Nonexistence of Totally Contact Umbilical Slant Lightlike Submanifolds of Indefinite Cosymplectic Manifolds

Abstract

We study totally contact umbilical slant lightlike submanifolds of indefinite cosymplectic manifolds. We prove that there do not exist totally contact umbilical proper slant lightlike submanifolds in indefinite cosymplectic manifolds other than totally contact geodesic proper slant lightlike submanifolds. We also prove that there do not exist totally contact umbilical proper slant lightlike submanifolds of indefinite cosymplectic space forms. Finally we give characterization theorems on minimal slant lightlike submanifolds.

1. Introduction

The notion of slant submanifolds was initiated by Chen [1, 2], as a generalization of both holomorphic and totally real submanifolds in complex geometry. Since then such submanifolds have been studied by many authors. Lotta [3, 4] defined and studied slant submanifolds in contact geometry. Cabrerizo et al. studied slant, semislant, and bislant submanifolds in contact geometry [5, 6]. They all studied the geometry of slant submanifolds with positive definite metric. Therefore this geometry may not be applicable to the other branches of mathematics and physics, where the metric is not necessarily definite. Thus the geometry of slant submanifolds with indefinite metric became a topic of chief discussion, and Şahin [7] played a very crucial role in this direction by introducing the notion of slant lightlike submanifolds of indefinite Hermitian manifolds. Recently Gupta et al. [8] introduced the notion of slant lightlike submanifolds of an indefinite cosymplectic manifold and obtained necessary and sufficient conditions for the existence of slant lightlike submanifolds. In [9], Jain et al. studied GCR-lightlike submanifolds of indefinite cosymplectic manifolds and proved that every totally contact umbilical GCR-lightlike submanifold of an indefinite cosymplectic manifold is a totally contact geodesic GCR-lightlike submanifold.

In this paper we prove that there do not exist totally contact umbilical proper slant lightlike submanifolds of indefinite cosymplectic manifolds other than totally contact geodesic. We also prove that there do not exist totally contact umbilical proper slant lightlike submanifolds of indefinite cosymplectic space forms. Finally, we give characterization theorems on minimal slant lightlike submanifolds.

2. Preliminaries

An odd-dimensional semi-Riemannian manifold 𝑀 is said to be an indefinite almost contact metric manifold if there exist structure tensors (𝜙,𝑉,𝜂,𝑔), where 𝜙 is a (1,1) tensor field, 𝑉 is a vector field, called structure vector field, 𝜂 is a 1-form, and 𝑔 is the semi-Riemannian metric on 𝑀 satisfying (see [10]) the following: 𝜙²𝑋=−𝑋+𝜂(𝑋)𝑉,𝜂∘𝜙=0,𝜙𝑉=0,𝜂(𝑉)=1,𝑔𝜙𝑋,𝜙𝑌=𝑔(𝑋,𝑌)−𝜂(𝑋)𝜂(𝑌),𝑔(𝑋,𝑉)=𝜂(𝑋), (1)(2) for 𝑋,𝑌∈Γ(𝑇𝑀), where Γ(𝑇𝑀) denotes the Lie algebra of vector fields on 𝑀.

An indefinite almost contact metric manifold 𝑀 is called an indefinite cosymplectic manifold if (see [11]) ∇_𝑋𝜙𝑌=0,∇_𝑋𝑉=0, (3) for any 𝑋,𝑌∈Γ(𝑇𝑀), where ∇ denote the Levi-Civita connection on 𝑀.

Let (𝑀,𝑔) be a real (𝑚+𝑛)-dimensional semi-Riemannian manifold of constant index 𝑞 such that 𝑚,𝑛≥1, 1≤𝑞≤𝑚+𝑛−1, let (𝑀,𝑔) be an 𝑚-dimensional submanifold of 𝑀, and let 𝑔 be the induced metric of 𝑔 on 𝑀. If 𝑔 is degenerate on the tangent bundle 𝑇𝑀 of 𝑀, then 𝑀 is called a lightlike submanifold of 𝑀 (see [12]). For a degenerate metric 𝑔 on 𝑀, 𝑇𝑀^⟂ is a degenerate 𝑛-dimensional subspace of 𝑇_𝑥𝑀. Thus both 𝑇_𝑥𝑀 and 𝑇_𝑥𝑀^⟂ are degenerate orthogonal subspaces but no longer complementary. In this case, there exists a subspace Rad𝑇_𝑥𝑀=𝑇_𝑥𝑀∩𝑇_𝑥𝑀^⟂ which is known as radical (null) subspace. If the mapping Rad𝑇𝑀∶𝑥∈𝑀→Rad𝑇_𝑥𝑀 defines a smooth distribution on 𝑀 of rank 𝑟>0 then the submanifold 𝑀 of 𝑀 is called an 𝑟-lightlike submanifold and Rad𝑇𝑀 is called the radical distribution on 𝑀.

Screen distribution 𝑆(𝑇𝑀) is a semi-Riemannian complementary distribution of Rad(𝑇𝑀) in 𝑇𝑀; that is, 𝑇𝑀=Rad𝑇𝑀⟂𝑆(𝑇𝑀) and 𝑆(𝑇𝑀^⟂) is a complementary vector subbundle to Rad𝑇𝑀 in 𝑇𝑀^⟂. Let tr(𝑇𝑀) and ltr(𝑇𝑀) be complementary (but not orthogonal) vector bundles to 𝑇𝑀 in 𝑇𝑀|_𝑀 and to Rad𝑇𝑀 in 𝑆(𝑇𝑀^⟂)^⟂, respectively. Then we have tr(𝑇𝑀)=ltr(𝑇𝑀)⟂𝑆𝑇𝑀^⟂𝑇,𝑀|_𝑀()=((()=𝑇𝑀⊕tr𝑇𝑀Rad𝑇𝑀⊕ltr𝑇𝑀))⟂𝑆𝑇𝑀⟂𝑆𝑇𝑀^⟂. (4)(5) For a quasi-orthonormal fields of frames on 𝑇𝑀, we have the following.

Theorem 1 (see [12]). Let (𝑀,𝑔,𝑆(𝑇𝑀),𝑆(𝑇𝑀^⟂)) be an 𝑟-lightlike submanifold of a semi-Riemannian manifold (𝑀,𝑔). Then there exists a complementary vector bundle ltr(𝑇𝑀) of Rad𝑇𝑀 in 𝑆(𝑇𝑀^⟂)^⟂ and a basis of Γ(ltr(𝑇𝑀)|_𝑢) consisting of smooth section {𝑁_𝑖} of 𝑆(𝑇𝑀^⟂)^⟂|_𝑢, where 𝑢 is a coordinate neighborhood of 𝑀 such that 𝑔𝑁_𝑖,𝜉_𝑗=𝛿_𝑖𝑗,𝑔𝑁_𝑖,𝑁_𝑗{},=0,𝑓𝑜𝑟𝑎𝑛𝑦𝑖,𝑗∈1,2,…,𝑟 (6) where {𝜉₁,…,𝜉_𝑟} is a lightlike basis of Γ(Rad(𝑇𝑀)).

Let ∇ be the Levi-Civita connection on 𝑀, then according to the decomposition (5), the Gauss and Weingarten formulas are given by ∇_𝑋𝑌=∇_𝑋𝑌+ℎ(𝑋,𝑌),∇_𝑋𝑈=−𝐴_𝑈𝑋+∇^⟂_𝑋𝑈, (7) for any 𝑋,𝑌∈Γ(𝑇𝑀) and 𝑈∈Γ(tr(𝑇𝑀)), where {∇_𝑋𝑌,𝐴_𝑈𝑋} and {ℎ(𝑋,𝑌), ∇^⟂_𝑋𝑈} belong to Γ(𝑇𝑀) and Γ(tr(𝑇𝑀)), respectively. Here ∇ is a torsion-free linear connection on 𝑀, ℎ is a symmetric bilinear form on Γ(𝑇𝑀) which is called the second fundamental form, and 𝐴_𝑈 is a linear operator on 𝑀 and known as a shape operator.

According to (4), considering the projection morphisms 𝐿 and 𝑆 of tr(𝑇𝑀) on ltr(𝑇𝑀) and 𝑆(𝑇𝑀^⟂), respectively, then (7) becomes ∇_𝑋𝑌=∇_𝑋𝑌+ℎ^𝑙(𝑋,𝑌)+ℎ^𝑠(𝑋,𝑌),∇_𝑋𝑈=−𝐴_𝑈𝑋+𝐷^𝑙_𝑋𝑈+𝐷^𝑠_𝑋𝑈, (8) where we put ℎ^𝑙(𝑋,𝑌)=𝐿(ℎ(𝑋,𝑌)),ℎ^𝑠(𝑋,𝑌)=𝑆(ℎ(𝑋,𝑌)),𝐷^𝑙_𝑋𝑈=𝐿(∇^⟂_𝑋𝑈), and 𝐷^𝑠_𝑋𝑈=𝑆(∇^⟂_𝑋𝑈). As ℎ^𝑙 and ℎ^𝑠 are Γ(ltr(𝑇𝑀)) valued, and Γ(𝑆(𝑇𝑀^⟂)) valued, respectively, therefore they are called as the lightlike second fundamental form and the screen second fundamental form on 𝑀. In particular ∇_𝑋𝑁=−𝐴_𝑁𝑋+∇^𝑙_𝑋𝑁+𝐷^𝑠(𝑋,𝑁),∇_𝑋𝑊=−𝐴_𝑊𝑋+∇^𝑠_𝑋𝑊+𝐷^𝑙(𝑋,𝑊), (9) where 𝑋∈Γ(𝑇𝑀),𝑁∈Γ(ltr(𝑇𝑀)) and 𝑊∈Γ(𝑆(𝑇𝑀^⟂)). Using (7) and (8) we obtain 𝑔ℎ^𝑠(𝑋,𝑌),𝑊+𝑔𝑌,𝐷^𝑙(𝑋,𝑊)=𝑔𝐴_𝑊𝑋,𝑌, (10)

Let 𝑃 be a projection of 𝑇𝑀 on 𝑆(𝑇𝑀) then using the decomposition 𝑇𝑀=Rad𝑇𝑀⟂𝑆(𝑇𝑀), we can write ∇_𝑋𝑃𝑌=∇^∗_𝑋𝑃𝑌+ℎ^∗𝑋,,∇𝑃𝑌_𝑋𝜉=−𝐴^∗_𝜉𝑋+∇_𝑋^∗𝑡𝜉, (11) for any 𝑋,𝑌∈Γ(𝑇𝑀) and 𝜉∈Γ(Rad𝑇𝑀), where {∇^∗_𝑋𝑃𝑌, 𝐴^∗_𝜉𝑋} and {ℎ^∗(𝑋,𝑃𝑌), ∇_𝑋^∗𝑡𝜉} belong to Γ(𝑆(𝑇𝑀)) and Γ(Rad𝑇𝑀), respectively. Here ∇^∗ and ∇_𝑋^∗𝑡 are linear connections on 𝑆(𝑇𝑀) and Rad𝑇𝑀, respectively. By using (8), (9) and (11), we obtain 𝑔ℎ^𝑙𝑋,𝑃𝑌,𝜉=𝑔𝐴^∗_𝜉𝑋,𝑃𝑌,𝑔ℎ^∗𝑋,𝑃𝑌,𝑁=𝑔𝐴_𝑁𝑋,𝑃𝑌. (12)

3. Slant Lightlike Submanifolds

A lightlike submanifold has two distributions, namely, the radical distribution and the screen distribution. The radical distribution is totally lightlike and it is not possible to define angle between two vector fields of radical distribution where the screen distribution is nondegenerate. There are some definitions for the angle between two vector fields in Lorentzian setup [13], but they are not appropriate for our goal. Therefore to introduce the notion of slant lightlike submanifolds, one needs a Riemannian distribution, and Gupta et al. [8] played a crucial role in the development of the theory of slant lightlike submanifolds of indefinite cosymplectic manifolds.

Definition 2 (see [8]). Let 𝑀 be an 𝑟-lightlike submanifold of an indefinite cosymplectic manifold 𝑀 of index 2𝑟 with structure vector field 𝑉 tangent to 𝑀. Then one say that 𝑀 is a slant lightlike submanifold of 𝑀 if the following conditions are satisfied. (a)Rad𝑇𝑀 is a distribution on 𝑀 such that 𝜙Rad𝑇𝑀∩Rad(𝑇𝑀)={0}. (b)For all 𝑥∈𝑈⊂𝑀 and for each non zero vector field 𝑋 tangent to 𝐷=𝐷⟂{𝑉}, if 𝑋 and 𝑉 are linearly independent, then the angle 𝜃(𝑋) between 𝜙𝑋 and the vector space 𝐷_𝑥 is constant, where 𝐷 is complementary distribution to 𝜙ltr(𝑇𝑀)⊕𝜙Rad𝑇𝑀 in screen distribution 𝑆(𝑇𝑀). The constant angle 𝜃(𝑋) is called the slant angle of 𝐷. A slant lightlike submanifold 𝑀 is said to be proper if 𝐷≠{0} and 𝜃≠0,𝜋/2.

Since a submanifold 𝑀 is invariant (anti-invariant, resp.), if 𝜙𝑇_𝑝𝑀⊂𝑇_𝑝𝑀, (𝜙𝑇_𝑝𝑀⊂𝑇_𝑝𝑀^⟂,resp.), for any 𝑝∈𝑀, Therefore from the previous definition, it is clear that 𝑀 is invariant (anti-invariant, resp.) if 𝜃(𝑋)=0(𝜃(𝑋)=𝜋/2, resp.).

Then the tangent bundle 𝑇𝑀 of 𝑀 is decomposed as ()()⟂𝑇𝑀=Rad𝑇𝑀⟂𝑆𝑇𝑀=Rad𝑇𝑀⟂𝜙Rad𝑇𝑀⊕𝜙ltr𝑇𝑀𝐷, (13) where 𝐷=𝐷⟂{𝑉}. For any 𝑋∈Γ(𝑇𝑀), we write 𝜙𝑋=𝑇𝑋+𝐹𝑋, (14) where 𝑇𝑋 is the tangential component of 𝜙𝑋 and 𝐹𝑋 is the transversal component of 𝜙𝑋. Similarly for any 𝑈∈Γ(tr(𝑇𝑀)), we write 𝜙𝑈=𝐵𝑈+𝐶𝑈, (15) where 𝐵𝑈 is the tangential component of 𝜙𝑈 and 𝐶𝑈 is the transversal component of 𝜙𝑈. Using the decomposition in (13), we denote by 𝑃₁,𝑃₂,𝑄₁, 𝑄₂, and 𝑄₂ the projections on the distributions Rad𝑇𝑀, 𝜙Rad𝑇𝑀, 𝜙ltr(𝑇𝑀), 𝐷 and 𝐷=𝐷⟂𝑉, respectively. Then for any 𝑋∈Γ(𝑇𝑀), we can write 𝑋=𝑃₁𝑋+𝑃₂𝑋+𝑄₁𝑋+𝑄₂𝑋, (16) where 𝑄₂𝑋=𝑄₂𝑋+𝜂(𝑋)𝑉.

Applying 𝜙 to (16), we obtain 𝜙𝑋=𝜙𝑃₁𝑋+𝜙𝑃₂𝑋+𝐹𝑄₁𝑋+𝑇𝑄₂𝑋+𝐹𝑄₂𝑋. (17) Then using (14) and (15), we get 𝜙𝑃₁𝑋=𝑇𝑃₁𝑋∈Γ𝜙Rad𝑇𝑀,𝜙𝑃₂𝑋=𝑇𝑃₂𝑋∈Γ(Rad𝑇𝑀),𝐹𝑃₁𝑋=𝐹𝑃₂𝑋=0,𝑇𝑄₂𝑋∈Γ(𝐷),𝐹𝑄₁𝑋∈Γ(ltr(𝑇𝑀)). (18) Differentiating (17) and using (8)-(9), (14) and (15), for any 𝑋,𝑌∈Γ(𝑇𝑀), we have ∇_𝑋𝑇𝑌=𝐴_{𝐹𝑄1𝑌}𝑋+𝐴_{𝐹𝑄2𝑌}(),𝐷𝑋+𝐵ℎ𝑋,𝑌_𝑙𝑋,𝐹𝑄₂𝑌=𝐹𝑄₁∇_𝑋𝑌−ℎ^𝑙(𝑋,𝑇𝑌)−∇^𝑙_𝑋𝐹𝑄₁𝐷𝑌,_𝑠𝑋,𝐹𝑄₁𝑌=𝐹𝑄₂∇_𝑋𝑌−ℎ^𝑠(𝑋,𝑇𝑌)+𝐶ℎ^𝑠(𝑋,𝑌)−∇^𝑠_𝑋𝐹𝑄₂𝑌. (19)(20)(21) By using cosymplectic property of ∇ with (7), we have the following lemmas.

Lemma 3. Let 𝑀 be a slant lightlike submanifold of an indefinite cosymplectic manifold 𝑀, then on has ∇_𝑋𝑇𝑌=𝐴_𝐹𝑌(),∇𝑋+𝐵ℎ𝑋,𝑌_𝑡𝑋𝐹𝑌=𝐶ℎ(𝑋,𝑌)−ℎ(𝑋,𝑇𝑌), (22) where 𝑋,𝑌∈Γ(𝑇𝑀) and ∇_𝑋𝑇𝑌=∇_𝑋𝑇𝑌−𝑇∇_𝑋𝑌,∇^𝑡_𝑋𝐹𝑌=∇^𝑡_𝑋𝐹𝑌−𝐹∇_𝑋𝑌. (23)

Lemma 4. Let 𝑀 be a slant lightlike submanifold of an indefinite cosymplectic manifold 𝑀 then we have ∇_𝑋𝐵𝑈=𝐴_𝐶𝑈𝑋−𝑇𝐴_𝑈∇𝑋,_𝑡𝑋𝐶𝑈=−𝐹𝐴_𝑈𝑋−ℎ(𝑋,𝐵𝑈), (24) where 𝑋∈Γ(𝑇𝑀) and 𝑈∈Γ(tr(𝑇𝑀)) and ∇_𝑋𝐵𝑈=∇_𝑋𝐵𝑈−𝐵∇^𝑡_𝑋𝑈,∇^𝑡_𝑋𝐶𝑈=∇^𝑡_𝑋𝐶𝑈−𝐶∇^𝑡_𝑋𝑈. (25)

Theorem 5. Let 𝑀 be a slant lightlike submanifold of an indefinite cosymplectic manifold 𝑀; then (a)the distribution 𝐷 is integrable, if and only if, ℎ(𝑋,𝑇𝑌)=ℎ(𝑌,𝑇𝑋), 𝐷^𝑙(𝑋,𝐹𝑄₂𝑌)=𝐷^𝑙(𝑌,𝐹𝑄₂𝑋), and ∇^𝑠_𝑋𝐹𝑄₂𝑌=∇^𝑠_𝑌𝐹𝑄₂𝑋, for any 𝑋,𝑌∈Γ(𝐷),(b)the distribution 𝜙(ltr(𝑇𝑀)) is integrable, if and only if 𝐴_{𝐹𝑄1𝑌}𝑋=𝐴_{𝐹𝑄1𝑋}𝑌, for any 𝑋,𝑌∈Γ(𝜙(ltr(𝑇𝑀))).

Proof. Using (20) and (21), we have 𝐹∇_𝑋𝑌=𝐷^𝑙(𝑋,𝐹𝑄₂𝑌)+ℎ(𝑋,𝑇𝑌)+∇^𝑠_𝑋𝐹𝑄₂𝑌−𝐶ℎ^𝑠(𝑋,𝑌), for any 𝑋,𝑌∈Γ(𝐷). Here replacing 𝑋 by 𝑌 and then subtracting the resulting equation from this equation, we get (𝐴). Next from (22) and (23), we have −𝑇∇_𝑋𝑌=𝐴_{𝐹𝑄1𝑌}𝑋+𝐵ℎ(𝑋,𝑌), (26) for all 𝑋,𝑌∈Γ(𝜙(ltr(𝑇𝑀))). Then, similarly to the above, we have 𝑇[𝑋,𝑌]=𝐴_{𝐹𝑄1𝑋}𝑌−𝐴_{𝐹𝑄1𝑌}𝑋, and this completes the proof of (𝑏).

Corollary 6 (see [8]). Let 𝑀 be a slant lightlike submanifold of an indefinite cosymplectic manifold 𝑀 with structure vector field 𝑉 tangent to 𝑀. Then one has 𝑔𝑇𝑄₂𝑋,𝑇𝑄₂𝑌=cos²𝜃𝑔𝑄₂𝑋,𝑄₂𝑌−𝜂𝑄₂𝑋𝜂𝑄₂𝑌,𝑔𝐹𝑄₂𝑋,𝐹𝑄₂𝑌=sin²𝜃𝑔𝑄₂𝑋,𝑄₂𝑌−𝜂𝑄₂𝑋𝜂𝑄₂𝑌 (27)(28) for any 𝑋,𝑌∈Γ(𝑇𝑀).

4. Totally Contact Umbilical Slant Lightlike Submanifolds

Definition 7 (see [14]). If the second fundamental form ℎ of a submanifold tangent to characteristic vector field 𝑉, of an indefinite Sasakian manifold 𝑀 is of the form ℎ()=𝑔()(𝑋)𝜂(𝑌𝛼(𝑋)ℎ()(𝑌)ℎ(),𝑋,𝑌𝑋,𝑌−𝜂)+𝜂𝑌,𝑉+𝜂𝑋,𝑉 (29) for any 𝑋,𝑌∈Γ(𝑇𝑀), where 𝛼 is a vector field transversal to 𝑀, then 𝑀 is called a totally contact umbilical and totally contact geodesic if 𝛼=0.
The above definition also holds for a lightlike submanifold 𝑀. For a totally contact umbilical lightlike submanifold 𝑀, we have ℎ^𝑙(𝑋,𝑌)=𝑔(𝑋,𝑌)−𝜂(𝑋)𝜂(𝑌)𝛼_𝐿+𝜂(𝑋)ℎ^𝑙(𝑌,𝑉)+𝜂(𝑌)ℎ^𝑙(),ℎ𝑋,𝑉_𝑠(𝑋,𝑌)=𝑔(𝑋,𝑌)−𝜂(𝑋)𝜂(𝑌)𝛼_𝑆+𝜂(𝑋)ℎ^𝑠(𝑌,𝑉)+𝜂(𝑌)ℎ^𝑠(𝑋,𝑉), (30)(31) where 𝛼_𝐿∈Γ(ltr(𝑇𝑀)) and 𝛼_𝑆∈Γ(𝑆(𝑇𝑀^⟂)).

Lemma 8. Let 𝑀 be a slant lightlike submanifold of an indefinite cosymplectic manifold 𝑀; then 𝐹𝑄₂𝑋∈Γ(𝑆(𝑇𝑀^⟂)), for any 𝑋∈Γ(𝑇𝑀).

Proof. Using (4) and (6), it is clear that 𝐹𝑄₂𝑋∈Γ(𝑆(𝑇𝑀^⟂)) if 𝑔(𝐹𝑄₂𝑋,𝜉)=0, for any 𝜉∈Γ(Rad(𝑇𝑀)). Therefore 𝑔𝐹𝑄₂𝑋,𝜉=𝑔𝜙𝑄₂𝑋−𝑇𝑄₂𝑋,𝜉=𝑔𝜙𝑄₂𝑋,𝜉=−𝑔𝑄₂𝑋,𝜙𝜉=0. (32) Hence the result follows.
Thus from Lemma 8 it follows that 𝐹(𝐷_𝑝) is a subspace of 𝑆(𝑇𝑀^⟂). Therefore there exists an invariant subspace 𝜇_𝑝 of 𝑇_𝑝𝑀 such that 𝑆𝑇_𝑝𝑀^⟂=𝐹𝐷_𝑝⟂𝜇_𝑝. (33) Thus 𝑇_𝑝𝑀=𝑆𝑇_𝑝𝑀⟂Rad𝑇_𝑝𝑀⊕ltr𝑇_𝑝𝑀⟂𝐹𝐷_𝑝⟂𝜇_𝑝. (34)

Theorem 9. Let 𝑀 be a totally contact umbilical slant lightlike submanifold of an indefinite cosymplectic manifold 𝑀. Then at least one of the following statements is true: (i)𝑀 is an anti-invariant submanifold, (ii)𝐷={0},(iii)if 𝑀 is a proper slant submanifold, then 𝛼_𝑆∈Γ(𝜇).

Proof. Let 𝑀 be a totally contact umbilical slant lightlike submanifold of an indefinite cosymplectic manifold 𝑀; then for any 𝑋=𝑄₂𝑋∈Γ(𝐷) with (29), we have ℎ𝑇𝑄₂𝑋,𝑇𝑄₂𝑋=𝑔𝑇𝑄₂𝑋,𝑇𝑄₂𝑋𝛼. (35) Therefore from (7), (27), and the above equation, we get ∇_{𝑇𝑄2𝑋}𝑇𝑄₂𝑋−∇_{𝑇𝑄2𝑋}𝑇𝑄₂𝑋=cos²𝜃𝑔𝑄₂𝑋,𝑄₂𝑋𝛼. (36) Using (14) and the fact that 𝑀 is cosymplectic manifold, we obtain 𝜙∇_{𝑇𝑄2𝑋}𝑄₂𝑋−∇_{𝑇𝑄2𝑋}𝐹𝑄₂𝑋−∇_{𝑇𝑄2𝑋}𝑇𝑄₂𝑋=cos²𝜃𝑔𝑄₂𝑋,𝑄₂𝑋𝛼. (37) Then using (8) and (9), we get 𝜙∇_{𝑇𝑄2𝑋}𝑄₂𝑋+𝜙ℎ^𝑙𝑇𝑄₂𝑋,𝑋+𝜙ℎ^𝑠𝑇𝑄₂𝑋,𝑋+𝐴_{𝐹𝑄2𝑋}𝑇𝑄₂𝑋−∇^𝑠_{𝑇𝑄2𝑋}𝐹𝑄₂𝑋−𝐷^𝑙𝑇𝑄₂𝑋,𝐹𝑄₂𝑋−∇_{𝑇𝑄2𝑋}𝑇𝑄₂𝑋=cos²𝜃𝑔𝑄₂𝑋,𝑄₂𝑋𝛼. (38) Thus using (14), (15), (30), and (31), we have 𝑇∇_{𝑇𝑄2𝑋}𝑄₂𝑋+𝐹∇_{𝑇𝑄2𝑋}𝑄₂𝑋+𝑔𝑇𝑄₂𝑋,𝑋𝜙𝛼^𝑙+𝑔𝑇𝑄₂𝑋,𝑋𝐵𝛼^𝑠+𝑔𝑇𝑄₂𝑋,𝑋𝐶𝛼^𝑠+𝐴_{𝐹𝑄2𝑋}𝑇𝑄₂𝑋−∇^𝑠_{𝑇𝑄2𝑋}𝐹𝑄₂𝑋−𝐷^𝑙𝑇𝑄₂𝑋,𝐹𝑄₂𝑋−∇_{𝑇𝑄2𝑋}𝑇𝑄₂𝑋=cos²𝜃𝑔𝑄₂𝑋,𝑄₂𝑋𝛼. (39) Equating the transversal components, we get 𝐹∇_{𝑇𝑄2𝑋}𝑄₂𝑋+𝑔𝑇𝑄₂𝑋,𝑋𝐶𝛼^𝑠−∇^𝑠_{𝑇𝑄2𝑋}𝐹𝑄₂𝑋−𝐷^𝑙𝑇𝑄₂𝑋,𝐹𝑄₂𝑋=cos²𝜃𝑔𝑄₂𝑋,𝑄₂𝑋𝛼. (40) On the other hand, (28) holds for any 𝑋=𝑌∈Γ(𝐷), and by taking the covariant derivative with respect to 𝑇𝑄₂𝑋, we obtain 𝑔∇^𝑠_{𝑇𝑄2𝑋}𝐹𝑄₂𝑋,𝐹𝑄₂𝑋=sin²𝜃𝑔∇_{𝑇𝑄2𝑋}𝑄₂𝑋,𝑄₂𝑋. (41) Now taking the inner product in (40) with 𝐹𝑄₂𝑋, we obtain 𝑔𝐹∇_{𝑇𝑄2𝑋}𝑄₂𝑋,𝐹𝑄₂𝑋−𝑔∇^𝑠_{𝑇𝑄2𝑋}𝐹𝑄₂𝑋,𝐹𝑄₂𝑋=cos²𝜃𝑔𝑄₂𝑋,𝑄₂𝑋𝑔𝛼_𝑆,𝐹𝑄₂𝑋. (42) Then using (28) and (41), we get cos²𝜃𝑔𝑄₂𝑋,𝑄₂𝑋𝑔𝛼_𝑆,𝐹𝑄₂𝑋=0. (43) Thus from (43), it follows that; 𝜃=𝜋/2, 𝑄₂𝑋=0 or 𝛼_𝑆∈Γ(𝜇). This completes the proof.

Lemma 10. Let 𝑀 be a totally contact umbilical slant lightlike submanifold of an indefinite cosymplectic manifold 𝑀; then 𝑔(∇_𝑋𝑋,𝜙𝜉)=0, for any 𝑋∈Γ(𝐷) and 𝜉∈Γ(Rad𝑇𝑀).

Proof. Let 𝑋∈Γ(𝐷); therefore 𝑋=𝑄₂𝑋, and then using (3), (8), and (9) for a totally contact umbilical slant lightlike submanifold, we have 𝑔∇_𝑋𝑋,𝜙𝜉=𝑔∇_𝑋𝑋,𝜙𝜉=−𝑔∇_𝑋𝑇𝑄₂𝑋,𝜉−𝑔∇_𝑋𝐹𝑄₂𝑋,𝜉=−𝑔ℎ^𝑙𝑋,𝑇𝑄₂𝑋,𝜉−𝑔𝐷^𝑙𝑋,𝐹𝑄₂𝑋,𝜉=−𝑔𝐷^𝑙𝑋,𝐹𝑄₂𝑋,𝜉, (44) since for 𝑋∈Γ(𝐷), using (2), (14), and (30), we have ℎ^𝑙(𝑋,𝑇𝑄₂𝑋) = {𝑔(𝑋,𝑇𝑄₂𝑋)}𝛼_𝐿 = 0. Also for 𝑋∈Γ(𝐷), we have 𝜂(𝑄₂𝑋)=0, and 𝜂(𝜉)=0 therefore by replacing 𝑊 by 𝐹𝑄₂𝑋 and 𝑌 by 𝜉 in (10) and using the hypothesis that 𝑀 is totally contact umbilical slant lightlike submanifold, we obtain 𝑔𝐷^𝑙𝑋,𝐹𝑄₂𝑋,𝜉=−𝑔ℎ^𝑠(𝑋,𝜉),𝐹𝑄₂𝑋=−𝑔(𝑋,𝜉)𝑔𝛼_𝑆,𝐹𝑄₂𝑋=0. (45) Therefore from (44) and (45), the result follows.

Theorem 11. Every totally contact umbilical proper slant lightlike submanifold 𝑀 of an indefinite cosymplectic manifold 𝑀 is totally contact geodesic.

Proof. Since 𝑀 is a totally contact umbilical slant lightlike submanifold, therefore for any 𝑋=𝑄₂𝑋∈Γ(𝐷), using (29), we have ℎ(𝑇𝑄₂𝑋,𝑇𝑄₂𝑋)=𝑔(𝑇𝑄₂𝑋,𝑇𝑄₂𝑋)𝛼, and therefore using (27), we obtain ℎ𝑇𝑄₂𝑋,𝑇𝑄₂𝑋=cos²𝜃𝑔𝑄₂𝑋,𝑄₂𝑋−𝜂𝑄₂𝑋𝜂𝑄₂𝑋𝛼=cos²𝜃𝑔𝑄₂𝑋,𝑄₂𝑋𝛼. (46) Using (20) for 𝑋∈Γ(𝐷), we have 𝐹∇_{𝑇𝑄2𝑋}𝑋=ℎ𝑇𝑄₂𝑋,𝑇𝑄₂𝑋−𝐶ℎ𝑇𝑄₂𝑋,𝑋+∇^𝑠_{𝑇𝑄2𝑋}𝐹𝑄₂𝑋+𝐷^𝑙𝑇𝑄₂𝑋,𝐹𝑄₂𝑋. (47) Since 𝑀 is totally contact umbilical slant lightlike submanifold therefore 𝐶ℎ(𝑇𝑄₂𝑋,𝑋) = 𝑔(𝑇𝑄₂𝑋,𝑋)𝐶𝛼=0, therefore using (46) and (47), we get cos²𝜃𝑔𝑄₂𝑋,𝑄₂𝑋𝛼=𝐹∇_{𝑇𝑄2𝑋}𝑋−∇^𝑠_{𝑇𝑄2𝑋}𝐹𝑄₂𝑋−𝐷^𝑙𝑇𝑄₂𝑋,𝐹𝑄₂𝑋. (48) Taking the scalar product of both sides of (48) with respect to 𝐹𝑄₂𝑋 for 𝑋∈Γ(𝐷), we obtain cos²𝜃𝑔𝑄₂𝑋,𝑄₂𝑋𝑔𝛼_𝑆,𝐹𝑄₂𝑋=𝑔𝐹∇_{𝑇𝑄2𝑋}𝑋,𝐹𝑄₂𝑋−𝑔∇^𝑠_{𝑇𝑄2𝑋}𝐹𝑄₂𝑋,𝐹𝑄₂𝑋. (49) Here using (28), we get cos²𝜃𝑔𝑄₂𝑋,𝑄₂𝑋𝑔𝛼_𝑆,𝐹𝑄₂𝑋=sin²𝜃𝑔∇_{𝑇𝑄2𝑋}𝑋,𝑄₂𝑋−𝑔∇^𝑠_{𝑇𝑄2𝑋}𝐹𝑄₂𝑋,𝐹𝑄₂𝑋. (50) Since (28) holds for any 𝑋=𝑌∈Γ(𝐷) and by taking the covariant derivative with respect to ∇_{𝑇𝑄2𝑋}, we get 𝑔∇^𝑠_{𝑇𝑄2𝑋}𝐹𝑄₂𝑋,𝐹𝑄₂𝑋=sin²𝜃𝑔∇_{𝑇𝑄2𝑋}𝑄₂𝑋,𝑄₂𝑋. (51) Using (51) in (50), we obtain cos²𝜃𝑔𝑄₂𝑋,𝑄₂𝑋𝑔𝛼_𝑆,𝐹𝑄₂𝑋=0. (52) Since 𝑀 is a proper slant lightlike submanifold and 𝑔 is a Riemannian metric on 𝐷, therefore we have 𝑔(𝛼_𝑠,𝐹𝑄₂𝑋)=0. Thus using Lemma 8 and (33), we obtain 𝛼_𝑆∈Γ𝜇. (53) Now using the cosymplectic property of 𝑀, we have ∇_𝑋𝜙𝑌=𝜙∇_𝑋𝑌, for any 𝑋,𝑌∈Γ(𝐷), and then using (9), (14), and (29), we obtain ∇_𝑋𝑇𝑄₂𝑌+𝑔𝑋,𝑇𝑄₂𝑌𝛼−𝐴_{𝐹𝑄2𝑌}𝑋+∇^𝑠_𝑋𝐹𝑄₂𝑌+𝐷^𝑙𝑋,𝐹𝑄₂𝑌=𝑇∇_𝑋𝑌+𝐹∇_𝑋𝑌+𝑔(𝑋,𝑌)𝜙𝛼. (54) Taking the scalar product of both sides of (54), with respect to 𝜙𝛼_𝑆 and then using (2), (34) and (53), we obtain 𝑔∇^𝑠_𝑋𝐹𝑄₂𝑌,𝜙𝛼_𝑆=𝑔𝑄₂𝑋,𝑄₂𝑌𝑔𝛼_𝑆,𝛼_𝑆. (55) Since 𝜇 is an invariant subspace, therefore using the cosymplectic character of 𝑀, we have ∇_𝑋𝜙𝛼_𝑆=𝜙∇_𝑋𝛼_𝑆, and this implies that −𝐴_𝜙𝛼𝑆𝑋+∇^𝑠_𝑋𝜙𝛼_𝑆+𝐷^𝑙𝑋,𝜙𝛼_𝑆=−𝑇𝐴_𝛼𝑆𝑋−𝐹𝐴_𝛼𝑆𝑋+𝐵∇^𝑠_𝑋𝛼_𝑆+𝐶∇^𝑠_𝑋𝛼_𝑆+𝜙𝐷^𝑙𝑋,𝛼_𝑆. (56) Taking the scalar product of both sides of the previous equation with respect to 𝐹𝑄₂𝑌, we obtain 𝑔∇^𝑠_𝑋𝜙𝛼_𝑆,𝐹𝑄₂𝑌=−𝑔𝐹𝐴_𝛼𝑆𝑋,𝐹𝑄₂𝑌+𝑔𝐶∇^𝑆_𝑋𝛼_𝑆,𝐹𝑄₂𝑌. (57) From (15), we know that 𝐵𝑈 and 𝐶𝑈 are tangential and transversal components of 𝜙(𝑈), respectively, for any 𝑈∈Γ(tr(𝑇𝑀)). Therefore if 𝑈∈Γ(ltr(𝑇𝑀)), then 𝜙𝑈=𝐵𝑈∈Γ(𝜙ltr(𝑇𝑀))⊂𝑆(𝑇𝑀) and 𝐶𝑈=0. Moreover, since 𝑆(𝑇𝑀^⟂)=𝐹(𝐷)⟂𝜇, therefore for any 𝑈∈Γ(𝑆(𝑇𝑀^⟂)), 𝐵𝑈∈Γ(𝐷)⊂𝑆(𝑇𝑀) and 𝐶𝑈∈Γ(𝜇)⊂tr(𝑇𝑀). Since ∇^𝑠_𝑋𝛼_𝑆∈Γ(𝑆(𝑇𝑀^⟂)), and therefore 𝐶∇^𝑠_𝑋𝛼_𝑆∈Γ(𝜇), and using this with (28), we get 𝑔∇^𝑠_𝑋𝜙𝛼_𝑆,𝐹𝑄₂𝑌=−𝑔𝐹𝐴_𝛼𝑆𝑋,𝐹𝑄₂𝑌=−sin²𝜃𝑔𝐴_𝛼𝑆𝑋,𝑄₂𝑌. (58) Since ∇ is a metric connection, therefore (∇_𝑋𝑔)(𝐹𝑄₂𝑌,𝜙𝛼_𝑆)=0. This further implies that 𝑔(∇^𝑠_𝑋𝐹𝑄₂𝑌,𝜙𝛼_𝑆)=𝑔(∇^𝑠_𝑋𝜙𝛼_𝑆,𝐹𝑄₂𝑌), and therefore using (58), we obtain 𝑔∇^𝑠_𝑋𝐹𝑄₂𝑌,𝜙𝛼_𝑆=−sin²𝜃𝑔𝐴_𝛼𝑆𝑋,𝑄₂𝑌. (59) From (55) and (59), we have 𝑔𝑄₂𝑋,𝑄₂𝑌𝑔𝛼_𝑆,𝛼_𝑆=−sin²𝜃𝑔𝐴_𝛼𝑆𝑋,𝑄₂𝑌. (60) Then using (10) in (60), we obtain 𝑔𝑄₂𝑋,𝑄₂𝑌𝑔𝛼_𝑆,𝛼_𝑆=−sin²𝜃𝑔ℎ^𝑠𝑄₂𝑋,𝑄₂𝑌,𝛼_𝑆=−sin²𝜃𝑔𝑄₂𝑋,𝑄₂𝑌𝑔𝛼_𝑆,𝛼_𝑆, (61) this implies that 1+sin²𝜃𝑔𝑄₂𝑋,𝑄₂𝑌𝑔𝛼_𝑆,𝛼_𝑆=0. (62) Since 𝑔 is a Riemannian metric on 𝐷, thus we obtain 𝛼_𝑆=0. (63) Next, for 𝑋∈Γ(𝐷), using the cosymplectic character of 𝑀, we have ∇_𝑋𝜙𝑋=𝜙∇_𝑋𝑋, and this implies that ∇_𝑋𝑇𝑄₂𝑋+ℎ(𝑋,𝑇𝑄₂𝑋)−𝐴_{𝐹𝑄2𝑋}𝑋+∇^𝑠_𝑋𝐹𝑄₂𝑋+𝐷^𝑙(𝑋,𝐹𝑄₂𝑋)=𝑇∇_𝑋𝑋+𝐹∇_𝑋𝑋+𝐵ℎ(𝑋,𝑋)+𝐶ℎ(𝑋,𝑋). Since 𝑀 is totally contact umbilical slant lightlike manifold, therefore using ℎ(𝑋,𝑇𝑄₂𝑋)=0 in the previous equation and then comparing the tangential components, we obtain ∇_𝑋𝑇𝑄₂𝑋−𝐴_{𝐹𝑄2𝑋}𝑋=𝑇∇_𝑋𝑋+𝐵ℎ(𝑋,𝑋). (64) Taking the scalar product of both sides of (64) with respect to 𝜙𝜉∈Γ(𝜙Rad(𝑇𝑀)) and using Lemma 10, we get 𝑔𝐴_{𝐹𝑄2𝑋}𝑋,𝜙𝜉+𝑔ℎ^𝑙𝑄₂𝑋,𝑄₂𝑋,𝜉=0. (65) Now using (9), we have 𝑔ℎ^𝑠𝑋,𝜙𝜉,𝐹𝑄₂𝑋+𝑔𝜙𝜉,𝐷^𝑙𝑋,𝐹𝑄₂𝑋=𝑔𝐴_{𝐹𝑄2𝑋}𝑋,𝜙𝜉. (66) Since 𝑀 is a totally contact umbilical slant lightlike submanifold therefore using (63) in the previous equation, we obtain 𝑔𝐴_{𝐹𝑄2𝑋}𝑋,𝜙𝜉=0. (67) Using (67) in (65), we obtain that 𝑔(ℎ^𝑙(𝑄₂𝑋,𝑄₂𝑋),𝜉)=0 and then using (30), we get 𝑔𝑄₂𝑋,𝑄₂𝑋𝑔𝛼_𝐿,𝜉=0. (68) Since 𝑔 is a Riemannian metric on 𝐷, therefore 𝑔(𝛼_𝐿,𝜉)=0, then using (6), we obtain that 𝛼_𝐿=0. (69) Thus from (63) and (69), the proof is complete.

Denote by 𝑅 and 𝑅 the curvature tensors of ∇ and ∇ respectively, and then by straightforward calculations [12], we have 𝑅()𝑍()𝑋,𝑌=𝑅𝑋,𝑌𝑍+𝐴_ℎ^𝑙_(𝑋,𝑍)𝑌−𝐴_ℎ^𝑙_(𝑌,𝑍)𝑋+𝐴_ℎ^𝑠_(𝑋,𝑍)𝑌−𝐴_ℎ^𝑠_(𝑌,𝑍)𝑋+∇_𝑋ℎ^𝑙(𝑌,𝑍)−∇_𝑌ℎ^𝑙(𝑋,𝑍)+𝐷^𝑙𝑋,ℎ^𝑠(𝑌,𝑍)−𝐷^𝑙𝑌,ℎ^𝑠(𝑋,𝑍)+∇_𝑋ℎ^𝑠()−∇𝑌,𝑍_𝑌ℎ^𝑠(𝑋,𝑍)+𝐷^𝑠𝑋,ℎ^𝑙(𝑌,𝑍)−𝐷^𝑠𝑌,ℎ^𝑙(𝑋,𝑍), (70) where ∇_𝑋ℎ^𝑠(𝑌,𝑍)=∇^𝑠_𝑋ℎ^𝑠(𝑌,𝑍)−ℎ^𝑠∇_𝑋𝑌,𝑍−ℎ^𝑠𝑌,∇_𝑋𝑍,∇_𝑋ℎ^𝑙(𝑌,𝑍)=∇^𝑙_𝑋ℎ^𝑙(𝑌,𝑍)−ℎ^𝑙∇_𝑋𝑌,𝑍−ℎ^𝑙𝑌,∇_𝑋𝑍. (71)(72) An indefinite cosymplectic space form is a connected indefinite cosymplectic manifold of constant holomorphic sectional curvature 𝑐 and is denoted by 𝑀(𝑐). Then the curvature tensor 𝑅 of 𝑀(𝑐) is given by 𝑅()𝑍=𝑐𝑋,𝑌4𝑔(𝑌,𝑍)𝑋−𝑔()(𝑋)𝜂(𝑍)𝑌(𝑌)𝜂(𝑍)𝑋,𝑍𝑌+𝜂−𝜂𝑋+𝑔()𝜂(𝑌)()𝜂(𝑋)𝑉+𝑋,𝑍𝑉−𝑔𝑌,𝑍𝑔𝜙𝑌,𝑍𝜙𝑋+𝑔𝜙𝑍,𝑋𝜙𝑌−2𝑔𝜙𝑋,𝑌𝜙𝑍 (73) for 𝑋,𝑌,𝑍 vector fields on 𝑀.

Theorem 12. There do not exist totally contact umbilical proper slant lightlike submanifolds of an indefinite cosymplectic space form 𝑀(𝑐) such that 𝑐≠0.

Proof. Suppose 𝑀 be a totally contact umbilical proper lightlike submanifold of 𝑀(𝑐) such that 𝑐≠0. Then using (73), for any 𝑋∈Γ(𝐷), 𝑍∈Γ(𝜙ltr(𝑇𝑀)), and 𝜉∈Γ(Rad(𝑇𝑀)), we obtain 𝑔𝑅𝑐𝑋,𝜙𝑋𝑍,𝜉=−2𝑔𝜙𝑋,𝜙𝑋𝑔𝜙𝑍,𝜉. (74) Then using (2), we get 𝑔𝑅𝑐𝑋,𝜙𝑋𝑍,𝜉=−2𝑔𝑄₂𝑋,𝑄₂𝑋𝑔𝜙𝑍,𝜉. (75) On the other hand using (31) and (63) in (70), we get 𝑔𝑅𝑋,𝜙𝑋𝑍,𝜉=𝑔∇_𝑋ℎ^𝑙−𝜙𝑋,𝑍,𝜉𝑔∇_𝜙𝑋ℎ^𝑙(𝑋,𝑍),𝜉. (76) Using (30) and (72), we have ∇_𝑋ℎ^𝑙𝜙𝑋,𝑍=−𝑔∇_𝑋𝑇𝑄₂𝑋,𝑍𝛼_𝐿−𝑔𝑇𝑄₂𝑋,∇_𝑋𝑍𝛼_𝐿. (77) Similarly ∇_𝜙𝑋ℎ^𝑙(𝑋,𝑍)=−𝑔∇_𝜙𝑋𝑋,𝑍𝛼_𝐿−𝑔𝑋,∇_𝜙𝑋𝑍𝛼_𝐿. (78) Using (77) and (78) in (76), we obtain 𝑔𝑅∇𝑋,𝜙𝑋𝑍,𝜉=−𝑔_𝑋𝑇𝑄₂𝑋,𝑍𝑔𝛼_𝐿,𝜉−𝑔𝑇𝑄₂𝑋,∇_𝑋𝑍𝑔𝛼_𝐿∇,𝜉+𝑔_𝜙𝑋𝑋,𝑍𝑔𝛼_𝐿,𝜉+𝑔𝑋,∇_𝜙𝑋𝑍𝑔𝛼_𝐿,𝜉. (79) Now using (69), we have 𝑔𝑅𝑋,𝜙𝑋𝑍,𝜉=0. (80) Thus using (80) in (75), we have 𝑐2𝑔𝑄₂𝑋,𝑄₂𝑋𝑔𝜙𝑍,𝜉=0. (81) Since 𝑔 is a Riemannian metric on 𝐷 and (81) implies that 𝑔(𝜙𝑍,𝜉)≠0, therefore 𝑐=0. This contradiction completes the proof.

In [12], a minimal lightlike submanifold 𝑀 is defined when 𝑀 is a hypersurface of a 4-dimensional Minkowski space. Then in [15], a general notion of minimal lightlike submanifold of a semi-Riemannian manifold 𝑀 is introduced as follows.

Definition 13. A lightlike submanifold (𝑀,𝑔,𝑆(𝑇𝑀)) isometrically immersed in a semi-Riemannian manifold (𝑀,𝑔) is minimal if (i)ℎ^𝑠=0 on Rad(𝑇𝑀),(ii)traceℎ=0, where trace is written with respect to 𝑔 restricted to 𝑆(𝑇𝑀). We use the quasi-orthonormal basis of 𝑀 given by 𝜉₁,…,𝜉_𝑟,𝜙𝜉₁,…,𝜙𝜉_𝑟,𝑉,𝑒₁,…,𝑒_𝑞,𝜙𝑁₁,…,𝜙𝑁_𝑟, (82) such that {𝜉₁,…,𝜉_𝑟}, {𝜙𝜉₁,…,𝜙𝜉_𝑟}, {𝑒₁,…,𝑒_𝑞}, and {𝜙𝑁₁,…,𝜙𝑁_𝑟} form a basis of Rad(𝑇𝑀), 𝜙(Rad(𝑇𝑀)), 𝐷, and 𝜙(ltr(𝑇𝑀)), respectively.

Definition 14 (see [16]). A lightlike submanifold is called irrotational if and only if ∇_𝑋𝜉∈Γ(𝑇𝑀) for all 𝑋∈Γ(𝑇𝑀) and 𝜉∈Γ(Rad(𝑇𝑀)).

Theorem 15. Let 𝑀 be an irrotational slant lightlike submanifold of an indefinite cosymplectic manifold 𝑀. Then 𝑀 is minimal if and only if trace𝐴_{𝑊𝑘𝑆(𝑇𝑀)}=0,trace𝐴^∗_{𝜉𝑖𝑆(𝑇𝑀)}=0, (83) where {𝑊_𝑘}^𝑙_𝑘=1 is a basis of 𝑆(𝑇𝑀^⟂) and {𝜉_𝑖}^𝑟_𝑖=1 is a basis of Rad(𝑇𝑀).

Proof. Since ∇_𝑉𝑉=0, therefore using (8), we get ℎ^𝑙(𝑉,𝑉)=0 and ℎ^𝑠(𝑉,𝑉)=0. Moreover, irrotational M implies ℎ^𝑠(𝑋,𝜉)=0 for 𝑋∈Γ(𝑇𝑀) and 𝜉∈Γ(Rad(𝑇𝑀)). Thus ℎ^𝑠 vanishes on Rad(𝑇𝑀). Hence 𝑀 is minimal if and only if trace ℎ=0 on 𝑆(𝑇𝑀); that is, 𝑀 is minimal if and only if ^𝑟_𝑖=1ℎ𝜙𝜉_𝑖,𝜙𝜉_𝑖+^𝑟_𝑖=1ℎ𝜙𝑁_𝑖,𝜙𝑁_𝑖+^𝑞_𝑗=1ℎ𝑒_𝑗,𝑒_𝑗=0. (84) Using (10) and (12), we obtain ^𝑟_𝑖=1ℎ𝜙𝜉_𝑖,𝜙𝜉_𝑖=^𝑟_𝑖=11𝑟^𝑟_𝑎=1𝑔𝐴^∗_𝜉𝑎𝜙𝜉_𝑖,𝜙𝜉_𝑖𝑁_𝑎+1𝑙^𝑙_𝑘=1𝑔𝐴_𝑊𝑘𝜙𝜉_𝑖,𝜙𝜉_𝑖𝑊_𝑘. (85) Similarly, we have ^𝑟_𝑖=1ℎ𝜙𝑁_𝑖,𝜙𝑁_𝑖=^𝑟_𝑖=11𝑟^𝑟_𝑎=1𝑔𝐴^∗_𝜉𝑎𝜙𝑁_𝑖,𝜙𝑁_𝑖𝑁_𝑎+1𝑙^𝑙_𝑘=1𝑔𝐴_𝑊𝑘𝑒_𝑗,𝑒_𝑗𝑊_𝑘,_𝑞𝑗=1ℎ𝑒_𝑗,𝑒_𝑗=^𝑞_𝑗=11𝑟^𝑟_𝑖=1𝑔𝐴^∗_𝜉𝑖𝑒_𝑗,𝑒_𝑗𝑁_𝑖+1𝑙^𝑙_𝑘=1𝑔𝐴_𝑊𝑘𝑒_𝑗,𝑒_𝑗𝑊_𝑘. (86) Thus our assertion follows from (85) and (86).