Multiply Warped Product Generalized Semi-Invariant Submanifolds

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DOI:HTTPS://DOI.ORG/10.36890/IEJG.845483

Multiply Warped Product Generalized Semi-Invariant Submanifolds

Moctar Traore, Hakan Mete Ta¸stan and Sibel Gerdan Aydın

^*

(Communicated by Bang-Yen Chen)

ABSTRACT

We define generalized semi-invariant submanifolds in locally product Riemannian manifolds.

Then we study multiply warped product generalized semi-invariant submanifolds in the same structure. We give an existence theorem for such submanifolds. We also give necessary and sufficient conditions for such a submanifold to be a multiply direct product submanifold.

Moreover, we establish a general inequality for such submanifolds.

Keywords: Multiply warped product submanifold, slant distribution, invariant distribution, anti-invariant distribution, generalized semi-invariant submanifold, locally product Riemannian manifold.

AMS Subject Classification (2020): Primary: 53C15 ; Secondary: 53B20

1. Introduction

Multiply warped product manifolds [12] are natural generalization of the warped product manifolds [8].

These notions play very important roles in physics as well as in differential geometry, especially in the theory of relativity. Indeed, the standard spacetimes models such as Roberston-Walker, Schwarschild, static and Kruskal are warped products. Also, the simplest models of neighborhoods of stars and black holes are warped product [16].

On the other hand, warped or multiply warped product submanifolds have been studying very actively since Chen [9] studied the warped product CR-submanifolds in Kaehler structures. The most of the studies related to the warped or multiply warped product submanifolds can be found in the book [11] and its list of references.

In this paper, motivated by the papers placed in [11], especially Chen and Dillen’s paper [10], we study a certain type of multiply warped product submanifolds in locally product Riemannian manifolds. In particular, we consider the multiply warped product submanifolds in the form M^θ×f₁M₁^T× . . . ×f_kM_k^T ×σ₁M₁^⊥× . . . ×σ_lM_l^⊥, where M^θ is a proper slant, M_i^T is an invariant submanifold and M_j^⊥ is an anti-invariant submanifold of the locally product Riemannian manifold for1 ≤ i ≤ kand1 ≤ j ≤ l. We give necessary and sufficient conditions for a generalized semi-invariant submanifold to be a locally multiply warped product in the main theorem. Also, we investigate the behavior of the second fundamental form of such submanifolds and as results, we give necessary and sufficient conditions for such submanifolds to be locally multiply direct or usual product and get an inequality for the squared norm of the second fundamental form in terms of the warping functions for such submanifolds.

Received : 23–12–2020, Accepted : 13–09–2021

* Corresponding author

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2. Preliminaries

In this section, we give the fundamental definitions and notions needed for further study. In subsection 2.1, we will recall the definition of the multiply warped product manifolds. In subsection 2.2, we give the basic background for submanifolds of Riemannian manifolds. The definition of a locally product Riemannian manifold is placed in the last subsection.

2.1. Multiply warped product manifolds

Let (M0, g0), (M1, g1), . . . , (Mk, gk) be Riemannian manifolds and let f1, f2, . . . , fk be positive smooth functions onM0. Then the multiply warped product manifold [12]M0×f1M1× . . . ×f_kMkis the multiply product manifold^M0× M1× . . . × Mkfurnished with the metric

g = π₀^∗(g0) ⊕ (f1◦ π0)²π₁^∗(g1) ⊕ . . . ⊕ (fk◦ π0)²π_k^∗(gk).

More precisely, for any vector fieldsX^¯ andY^¯ onM^¯, we have

g( ¯X, ¯Y ) = g0(π0_∗X, π¯ 0_∗Y ) +¯

k

X

i=1

(fi◦ π0)²gi(πi_∗X, π¯ i_∗Y ),¯ (2.1)

where πi : ¯M = M0× M1× . . . × Mk → Mi,i = 0, 1, . . . , k is the canonical projection,π^∗_i(gi)is the pullback of giviaπi and the subscript * denotes the derivative map ofπi. The functionsf1, . . . , fk are called the warping functions ofM0×f₁M1× . . . ×f_kMk. The manifolds(M1, g1), . . . , (Mk, gk)are called the fibers and the manifold (M0, g0)is called the base manifold of the multiply warped product manifoldM0×f₁M1× . . . ×f_kMk. It is well known that the base manifold is totally geodesic and the fibers are totally umbilic inM0×f₁M1× . . . ×f_kMk.

As mentioned in the previous section, the notion of the multiply warped product is a generalization of direct product as well as warped product manifolds. Indeed, if we choosek = 1in the definition above, then we get a warped product [8] and if each warping functionfiis constant in the definition above, then we get a multiply direct product [11].

Let^M0×_f₁M₁× . . . ×_f_kM_kbe a multiply warped product manifold with the Levi-Civita connection^∇^¯ with respect to the metric^ggiven in (2.1) and^∇ⁱdenote the Levi-Civita connection of^(Mi, g_i)fori ∈ {0, 1, . . . , k}. By usual convenience, we denote the set of lifts of vector fields on^Miby^L(Mi)and use the same notation for a vector field (resp. warping function) and its lift (resp. its pulback). On the other hand, since the mapπ0is an isometry andπ1, . . . , πk are positive homotheties, they preserve the Levi-Civita connections. Thus there is no confusion using the same symbol for a connection on Mi and for its pullback viaπi. Then, the covariant derivative formulas [23] of the multiply warped product manifoldM0×f₁M1× . . . ×f_kMkare given by

∇¯_ZW = ∇⁰_ZW (2.2)

∇¯ZX = ∇¯XZ = Z(ln f_i)X (2.3)

∇¯_XY =

0 if i 6= j,

∇ⁱ_XY − g(X, Y )∇⁰(ln fi) if i = j, (2.4) where^{Z, W ∈ L(M}0),^{X ∈ L(M}i)and^{Y ∈ L(M}j)fori, j ∈ {1, 2, . . . , k}.

2.2. Submanifolds of Riemannian manifolds

LetM be a Riemannian manifold isometrically immersed in a Riemannian manifold( ¯M , g) and∇^¯ be the Levi-Civita connection of M^¯ with respect to the metric g. Also, let ∇ and ∇^⊥ be the induced and induced normal connection onM, respectively. Then the Gauss and Weingarten formulas [25] are given respectively by

∇¯VW = ∇_VW + h(V, W ) and ∇¯VZ = −A_ZV + ∇^⊥_VZ, (2.5)

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where the vector fieldsV, W are tangent toM andZis normal toM. In addition,his the second fundamental form ofM andAZ is the Weingarten endomorphism associated withZ. The second fundamental formhand the shape operatorAare related by

g(h(V, W ), Z) = g(AZV, W ). (2.6)

The mean curvature vector^H for an orthonormal frame ^{e1, . . . , e_m} of tangent space ^TpM, ^{p ∈ M} on ^M is defined by

H = 1

mtrace(h) = 1 m

m

X

i=1

h(e_i, e_i) (2.7)

wherem = dim(M ). Also, we set

h^r_ij = g(h(ei, ej), er) and k h k²=

m

X

i,j=1

g(h(ei, ej), h(ei, ej)) (2.8)

r = n − m, wheren = dim( ¯M )andm = dim(M ). 2.3. Locally product Riemannian manifolds

LetM^¯ be any manifold equipped with a tensor field of type(1, 1)such that

F²= I, (F 6= ∓I) (2.9)

where^Iis the identity endomorphism on the tangent bundle^{T ¯}^M of^M^¯. Then we say that^{( ¯}^{M , F )}is an almost product manifold with almost product structure^F. If the almost product manifold^{( ¯}^{M , F )}admits a metric tensor gsuch that

g(F ¯X, F ¯Y ) = g( ¯X, ¯Y ) (2.10)

for allX, ¯^¯ Y ∈ Γ(T ¯M ), then( ¯M , F, g)is called an almost product Riemannian manifold. Let∇^¯ be the Levi-Civita connection of( ¯M , F, g), then we say that( ¯M , F, g)is a locally product Riemannian manifold (briefly, l.p.R. manifold) or locally decomposable Riemannian manifold ifFis parallel with respect to∇^¯, i.e.

∇¯X¯F ≡ 0 (2.11)

for all^{X ∈ Γ(T ¯}^¯ ^{M )}[25].

3. Generalized semi-invariant submanifolds in locally product Riemannian manifolds In this section, we define the definition of the generalized semi-invariant submanifolds of a l.p.R. manifold and get some useful results for further study.

Let( ¯M , F, g)be a locally product Riemannian manifold and let M be a submanifold of M^¯. A distribution Don M is said to be a slant distribution if the angleθ between F V andDp is constant for V ∈ Dp, i.e., it is independent ofp ∈ M andV ∈ Dp.The constant angle θis called the slant angle of the slant distribution D. Thus, the invariant and anti-invariant distributions with respect toF are slant distributions with slant angle θ = 0andθ = π/2, respectively. A submanifoldM ofM^¯ is said to be a slant submanifold if the tangent bundle T M ofM is slant [14,17]. A slant submanifold that is neither invariant nor anti-invariant is called a proper slant submanifold.

Let M be a slant submanifold with slant angle θ of a l.p.R. manifold ( ¯M , g, F ), for any V ∈ Γ(T M ) and ξ ∈ Γ(T^⊥M ), we write

F V = T V + N V and F ξ = tξ + wξ. (3.1)

HereT V is the tangential part ofF V andN V is the normal part ofF V alsotξis the tangential part ofF ξand wξis the normal part ofF ξ. Then, using (2.10) and (3.1) we find

T²+ tN = I, N T + wN = 0, w²+ N t = I, T t + tw = 0. (3.2) Then, for anyU, V ∈ Γ(T M )we have [17]

T²V = cos²θV, (3.3)

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g(T U, T V ) = cos²θg(U, V ) and g(N U, N V ) = sin²θg(U, V ). (3.4) A submanifold ^M of a l.p.R. manifold^{( ¯}^{M , F, g)}is called a generalized semi-invariant submanifold if its tangent bundle^{T M} of^M has the form

T M = D^θ⊕ D₁^T⊕ . . . ⊕ D^T_k ⊕ D₁^⊥⊕ · · · ⊕ D_l^⊥, (3.5) where the distribution^D^Tαis an invariant for^{1 ≤ α ≤ k}, i.e.,^{F D}^Tα ⊆ D_α^T, the distribution^Da^⊥is an anti-invariant for^{1 ≤ a ≤ l}, i.e.^{F D}^⊥a ⊆ T^⊥Mand the distribution^D^θis slant with slant angle^θ. In that case, the normal bundle T^⊥M of^M decomposed as

T^⊥M = N (D^θ) ⊕ F (D^⊥₁) ⊕ . . . ⊕ F (D_l^⊥) ⊕ ¯D^T, (3.6) where D^¯^T is the orthogonal complementary distribution ofN (D^θ) ⊕ F (D^⊥₁) ⊕ · · · ⊕ F (D_l^⊥)in T^⊥M and it is invariant subbundle of^T^⊥^M with respect to^F. We say that a generalized semi-invariant submanifold is proper, neither^{θ = 0}nor^{θ =} ^θ₂.

Remark 3.1. The notion of generalized semi-invariant submanifold of a l.p.R. manifolds is a natural generalization of invariant, anti-invariant [1] semi-invariant [7], slant [17], semi-slant [15], hemi-slant [21] and skew semi- invariant submanifold of order 1 [20] of a l.p.R. manifold. Also, this notion is slightly different from the definition of the skew semi-invariant submanifold [14]. For more details, we refer to [2,4,6,24].

We need the following lemma.

Lemma 3.1. [20] Let^M be a generalized semi-invariant submanifold of a l.p.R. manifold^{( ¯}^{M , F, g)}. Then we have

g(∇_ZW, U_α) = − csc²θ

g(A_{N T W}Z, U_α) + g(A_{N W}Z, F U_α)

, (3.7)

g(∇_ZW, X_a) = sec²θ

g(A_{F X}_aZ, T W ) + g(A_{N T W}Z, X_a)

, (3.8)

g(∇_U_αV_α, Z) = csc²θ

g(A_{N T Z}U_α, V_α) + g(A_{N Z}U_α, F V_α)

, (3.9)

g(∇_U_αV_α, X_a) = g(A_{F X}_aU_α, F V_α), (3.10) g(∇X_aYa, Uα) = −g(AF Y_aXa, F Uα), (3.11)

g(∇_X_aZ, U_α) = − csc²θ

g(A_{N T Z}X_a, U_α) + g(A_{N Z}X_a, F U_α)

, (3.12)

g(∇ZXa, Uα) = −g(AF X_aZ, F Uα), (3.13)

g(∇_U_αX_a, Z) = − sec²θ

g(A_{F X}_aU_α, T Z) + g(A_{N T Z}U_α, X_a)

, (3.14)

for^Uα, V_α∈ Γ(D^T_α)with^{1 ≤ α ≤ k},^Xa, Y_a ∈ Γ(D_a^⊥)with^{1 ≤ a ≤ l}and^{Z, W ∈ Γ(D}^θ⁾.

Lemma 3.2. Let^Mbe a generalized semi-invariant submanifold of a l.p.R. manifold^{( ¯}^{M , F, g)}. Then we have

g(∇XaYa, Z) = − sec²θ

g(AF YaXa, T Z) + g(AN T ZXa, Ya)

, (3.15)

g(∇U_αVα, Uβ) = g(∇U_αF Vα, F Uβ), (3.16) g(∇X_aYa, Xb) = g(∇^⊥_X

aF Ya, F Xb), (3.17)

for Uα, Vα∈ Γ(D^T_α), Uβ∈ Γ(D_β^T) with 1 ≤ α 6= β ≤ k, Xa, Ya∈ Γ(D_a^⊥), Xb ∈ Γ(D_b^⊥) with 1 ≤ a 6= b ≤ l and Z ∈ Γ(D^θ).

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Proof. LetXa, Ya ∈ Γ(D^⊥_a)andZ ∈ Γ(D^θ). By using (2.5), (2.10) and (3.1), we have

g(∇X_aYa, Z) = g( ¯∇X_aF Ya, F Z) = g( ¯∇X_aF Ya, T Z) + g( ¯∇X_aF Ya, N Z).

Hence using (2.10) and (3.1) we have

g(∇X_aYa, Z) = −g(AF Y_aXa, T Z) + g( ¯∇X_aYa, tN Z) + g( ¯∇X_aYa, wN Z).

Again using (2.5), (3.3) and (3.2), we obtain

g(∇X_aYa, Z) = −g(AF Y_aXa, T Z) + sin²θg( ¯∇X_aYa, Z) − g(AN T ZXa, Ya).

According to direct calculating we find (3.15). Let Uα, Vα∈ Γ(D^T_α),Uβ, Vβ∈ Γ(D^T_β). Then using (2.5), we have g(∇_U_αV_α, U_β) = g( ¯∇_U_αV_α, U_β).By using (2.10), we obtain^g(∇UαV_α, U_β) = g(F ¯∇_U_αV_α, F U_β).Hence using (2.11), we getg(∇U_αVα, Uβ) = g( ¯∇U_αF Vα, F Uβ),sinceF Uβ∈ Γ(T M ). With the help of (2.5) we obtain (3.16)

g(∇U_αVα, Uβ) = g(∇U_αF Vα, F Uβ).

LetXa, Ya ∈ Γ(D^⊥_a),Xb, Yb∈ Γ(D^⊥_b). Then using (2.5) we have g(∇X_aYa, Xb) = g( ¯∇X_aYa, Xb).By using (2.10), we obtaing(∇X_aYa, Xb) = g(F ¯∇U_αVα, F Xb).Hence by using (2.11), we getg(∇X_aYa, Xb) = g( ¯∇^⊥_X

aF Ya, F Xb), sinceF Xb∈ Γ(T M^⊥). With the help of (2.5) we obtain (3.16)g(∇X_aYa, Xb) = g(∇^⊥_X

aF Ya, F Xb).

Theorem 3.1. LetM be a generalized proper semi-invariant submanifold of a l.p.R manifold( ¯M , F, g). Then the slant distributionD^θis totally geodesic if and only if the following equations hold

g(A_{N T W}Z, U_α) = −g(A_{N W}Z, F U_α), (3.18) g(AF X_aZ, T W ) = −g(AN T WZ, Xa) (3.19) for^{Z, W ∈ Γ(D}^θ⁾,^Uα∈ Γ(D^T_α)and^Xa∈ Γ(D^⊥_a).

Proof. Let ^M be a generalized semi-invariant submanifold of a l.p.R manifold ^{( ¯}^{M , F, g)}. Then the slant distribution^D^θ is totally geodesic if and only if^g(∇ZW, X_a) = 0 and ^g(∇ZW, U_α) = 0 for all ^{Z, W ∈ Γ(D}^θ⁾, Xa∈ Γ(D_a^⊥)andUα∈ Γ(D^T_α). Thus, the assertions (3.18) and (3.19) follow from (3.7) and (3.8), respectively.

Theorem 3.2. Let M be a generalized proper semi-invariant submanifold of a l.p.R manifold ( ¯M , F, g). Then the invariant distribution^D^Tα,^{1 ≤ α ≤ k}is integrable if and only if the following equations hold

g(AF X_aUα, F Vα) = g(AF X_aVα, F Uα), (3.20) g(A_{N T Z}U_α, V_α) + g(A_{N Z}U_α, F V_α) = g(A_{N T Z}V_α, U_α) + g(A_{N Z}V_α, F U_α), (3.21) g(∇U_αF Vα, F Uβ) = g(∇V_αF Uα, F Uβ), (3.22) for^{Z ∈ Γ(D}^θ⁾,^Uα, V_α∈ Γ(D^T_α),^Uβ ∈ Γ(D_β^T),1 ≤ α 6= β ≤ kand^Xa ∈ Γ(D_a^⊥).

Proof. LetM be a generalized semi-invariant submanifold of a l.p.R manifold ( ¯M , F, g). Then the invariant distribution^Dα^T is integrable if and only if^g([Uα, V_α], X_a) = 0,^g([Uα, V_α], Z) = 0and^g([Uα, V_α], U_β) = 0for all Z ∈ Γ(D^θ),^Xa∈ Γ(D_a^⊥)and^Uα, V_α∈ Γ(D_α^T),^Uβ∈ Γ(D_β^T)with1 ≤ α 6= β ≤ k. Thus, the assertions (3.20), (3.21) and (3.22) follow from (3.9), (3.10) and (3.16), respectively.

Theorem 3.3. LetM be a generalized proper semi-invariant submanifold of a l.p.R manifold( ¯M , F, g). Then the anti- invariant distributionD^⊥_a,1 ≤ a ≤ lis integrable if and only if the following equations hold

g(A_{F X}_aY_a, F U_α) = g(A_{F Y}_aX_a, F U_α), (3.23) g(AF Y_aXa, T Z) = g(AF X_aYa, T Z), (3.24) g(∇^⊥_X_aF Ya, Xb) = g(∇^⊥_Y_aF Xa, Xb), (3.25) forZ ∈ Γ(D^θ),Uα∈ Γ(D^T_α)andXa, Ya ∈ Γ(D^⊥_a),Xb∈ Γ(D^⊥_b),1 ≤ a 6= b ≤ l.

Proof. LetM be a generalized semi-invariant submanifold of a l.p.R manifold( ¯M , F, g). Then the anti-invariant distributionD_a^⊥is integrable if and only ifg([Xa, Ya], Z) = 0,g([Xa, Ya], Uα) = 0andg([Xa, Ya], Xb) = 0for all Z ∈ Γ(D^θ), ^Uα∈ Γ(D^T_α) and ^Xa, Y_a∈ Γ(D^⊥_a), ^Xb∈ Γ(D^⊥_b) with 1 ≤ a 6= b ≤ l. Thus, the assertions (3.23) and (3.25) follow from (3.11), (3.15) and (3.17), respectively.

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4. Certain Types of Multiply Warped Product Submanifolds in Locally Product Riemannian Manifolds

In this section, we check that the existence of certain types of multiply warped product generalized semi-invariant submanifolds in the form,

I. ^M^T ×σ1M₁^⊥× . . . ×σlM₁^⊥×λ1M₁^θ¹× . . . ×λmM_m^θ^m, II. M^⊥×f₁M₁^T × . . . ×f_kM_k^T ×λ₁M^θ¹× . . . ×λ_mM_m^θ^m, III. ^M^θ^×f₁M₁^T × . . . ×f_kM_k^T ×σ₁M₁^⊥× . . . ×σ_lM_l^⊥,

whereM_α^T,1 ≤ α ≤ kis an invariant,M_a^⊥,1 ≤ a ≤ lis an anti-invariant andM_β^θ^βis a proper slant submanifold with slant angle^θβ,^{1 ≤ β ≤ m}of a l.p.R manifold^{( ¯}^{M , F, g)}.

M. Atçeken and B. S.ahin independently proved that there do not exist (non-trivial) warped product semi- invariant submanifolds in the form ^M^T ^×fM^⊥ in a l.p.R. manifold^{( ¯}^{M , F, g)}, such that ^M^T is an invariant submanifold and^M⊥ is an anti-invariant submanifold of^{( ¯}^{M , F, g)}in Theorem 3.1([5]) and Theorem 3.1([19]), respectively. Again, M. Atçeken and B. S.ahin independently proved that there do not exist (non-trivial) warped product semi-slant submanifolds in the formM^T ×fM^θ in a l.p.R. manifoldM^¯, such thatM^T is an invariant submanifold andM^θis a proper slant submanifold ofM^¯ in Theorem 3.3([3]) and Theorem 3.1([18]), respectively. Thus, we obtain the following result.

Corollary 4.1. There do not exist (non-trivial) multiply warped product generalized semi-invariant submanifold in the form I of a l.p.R. manifold( ¯M , F, g).

On the other hand, it was proved that there do not exist (non-trivial) warped product semi-invariant submanifold in the form ^M^⊥×fM^θ in a l.p.R. manifold^M^¯ such that^M^⊥ is an anti-invariant submanifold and^M^θis a proper slant submanifold of^M^¯ in Theorem 3.4 of [3]. Thus, we deduce the following result.

Corollary 4.2. There do not exist (non-trivial) multiply warped product generalized semi-invariant submanifold in the form II of a l.p.R. manifold( ¯M , F, g).

Now, we consider (non-trivial) multiply warped product generalized semi-invariant submanifolds in the formM^θ×f₁M₁^T × . . . ×f_kM_k^T ×σ₁M₁^⊥× . . . ×σ_lM_l^⊥ in a l.p.R. manifold( ¯M , F, g)such thatM_α^T,1 ≤ α ≤ kis an invariant,M_a^⊥,1 ≤ a ≤ lis an anti-invariant andM^θis a proper slant submanifold ofM^¯. We first present an example of such a submanifold.

Example 4.1. Consider the^{4k + 4l}-dimensional Euclidean space^R^4k+4lwith usual metric^gand almost product structure^F defined by

F ∂i = ∂i, 1 ≤ i ≤ 2k, F ∂i= −∂i, 2k + 1 ≤ i ≤ 4k, F ∂j = ∂j+1, F ∂j+1= ∂j 4k + 1 ≤ j ≤ 4k + 4l, where^∂s= _∂x^∂

s and^{xs}_{1≤s≤4k+4l}are natural coordinates of^R^4k+4l. Upon straightforward calculation, we see that(R^4k+4l, F, g)is a l.p.R. manifold. LetM be a submanifold of(R^4k+4l, F, g)given by

x1= t sin u1, x2= t cos u1, x3= 2t sin u2, x4= 2t cos u2,

. . . ,

x2k−1= kt sin uk, x2k = kt cos uk, x_2k+1= t

√2cos v₁, x_2k+2= t

√2sin v₁,

x2k+3= 2t

√2cos v2, x2k+4= 2t

√2sin v2,

. . . ,

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x4k−1= kt

√

2cos vk, x4k= kt

√

2sin vk, x4k+1= 2t sin z1, x4k+2= 0, x_4k+3= 2t cos z₁, x_4k+4= 0, x_4k+5= 2t sin z₂, x_4k+6= 0, x_4k+7= 2t cos z₂, x_4k+8= 0,

. . . ,

x4k+4l−3= 2lt sin zl, x4k+4l−2= 0, x4k+4l−1= 2lt cos zl, x4k+4l= 0.

whereui, vi, zj∈ (0,^π₂)andt > 0. Then, the local frame ofT Mgiven by

T = sin u₁∂₁+ cos u₁∂₂+ 2 sin u₂∂₃+ 2 cos u₂∂₄+ · · · + k sin u_k∂_2k−1 + k cos u_k∂_2k

+^√¹

2{cos v₁∂_2k+1+ sin v₁∂_2k+2+ 2 cos v₂∂_2k+3+ 2 sin v₂∂_2k+4 + · · · + k cos vk∂4k−1+ k sin vk∂4k}

+ 2{sin z1∂4k+1+ cos z1∂4k+3+ 2 sin z2∂4k+5+ 2 cos z2∂4k+7

+ · · · + l sin zl∂4k+4l−3+ l cos zl∂4k+4l−1},

U1= t cos u1∂1− t sin u1∂2, U2= 2t cos u2∂3− 2t sin u2∂4,

. . . ,

U_k= kt cos u_k∂_2k−1− kt sin uk∂_2k, V₁= −^√^t₂sin v₁∂_2k+1+^√^t

2cos v₁∂_2k+2, V₂= −^√^2t

2sin v₂∂_2k+3+^√^2t

2cos v₂∂_2k+4, . . . ,

V_k = −^√^kt₂sin v_k∂_4k−1+^√^kt

2cos v_k∂_4k, Z1= 2t cos z1∂4k+1− 2t sin z1∂4k+3, Z2= 2t cos z2∂4k+5− 2t sin z2∂4k+7,

. . .

Zl= 2lt cos zl∂4k+4l−3− 2lt sin zl∂4k+4l−1.

By direct calculation, we see that^D^θ⁼span^{{T }}is a proper slant distribution with slant angle^{θ = cos}⁻¹

1 3+

k(k+1)(2k+1) 8l(l+1)(2l+1)

and D^T_i =span{Ui, Vi}, 1 ≤ i ≤ k is an invariant distribution and D_j^⊥=span{Zj}, 1 ≤ j ≤ l is an anti-invariant distribution. So far,M is a proper generalized semi-invariant submanifold. Moreover,D^θ is totally geodesic and bothD_i^T andD_j^⊥are integrables distributions. If we denote the integral manifolds ofD^θ, D^T_i andD_j^⊥byM^θ,M_i^T andM_j^⊥, respectively, then the induced metric tensor ofM is

ds² = g(T, T )dt²+Pk

i=1g(U_i, U_i)du²_i +Pk

i=1g(V_i, V_i)du²_j+Pl

j=1g(Z_j, Z_j)dz_j². Upon straightforward calculation, we have

ds² = ₁₂¹[3k(k + 1)(2k + 1) + 8l(l + 1)(2l + 1)]dt²+ t²(du²₁+¹₂dv₁²)+

(2t)²(du²₂+¹₂dv₂²) + . . . + (kt)²(du²_k+¹₂dv²_k) + (2t)²dz₁²+ (4t)²dz₂²+ . . . + (2lt)²dz_l²

= g_Mθ+ t²g_MT

1 + (2t)²g_MT

2 + . . . + (kt)²g_MT

k + (2t)²g_M⊥

1 + (4t)²g_M⊥ 2+ . . . + (2lt)²g_M^⊥

l

.

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Thus, M = M^θ×f₁M₁^T × . . . ×f_kM_k^T×σ₁M₁^⊥× . . . ×σ_lM_l^⊥ is a (non-trivial) multiply warped product generalized semi-invariant submanifold of (R^4k+4l, F, g)with warping functions f1= t, f2= 2t, . . . , fk = kt andσ1= 2t,σ2= 4t, . . . ,σl= 2lt.

5. Multiply warped product generalized semi-invariant submanifolds

In this section, we give a characterization for a multiply warped product proper generalized semi- invariant submanifold in the form ^M^θ×f1M₁^T × . . . ×fkM_k^T ×σ1M₁^⊥× . . . ×σlM_l^⊥, where ^M^θ is a proper slant submanifold, ^Mα^T, ^{1 ≤ α ≤ k} is an invariant and ^Ma^⊥, ^{1 ≤ a ≤ l} is an anti invariant submanifold of a l.p.R. manifold ^{( ¯}^{M , F, g)}. After that we investigate the behavior of the second fundemental form of such submanifolds and as a result, we give a necessary and sufficient condition for such submanifolds to be locally multiply warped product generalized. We first recall the following fact given in [12] to prove our theorem.

Remark 5.1. (Remark 2.1 [12]) Suppose that the tangent bundle of a Riemannian manifoldM splints into an orthogonal sum T M = D0⊕ D1⊕ . . . ⊕ Dk of non-trivial distributions such that each Dj is spherical and its complement in^{T M} is autoparallel forj ∈ {1, 2, . . . , k}. Then the manifold^M is locally isometric to a multiply warped product^M0×f1M₁× . . . ×fkM_k.

Now, we give one of the main theorems of this paper.

Theorem 5.1. LetM be a(D^θ, D^⊥_a)-mixed geodesic multiply warped product generalized semi-invariant submanifold of a l.p.R. manifold( ¯M , F, g). ThenM is a locally multiply warped product generalized submanifold of typeM^θ×f₁M₁^T × . . . ×f_kM_k^T ×σ₁M₁^⊥× . . . ×σ_lM_l^⊥if and only if we have

AN T ZXa = cos²θZ(λ)Xa, (5.1)

AN ZUα+ AN T ZF Uα= − sin²θZ(µ)F Uα, (5.2) for some functionsλandµsatisfyingXa(λ) = Uα(λ) = 0andXa(µ) = Uα(µ) = 0

g(AF X_aZ, T W ) = −g(AN T WZ, Xa), (5.3)

g(AF X_aUα, F Vα) = 0, (5.4)

g(AF Y_aXa, F Vα) = 0, (5.5)

g(A_{F X}_aZ, F U_α) = 0, (5.6)

g(AF X_aUα, T Z) = −g(AN T ZUα, Xa), (5.7)

g(∇U_βUγ, Uα) = 0, (5.8)

g(∇_X_bX_c, X_a) = 0 (5.9)

and (3.22) and (3.25) hold, whereZ, W ∈ Γ(D^θ),Uα, Vα∈ Γ(D_α^T),Uβ∈ Γ(D^T_β)andUγ ∈ Γ(D^T_γ)for1 ≤ α, β, γ ≤ k withα 6= βandα 6= γ,Xa, Ya∈ Γ(D^⊥_a),Xb∈ Γ(D_b^⊥)andXc∈ Γ(D^⊥_c)for1 ≤ a, b, c ≤ lwitha 6= banda 6= c. Proof. Let^M be a^(D^θ^{, D}^⊥a)-mixed geodesic multiply warped product generalized semi-invariant submanifold of a l.p.R. manifold^{( ¯}^{M , F, g)}in the form^M^θ^×f1M₁^T× . . . ×_f_kM_k^T ×_σ₁M₁^⊥× . . . ×_σ_lM_l^⊥. Since^M is^(D^θ^{, D}^⊥a)- mixed geodesic, forZ, W ∈ Γ(D^θ)andXa ∈ Γ(D^⊥_a)with1 ≤ a ≤ l, using (2.6), we find

g(AN T ZXα, W ) = g(h(Xα, W ), N T Z) = 0. (5.10) Moreover for anyUα∈ Γ(D_α^T)with1 ≤ α ≤ k, using (2.5) and (3.1),

g(AN T ZXa, Uα) = −g( ¯∇X_aN T Z, Uα) = −g( ¯∇X_aF T Z, Uα) + g( ¯∇X_aT²Z, Uα).

Then using (2.9)∼(2.11) and (3.3), we find

g(AN T ZXa, Uα) = −g( ¯∇X_aT Z, F Uα) + cos²θg( ¯∇X_aZ, Uα).

Here, using (2.5), we arrive

g(A_{N T Z}X_a, U_α) = −g(∇_X_aT Z, F U_α) + cos²θg(∇_X_aZ, U_α).

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So, using (2.3), we conclude

g(A_{N T Z}X_a, U_α) = −T Z(ln σ_a)g(X_a, F U_α) + cos²θZ(ln σ_a)g(X_a, U_α) = 0. (5.11) Next, by a similar argument, for^Ya∈ Γ(D_a^⊥), using (2.5) and (3.1) we have

g(h(X_a, Y_a), N Z) = g( ¯∇XaY_a, N Z) = g( ¯∇XaY_a, F Z) − g( ¯∇XaY_a, T Z).

Then using (2.10),(2.11) and (2.3), we find

g(h(Xa, Ya), N Z) = g( ¯∇X_aF Ya, Z) + T Z(ln σa)g(Xa, Ya).

Hence using (2.5) and (2.6), we arrive

g(h(Xa, Ya), N Z) = −g(AF Y_aXa, Z) + T Z(ln σa)g(Xa, Ya)

= −g(h(Xa, Z), F Ya) + T Z(ln σa)g(Xa, Ya).

In this equation, ifT Zis written instead ofZ, we have

g(h(Xa, Ya), N T Z) = −g(h(Xa, T Z), F Ya) + cos²θZ(ln σa)g(Xa, Ya).

Since^M is^(D^θ^{, D}^⊥a)-mixed geodesic, we conclude that

g(A_{N T Z}X_a, Y_a) = cos²θZ(ln σ_a)g(X_a, Y_a). (5.12) Moreover, we have ^Xa(ln σ_a) = U_α(ln σ_a) = 0, since^σa depends on only points of ^Mθ. So, we conclude that λ = ln σ_a. Thus, from (5.10)^∼(5.12), it follows that (5.1). Now, we prove (5.2). For^{Z ∈ Γ(D}^θ⁾,^Uα∈ Γ(D^T_α)and X_a∈ Γ(D_a^⊥), using (2.5) and (3.1), we have

g(AN ZUα+ AN T ZF Uα, Xa) = g(AN ZUα, Xa) + g(AN T ZF Uα, Xa)

= g(AN ZXa, Uα) + g(AN T ZXa, F Uα)

= −g( ¯∇XaN Z, Uα) − g( ¯∇XaN T Z, F Uα)

= −g( ¯∇XaN Z, U_α) − g( ¯∇XaF T Z, F U_α) +g( ¯∇XaT²Z, F U_α).

Using (2.10), (2.11), (3.1) and (3.3) and, we arrive

g(AN ZUα+ AN T ZF Uα, Xa) = −g( ¯∇X_aF Z, Uα) + g( ¯∇X_aT Z, Uα) − g( ¯∇X_aT Z, Uα) + cos²θg( ¯∇X_aZ, F Uα)

= −g( ¯∇X_aF Z, Uα) + cos²θg( ¯∇X_aZ, F Uα).

Then, using (2.3), (2.5), (2.9)∼(2.11), we find

g(AN ZUα+ AN T ZF Uα, Xa) = −g( ¯∇X_aZ, F Uα) + cos²θg(∇X_aZ, F Uα)

= − sin²θg(∇_X_aZ, F U_α)

= − sin²θZ(ln σa)g(Xa, F Uα).

Sinceg(Xa, F Uα) = 0, we conclude that

g(AN ZUα+ AN T ZF Uα, Xa) = − sin²θZ(ln σa)g(Xa, F Uα) = 0. (5.13) Similarly, for^{Z, W ∈ Γ(D}^θ⁾and^Uα∈ Γ(D^T_α), using (2.5) and (3.1), we have

g(AN ZUα+ AN T ZF Uα, W ) = g(AN ZUα, W ) + g(AN T ZF Uα, W )

= g(A_{N Z}W, U_α) + g(A_{N T Z}W, F U_α)

= −g( ¯∇WN Z, U_α) − g( ¯∇WN T Z, F U_α)

= −g( ¯∇WN Z, U_α) − g( ¯∇WF T Z, F U_α) +g( ¯∇_WT²Z, F U_α).

Using (2.10), (2.11), (3.1) and (3.3), we arrive

g(AN ZUα+ AN T ZF Uα, W ) = −g( ¯∇WF Z, Uα) + g( ¯∇WT Z, Uα) − g( ¯∇WT Z, Uα) + cos²θg( ¯∇WZ, F Uα)

= −g( ¯∇WF Z, U ) + cos²θg( ¯∇WZ, F Uα).

Then, using (2.3), (2.5), (2.9)∼(2.11), we find

g(AN ZUα+ AN T ZF Uα, W ) = −g( ¯∇WZ, F Uα) + cos²θg(∇WZ, F Uα)

= −g(∇WZ, F Uα) + cos²θg(∇WZ, F Uα)

= − sin²θg(∇_WZ, F U_α) = + sin²θg(Z, ∇^θ_WF U_α).

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Sinceg(Z, ∇^θ_WF Uα) = 0, we conclude

g(AN ZUα+ AN T ZF Uα, W ) = sin²θg(Z, ∇^θ_WF Uα) = 0. (5.14) On the other hand, forZ ∈ Γ(D^θ)andUα, Vα∈ Γ(D^T_α), using (2.5) and we have

g(A_{N Z}U_α+ A_{N T Z}F U_α, V_α) = g(A_{N Z}U_α, V_α) + g(A_{N T Z}F U_α, V_α)

= g(A_{N Z}V_α, U_α) + g(A_{N T Z}V_α, F U_α)

= −g( ¯∇_V_αN Z, U_α) − g( ¯∇_V_αN T Z, F U_α)

= −g( ¯∇_V_αN Z, U_α) − g( ¯∇_V_αF T Z, F U_α) +g( ¯∇V_αT²Z, F Uα).

Using (2.10), (2.11), (3.1) and (3.3), we arrive

g(A_{N Z}U_α+ A_{N T Z}F U_α, V_α) = −g( ¯∇VαF Z, U_α) + g( ¯∇VαT Z, U_α) − g( ¯∇VαT Z, U_α) + cos²θg( ¯∇VαZ, F U_α)

= −g( ¯∇VαF Z, U_α) + cos²θg( ¯∇VαZ, F U_α).

Using (2.3), (2.5), (2.9)∼(2.11), we find

g(AN ZUα+ AN T ZF Uα, Vα) = −g( ¯∇V_αZ, F Uα) + cos²θg(∇V_αZ, F Uα)

= −g(∇V_αZ, F Uα) + cos²θg(∇V_αZ, F Uα)

= − sin²θg(∇V_αZ, F Uα)

= − sin²θZ(ln f_α)g(V_α, F U_α).

So, we conclude that

g(A_{N Z}U_α+ A_{N T Z}F U_α, V_α) = − sin²θZ(ln f_α)g(F U_α, V_α). (5.15) Moreover, we have ^Xa(ln f_α) = U_α(ln f_α) = 0, since ^f depends on only points of ^Mθ. So, we conclude that µ = ln f_α. Thus from (5.13)∼(5.15), we get (5.2).

Next, we prove (5.3)^∼(5.9). We know^Mis a multiply warped product generalized semi-invariant submanifold of a l.p.R. manifold ^{( ¯}^{M , F, g)}. Then, for^{Z, W ∈ Γ(D}^θ⁾, using (2.2), we get^∇ZW = ∇^θ_ZW and for ^Xa∈ Γ(D_a^⊥), we have

g(∇_ZW, X_a) = sec²θ{g(A_{F X}_aZ, T W ) + g(A_{N T W}Z, X_a)} = g(∇^θ_ZW, X_a) = 0 from (3.8). SinceMθis a proper slant submanifold, it follows that

g(AF X_aZ, T W ) + g(AN T WZ, Xa) = 0.

Which is (5.3). For Uα, Vα∈ Γ(D_α^T) and Xa ∈ Γ(D^⊥_a), using (2.4), we get g(∇U_αVα, Xa) = g(∇^T_U

αVα− g(Uα, Vα)∇(ln fα), Xa) = 0.Then from (3.10) we find

g(∇U_αVα, Xa) = g(AF X_aUα, F Vα) = 0.

Therefore, we get (5.4). ForUα∈ Γ(D_α^T)andXa, Ya∈ Γ(D_a^⊥), using (2.4), we haveg(∇X_aYa, Uα) = g(∇^⊥X_aYa− g(Xa, Ya)∇(ln σa), Uα) = 0. Then from (3.11) we find,

g(∇X_aYa, Uα) = −g(AF Y_aXa, F Uα) = 0.

Hence, we conclude that (5.5). For Xa ∈ Γ(D_a^⊥), Z ∈ Γ(D^θ) and Uα∈ Γ(D^T_α), using (2.3), we write g(∇_ZX_a, F U_α) = g(Z(ln σ_a)X_a, F U_α) = Z(ln σ_a)g(X_a, F U_α) = 0. On the other hand, from (3.13) we find

g(∇_ZX_a, F U_α) = −g(A_{F X}_aZ, F U_α) = 0.

Thus, we get (5.6). For^Xa∈ Γ(D^⊥_a),^{Z ∈ Γ(D}^θ⁾and^Uα∈ Γ(D_α^T), using (2.4), we have^g(∇UαX_a, Z) = 0. Then, from (3.14) we find,

g(∇U_αXa, Z) = − sec²θ{g(AF X_aUα, T Z) + g(AN T ZUα, Xa)} = 0.

It follows that (5.7).

For Uα, Vα∈ Γ(D_α^T), Uβ∈ Γ(D^T_β) and Uγ ∈ Γ(D^T_γ) for 1 ≤ α, β, γ ≤ k with α 6= β and α 6= γ then we have g(∇U_βUγ, Uα) = 0 from (2.4). Hence, we get (5.8). For Xa, Ya ∈ Γ(D_a^⊥), Xb∈ Γ(D^⊥_b) and Xc∈ Γ(D^⊥_c) for 1 ≤ a, b, c ≤ l with ^{a 6= b} and ^{a 6= c} then we have ^g(∇XbX_c, X_a) = 0 from (2.4). Thus, we get (5.9). Since ^M is a multiply warped product generalized semi-invariant submanifold then all distrubutions involve in the

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definition must be integrable. Thus (3.22) and (3.25) respectively hold.

Conversely, assume that M is a (D^θ, D^⊥_a)-mixed geodesic multiply warped product generalized semi- invariant submanifold of l.p.R manifold ( ¯M , F, g) such that (5.1)∼(5.9) and (3.22)∼(3.25) hold. From (5.3), we satisfy (3.19). On the other hand if we write F Uα instead of Uα and W instead of Z in (5.2), we find AN WF Uα+ AN T WUα= − sin²θW (µ)Uα. If we take inner product of this equation withZ ∈ Γ(D^θ), we get

g(A_{N W}F U_α+ A_{N T W}U_α, Z) = g(A_{N W}Z, F U_α) + g(A_{N T W}Z, U_α)

= − sin²θW (µ)g(Uα, Z) = 0.

So, (3.18) holds. Thus, the slant distribution ^D^θ is totally geodesic and as a result it is integrable. On the other hand, from (5.4), for all ^Uα, V_α∈ Γ(D^T_α) and ^Xa∈ Γ(D_a^⊥), we write ^g(AF XaU_α, F V_α) = 0. Thus, g(A_{F X}_aU_α, F V_α) = g(A_{F X}_aU_α, F V_α). Which is (3.20). On the other hand, in (5.2), if we write^{F U}α instead of Uα, we findAN ZF Uα+ AN T ZUα= − sin²θZ(µ)Uα. If we take inner product of this equation withVα∈ Γ(D^T_α), we arrive at

g(A_{N Z}F U_α+ A_{N T Z}U_α, V_α) = g(A_{N Z}F U_α, V_α) + g(A_{N T Z}U_α, V_α)

= − sin²θZ(µ)g(Uα, Vα). (5.16)

Here, if we interchange^Uαand^Vαin (5.16), we find

g(AN ZF Uα+ AN T ZUα, Vα) = g(AN ZF Uα, Vα) + g(AN T ZUα, Vα)

= − sin²θZ(µ)g(Uα, Vα). (5.17)

From (5.16) and (5.17), we get

g(A_{N Z}U_α, F V_α) + g(A_{N T Z}U_α, V_α) = g(A_{N Z}V_α, F U_α) + g(A_{N T Z}V_α, U_α).

This is (3.21). We have already (3.22). Thus, by Theorem 3.2, the invariant distribution ^Dα^T, ^{1 ≤ α ≤ k} is integrable. On the other hand, for all^Xa, Y_a ∈ Γ(D^⊥_a)and^Uα∈ Γ(D^T_α), we have^g(AF YaX_a, F U_α) = 0from (5.5).

It follows that

g(AF Y_aXa, F Uα) = g(AF X_aYa, F Uα) = 0.

That is (3.23). Also, we get

g(∇_X_aY_a, Z) = − sec²θ{g(h(Y_a, T Z), X_a) + g(A_{N T Z}X_a, Y_a)}from (3.15). Since^M is^(D^θ^{, D}a^⊥)-mixed geodesic, it follows that^g(h(Ya, T Z), F X_a) = 0. Then, we find

g(∇_X_aY_a, Z) = g(∇_Y_aX_a, Z).

Thus (3.24) follows. We have already (3.25). Thus by Theorem3.3, the totally real distributions^Da^⊥,^{1 ≤ a ≤ l} is integrable. Let^Mθ, M_α^T and^Ma^⊥ be the integral manifolds of^D^θ^{, D}^Tα and^Da^⊥ respectively. If we denote the second fundamental form of^Mα^T in^M by^h^Tα, for^Uα, V_α∈ Γ(D^T_α)and^Xa ∈ Γ(D_a^⊥), using (2.5), (3.10) and (5.4), we have

g(h^T_α(Uα, Vα), Xa) = g(∇U_αVα, Xa) = g(AF X_aUα, F Vα) = 0. (5.18) For any^Uα, V_α∈ Γ(D^T_α)and^{Z ∈ Γ(D}^θ⁾, using (2.5) and (3.9), we get

g(h^T_α(U_α, V_α), Z) = g(∇_U_αV_α, Z) = csc²θg(A_{N T Z}U_α, V_α) + g(A_{N Z}U_α, F V_α).

At this equation, if we use (5.2), we have

g(h^T_α(U_α, V_α), Z) = csc²θ{g(A_{N T Z}V_α+ A_{N Z}F V_α, U_α)} = −Z(µ)g(V_α, U_α).

After some calculation, we obtain

g(h^T_α(U_α, V_α), Z) = g(−g(U_α, V_α)∇µ, Z) (5.19) where∇µis the gradient of^µ. Thus, from (5.18) and (5.19), we conclude that

h^T_α(U_α, V_α) = −g(U_α, V_α)∇µ.

This equation says that^Mα^T is totally umbilic in^M with the mean curvature vector field−∇µ. Now, we show that−∇µis parallel. We have to satisfy^g(∇Uα∇µ, E) = 0 for^Uα∈ Γ(D_α^T)and ^{E ∈ (D}α^T)^⊥= D^θ⊕ D₁^⊥⊕ . . . ⊕