Access the full text.
Sign up today, get DeepDyve free for 14 days.
Loading next page...
/lp/hindawi-publishing-corporation/a-study-of-doubly-warped-product-immersions-in-a-nearly-trans-sasakian-G2fSVtCeiw
References
References for this paper are not available at this time. We will be adding them shortly, thank you for your patience.
- Publisher
- Hindawi Publishing Corporation
- Copyright
- Copyright © 2021 Ali H. Alkhaldi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
- ISSN
- 1687-9120
- eISSN
- 1687-9139
- DOI
- 10.1155/2021/5065333
- Publisher site
- See Article on Publisher Site
Abstract
Hindawi Advances in Mathematical Physics Volume 2021, Article ID 5065333, 9 pages https://doi.org/10.1155/2021/5065333 Research Article A Study of Doubly Warped Product Immersions in a Nearly Trans-Sasakian Manifold with Slant Factor 1 2 2 1 Ali H. Alkhaldi, Aliya Naaz Siddiqui , Kamran Ahmad, and Akram Ali Department of Mathematics, College of Science, King Khalid University, 9004 Abha, Saudi Arabia M.M. Engineering College, Maharishi Markandeshwar (Deemed to be University), Mullana, Ambala 133207, India Correspondence should be addressed to Akram Ali; akramali133@gmail.com Received 10 July 2021; Accepted 14 September 2021; Published 4 October 2021 Academic Editor: Zine El Abiddine Fellah Copyright © 2021 Ali H. Alkhaldi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In this article, we discuss the de Rham cohomology class for bislant submanifolds in nearly trans-Sasakian manifolds. Moreover, we give a classification of warped product bislant submanifolds in nearly trans-Sasakian manifolds with some nontrivial examples in the support. Next, it is of great interest to prove that there does not exist any doubly warped product bislant submanifolds other than warped product bislant submanifolds in nearly trans-Sasakian manifolds. Some immediate consequences are also obtained. 1. Introduction and Motivations obtained several results on warped products and doubly warped products [7–12]. The most inventive topic in the field of differential geometry The concept of bislant submanifolds is defined by Cab- currently is the theory of warped product manifolds. These rerizo et al. [13] as the natural generalization of contact manifolds are the most fruitful and natural generalization CR-, slant, and semislant submanifolds. Such submanifolds of Riemannian product manifolds. Due to the important generalize invariant, anti-invariant, and pseudoslant subma- roles of the warped product in mathematical physics and nifolds as well. Recently, the warped product bislant subma- geometry, it has become the most active and interesting nifolds in nearly trans-Sasakian manifolds is studied by topic for researchers, and many nice results are available in Siddiqui et al. in [1]. They obtain several inequalities for the literature (see [1–3]). the squared norm of the second fundamental form in terms Chen [4, 5] initiates the concept of warped product sub- of a warping function f . manifolds by proving the nonexistence result of warped In this paper, firstly, we discuss the de Rham cohomol- product CR-submanifolds of type N × N in Kähler ogy class for closed bislant submanifolds in a nearly trans- ⊥ f T Sasakian manifold. Secondly, in view of embedding theorem manifolds, where N and N are anti-invariant and invari- ⊥ T ant submanifolds, respectively. Moreover, he considers of Nash [14], we study an isometric immersion of a warped product bislant submanifold into an arbitrary nearly trans- warped product CR-submanifolds of type N × N and T f ⊥ Sasakian manifold. Then, we investigate the existence of gives an inequality involving a warping function f and the doubly warped products in the same ambient. squared norm of the second fundamental form khk . On the other hand, the concept of ordinary warped 2. Nearly Trans-Sasakian Manifolds and products can be extended to doubly warped products. By their Submanifolds using this generalization, Sahin [6] shows that there exist no doubly warped product CR-submanifolds in Kähler man- Definition 1 (see [15]). A ð2m +1Þ-dimensional differentia- ifolds other than warped product CR-submanifolds. He also investigates the existence of doubly twisted product CR- ble manifold N is said to have an almost contact structure submanifolds in the same ambient. Many geometers have ðϕ, ξ, η, gÞ if there exists on N , where 2 Advances in Mathematical Physics (i) a tensor field ϕ of type ð1, 1Þ and set of all normal vector fields on N , respectively. The operator of covariant differentiation with respect to the (ii) a vector field ξ Levi-Civita connection in N and N is denoted by ∇ and ∇ , respectively. The Gauss and Weingarten formulae are (iii) a 1-form η respectively given as [15] (iv) a Riemannian metric g ∇ Y = ∇ Y +hX, Y , ð4Þ ðÞ X X such that ∇ V = −A X + ∇ Y, ð5Þ ðÞ ϕ = −I + η ⊗ ξ, ϕξ =0,ηξ =1, η ∘ ϕ =0, η X =gX, ξ , X V X ðÞ ðÞ ðÞ for any X, Y ∈ TN and V ∈ T N . Here, h is the second g ϕX, ϕY =gX, Y − η X η Y , g ϕX, Y +gX, ϕY =0, ðÞ ðÞ ðÞ ðÞ ðÞ ðÞ fundamental form, A is the shape operator, and ∇ is the ð1Þ operator of covariant differentiation with respect to the lin- ear connection induced in the normal bundle T N . for any X, Y ∈ TN . The second fundamental form and the shape operator are related as [15] The covariant derivative of the tensor field ϕ is given by gh X, Y , V =gA X , Y , ð6Þ ðÞ ðÞ ðÞ ðÞ ∇ ϕ Y = ∇ ϕY − ϕ∇ Y, ð2Þ X X X for any X, Y ∈ TN and V ∈ T N . Here, g denote the for any X, Y ∈ TN . induced metric on N as well as the Riemannian metric on In 2000, Gherghe introduced a notion of nearly trans- N . Sasakian structure of type ðα, βÞ, which generalizes the Let x ∈ N and fE , ⋯, E g be a local orthonormal 1 n trans-Sasakian structure. A nearly trans-Sasakian structure frame of T N and fE , ⋯, E g be a local orthonormal x n+1 2m+1 of type ðα, βÞ is called nearly α-Sasakian (resp. nearly β-Ken- frame of T N . The mean curvature vector H of a submani- motsu) if β =0 (resp. α =0). fold N at x is given by [15] Definition 2 (see [16]). An almost contact metric structure ðϕ, ξ, η, gÞ on N is called a nearly trans-Sasakian structure if H = 〠 hðÞ E , E : ð7Þ i i i=1 ∇ ϕ Y + ∇ ϕ X = αðÞ 2gXðÞ , Y ξ − ηðÞ Y X − ηðÞ X Y −βηðÞ ðÞ Y ϕX + ηðÞ X ϕY , X Y A submanifold N of N is said to be [15] ð3Þ (i) totally umbilical if hðX, YÞ = gðX, YÞH, for any X, for any X, Y ∈ TN . Y ∈ TN (ii) totally geodesic if hðX, YÞ =0, for any X, Y ∈ TN Remark 3. (iii) minimal if H =0, that is, trace h ≡ 0 (i) A nearly trans-Sasakian structure of type ðα, βÞ is For any X ∈ TN , we put [15] ϕX =PX +FX, ð8Þ (a) nearly Sasakian if β =0, α =1 [17] where PX = tangentðϕXÞ and FX = normalðϕXÞ. Then P (b) nearly Kenmotsu if α =0, β =1 [18] is an endomorphism of TN , and F is the normal bundle val- (c) nearly cosymplectic if α = β =0 [19] ⊥ ued 1-form on TN . In the same way, for any V ∈ T N ,we put [15] ϕV = BV +CV , ð9Þ (ii) Remark that every Kenmotsu manifold is a nearly Kenmotsu manifold but the converse is not true. where BV = tangentðϕV Þ and CV = normalðϕV Þ.Itis Also, a nearly Kenmotsu manifold is not a Sasakian easy to see that P and C are skew-symmetric and manifold. On another hand, every nearly Sasakian manifold of dimension greater than five is a Sasakian gðÞ FX, V = −gXðÞ , BV , ð10Þ manifold. for any X ∈ TN and V ∈ T N . We put dim N = n and dim N =2m +1. The Riemann- ian metric for N and N is denoted by the same symbol g. Definition 4. A submanifold N of an almost contact metric Let TN and T N denote the Lie algebra of the vector field manifold N is said to be invariant if F ≡ 0, that is, ϕX ∈ T Advances in Mathematical Physics 3 N , and anti-invariant if P ≡ 0, that is, ϕX ∈ T N , for any N such that X ∈ TN . TN = D ⊕ D ⊕ ξ : ð13Þ fg θ θ 1 2 In contact geometry, Lotta introduced slant immersions as follows [20]. (i) PD ⊥D and PD ⊥D θ θ θ θ 1 2 2 1 Definition 5. Let N be a submanifold of an almost contact metric manifold N . For each nonzero vector X ∈ T N − f (ii) Each distribution D is slant with slant angle θ for θ i ξ g and x ∈ N , the angle θðpÞ ∈ ½0, π/2 between ϕX and P i =1,2 X is called slant angle of N . If slant angle is constant for each X ∈ T N − fξ g, then the submanifold is called the slant x x submanifold. Remark 8. If we assume For slant submanifolds, the following facts are known: (i) θ =0 and θ = π/2,then N is a CR-submanifold 1 2 (ii) θ =0 and θ ≠ 0, π/2,then N is a semislant 1 2 submanifold 2 2 P ðÞ X = cos θðÞ −X + ηðÞ X ξ , ð11Þ (iii) θ = π/2 and θ ≠ 0, π/2, then N is a pseudoslant 1 2 gðÞ PX,PY = cos θðÞ gXðÞ , Y − ηðÞ Y ηðÞ X , submanifold (iv) θ , θ ∈ ð0, π/2Þ, then N is a proper bislant 1 2 submanifold ð12Þ gðÞ FX,FY = sin θðÞ gXðÞ , Y − ηðÞ Y ηðÞ X , For a bislant submanifold N of an almost contact metric manifold, the normal bundle of N is decomposed as for any X, Y ∈ TN . Here, θ is slant angle of N . T N =FD ⊕ FD ⊕ μ, ð14Þ θ θ 1 2 Remark 6. If we assume where μ is a ϕ-invariant normal subbundle of N . (i) θ =0, then N is an invariant submanifold 3. Cohomology Class for Bislant (ii) θ = π/2, then N is an anti-invariant submanifold Submanifolds of Nearly Trans- (iii) θðpÞ ∈ ð0, π/2Þ, then N is a proper slant submanifold Sasakian Manifolds Chen [21] introduces a canonical de Rham cohomology There are some other important classes of submanifolds class for a closed CR-submanifold in a Kähler manifold. which are determined by the behavior of tangent bundle of So, in this section, we discuss the de Rham cohomology class the submanifold under the action of an almost contact met- for a closed bislant submanifold of a nearly trans-Sasakian ric structure ϕ of N [1]: manifold ðN , ϕ, ξ, η, gÞ with minimal horizontal distribu- tion ðD ⊕ fξgÞ. We put dimðN Þ = m and (i) A submanifold N of N is called a contact CR-sub- dimðD ⊕ fξgÞ =2a +1. Let us assume the following: manifold of N if there exists a differentiable distri- bution D on N whose orthogonal complementary (i) fE , ⋯,E , E = sec θ PE , ⋯, E = sec θ PE , 1 a a+1 1 1 2a 1 a ∗ ∗ distribution D is anti-invariant E = ξ, E = E , ⋯, E = E , E = 2a+1 2a+2 1 2a+b+1 b 2a+b+2 ∗ ∗ ∗ E = sec θ PE , ⋯, E = E = E = sec θ P 2 m 2a+2b+1 2 b+1 1 2b (ii) A submanifold N of N is called a semislant subma- E g is a local orthonormal frame of N nifold of N if there exists a pair of orthogonal distri- butions D and D such that D is invariant and D is θ θ (ii) fE , ⋯,E , E = sec θ PE , ⋯, E = sec θ PE , 1 a a+1 1 1 2a 1 a proper slant E = ξg is a local orthonormal frame of ðD ⊕ f 2a+1 θ ξgÞ (iii) A submanifold N of N is called pseudoslant subma- ∗ ∗ ∗ nifold of N if there exists a pair of orthogonal distri- (iii) fE = E , ⋯, E = E , E = E = sec 2a+2 1 2a+b+1 b 2a+b+2 b+1 ⊥ ⊥ ∗ ∗ ∗ butions D and D such that D is anti-invariant θ θ PE , ⋯, E = E = E = sec θ PE g is a 2 1 m 2a+2b+1 2b 2 b and D is proper slant local orthonormal frame of D 1 2a+1 2a+2 m We choose ς , ⋯, ς , ς , ⋯, ς as the dual frame of Definition 7 (see [13]). A submanifold N of an almost con- 1-forms to the above local orthonormal frame. Then, we 1 2 2a+1 tact metric manifold M is said to be a bislant submanifold if define a ð2a +1Þ-form ϖ on N by ϖ = ς ∧ ς ∧⋯∧ς .It there exists a pair of orthogonal distributions D and D of is globally defined on N . In the same way, we again define θ θ 1 2 4 Advances in Mathematical Physics 2a+2 2a+3 m a ðm − 2a − 1Þ-form Ω on N by Ω = ς ∧ ς ∧⋯∧ς , for any Z ∈ ðD ⊕ fξgÞ. Thus, the assertion follows from which is globally defined on N . the fact that FD and FD are mutually perpendicular. In θ θ 1 2 We prepare some preliminary lemmas. this way, we proved the integrability condition of slant dis- tribution D .☐ Lemma 9. Let N be a submanifold of an arbitrary nearly trans-Sasakian manifold N , then We prove the following. ∇ PY − A X − P∇ X − 2BhðÞ X, Y + ∇ PX − A Y − P∇ Y X FY Y Y FX X Theorem 11. For any closed bislant submanifold N of an = α 2g X, Y ξ − η Y X − η X Y −βη Y PX + η X PY , ðÞ ðÞ ðÞ ðÞ ðÞ ðÞ ðÞ arbitrary nearly trans-Sasakian manifold ðN , ϕ, ξ, η, gÞ with minimal ðD ⊕ fξgÞ and ð15Þ ⊥ ⊥ hX, PY + ∇ FY − F∇ Y − 2ChX, Y +hY, PX +∇Y FX − F∇ X ⊥ ⊥ ðÞ ðÞ ðÞ X Y −2F∇ X +hXðÞ , PY +hYðÞ , PX − 2ChXðÞ , Y + ∇ FY + ∇ FX ∈ FD , Y X Y θ = −βη Y FX + η X FY , ðÞ ðÞ ðÞ ð21Þ ð16Þ for any X, Y ∈ D , the ð2a + 1Þ-form ϖ is closed and for any X, Y ∈ TN . 2a+1 defines a canonical de Rham cohomology class ½ϖ ∈ H ð N , ℝÞ, where dimðD ⊕ fξgÞ = 2a + 1. Proof. For any vector fields X, Y ∈ TN , making use of the 2a+1 structure equation and (2), we obtain Moreover, the cohomology group H ðN , ℝÞ is nontriv- ial if D is minimal and ðD ⊕ fξgÞ is integrable. θ θ 2 1 ∇ ϕY − ϕ∇ Y + ∇ ϕX − ϕ∇ X = α 2gX, Y ξ − η Y X − η X Y ðÞ ðÞ ðÞ ðÞ X X Y Y 2a+1 i−1 −βηðÞ ðÞ Y ϕX + ηðÞ X ϕY , Proof. From the definition of ϖ, we have dϖ = ∑ ð−1Þ i=1 1 i 2a+1 ð17Þ ς ∧⋯∧dς ∧⋯∧ς , which implies that dϖ =0 if and only if which gives dϖ X , Y , X , ⋯, X =0, ð22Þ ðÞ 2 2 1 2a ∇ PY + hðÞ PY, X − A X + ∇ FY − P∇ Y − 2BhXðÞ , Y X FY X − F∇ Y − 2ChXðÞ , Y + ∇ PY + hðÞ PX, Y − A Y X X FX dϖðÞ X , X , ⋯, X =0, ð23Þ 2 1 2a+1 + ∇ FX − P∇ X − F∇ X Y Y for any X , Y ∈ D and X , ⋯, X ∈ ðD ⊕ fξgÞ. 2 2 θ 1 2a+1 θ 2 1 = αðÞ 2gXðÞ , Y ξ − ηðÞ Y X − ηðÞ X Y Thus, by simple computation, we find that (22) is satisfied −βηðÞ ðÞ Y PX + ηðÞ X PY + ηðÞ Y FX + ηðÞ X FY : if and only if D is integrable. On the other hand, (23) is sat- isfied if and only if ðD ⊕ fξgÞ is minimal. However, the ð18Þ integrability condition of D holds due to Lemma 10, and Comparing the tangential and normal components of by the hypothesis of the theorem, we have ðD ⊕ fξgÞ is the above equation, we get the desired relations (15) and minimal. Hence, the form ϖ is closed. It defines a canonical (16). 2a+1 de Rham cohomology class ½ϖ ∈ H ðN , ℝÞ. The next lemma gives the integrability condition of slant Next, we prove that the cohomology class ½ϖ is nontriv- distribution D .☐ ial. Since D is minimal and ðD ⊕ fξgÞ is integrable, then θ θ 2 1 in this case, we need to show that ϖ is harmonic. By defini- Lemma 10. Let N be a bislant submanifold of an arbitrary tion of Ω and the similar argument for ϖ, we see that dΩ =0, nearly trans-Sasakian manifold ðN , ϕ, ξ, η, gÞ. Then, slant that is, Ω is closed, if ðD ⊕ fξgÞ is integrable and D is θ θ 1 2 distribution D is integrable if and only if minimal. This further proves that δϖ =0, that is, ϖ is coclosed. From dϖ =0, δϖ =0, and N is a closed submani- ⊥ ⊥ −2F∇ X +hXðÞ , PY +hYðÞ , PX − 2ChXðÞ , Y + ∇ FY + ∇ FX ∈ FD , Y X Y θ fold, we deduce that ϖ is harmonic ð2a +1Þ-form. Hence, 2a+1 ð19Þ the cohomology group H ðN , ℝÞ is nontrivial if D is minimal and ðD ⊕ fξgÞ is integrable.☐ for any X, Y ∈ D . 4. Warped Product Bislant Submanifolds Proof. Making use of Lemma 9, we obtain Definition 12 (see [22]). Let ðN , g Þ and ðN , g Þ be two 1 1 2 2 gðÞ F½ X, Y ,FZ = −2fgðÞ F∇ X,FZ +ghXðÞ ðÞ ,PY ,FZ Riemannian manifolds and f >0 be a differentiable function on N . Consider two projections on N × N , ρ : N × +ghðÞ ðÞ Y,PX ,FZ − gðÞ 2ChXðÞ , Y ,FZ 1 1 2 1 ⊥ ⊥ N ⟶ N and δ : N × N ⟶ N . The projection maps 2 1 1 2 2 + g ∇ FY,FZ + g ∇ FX,FZ , X Y given by ρðp, qÞ = p and δðp, qÞ = q for ðp, qÞ ∈ N × N . 1 2 ð20Þ Then, the warped product N = N × N is the product 1 f 2 Advances in Mathematical Physics 5 manifold N × N equipped with the Riemannian structure (i) N is a warped product pseudoslant submanifold 1 2 such that such that N is a totally real submanifold N of gX, Y = g ρ X, ρ Y + f ∘ ρ g δ X, δ Y , ð24Þ ðÞ ðÞ ðÞ ðÞ (ii) If N is nearly Sasakian manifold, that is, β = 0, then 1 ∗ ∗ 2 ∗ ∗ N is a Riemannian product for any X, Y ∈ TN , where ∗ is the symbol for the tangent (iii) If β ≠ 0, then βηðX Þ = −ðX ln f Þ 1 1 maps. The function f is called the warping function of N . Proof. For any vector fields X ∈ TN and X , Y ∈ TN ,we 1 1 2 2 2 Example 13. A surface of revolution is a warped product have manifold. gh X , X ,FY = g ∇ X , ϕY − g ∇ X ,PY Example 14. The standard space-time models of the universe ðÞ ðÞ 1 2 2 X 2 2 X 2 2 1 1 are warped products as the simplest models of neighbour- = g ∇ ϕ X , Y − g ∇ ϕX , Y X 2 2 X 2 2 1 1 hoods of stars and black holes. −ðÞ X ln f gXðÞ ,PY : 1 2 2 Remark 15. In particular, a warped product manifold is said ð27Þ to be trivial if its warping function is constant. In such a case, we call the warped product manifold a Riemannian product On the other hand, we have manifold. If N = N × N is a warped product manifold, 1 f 2 then N is totally geodesic and N is totally umbilical sub- 1 2 ghðÞ ðÞ X , X ,FY = g ∇ X , ϕY − g ∇ X ,PY 1 2 2 X 1 2 X 1 2 2 2 manifold of N [22]. = g ∇ ϕ X , Y − g ∇ ϕX , Y Let N = N × N be a warped product manifold with a X 1 2 X 1 2 1 f 2 2 2 warping function f . Then, − X ln f gX ,PY : ðÞðÞ 1 2 2 ð28Þ ∇ Z = ∇ X = X ln f Z, ð25Þ ðÞ X Z By adding (27) and (28), we get for any X ∈ TN and Z ∈ TN , where ∇ln f is the gradi- 1 2 2ghXðÞ ðÞ , X ,FY =ghXðÞ ðÞ , Y ,FX +ghXðÞ ðÞ , Y ,FX 1 2 2 1 2 2 2 2 1 ent of ln f and ∇ and ∇ denote the Levi-Civita connec- − Plnf gX , Y − X ln f gX ,PY ðÞðÞ ðÞðÞ 2 2 1 2 2 tions on N and N , respectively. − αηðÞ X gXðÞ , Y + βηðÞ X gðÞ PX , Y : 1 2 2 1 2 2 The definition of warped product bislant submanifolds in a ð29Þ nearly trans-Sasakian manifold is as follows. Interchanging X by Y in (29), we find 2 2 Definition 16. A warped product N × N of two slant sub- 1 f 2 manifolds N and N of a nearly trans-Sasakian manifold 1 2 2ghXðÞ ðÞ , Y ,FX =ghXðÞ ðÞ , X ,FY +ghXðÞ ðÞ , Y ,FX 1 2 2 1 2 2 2 2 1 N is called a warped product bislant submanifold. − Plnf gX , Y − X ln f gY ,PX ðÞðÞ ðÞðÞ 2 2 1 2 2 − αηðÞ X gXðÞ , Y + βηðÞ X gðÞ PY , X : 1 2 2 1 2 2 Remark 17. A warped product bislant submanifold N × 1 f ð30Þ N is called proper if N and N are proper slant in N . 2 1 2 Otherwise, the warped product bislant submanifold N × 1 f By subtracting (30) from (29) and by applying our N is called nonproper. assumption, we obtain For a warped product bislant submanifold in a nearly gðÞ PX , Y½ ðÞ X ln f + βηðÞ X =0: ð31Þ trans-Sasakian manifold such that ξ ∈ TN , we have the fol- 2 2 1 1 lowing result. For Y =PY , we get 2 2 Theorem 18. Let N = N × N be a warped product bislant 1 f 2 submanifold with bislant angles fθ , θ g in a nearly trans- cos θ gYðÞ , X½ ðÞ X ln f + βηðÞ X =0: ð32Þ 1 2 2 2 2 1 1 Sasakian manifold N such that ξ ∈ TN . If, for any X ∈ T 1 1 N and X , Y ∈ TN , 1 2 2 2 From the last expression, any one of the following holds: if β =0, then f is constant, or if β ≠ 0, then βηðX Þ = −ðX 1 1 ln f Þ or θ = π/2. Thus, our assertions follow. ghX , X , FY =ghX , Y , FX , ð26Þ ðÞ ðÞ ðÞ ðÞ 1 2 2 1 2 2 Now, we have the following theorem for a warped prod- uct bislant submanifold in a nearly trans-Sasakian manifold holds, then one of the following cases must occur: such that ξ ∈ TN .☐ 2 6 Advances in Mathematical Physics Theorem 19. Let N = N × N be a warped product bislant bislant angles θ ≠ 0, π/2 and θ = π/2. Such warped product 1 f 2 1 2 bislant submanifolds are called pseudoslant submanifolds. submanifold with bislant angles fθ , θ g in a nearly trans- 1 2 Sasakian manifold N such that ξ ∈ TN . If, for any X ∈ T 2 1 Example 20. Let ℂ be the complex Euclidean space with its N and X , Y ∈ TN , 1 2 2 2 usual Kähler structure and the real global coordinates ðx , y , x , y , x , y , x , y Þ and N = ℝ × ℂ be a warped prod- ghXðÞ ðÞ , X , FY =ghXðÞ ðÞ , Y , FX , ð33Þ 1 2 2 1 2 2 1 2 2 3 3 4 4 f uct manifold between the product real line of ℝ and the holds, then one of the following cases must occur: complex space ℂ . Let <, > be the Euclidean metric tensor of ℝ . An almost contact structure ϕ of N is defined by (i) N is a warped product pseudoslant submanifold such that N is a totally real submanifold N of N 2 ! ∂ ∂ ∂ ∂ ∂ (ii) N is a Riemannian product ϕ = , ϕ = − , ϕ =0, 1 ≥ i, j ≥ 4 ∂x ∂y ∂y ∂x ∂t i j i j ð40Þ Proof. For any vector fields X ∈ TN and X , Y ∈ TN ,we 1 1 2 2 2 have such that ghXðÞ ðÞ , X ,FY = g ∇ X , ϕY − g ∇ X ,PY 1 2 2 X 2 2 X 2 2 1 1 = g ∇ ϕ X , Y − g ∇ ϕX , Y : X 2 2 X 2 2 1 1 t t t ξ = e , η = e dt, g = e <, > : ð41Þ ð34Þ ∂t On the other hand, we have On the other hand, we define a submanifold N by immersion g as follows: ghXðÞ ðÞ , X ,FY = g ∇ X , ϕY − g ∇ X ,PY 1 2 2 X 1 2 X 1 2 2 2 = g ∇ ϕ X , Y − g ∇ ϕX , Y : X 1 2 X 1 2 2 2 guðÞ , v, w, s, t =ðÞ u, v,0,0, v cos r, v sin r, s cos w, s sin w, t : ð35Þ ð42Þ By adding (34) and (35), we get Therefore, it is easy to choose tangent bundle of N 2ghX , X ,FY =ghX , Y ,FX +ghX , Y ,FX ðÞ ðÞ ðÞ ðÞ ðÞ ðÞ 1 2 2 1 2 2 2 2 1 which is spanned by the following: −ðÞ Plnf gXðÞ , Y −ðÞ X ln f gXðÞ ,PY : 2 2 1 2 2 ð36Þ ∂ ∂ ∂ X = , X = cos r + sin r , 1 2 Interchanging X by Y in (36), we find ∂x ∂y ∂y 2 2 1 1 2 ð43Þ ∂ ∂ ∂ 2ghX , Y ,FX ð37Þ =ghX , X ,FY +ghX , Y ,FX ðÞ ðÞ ðÞ ðÞ ðÞ ðÞ 1 2 2 1 2 2 2 2 1 X = cos w + sin w , X = : 3 5 ∂y ∂y ∂z 3 4 −ðÞ Plnf gXðÞ , Y −ðÞ X ln f gYðÞ ,PX : 2 2 1 2 2 ð37Þ Thus, D = SpanfX , X g is a slant distribution with θ 1 2 By subtracting (37) from (36) and by applying our slant angle π/4. Also, it is easy to verify that D = SpanfX θ 3 assumption, we obtain , X g is a totally real distribution. Hence, the submanifold N defined by f is a bislant submanifold, which is tangent X ln f g PX , Y =0: ð38Þ ðÞðÞ 1 2 2 to the structure vector ξ and whose bislant angles satisfy θ ≠ 0, π/2 and θ = π/2. It is easy to check that the distribu- For Y =PY , we get 2 2 tions D and D are integrable. Then, it can be verified that θ θ 1 2 N = N × N is a warped product bislant submanifold of 2 θ f ⊥ cos θðÞ X ln f½ gYðÞ , X − ηðÞ X ηðÞ Y =0: ð39Þ 2 1 2 2 2 2 N with warping function f = e , t ∈ ℝ. Therefore, either f is constant or cos θ =0 holds. Con- Example 21. We consider any submanifold N in a nearly sequently, either N is a Riemannian product or θ = π/2. trans-Sasakian manifold ℝ In the latter case, N is a warped product pseudoslant sub- manifold.☐ f u, v, w, q = u cos v, w cos v, u sin v, w sin v, w − u, w + u, q : ðÞðÞ We give some nontrivial examples of warped product ð44Þ bislant submanifold of the form N = N × N whose θ f ⊥ Advances in Mathematical Physics 7 The tangent bundle of N is spanned by Definition 22 (see [23, 24]). Let ðN , g Þ and ðN , g Þ be 1 1 2 2 Riemannian manifolds. A doubly warped product ðN , gÞ is ∂ ∂ ∂ ∂ a product manifold which is of the form N = N × N f 1 f 2 1 E = cos v + sin v − + , 2 2 ∂x ∂x ∂x ∂y with the metric g = f g ⊕ f g , where f : N × N ⟶ ð0 1 2 3 3 1 1 2 2 1 1 2 ,∞Þ and f : N × N ⟶ ð0,∞Þ are smooth maps. More 2 1 2 ∂ ∂ ∂ ∂ E = −u sin v + u cos v − w sin v + w cos v , precisely, if ρ : N × N ⟶ N and δ : N × N ⟶ N 1 2 1 1 2 2 ∂x ∂x ∂y ∂y 1 2 1 2 are natural projections, the metric g is defined by ∂ ∂ ∂ ∂ E = + cos v + sin v + , 3 2 2 ∂x ∂y ∂y ∂y gX, Y = f ∘ δ g ρ X, ρ Y + f ∘ ρ g δ X, δ Y , ðÞ ðÞ ðÞ ðÞ ðÞ 3 1 2 3 2 1 ∗ ∗ 1 2 ∗ ∗ ð48Þ E = : ∂q for any X, Y ∈ TN , where ∗ is the symbol for the tangent ð45Þ maps. The functions f and f are called the warping func- 1 2 tions of N . Furthermore, we have Remark 23. If we assume ∂ ∂ ∂ ∂ ϕE = cos v + sin v − − , ∂y ∂y ∂y ∂x 1 2 3 3 (i) either f ≡ 1 or f ≡ 1, but not both, then we obtain a 1 2 ∂ ∂ ∂ ∂ warped product ϕE = −u sin v + u cos v + w sin v − w cos v , ∂y ∂y ∂x ∂x 1 2 1 2 (ii) both f ≡ 1 and f ≡ 1, then we have a product 1 2 ∂ ∂ ∂ ∂ manifold ϕE = − cos v − sin v − , ∂y ∂x ∂x ∂x 3 1 2 3 (iii) neither f nor f is constant, then we have a non- 1 2 ϕE =0: trivial doubly warped product ð46Þ For doubly warped product manifold N = N × N f 1 f It is easy to check that ϕE is orthogonal to TN . Then, 2 1 with warping functions f and g, we have the following: the proper slant and anti-invariant distributions of N are respectively defined by D = SpanfE , E g with slant angle θ 1 3 ∇ X = ∇ Y = Y ln f X + X ln f Y, ð49Þ θ = arccos ð1/3Þ and D = SpanfE g. Also, E = ξ is tangent ðÞ ðÞ ⊥ 2 4 Y X 1 2 to D . Hence, f defines a proper 4-dimensional pseudoslant submanifold (bislant submanifold with bislant angles f for any X ∈ TN and Y ∈ TN . 1 2 arccos ð1/3Þ, π/2g) N in ℝ . It is easy to check that the dis- Now, we define the notion of doubly warped product tributions D ⊕ fξg and D are integrable. θ ⊥ bislant submanifolds in nearly trans-Sasakian manifolds as follows. Now, we assume that N and N are the integral man- θ ⊥ ifolds of D and D , respectively. Then, it follows from Def- Definition 24. The doubly warped product of two slant sub- θ ⊥ inition 12 and (44) that the induced metric tensor g of N is manifolds, N × N , is called the doubly warped product f 1 f 2 2 1 given by bislant submanifold of slant submanifolds N and N with 1 2 slant angles θ and θ , respectively, of a nearly trans- 1 2 2 2 2 2 2 2 2 2 2 2 2 2 g = cos v + sin v +2 du + u sin v + u cos v + w sin v + w cos v dv Sasakian manifold with warping functions f and f if only 1 2 2 2 2 2 2 2 2 2 2 2 + cos v + sin v +2 dw + dq =3 du + dw + dq + u + w dv depend on the points of N and N , respectively. 1 2 = g + g , 1 2 First we have the following theorem for doubly warped ð47Þ product submanifolds N = N × N in nearly trans- f 1 f 2 2 1 Sasakian manifolds such that ξ ∈ TN . 2 2 2 2 2 2 where g =3ðdu + dw Þ + dq and g = ðu + w Þdv 1 2 are respectively the metric tensors of N and N . As a con- θ ⊥ Theorem 25. Let N = N × N be a doubly warped prod- f 1 f 2 2 1 sequence, N = N × N is a warped product pseudoslant θ f ⊥ uct submanifold in a nearly trans-Sasakian manifold N , submanifold of ℝ with a warping function, that is, f = pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where N and N are Riemannian submanifolds of N and 1 2 2 2 u + w such that ξ is tangent to N . ξ ∈ TN . Then, N is a warped product bislant submanifold of type N × N if and only if 1 f 2 5. Doubly Warped Product Bislant Submanifolds ghX, Y , FX =ghX, X , FY , ð50Þ ðÞ ðÞ ðÞ ðÞ In general, doubly warped products can be considered as a generalization of warped products. for any X ∈ TN and Y ∈ TN . 1 2 8 Advances in Mathematical Physics Proof. From Lemma 9, we get Proof. For any vector fields X ∈ TN and Y ∈ TN , we have 2 1 ghðÞ ðÞ PX, Y ,FY = g ∇ PX, ϕY = −g ϕ∇ PX, Y ∇ PY − A X − P∇ X − 2BhX, Y + ∇ PX − A Y − P∇ Y ðÞ Y Y X FY Y Y FX X = g ∇ ϕ PX, Y − g ∇ ϕPX, Y = −αηðÞ X Y − βηðÞ X PY, Y Y = −g ∇ ϕ Y,PX − g ∇ P X, Y − g ∇ FP X, Y Y Y Y ð51Þ = cos θ gðÞ ∇ X, Y +ghðÞ ðÞ Y, Y ,FP X 1 Y 2 2 = cos θ X ln f Y +gh Y, Y ,FP X : ðÞkk ðÞ ðÞ for any X ∈ TN and Y ∈ TN . Applying (49), we derive 1 2 ð57Þ ðÞ PY ln f X −ðÞ Y ln f PX −ðÞ X ln f PY +PðÞ X ln f Y 2 2 1 1 Replacing X by PX in the last relation, we obtain − A X − 2BhXðÞ , Y − A Y = −αηðÞ X Y − βηðÞ X PY : FY FX ð52Þ PX ln f Y =ghY, Y ,FX −gh X, Y ,FY : ð58Þ ðÞkk ðÞ ðÞ ðÞ ðÞ Taking the inner product with X ∈ TN , we obtain Thus, from (54), we conclude that ðPX ln f Þ =0 if and only if ðÞ PY ln f kk X −ghXðÞ ðÞ , X ,FY − 2gðÞ BhXðÞ , Y , X −ghYðÞ ðÞ , X ,FX =0: gh Y, Y ,FX =ghX, Y ,FY , ð59Þ ð53Þ ðÞ ðÞ ðÞ ðÞ for any X ∈ TN and Y ∈ TN . Using relation (10) in the above equation, we get 2 1 ðPX ln f Þ =0 implies that f is constant, that is, f 2 2 2 depends only on the points of N . Hence, N is a warped ðÞ PY ln f kk X =ghXðÞ ðÞ , X ,FY −ghYðÞ ðÞ , X ,FX =0: product bislant submanifold of type N × N . This proves 1 f 2 the theorem completely.☐ ð54Þ 6. Conclusion Thus, from (54), we conclude that ðPY ln f Þ =0 if and From Theorems 25 and 26, we conclude that there exist no only if doubly warped product bislant submanifolds in nearly trans-Sasakian manifolds, other than warped product bislant ghXðÞ ðÞ , Y ,FX =ghXðÞ ðÞ , X ,FY , ð55Þ submanifolds, under some additional conditions. 7. Some Applications of Theorem 25 for for any X ∈ TN and Y ∈ TN . ðPY ln f Þ =0 shows 1 2 Different Kinds of Ambient Manifolds that f is constant, that is, f depends only on the points of 2 2 N . Thus, it follows that N is a warped product bislant sub- 1 Let N = N × N be a doubly warped product submani- f 1 f 2 2 1 manifold of type N × N . This proves the theorem 1 f 2 fold, where N and N are Riemannian submanifolds of 1 2 completely.☐ N and ξ ∈ TN . The following corollaries are the immediate consequences of Theorem 25. Secondly, we prove the following theorem for doubly warped product bislant submanifolds N = N × N in f 1 f 2 Corollary 27. There does not exist any doubly warped prod- 2 1 nearly trans-Sasakian manifolds such that ξ ∈ TN . 2 uct submanifold N = N × N in a nearly Sasakian man- f 1 f 2 2 1 ifold N , other than the warped product bislant submanifold, Theorem 26. Let N = N × N be a doubly warped prod- f 1 f 2 2 1 if and only if (50) holds. uct bislant submanifold in a nearly trans-Sasakian manifold N , where N and N are proper slant submanifolds with 1 2 Corollary 28. There does not exist a doubly warped product respect to θ and θ , respectively, and ξ ∈ TN . Then, N is submanifold N = N × N in a nearly Kenmotsu manifold 1 2 2 f 1 f 2 2 1 a warped product bislant submanifold of type N × N if 1 f 2 N , other than the warped product bislant submanifold, if and and only if only if (50) holds. Corollary 29. There does not exist a doubly warped product ghXðÞ ðÞ , Y , FY =ghYðÞ ðÞ , Y , FX , ð56Þ submanifold N = N × N in a nearly cosymplectic mani- f 1 f 2 2 1 fold N , other than the warped product bislant submanifold, if for any X ∈ TN and Y ∈ TN . and only if (50) holds. 2 1 Advances in Mathematical Physics 9 8. Some Applications of Theorem 26 for [5] B.-Y. Chen, “Geometry of warped product CR-submanifolds in Kaehler manifolds, II,” Monatshefte für Mathematik, Different Kinds of Ambient Manifolds vol. 134, no. 2, pp. 103–119, 2001. Let N = N × N be a doubly warped product bislant [6] B. Sahin, “Notes on doubly warped and doubly twisted prod- f 1 f 2 2 1 uct CR-submanifolds of Kaehler manifolds,” Matematiqki submanifold, where N and N are proper slant submani- 1 2 Vesnik, vol. 59, pp. 205–210, 2007. folds with respect to θ and θ , respectively, and ξ ∈ TN . 1 2 2 [7] A. N. Siddiqui, “Warped product and doubly warped product The following corollaries are the immediate consequences bi-slant submanifolds in trans-Sasakian manifolds,” JMI Inter- of Theorem 26. national Journal of Mathematical Sciences, vol. 9, pp. 15–27, Corollary 30. There is no doubly warped product bislant sub- [8] A. Olteanu, “Contact CR-doubly warped product submani- manifold N = N × N in a nearly Sasakian manifold N , f 1 f 2 folds in Kenmotsu space forms,” Journal of Inequalities in Pure 2 1 other than the warped product bislant submanifold, if and and Applied Mathematics, vol. 10, no. 4, 2009. only if (56) holds. [9] F. R. Al-Solamy, M. F. Naghi, and S. Uddin, “Geometry of warped product pseudo-slant submanifolds of Kenmotsu manifolds,” Quaestiones Mathematicae, vol. 42, no. 3, Corollary 31. There is no doubly warped product bislant sub- pp. 373–389, 2019. manifold N = N × N in a nearly Kenmotsu manifold N f 1 f 2 2 1 [10] M. I. Munteanu, “Doubly warped product CR-submanifolds in , other than the warped product bislant submanifold, if and locally conformal Kähler manifolds,” Monatshefte für Mathe- only if (56) holds. matik, vol. 150, no. 4, pp. 333–342, 2007. [11] S. Uddin, “On doubly warped and doubly twisted product sub- Corollary 32. There is no doubly warped product bislant sub- manifolds,” International Electronic Journal of Geometry, manifold N = N × N in a nearly cosymplectic manifold f 1 f 2 vol. 3, no. 1, pp. 35–39, 2010. 2 1 N , other than the warped product bislant submanifold, if [12] S. Uddin, I. Mihai, and A. Mihai, “On warped product bi-slant and only if (56) holds. submanifolds of Kenmotsu manifolds,” Arab Journal of Math- ematical Sciences, vol. 27, no. 1, pp. 2–14, 2021. [13] J. L. Cabrerizo, A. Carriazo, L. M. Fernandez, and Data Availability M. Fernandez, “Semi-slant submanifolds of a Sasakian mani- There is no data used for this manuscript. fold,” Geometriae Dedicata, vol. 78, no. 2, pp. 183–199, 1999. [14] J. F. Nash, “The imbedding problem for Riemannian mani- folds,” Annals of Mathematics, vol. 63, no. 1, pp. 20–63, 1956. Conflicts of Interest [15] K. Yano and M. Kon, Structures on Manifolds, Series in Pure The authors declare no competing of interest. Mathematics, Worlds Scientific Publishing Co., Singapore, [16] C. Gherghe, “Harmonicity on nearly trans-Sasaki manifolds,” Authors’ Contributions Demonstratio Mathematica, vol. 33, no. 1, pp. 151–157, 2000. All authors have equal contribution and finalized. [17] D. E. Blair, D. K. Showers, and K. Yano, “Nearly Sasakian structure,” Kodai Mathematical Seminar Reports, vol. 27, pp. 175–180, 1976. Acknowledgments [18] M. M. Tripathi and S. S. Shukla, “Semi-invariant submanifolds The authors extend their appreciation to the Deanship of of nearly Kenmotsu manifolds,” Bulletin of the Calcutta Math- Scientific Research at King Khalid University for funding ematical Society, vol. 95, pp. 17–30, 2003. this work through a research group program under grant [19] D. E. Blair, “Almost contact manifolds with killing structure number R.G.P.2/74/42. tensors,” Pacific Journal of Mathematics, vol. 39, no. 2, pp. 285–292, 1971. [20] A. Lotta, “Slant submanifolds in contact geometry,” Bulletin of References Mathematical Society, Romania, vol. 39, pp. 183–198, 1996. [1] A. N. Siddiqui, M. H. Shahid, and J. W. Lee, “Geometric [21] B.-Y. Chen, “Cohomology of CR-submanifolds,” Annales de la inequalities for warped product bi-slant submanifolds with a Faculté des sciences de Toulouse: Mathématiques, vol. 3, warping function,” Journal of Inequalities and Applications, pp. 167–172, 1981. vol. 2018, no. 1, 2018. [22] R. L. Bishop and B. O’Neill, “Manifolds of negative curvature,” [2] B. Sahin, “Non-existence of warped product semi-slant sub- Transactions of the American Mathematical Society, vol. 145, manifolds of Kaehler manifolds,” Geometriae Dedicata, pp. 1–49, 1969. vol. 117, no. 1, pp. 195–202, 2006. [23] B. Unal, Doubly warped products, PhD Thesis, University of [3] S. Uddin, B.-Y. Chen, and F. R. Al-Solamy, “Warped product Missouri-Columbia, 2000. bi-slant immersions in Kaehler manifolds,” Mediterranean [24] B. Unal, “Doubly warped products,” Differential Geometry and Journal of Mathematics, vol. 14, no. 2, 2017. its Applications, vol. 15, no. 3, pp. 253–263, 2001. [4] B.-Y. Chen, “Geometry of warped product CR-submanifolds in Kaehler manifolds,” Monatshefte für Mathematik, vol. 133, no. 3, pp. 177–195, 2001.
Journal
Advances in Mathematical Physics – Hindawi Publishing Corporation
Published: Oct 4, 2021