉}) and 𝑌∈Γ(𝐷^⟂).

Proof. Assume that 𝑀 is mixed totally geodesic. Then 𝑔(ℎ^𝑙(𝑋,𝑌),𝜉)=0,𝑔(ℎ^𝑙(𝑋,𝑌),𝑊)=0, for any 𝑋∈Γ(𝐷⟂{𝑉}) and 𝑌∈Γ(𝐷^⟂).
Moreover, using (2.6), (2.8), (2.15), (2.18), and (3.5) a direct calculation shows that 𝑔ℎ^𝑙(𝑋,𝑌),𝜉=𝑔𝐷^𝑙(𝑋,𝜔𝑌),𝜙𝜉. (3.11) From (2.9) and (3.11), we have 𝑔ℎ^𝑙(𝑋,𝑌),𝜉=−𝑔ℎ^𝑠𝑋,𝜙𝜉,𝜔𝑌. (3.12) On the other hand, using (2.6), (2.8), (2.15), (2.18), (3.5), and (3.6), we derive 𝑔ℎ^𝑠(𝑋,𝑌),𝑊=−𝑔𝐴_𝜔𝑌𝑋,𝐵𝑊+𝑔∇^𝑠_𝑋𝜔𝑌,𝐶𝑊. (3.13) Thus, (i) and (ii) follow from (3.12) and (3.13) along with 𝑔(ℎ^𝑙(𝑋,𝑌),𝜉)=0,𝑔(ℎ^𝑙(𝑋,𝑌),𝑊)=0.

Converse part directly follows from (3.12) and (3.13).

4. Screen Transversal Lightlike Submanifolds

We begin this section by recalling the following definitions from [6].

Definition 4.1. An 𝑟-lightlike submanifold 𝑀 of an indefinite Kenmotsu manifold 𝑀 is said to be screen transversal (𝑆𝑇) lightlike submanifold of 𝑀 if there exists a screen transversal bundle 𝑆(𝑇𝑀^⟂) such that 𝜙(Rad𝑇𝑀)⊂𝑆𝑇𝑀^⟂. (4.1)

Definition 4.2. An 𝑆𝑇-lightlike submanifold of an indefinite Kenmotsu manifold 𝑀 is said to be (i)radical 𝑆𝑇-lightlike submanifold if 𝑆(𝑇𝑀) is invariant with respect to 𝜙.(ii)𝑆𝑇-anti-invariant lightlike submanifold if 𝑆(𝑇𝑀) is screen transversal with respect to 𝜙, that is, 𝜙((𝑆(𝑇𝑀)))⊂𝑆𝑇𝑀^⟂. (4.2)

For a radical screen transversal lightlike submanifolds 𝑀 immersed in indefinite Kenmotsu manifold 𝑀, we will denote the projection morphisms of 𝑆(𝑇𝑀) and Rad𝑇𝑀 by 𝑃 and 𝑄, respectively. Then for 𝑋∈Γ(𝑇𝑀), we can write 𝑋=𝑃𝑋+𝑄𝑋. (4.3) We apply 𝜙 to (4.3) and then using (2.19), we obtain 𝜙𝑋=𝑇𝑃𝑋+𝜔𝑄𝑋. (4.4) Denoting 𝑇𝑃𝑋 by 𝑇𝑋 and 𝜔𝑄𝑋 by 𝜔𝑋, (4.4) can be we rewritten as 𝜙𝑋=𝑇𝑋+𝜔𝑋, (4.5) where 𝑇𝑋∈Γ(𝑆(𝑇𝑀)) and 𝜔𝑋∈Γ(𝜙(Rad𝑇𝑀))⊂𝑆(𝑇𝑀^⟂).

For 𝑊∈Γ(𝑆(𝑇𝑀^⟂)), we write 𝜙𝑊=𝐵𝑊+𝐶₁𝑊+𝐶₂𝑊, (4.6) where 𝐵𝑊∈Γ(Rad𝑇𝑀),𝐶₁𝑊∈Γ(ltr(𝑇𝑀)), and 𝐶₂𝑊∈Γ(𝜇) (𝜇 is the orthogonal complement of 𝜙(Rad𝑇𝑀)⊕𝜙(ltr(𝑇𝑀)) in 𝑆(𝑇𝑀^⟂)).

The geometric conditions under which the distribution Rad𝑇𝑀 is totally geodesic is given by the following theorem.

Theorem 4.3. Let 𝑀 be a radical screen transversal lightlike submanifold of an indefinite Kenmotsu manifold 𝑀. Then Rad𝑇𝑀is totally geodesic if and only if ℎ^𝑠(𝜉₁,𝐵𝑊)+𝐷^𝑠(𝜉₁,𝐶₁𝑊)+∇^𝑠_𝜉1𝐶₂𝑊 has no component in 𝜙(ltr(𝑇𝑀))⟂𝜇.

Proof. If the distribution Rad𝑇𝑀 is totally geodesic, then 𝑔ℎ^𝑙𝜉₁,𝜉₂,𝜉₃=0,𝑔ℎ^𝑠𝜉₁,𝜉₂,𝑊=0 (4.7) for any 𝜉₁,𝜉₂,𝜉₃∈Γ(Rad𝑇𝑀),𝑊∈Γ(𝑆(𝑇𝑀^⟂)) and ℎ^𝑙=0 on Rad𝑇𝑀 [9]. On the other hand, using (2.6), (2.15), (2.18), and (4.6), we arrive at 𝑔ℎ^𝑠𝜉₁,𝜉₂,𝑊=−𝑔𝜙𝜉₂,ℎ^𝑠𝜉₁,𝐵𝑊+𝐷^𝑠𝜉₁,𝐶₁𝑊+∇^𝑠_𝜉1𝐶₂𝑊. (4.8) Thus, our assertion follows from (4.8) and (4.7).
Converse part directly follows from (4.8) and (4.7).

For the screen distribution 𝑆(𝑇𝑀) to be totally geodesic in 𝑀, we have the following.

Theorem 4.4. Let 𝑀 be a radical screen transversal lightlike submanifold of an indefinite Kenmotsu manifold 𝑀. Then, 𝑆(𝑇𝑀) is totally geodesic if and only if 𝐴_𝜔𝜉𝑋,𝐴^∗_𝐵𝑊𝑋+𝐴_𝐶1𝑊𝑋+𝐴_𝐶2𝑊𝑋∉𝑆(𝑇𝑀)  for all  𝑋∈Γ(𝑆(𝑇𝑀)),𝜉∈Γ(Rad𝑇𝑀)  and  𝑊∈Γ(𝑆(𝑇𝑀^⟂)).

Proof. Suppose that the distribution 𝑆(𝑇𝑀) is totally geodesic. Then 𝑔ℎ^𝑙(𝑋,𝑌),𝜉=0,𝑔ℎ^𝑠(𝑋,𝑌),𝑊=0 (4.9) for any 𝑋,𝑌∈Γ(𝑆(𝑇𝑀)),𝜉∈Γ(Rad𝑇𝑀), and 𝑊∈Γ(𝑆(𝑇𝑀^⟂)). Using (2.6), (2.8), (2.15), (2.18), and (4.5), a direct calculation shows that 𝑔ℎ^𝑙(𝑋,𝑌),𝜉=𝑔𝑇𝑌,𝐴_𝜔𝜉𝑋. (4.10) On the other hand, from (2.6), (2.7), (2.12), (2.15), (2.18), (4.5), and (4.6), we obtain 𝑔ℎ^𝑠(𝑋,𝑌),𝑊=𝑔𝑇𝑌,𝐴^∗_𝐵𝑊𝑋+𝐴_𝐶1𝑊𝑋+𝐴_𝐶2𝑊𝑋. (4.11) Thus, our assertion follows from (4.9),(4.10),  and (4.11).
Converse part directly follows from (4.10) and (4.11).

In respect of a radical screen transversal lightlike submanifold to be mixed totally geodesic and totally geodesic, we have the following two theorems.

Theorem 4.5. Let 𝑀 be a radical screen transversal lightlike submanifold of 𝑀. Then 𝑀 is mixed totally geodesic if and only if 𝐴_𝜔𝜉𝑋∉(Rad𝑇𝑀),∇^𝑠_𝑋𝜔𝜉∉𝜇 and 𝐷^𝑙(𝑋,𝜔𝜉)=0forall,𝑋∈Γ(𝑆(𝑇𝑀)),𝜉∈Γ(Rad𝑇𝑀).

Proof. Assume that the submanifold 𝑀 is mixed geodesic. Then 𝑔ℎ^𝑙(𝑋,𝜉),𝜉=0,𝑔ℎ^𝑠(𝑋,𝜉),𝑊=0 (4.12) for any 𝑋∈Γ(𝑆(𝑇𝑀)),𝜉∈Γ(Rad𝑇𝑀), and 𝑊∈Γ(𝑆(𝑇𝑀^⟂)). Also, from (2.14) we have 𝑔ℎ^𝑙(𝑋,𝜉),𝜉=0. (4.13) On the other hand, by the use of (2.6), (2.8), (2.15), (2.18), and (4.6), we obtain 𝑔ℎ^𝑠(𝑋,𝜉),𝑊=−𝑔𝐴_𝜔𝜉𝑋,𝐶₁𝑊+𝑔∇^𝑠_𝑋𝜔𝜉,𝐶₂𝑊+𝑔𝐷^𝑙(𝑋,𝜔𝜉),𝐵𝑊. (4.14) Thus, our assertion follows from (4.12) and (4.14).
Converse part directly follows from (4.14) and the fact that 𝑔(ℎ^𝑙(𝑋,𝜉),𝜉)=0.

Theorem 4.6. Let 𝑀 be a radical screen transversal lightlike submanifold of 𝑀. Then 𝑀 is totally geodesic if and only if(i)𝐴_𝜙𝜉𝑋∉Γ(𝑆(𝑇𝑀)) and ∇^𝑠_𝑋𝜙𝜉∉𝜙(ltr(𝑇𝑀))⟂𝜇,(ii)ℎ^𝑙(𝑋,𝑇𝑌)+𝐷^𝑙(𝑋,𝜔𝑌)∉(ltr(𝑇𝑀)),ℎ^∗(𝑋,𝑇𝑌)−𝐴_𝜔𝑌𝑋∉Rad𝑇𝑀  and  ℎ^𝑠(𝑋,𝑇𝑌)+∇^𝑠_𝑋𝜔𝑌∉𝜇for any 𝑋,𝑌∈Γ(𝑇𝑀) and 𝜉∈Γ(Rad𝑇𝑀).

Proof. If the submanifold 𝑀 is totally geodesic, then 𝑔ℎ^𝑙(𝑋,𝑌),𝜉=0,𝑔ℎ^𝑠(𝑋,𝑌),𝑊=0 (4.15) for any 𝑋,𝑌∈Γ(𝑇𝑀),𝜉∈Γ(Rad𝑇𝑀), and 𝑊∈Γ(𝑆(𝑇𝑀^⟂)). Also, making use of (2.6), (2.8), (2.15), (2.17), (2.18), and (4.5), a direct calculation shows that 𝑔ℎ^𝑙(𝑋,𝑌),𝜉=𝑔𝑇𝑌,𝐴_𝜙𝜉𝑋−𝑔∇^𝑠_𝑋𝜙𝜉,𝜔𝑌. (4.16) On the other hand, from (2.6), (2.8), (2.11), (2.15), (2.18), (4.5), and (4.6), we obtain 𝑔ℎ^𝑠(𝑋,𝑌),𝑊=𝑔ℎ^𝑙(𝑋,𝑇𝑌)+𝐷^𝑙()ℎ𝑋,𝜔𝑌,𝐵𝑊+𝑔^∗(𝑋,𝑇𝑌)−𝐴_𝜔𝑌𝑋,𝐶₁𝑊+𝑔ℎ^𝑠(𝑋,𝑇𝑌)+∇^𝑠_𝑋𝜔𝑌,𝐶₂𝑊. (4.17) Thus, (i) and (ii) follow from (4.16), (4.17), and (4.15).
Converse part directly follows from (4.16) and (4.17).
For a 𝑆𝑇-anti-invariant lightlike submanifold 𝑀 immersed in 𝑀, if we denote the projection morphism of 𝑆(𝑇𝑀) and Rad𝑇𝑀 by 𝑃 and 𝑄, respectively, then for any vector field tangent to 𝑀 we can write 𝑋=𝑃𝑋+𝑄𝑋. (4.18) By applying 𝜙 to (4.18) and then using (2.19), we obtain 𝜙𝑋=𝜔𝑃𝑋+𝜔𝑄𝑋. (4.19)
Denoting 𝜔𝑃𝑋 by 𝜔₂ and 𝜔𝑄𝑋 by 𝜔₁. Then (4.19) can be rewritten as 𝜙𝑋=𝜔₁𝑋+𝜔₂𝑋, (4.20) where 𝜔₁𝑋∈Γ(ltr(𝑇𝑀)) and 𝜔₂𝑋∈Γ(𝑆(𝑇𝑀^⟂)).
For 𝑊∈Γ(𝑆(𝑇𝑀^⟂)), writing 𝜙𝑊=𝐵₁𝑊+𝐵₂𝑊+𝐶₁𝑊+𝐶₂𝑊, (4.21) where 𝐵₁𝑊∈Γ(Rad𝑇𝑀),𝐵₂𝑊∈Γ(𝑆(𝑇𝑀)),𝐶₁𝑊∈Γ(ltr(𝑇𝑀)), and 𝐶₂𝑊∈Γ(𝜇) (𝜇 is the orthogonal complement of {𝜙(Rad𝑇𝑀)⊕𝜙(ltr(𝑇𝑀))}⊕_ortho𝜙(𝑆(𝑇𝑀) in 𝑆(𝑇𝑀^⟂)).

In view of the above discussions, the conditions under which the distribution Rad𝑇𝑀 of a ST-anti-invariant lightlike submanifold immersed in indefinite Kenmotsu manifolds to be totally geodesic is given by the following result.

Theorem 4.7. Let 𝑀 be a ST-anti-invariant lightlike submanifold of an indefinite Kenmotsu manifold 𝑀. Then the distribution Rad𝑇𝑀 is totally geodesic if and only if 𝐴_𝜔𝜉2𝜉₁=0,∇^𝑠_𝜉1𝜔𝜉₂∉𝜇 and 𝐷^𝑙(𝜉₁,𝜙𝜉₂)=0, for any 𝜉₁,𝜉₂∈Γ(Rad𝑇𝑀).

Proof. Suppose that the distribution Rad𝑇𝑀 is totally geodesic. Then we see that 𝑔ℎ^𝑙𝜉₁,𝜉₂,𝜉=0,𝑔ℎ^𝑠𝜉₁,𝜉₂,𝑊=0 (4.22) for any 𝜉₁,𝜉₂∈Γ(Rad𝑇𝑀) and 𝑊∈Γ(𝑆(𝑇𝑀^⟂). Recall that ℎ^𝑙=0 on Rad𝑇𝑀 [9]. On the other hand, by the use of (2.6), (2.8), (2.15), (4.20), and (4.21), we obtain 𝑔ℎ^𝑠𝜉₁,𝜉₂,𝑊=−𝑔𝐴_𝜔𝜉2𝜉₁,𝐵₂𝑊+𝐶₁𝑊+𝑔∇^𝑠_𝜉2𝜔𝜉₂,𝐶₂𝑊+𝑔𝐷^𝑙𝜉₁,𝜙𝜉₂,𝐵₁𝑊. (4.23) Thus, our assertion follows from (4.22) and (4.23).
Converse part directly follows from (4.22) and (4.23).

For the screen distribution 𝑆(𝑇𝑀) of a ST-anti-invariant lightlike submanifold to be totally geodesic, we have the following.

Theorem 4.8. Let 𝑀 be a ST-anti-invariant lightlike submanifold of an indefinite Kenmotsu manifold 𝑀. Then the distribution 𝑆(𝑇𝑀) is totally geodesic if and only if ∇^𝑠_𝑋𝜔𝑌∉𝜙(ltr(𝑇𝑀))⟂𝜇,𝐴_𝜔𝑌𝑋=0 and 𝐷^𝑙(𝑋,𝜔𝑌)=0, for any 𝑋,𝑌∈Γ(𝑆(𝑇𝑀)).

Proof. If 𝑆(𝑇𝑀) is totally geodesic, then 𝑔ℎ^𝑙(𝑋,𝑌),𝜉=0,𝑔ℎ^𝑠(𝑋,𝑌),𝑊=0, (4.24) for any 𝑋,𝑌∈Γ(𝑆(𝑇𝑀)),𝜉∈Γ(Rad𝑇𝑀) and 𝑊∈Γ(𝑆(𝑇𝑀^⟂)).
On the other hand, using (2.6), (2.15), (2.18), and (4.20), we obtain 𝑔ℎ^𝑙(𝑋,𝑌),𝜉=𝑔∇^𝑠_𝑋𝜔𝑌,𝜔𝜉. (4.25) Also, 𝑔ℎ^𝑠()=𝑋,𝑌,𝑊𝑔−𝐴_𝜔𝑌𝑋,𝐵₂𝑊+𝐶₁𝑊+𝑔∇^𝑠_𝑋𝜔𝑌,𝐶₂𝑊+𝑔𝐷^𝑙(𝑋,𝜔𝑌),𝐵₁𝑊, (4.26) where we have used (2.6), (2.8), (2.15), (2.18), (4.20), and (4.21). Thus, our assertion follows from (4.25), (4.26), and (4.24).
Converse part directly follows from (4.25) and (4.26).

The necessary and sufficient conditions for a ST-anti-invariant lightlike submanifold to be mixed totally geodesic, we have the following.

Theorem 4.9. Let 𝑀 be a ST-anti-invariant lightlike submanifold of an indefinite Kenmotsu manifold 𝑀. Then 𝑀 is mixed totally geodesic if and only if 𝐴_𝜔𝜉𝑋=0,𝐷^𝑙(𝑋,𝜔𝜉)=0 and ∇^𝑠_𝑋𝜔𝜉∉𝜇. for any 𝑋∈Γ(𝑆(𝑇𝑀)) and 𝜉∈Γ(Rad𝑇𝑀).

Proof. Assume that the submanifold 𝑀 is mixed gedesic. Then 𝑔ℎ^𝑙(𝑋,𝜉),𝜉=0,𝑔ℎ^𝑙(𝑋,𝜉),𝑊=0 (4.27) for any 𝑋∈Γ(𝑆(𝑇𝑀)),𝜉∈Γ(Rad𝑇𝑀), and 𝑊∈Γ(𝑆(𝑇𝑀^⟂)). By virtue of (2.14), we have 𝑔ℎ^𝑙(𝑋,𝜉),𝜉=0. (4.28) On the other hand, using (2.6), (2.8), (2.15), (2.18), (4.20), and (4.21), we get 𝑔ℎ^𝑠()=𝑋,𝜉,𝑊𝑔−𝐴_𝜔𝜉𝑋,𝐵₂𝑊+𝐶₁𝑊+𝑔∇^𝑠_𝑋𝜔𝜉,𝐶₂𝑊+𝑔𝐷^𝑙(𝑋,𝜔𝜉),𝐵₁𝑊. (4.29) Thus, our assertion follows from (4.27) and (4.29).
Converse part directly follows from (4.29).

Now, we prove the following.

Theorem 4.10. Let 𝑀 be a ST-anti-invariant lightlike submanifold of an indefinite Kenmotsu manifold 𝑀. Then 𝑀 is totally geodesic if and only if ∇^𝑠_𝑋𝜔𝑌∉𝜇,𝐴_𝜔𝑌𝑋=0 and 𝐷^𝑙(𝑋,𝜔𝑌)=0, for any 𝑋,𝑌∈Γ(𝑇𝑀).

Proof. The submanifold 𝑀 is totally geodesic if and only if 𝑔ℎ^𝑙(𝑋,𝑌),𝜉=0,𝑔ℎ^𝑠(𝑋,𝑌),𝑊=0 (4.30) for any 𝑋,𝑌∈Γ(𝑇𝑀),𝜉∈Γ(Rad𝑇𝑀) and 𝑊∈Γ(𝑆(𝑇𝑀^⟂)). By the use of (2.6), (2.8), (2.15), (2.18), and (4.20), we obtain 𝑔ℎ^𝑙(𝑋,𝑌),𝜉=𝑔∇^𝑠_𝑋𝜔𝑌,𝜔𝜉. (4.31) On the other hand, from (2.6), (2.8), (2.15), (4.20), and (4.21), we have 𝑔ℎ^𝑠(𝑋,𝑌),𝑊=−𝑔𝐴_𝜔𝑌𝑋,𝐵₂𝑊+𝐶₁𝑊+𝑔∇^𝑠_𝑋𝜔𝑌,𝐶₂𝑊+𝑔𝐷^𝑙(𝑋,𝜔𝑌),𝐵₁𝑊. (4.32) Thus, our assertion follows from (4.31), (4.32), and (4.30).
Converse part directly follows from (4.31) and (4.32).

5. Transversal Lightlike Submanifolds

The purpose of this section is to study transversal and radical transversal lightlike submanifolds in an indefinite Kenmotsu manifold. We recall here the definitions of these submanifolds given by Yıldırım and Sahin [10].

Definition 5.1. A lightlike submanifold 𝑀 tangent to structure vector field 𝑉 immersed in an indefinite Kenmotsu manifold 𝑀 is said to be(i)radical transversal lightlike submanifold of 𝑀 if 𝜙(Rad𝑇𝑀)=ltr(𝑇𝑀),𝜙(𝑆(𝑇𝑀))=𝑆(𝑇𝑀), (5.1)(ii)transversal lightlike submanifold of 𝑀 if 𝜙(Rad𝑇𝑀)=ltr(𝑇𝑀),𝜙(𝑆(𝑇𝑀))⊆𝑆𝑇𝑀^⟂. (5.2)
For a radical transversal lightlike submanifold 𝑀 of an indefinite Kenmotsu manifold 𝑀, if 𝑃 and 𝑄 are the projection morphism on 𝑆(𝑇𝑀) and Rad𝑇𝑀, respectively, then any vector field 𝑋 tangent to 𝑀 can be written as 𝑋=𝑃𝑋+𝑄𝑋. (5.3) We apply 𝜙 to (5.3) and then using (2.19), we get 𝜙𝑋=𝑇𝑃𝑋+𝜔𝑄𝑋. (5.4) If we denote 𝑇𝑃𝑋 by 𝑇𝑋 and 𝜔𝑄𝑋 by 𝜔𝑋, then (5.4) can be rewritten as 𝜙𝑋=𝑇𝑋+𝜔𝑋, (5.5) where 𝑇𝑋∈Γ(𝑆(𝑇𝑀)) and 𝜔𝑋∈Γ(ltr(𝑇𝑀)).
Moreover, if 𝑊∈Γ(𝑆(𝑇𝑀^⟂)), then 𝜙𝑊=𝐶𝑊, (5.6) from which we observe that 𝜙𝑊∈Γ(𝑆(𝑇𝑀^⟂)).

Using the above notations, one can prove the following.

Theorem 5.2. Let 𝑀 be a radical transversal lightlike submanifold of an indefinite Kenmotsu manifold 𝑀. Then the distribution Rad𝑇𝑀 is totally geodesic if and only if 𝐴_𝐶𝑊𝜉₁∉Γ(Rad𝑇𝑀) for any 𝜉₁∈Γ(Rad𝑇𝑀) and 𝑊∈Γ(𝑆(𝑇𝑀^⟂)).

Proof. Since ℎ^𝑙=0 on Rad𝑇𝑀 [9], we observe that the distribution Rad𝑇𝑀 is totally geodesic if and only if 𝑔ℎ^𝑙𝜉₁,𝜉₂,𝜉=0,𝑔ℎ^𝑠𝜉₁,𝜉₂,𝑊=0 (5.7) for any 𝜉₁∈Γ(Rad𝑇𝑀) and 𝑊∈Γ(𝑆(𝑇𝑀^⟂)). On the other hand, using (2.6), (2.8), (2.15), (2.18), (5.5), and (5.6), we arrive at 𝑔ℎ^𝑠𝜉₁,𝜉₂,𝑊=𝑔𝜔𝜉₂,𝐴_𝐶𝑊𝜉₁. (5.8) Thus, our assertion follows from (5.7) and (5.8).
Converse part directly follows from (5.8).

A screen distribution 𝑆(𝑇𝑀) of a radical transversal lightlike submanifold in indefinite Kenmotsu manifolds to be totally geodesic, we have the following.

Theorem 5.3. Let 𝑀 be a radical transversal lightlike submanifold of an indefinite Kenmotsu manifold 𝑀. Then the distribution 𝑆(𝑇𝑀) is totally geodesic if and only if ℎ^∗(𝑋,𝑇𝑌)=0 and 𝐴_𝐶𝑊𝑋∉Γ(𝑆(𝑇𝑀)) for any 𝑋,𝑌∈Γ(𝑆(𝑇𝑀)) and 𝑊∈Γ(𝑆(𝑇𝑀^⟂)).

Proof. We note that the distribution 𝑆(𝑇𝑀) is totally geodesic if and only if 𝑔ℎ^𝑙(𝑋,𝑌),𝜉=0,𝑔ℎ^𝑠(𝑋,𝑌),𝑊=0 (5.9) for any 𝑋,𝑌∈Γ(𝑆(𝑇𝑀)),𝜉∈Γ(Rad𝑇𝑀), and 𝑊∈Γ(𝑆(𝑇𝑀^⟂)). Making use of (2.6), (2.11), (2.15), (2.18), and (5.5), we get 𝑔ℎ^𝑙(𝑋,𝑌),𝜉=𝑔ℎ^∗(𝑋,𝑇𝑌),𝜔𝜉. (5.10) On the other hand, from (2.6), (2.8), (2.15), (2.18), (5.5), and (5.6), we have 𝑔ℎ^𝑠(𝑋,𝑌),𝑊=𝑔𝑇𝑌,𝐴_𝐶𝑊𝑋. (5.11) Thus, our assertion follows from (5.10), (5.11), and (5.9).
Converse part directly follows from (5.10) and (5.11).

The conditions under which a radical transversal lightlike submanifold immersed in indefinite Kenmotsu manifolds to be mixed totally geodesic is given by the following theorem.

Theorem 5.4. Let 𝑀 be a radical transversal lightlike submanifold of an indefinite Kenmotsu manifold 𝑀. Then 𝑀 is mixed totally geodesic if and only if 𝐴_𝐶𝑊𝑋∉Γ(Rad𝑇𝑀) for any 𝑋∈Γ(𝑆(𝑇𝑀)) and 𝑊∈Γ(𝑆(𝑇𝑀^⟂)).

Proof. The submanifold 𝑀 is mixed totally geodesic if and only if 𝑔ℎ^𝑙(𝑋,𝑌),𝜉=0,𝑔ℎ^𝑠(𝑋,𝜉),𝑊=0 (5.12) for any 𝑋∈Γ(𝑆(𝑇𝑀)),𝜉∈Γ(Rad𝑇𝑀), and 𝑊∈Γ(𝑆(𝑇𝑀^⟂)). From (2.14), we have 𝑔ℎ^𝑙(𝑋,𝜉),𝑊=0. (5.13) On the other hand, using (2.6), (2.8), (2.15), (2.18), (5.5), and (5.6), we obtain 𝑔ℎ^𝑠(𝑋,𝑌),𝑊=𝑔𝜔𝜉,𝐴_𝐶𝑊𝑋. (5.14) Thus, our assertion follows from (5.12) and (5.14).
Converse part directly follows from (5.14).

Theorem 5.5. Let 𝑀 be a radical transversal lightlike submanifold of an indefinite Kenmotsu manifold 𝑀. Then the submanifold 𝑀 is totally geodesic if and only if (i)ℎ^∗(𝑋,𝑇𝑌)−𝐴_𝜔𝑌𝑋∉Γ(Rad𝑀).(ii)𝐴_𝐶𝑊𝑋=0
for any 𝑋,𝑌∈Γ(𝑇𝑀).

Proof. We observe that the submanifold 𝑀 is totally geodesic if and only if 𝑔ℎ^𝑙(𝑋,𝑌),𝜉=0,𝑔ℎ^𝑠(𝑋,𝑌),𝑊=0 (5.15) for any 𝑋,𝑌∈Γ(𝑇𝑀),𝜉∈Γ(Rad𝑇𝑀), and 𝑊∈Γ(𝑆(𝑇𝑀^⟂)).
By the use of (2.6), (2.8), (2.11), (2.15), (2.18), (5.5), and (5.6), we arrive at 𝑔ℎ^𝑙(𝑋,𝑌),𝜉=𝑔ℎ^∗(𝑋,𝑇𝑌),𝜔𝜉−𝑔𝐴_𝜔𝑌𝑋,𝜔𝜉. (5.16) On the other hand, from (2.6), (2.8), (2.15), (2.18), (5.5), and (5.6), we have 𝑔ℎ^𝑠(𝑋,𝑌),𝑊=𝑔𝑇𝑌,𝐴_𝐶𝑊𝑋+𝑔𝜔𝑌,𝐴_𝐶𝑊𝑋. (5.17) Thus, our assertion follows from (5.15), (5.16), and (5.17).
Converse part directly follows from (5.16), and (5.17).

If we denote the projections on the distributions 𝑆(𝑇𝑀) and Rad𝑇𝑀 involved with the definition of a transversal lightlike submanifold 𝑀 immersed in indefinite Kenmotsu manifold 𝑀 by 𝑃 and 𝑄, respectively, then any vector field 𝑋 tangent to 𝑀 can be written as 𝑋=𝑃𝑋+𝑄𝑋. (5.18) Applying 𝜙 to (5.18) and then using (2.19), we get 𝜙𝑋=𝜔𝑃𝑋+𝜔𝑄𝑋. (5.19) If we denote 𝜔𝑄𝑋 by 𝜔₁ and 𝜔𝑃𝑋 by 𝜔₂, then (5.19) can be written as 𝜙𝑋=𝜔₂𝑋+𝜔₁𝑋, (5.20) where 𝜔₁∈Γ(ltr(𝑇𝑀)) and 𝜔₂𝑋∈Γ(𝑆(𝑇𝑀^⟂)).

For 𝑊∈Γ(𝑆(𝑇𝑀^⟂)), we have 𝜙𝑊=𝐵𝑊+𝐶𝑊, (5.21) where 𝐵𝑊∈Γ(𝑆(𝑇𝑀)) and 𝐶𝑊∈Γ(𝜇) (𝜇 is the orthogonal complement of 𝜙(𝑆(𝑇𝑀)) in 𝑆(𝑇𝑀^⟂)).

Theorem 5.6. Let 𝑀 be a transversal lightlike submanifold of an indefinite Kenmotsu manifold 𝑀. Then the distribution Rad𝑇𝑀 is totally geodesic if and only if 𝐴_𝜔𝜉2𝜉₁∉𝑆(𝑇𝑀) and 𝐷^𝑠(𝜉₁,𝜙𝜉₂)∈(𝜇) for any 𝜉₁,𝜉₂∈Γ(Rad𝑇𝑀).

Proof. The distribution Rad𝑇𝑀 is totally geodesic if and only if 𝑔ℎ^𝑙𝜉₁,𝜉₂,𝜉=0,𝑔ℎ^𝑠𝜉₁,𝜉₂,𝜉=0 (5.22) for any 𝜉₁,𝜉₂,𝜉∈Γ(Rad𝑇𝑀). In view of ℎ^𝑙=0 on Rad𝑇𝑀[9], we have 𝑔ℎ^𝑙𝜉₁,𝜉₂,𝜉=0. (5.23) On the other hand, making use of (2.6), (2.7), (2.15), (2.18), (5.20), and (5.21), we get 𝑔ℎ^𝑠𝜉₁,𝜉₂,𝑊=−𝑔𝐴_𝜔𝜉2𝜉₁,𝐵𝑊+𝑔𝐷^𝑠𝜉₁,𝜙𝜉₂,𝐶𝑊. (5.24) Thus, our assertion follows from (5.22), (5.23), and (5.24).
Converse part directly follows from (5.23) and (5.24).

Theorem 5.7. Let 𝑀 be a transversal lightlike submanifold of an indefinite Kenmotsu manifold 𝑀. Then 𝑆(𝑇𝑀) is totally geodesic if and only if 𝐷^𝑠𝑋,𝜔₁𝜉,∇^𝑠_𝑋𝜔₂𝑌∉𝜇,𝐴_𝜔2𝑌𝑋∉Γ(𝑆(𝑇𝑀)) (5.25) for any 𝑋,𝑌∈Γ(𝑆(𝑇𝑀)).

Proof. We note that the distribution 𝑆(𝑇𝑀) is totally geodesic if and only if 𝑔(ℎ^𝑙(𝑋,𝑌),𝜉)=0 and 𝑔(ℎ^𝑠(𝑋,𝑌),𝑊)=0 for any 𝑋,𝑌∈Γ(𝑆(𝑇𝑀)),𝜉∈Γ(Rad𝑇𝑀), and 𝑊∈Γ(𝑆(𝑇𝑀^⟂)).
Combining (2.6), (2.7), (2.15), (2.18), (5.20), and (5.21), we obtain 𝑔ℎ^𝑙(𝑋,𝑌),𝜉=−𝑔𝜔₂𝑌,𝐷^𝑠𝑋,𝜔₁𝜉. (5.26) On the other hand, from (2.6), (2.8), (2.15), (2.18), (5.20), and (5.21), we have 𝑔ℎ^𝑠(𝑋,𝑌),𝑊=−𝑔−𝐴_𝜔2𝑌𝑋,𝐵𝑊+𝑔∇^𝑠_𝑋𝜔₂𝑌,𝐶𝑊. (5.27) Thus, our assertion follows from (5.25), (5.26), and (5.27).
Converse part directly follows from (5.26) and (5.27).

Theorem 5.8. Let 𝑀 be a transversal lightlike submanifold of an indefinite Kenmotsu manifold 𝑀. Then 𝑀 is mixed totally geodesic if and only if 𝐴_𝜔1𝜉𝑋∉Γ(𝑆(𝑇𝑀)) and 𝐷^𝑠(𝑋,𝜔₁𝜉)∉(𝜇) for any 𝑋∈Γ(𝑆(𝑇𝑀)) and 𝜉∈Γ(Rad𝑇𝑀).

Proof. We observe that the submanifold 𝑀 is mixed totally geodesic if and only if 𝑔ℎ^𝑙(𝑋,𝜉),𝜉=0,𝑔ℎ^𝑠(𝑋,𝜉),𝑊=0 (5.28) for all 𝑋∈Γ(𝑆(𝑇𝑀)),𝜉∈Γ(Rad𝑇𝑀) and 𝑊∈Γ(𝑆(𝑇𝑀^⟂)).
From (2.14), we infer that 𝑔ℎ^𝑙(𝑋,𝜉),𝜉=0. (5.29) On the other hand, by the use of (2.6), (2.7), (2.15), (2.18), (5.20), and (5.21), we arrive at 𝑔ℎ^𝑠(𝑋,𝜉),𝑊=−𝑔𝐴_𝜔1𝜉𝑋,𝐵𝑊+𝑔𝐷^𝑠𝑋,𝜔₁𝜉,𝐶𝑊. (5.30) Thus, our assertion follows from (5.28), (5.29), and (5.30).
Converse part directly follows from (5.29) and (5.30).

Theorem 5.9. Let 𝑀 be a transversal lightlike submanifold of an indefinite Kenmotsu manifold 𝑀. Then 𝑀 is totally geodesic if and only 𝐴_𝜔𝑌𝑋=0 and 𝐷^𝑠(𝑋,𝜔₁𝑌)+∇^𝑠_𝑋𝜔₂𝑌∉Γ(𝜇)for all 𝑋,𝑌∈Γ(𝑇𝑀).

Proof. The submanifold 𝑀 is totally geodesic if and only if 𝑔ℎ^𝑙(𝑋,𝑌),𝜉=0,𝑔ℎ^𝑠(𝑋,𝑌),𝑊=0 (5.31) for any 𝑋,𝑌∈Γ(𝑇𝑀),𝜉∈Γ(Rad𝑇𝑀) and 𝑊∈Γ(𝑆(𝑇𝑀^⟂)).
By virtue of (2.6), (2.7), (2.8), (2.15), (2.18), (5.20) and (5.21), we have 𝑔ℎ^𝑙(𝑋,𝑌),𝜉=𝑔−𝐴_𝜔𝑌𝑋,𝜙𝜉. (5.32) On the other hand, by the use of (2.6), (2.7), (2.8), (2.15), (2.18), (5.20), and (5.21), we get 𝑔ℎ^𝑠(𝑋,𝑌),𝜉=𝑔−𝐴_𝜔𝑌𝑋,𝐵𝑊+𝑔𝐷^𝑠𝑋,𝜔₁𝑌+∇^𝑠_𝑋𝜔₂𝑌,𝐶𝑊. (5.33) Thus, our assertion follows from (5.31), (5.32), and (5.33).
Converse part directly follows from (5.32), and (5.33).