Research Article Generalized Transversal Lightlike Submanifolds of Indefinite Sasakian Manifolds
|
|
- Rosalind French
- 5 years ago
- Views:
Transcription
1 International Journal of Mathematics and Mathematical Sciences Volume 2012, Article ID , 17 pages doi: /2012/ Research Article Generalized Transversal Lightlike Submanifolds of Indefinite Sasakian Manifolds Yaning Wang 1 and Ximin Liu 2 1 Department of Mathematics, South China University of Technology, Guangdong, Guangzhou , China 2 School of Mathematical Sciences, Dalian University of Technology, Liaoning, Dalian , China Correspondence should be addressed to Yaning Wang, wyn051@163.com Received 7 August 2012; Accepted 22 October 2012 Academic Editor: Hernando Quevedo Copyright q 2012 Y. Wang and X. Liu. 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. We introduce and study generalized transversal lightlike submanifold of indefinite Sasakian manifolds which includes radical and transversal lightlike submanifolds of indefinite Sasakian manifolds as its trivial subcases. A characteristic theorem and a classification theorem of generalized transversal lightlike submanifolds are obtained. 1. Introduction The theory of submanifolds in Riemannian geometry is one of the most important topics in differential geometry for years. We see from 1 that semi-riemannian submanifolds have many similarities with the Riemannian counterparts. However, it is well known that the intersection of the normal bundle and the tangent bundle of a submanifold of a semi- Riemannian manifold may be not trivial, so it is more difficult and interesting to study the geometry of lightlike submanifolds than nondegenerate submanifolds. The two standard methods to deal with the above difficulties were developed by Kupeli 2 and Duggal- Bejancu 3, 4, respectively. The study of CR-lightlike submanifolds of an indefinite Kaehler manifold was initiated by Duggal-Bejancu 3. Since the book was published, many geometers investigated the lightlike submanifolds of indefinite Kaehler manifolds by generalizing the CR-lightlike submanifold 3, SCR-lightlike submanifolds 5 to GCR-lightlike submanifolds 6, and discussing the integrability and umbilication of these lightlike submanifolds. We also refer the reader to 7 for invariant lightlike submanifolds and to 8 for totally real lightlike submanifolds of indefinite Kaehler manifolds.
2 2 International Journal of Mathematics and Mathematical Sciences On the other hand, after Duggal-Sahin introduced screen real lightlike submanifolds and contact screen CR-lightlike submanifolds 9 of indefinite Sasakian manifolds by studying the integrability of distributions and the geometry of leaves of distributions as well as other properties of this submanifolds, the generalized CR-lightlike submanifold which contains contact CR and SCR-lightlike submanifolds were introduced in 4. However, all these submanifolds of indefinite Sasakian manifolds mentioned above have the same geometric condition φ Rad TM TM, where φ is the almost contact structure on indefinite Sasakian manifolds, Rad TM is the radical distribution, and TM is the tangent bundle. Until recently YıldırımandŞahin 10 introduced radical transversal and transversal lightlike submanifold of indefinite Sasakian manifolds for which the action of the almost contact structure on radical distribution of such submanifolds does not belong to the tangent bundle, more precisely, φ Rad TM ltr TM, where ltr TM is the lightlike transversal bundle of lightlike submanifolds. The purpose of this paper is to generalize the radical and transversal lightlike submanifolds of indefinite Sasakian manifolds by introducing generalized transversal lightlike submanifolds. The paper is arranged as follows. In Section 2, we give the preliminaries of lightlike geometry of Sasakian manifolds needed for this paper. In Section 3, we introduce the generalized transversal lightlike submanifolds and obtain a characterization theorem for such lightlike submanifolds. Section 4 is devoted to discuss the integrability and geodesic foliation of distributions of generalized transversal lightlike submanifolds. In Section 5, we investigate the geometry of totally contact umbilical generalized transversal lightlike submanifolds and obtain a classification theorem for such lightlike submanifolds. 2. Preliminaries In this section, we follow 4, 10 developed by Duggal-Sahin and Yıldırım-Şahin, respectively, for the notations and fundamental equations for lightlike submanifolds of indefinite Sasakian manifolds. A submanifold M, g of dimension m immersed in a semi-riemannian manifold M, g of dimension m n is called a lightlike submanifold if the metric g induced from ambient space is degenerate and its radical distribution Rad TM is of rank r, where m 2 and 1 r m. It is well known that the radical distribution is given by Rad TM TM TM, where TM is called normal bundle of M in M. Thus there exist the nondegenerate complementary distribution S TM and S TM of Rad TM in TM and TM, respectively, which are called the screen and screen transversal distribution on M, respectively. Thus we have TM Rad TM orth S TM, TM Rad TM orth S TM, where orth denotes the orthogonal direct sum. Considering the orthogonal complementary distribution S TM of S TM in TM, it is easy to see that TM is a subbundle of S TM.AsS TM is a nondegenerate subbundle of S TM, the orthogonal complementary distribution S TM of S TM in S TM is also a nondegenerate distribution. Clearly, Rad TM is a subbundle of S TM. Since for any local basis {ξ i } of Rad TM, there exists a local null frame {N i } of sections with values
3 International Journal of Mathematics and Mathematical Sciences 3 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 } see 3. Then we have that S TM Rad TM ltr TM. Lettr TM S TM orth ltr TM. We call {N i },ltr TM and tr TM the lightlike transversal vector fields, lightlike transversal vector bundle and transversal vector bundle of M with respect to the chosen screen distribution S TM and S TM, respectively. Then TM is decomposed as follows: TM TM tr TM Rad TM tr TM orth S TM Rad TM ltr TM orth S TM orth S TM. 2.3 A lightlike submanifold M, g, S TM,S TM of g is said to be as follows. Case 1: r-lightlike if r<min m, n. Case 2: coisotropic if r n<m, S TM 0. Case 3: isotropic if r m<n, S TM 0. Case 4: totally lightlike if r m n, S TM S TM 0. Let,, and t denote the linear connections on M, M, andtr TM, respectively. Then the Gauss and Weingarten formulas are given by X Y X Y h X, Y, X, Y Γ TM, 2.4 X U A U X t XU, X Γ TM, U Γ tr TM, 2.5 where { X Y, A U X} and {h X, Y, t X U} belong to Γ TM and Γ tr TM, respectively, A U is the shape operator of M with respective to U. Moreover, according to the decomposition 2.3, denoting by h l and h s the Γ ltr TM -valued and Γ S TM -valued lightlike second fundamental form and screen second fundamental form of M, respectively, we have X Y X Y h l X, Y h s X, Y, X, Y Γ TM, 2.6 X N A N X l X N Ds X, N, N Γ ltr TM, 2.7 X W A W X s X W Dl X, W, W Γ S TM. 2.8 Then by using 2.4, , and the fact that is a metric connection, we have g h s X, Y,W g D l X, W,Y g A W X, Y, g D s X, N,W g A W X, N. 2.9
4 4 International Journal of Mathematics and Mathematical Sciences Let P be the projection morphism of TM on S TM with respect to the decomposition 2.1, then we have X PY X PY h X, PY, X ξ A ξ X X ξ, 2.10 for any X, Y Γ TM and ξ Γ Rad TM, where h and A are the second fundamental form and shape operator of distribution S TM and Rad TM, respectively. Then we have the following: g h l X, PY,ξ g A X,PY, g h X, PY,N g A N X, PY. g h l X, ξ,ξ 0, A ξξ It follows from 2.6 that X g Y, Z g h l X, Y,Z g h l X, Z,Y, 2.12 for any X, Y, Z Γ TM. Thus the induced connection on M is torsion free but is not metric, the induced connection on S TM is metric. Finally, we recall some basic definitions and results of indefinite Sasakian manifolds following from 4, 11. An odd dimensional semi-riemannian manifold M, g of dimension 2n 1 is said to be with an almost contact structure φ, η, V if there exist a 1, 1 -type tensor φ, a vector field V called the characteristic vector field and a 1-form η such that φ 2 X X η X V, η V ɛ ± It follows that φ V 0, η φ 0 and rank φ 2n. A Riemannian metric g on M, g is called an associated or compatible metric of an almost constant structure φ, η, V of M if g φx, φy g X, Y ɛη X η Y A semi-riemannian manifold endowed with an almost contact structure is said to be an almost contact metric manifold if the semi-riemannian metric is associated or compatible to almost contact structure and is denoted by M, φ, η, V, g. It is known that g is a normal contact structure if N φ dη V 0, where N φ is the Nijenhuis tensor field of φ. A normal contact metric manifolds is called a Sasakian manifold. We know from 12 that an almost contact metric manifold is Sasakian if and only if X φ Y g X, Y V ɛη Y X. 2.15
5 International Journal of Mathematics and Mathematical Sciences 5 It follows from 2.13 and 2.15 that X V φx R 2m 1 q It is well known that R 2m 1 q,φ,v,η,g the manifold R 2m 1 i 1 q can be regarded as a Sasakian manifold. Denoting by with its usual Sasakian structure given by η 1 m dz y i dx, i V 2 z, 2 i 1 g η η 1 q/2 m dx i dx i dy i dy i dx i dx i dy i dy i, 4 i 1 i q 1 m φ X i x i Y i y i m Z z Y i x i X i y i m Y i y i z, where x i,y i,z are the Cartesian coordinates in R 2m 1 q. i 1 i Generalized Transversal Lightlike Submanifolds In this section, we define a class of lightlike submanifolds of indefinite Sasakian manifolds and study the geometry of such lightlike submanifolds. Firstly, we recall the following lemma. Lemma 3.1 see 4. Let M be a lightlike submanifold of an indefinite almost contact metric manifold M. IfV is tangent to M, then the structure vector field V does not belong to Rad TM. Definition 3.2 see 10. Let M, g, S TM,S TM be a lightlike submanifold, tangent to the structure vector field V, immersed in an indefinite Sasakian manifold M, g. We say that M is a radical transversal lightlike submanifold of M if the following conditions are satisfied: φ Rad TM ltr TM, φ S TM S TM. 3.1 It follows from Lemma 3.1 and 2.1 that the structure vector field V Γ S TM. Suppose that φ S TM S TM, then we have X Γ S TM such that V φ X. since 2.13 implies that φ V 0, using 2.13, we have X η X V. Substituting X η X V into V φ X, we have V 0. So it is impossible for φ S TM S TM in Definition 3.2. Thus, we modify the above definition as the following one. Definition 3.3. Let M, g, S TM,S TM be a lightlike submanifold, tangent to the structure vector field V, immersed in an indefinite Sasakian manifold M, g. M is said to be a radical transversal lightlike submanifold of M if there exists an invariant nondegenerate vector subbundle D such φ Rad TM ltr TM, φ D D, 3.2 where D is a subbundle of S TM and S TM D {V }.
6 6 International Journal of Mathematics and Mathematical Sciences Definition 3.4 see 10. Let M, g, S TM,S TM be a lightlike submanifold, tangent to the structure vector field V, immersed in an indefinite Sasakian manifold M, g. We say that M is a transversal lightlike submanifold of M if the following conditions are satisfied: φ Rad TM ltr TM, φ S TM S TM. 3.3 From Definitions 3.3 and 3.4, we introduce generalized lightlike submanifold as follows. Definition 3.5. Let M, g, S TM,S TM be a lightlike submanifold, tangent to the structure vector field V, immersed in an indefinite Sasakian manifold M, g. M is said to be a generalized transversal lightlike submanifold of M if there exist vector subbundle D and D such φ Rad TM ltr TM, φ D D, φ D S TM, 3.4 where D and D are nondegenerate subbundles of S TM and S TM D D {V }. Remark 3.6. It is easy to see that a generalized transversal lightlike submanifold is a radical transversal lightlike submanifold or a transversal lightlike submanifold 10 if and only if D {0} or D {0}, respectively. We say that M is a proper generalized transversal lightlike submanifold of M if D / {0} and D / {0}. We denote by μ the orthogonal complement of φ D in S TM.The following properties of a generalized transversal lightlike submanifolds are easy to obtain. 1 There do not exist 1-lightlike generalized transversal lightlike submanifolds of an indefinite Sasakian manifolds. The proof of the above assertion is similar to Proposition 3.1 of 10, so we omit it here. Then, dim Rad TM 2. From 3.4,we know that there exists no a generalized transversal lightlike hypersurface of M as dim ltr TM 2. 2 Since 3.4 implies that dim D 2, we have dim M 6ifM is a proper generalized transversal lightlike submanifold of M. It follows that any 5-dimensional generalized transversal lightlike submanifold of M must be a 2-lightlike submanifold. 3 dim M 9ifM is a proper generalized transversal lightlike submanifold of an indefinite Sasakian manifold M. In this paper, we assume that the characteristic vector field is a spacelike vector filed, that is, ɛ 1. If V is a timelike vector field, then one can obtain similar results. Proposition 3.7. There exist no isotropic, coisotropic, and totally lightlike proper generalized transversal lightlike submanifolds of indefinite Sasakian manifolds. Proof. If M is isotropic or totally lightlike submanifolds of M, we have D {0} as S TM {0}, which is a contradiction with definition of proper generalized transversal lightlike submanifolds. Similarly, if M is coisotropic or totally lightlike submanifolds of M, we have D {0} as S TM {0}, there is a contradiction, which proves the assertion.
7 International Journal of Mathematics and Mathematical Sciences 7 Lemma 3.8. Let M be a generalized transversal lightlike submanifold of indefinite Sasakian manifolds M, then we have g X Y, V g Y, φx, 3.5 where X, Y Γ S TM {V }. Proof. Noticing the fact that on M is a metric connection and Definition 3.4, we have g X Y, V g X Y, V g Y, X V X g Y, V g Y, X V. 3.6 Thus, the proof follows from 2.14, 2.16, and 3.9. Theorem 3.9. Let M be a generalized transversal lightlike submanifold of indefinite Sasakian manifolds M, thenμ is an invariant distribution with respect to φ. Proof. For any X Γ μ, ξ Γ Rad TM and N Γ ltr TM, it follows from 2.14 that g φx, N g X, φn 0, which implies that φx has no components in ltr TM. Similarly, from 3.4 and 2.14, we have g φx, ξ g X, φξ 0, which implies that φx has no components in Rad TM. For any W Γ D, we have g φx, W g X, φw 0as φw Γ φ D.Thus,φX has no components in D. Finally, suppose that φx αv, where α is a smooth function on M, then we get X 0 by replacing X in 2.13 by φx. Thus, we have φ μ Γ μ, which completes the proof. Next, we give a characterization theorem for generalized transversal lightlike submanifolds. Theorem Let M be a lightlike submanifold of indefinite Sasakian space form M c, g, c / 1. Then M a generalized transversal lightlike submanifolds of M if and only if 1 the maximal invariant subspaces of T p M p M define a nondegenerate distribution D with respect to φ; 2 g R X, Y Z, W 0, for all X, Y Γ D Rad TM and for all Z, W Γ D. Proof. Suppose that M is a lightlike submanifold of indefinite Sasakian space form M c, g. It follows from 5.1 that g R X, Y Z, W 1 c 2 g X, Y g φz, W, 3.7 for any X, Y Γ D Rad TM and Z, W Γ D. Noticing that φz Γ S TM implies g φz, W 0, then 2 is satisfied. Also, 1 holds naturally by using the definition of generalized transversal lightlike submanifolds. Conversely, since D is a nondegenerate distribution, we may choose X, Y Γ D such that g X, Y / 0. Thus, from 3.7, we have g φz, W 0 for any Z, W Γ D, which implies that φz have components in D. For any X Γ D, g φz, X g Z, φx 0 implies that φz have no components in D. Noticing condition 1, we also have g φz, ξ g Z, φξ 0and
8 8 International Journal of Mathematics and Mathematical Sciences g φz, N g Z, φn 0, respectively. Thus, we get φ D S TM, which completes the proof. For any X Γ TM, we denote by P 0, P 1 and P 2 and P 3 are the projection morphisms of TM on Rad TM, D and D and {V }, respectively. Let φx TX QX FX, where TX, QX,andFX are the components of φx on TM,ltr TM and S TM, respectively. Moreover, we have TX φp 1 Γ D, QX φp 0 Γ ltr TM and FX φp 2 Γ φ D. Similarly, for any U Γ tr TM, we denote by U 1, U 2 and U 3 are the components of U on ltr TM, φ D and μ, respectively. Using Theorem 3.9, then the components of φu on TM and tr TM are denoted by SU φu 1 φu 2 Γ Rad TM D and LU φu 3 Γ μ. Let M be a lightlike submanifold of indefinite Sasakian manifolds M, g. It follows from 2.15 that X φy φ X Y g X, Y V η Y X. Substituting X TX QX FX Γ TM and Y TY QY FY Γ TM into the above equation and taking the tangential, screen transversal, and lightlike transversal parts, respectively, yield that X T Y A QY X A FY X φh l X, Y Sh s X, Y g X, Y V η Y X, Q X Y h l X, TY l X QY Dl X, FY, F X Y h s X, TY D s X, QY s X FY Lhs X, Y, where X T Y X TY T X Y. Proposition Let M be a generalized transversal lightlike submanifold of indefinite Sasakian manifolds M. Then we have X V φx, h l X, V 0, h s X, V 0, X Γ D, 3.11 X V 0, h l X, V 0, h s X, V φx φ Lh s X, V, X Γ D, 3.12 X V 0, h l X, V φx, h s X, V 0, X Γ Rad TM, 3.13 where V is the characteristic vector filed. Proof. Replacing Y by V in 3.8, it follows from φv 0and 3.8 that T X V X φh l X, V Sh s X, V η X V Similarly, replacing Y by V in 3.9 and 3.10, respectively, we get Q X V 0andF X V Lh s X, V. Noticing that FX Γ φd for any X Γ TM and LW Γ μ for any W Γ tr TM, we have F X V Lh s X, V 0. Let X Γ D, sinceη X 0andSh s X, V Γ D, we have T X V X and φh l X, V Sh s X, V 0, which proves Let X Γ D,from 3.14, we have T X V φh l X, V 0andSh s X, V X, which proves Let X Γ Rad TM, it follows from 3.14 that T X V Sh s X, V 0and φh l X, V X, which proves 3.13.
9 International Journal of Mathematics and Mathematical Sciences 9 We know from 2.12 that the induced connection on M is not a metric connection, the following theorem gives a necessary and sufficient condition for the to be a metric connection. Theorem Let M be a generalized transversal lightlike submanifold of indefinite Sasakian manifold M, g. Then the following assertions are equivalent. 1 The induced connection on M is a metric connection. 2 A φx Y has no components in Rad TM for any X Γ S TM and Y Γ TM. 3 A φξ X has no components in D for any X Γ TM and D s X, N Γ μ for any X Γ TM. Proof. From 2.13 and 2.15, we have g X ξ, V g X ξ, V g φ X ξ, φv 0, 3.15 which implies that X ξ has not component in {V } for any X Γ TM For X 1 Γ D and X Γ TM, it follows from 2.6, 2.14, 2.15, andthe fact that is metric connection, we have g X ξ, X 1 g X ξ, X 1 g φ X ξ, φx 1 η X ξ η X 1 g X φξ η ξ X g X, ξ V, X 1 g φξ, X φx 1 g φξ, A φx1 X For X 2 Γ D, Together with 2.8, 2.14, and 2.15, we have g X ξ, X 2 g X ξ, X 2 g φ X ξ, φx 2 η X ξ η X 2 g X φξ η ξ X g X, ξ V, X 2 g φξ, X φx 2 g φξ, A φx2 X Thus, we prove the equivalence between 1 and Using the similar method shown in the above, from 3.16 and 3.17, we have g X ξ, X 1 g X ξ, X 1 g A φ ξx, φx 1, g X ξ, X 2 g X ξ, X 2 g D s X, φξ 3.18,φX 2. Thus, the proof follows from the above equations. 4. Integrability and Geodesic of Distributions Theorem 4.1. Let M be a generalized transversal lightlike submanifold of indefinite Sasakian manifold M, g. ThenD {V } is integrable if and only if h X, φy h Y, φx for any X, Y Γ D.
10 10 International Journal of Mathematics and Mathematical Sciences Proof. Let X, Y Γ D, we have φy TY and QY FY 0. Thus, it follows from 3.9 that Q X Y h l X, φy. Interchanging the roles of X and Y in the above equation and subtracting, we have Q X, Y h l X, φy h l Y, φx, X, Y Γ D. 4.1 Similarly, it follows from 3.10 that F X Y h s X, φy Lh s X, Y. Noticing that h is symmetric and interchanging the roles of X and Y in the above equation and subtracting, we have F X, Y h s X, φy h s Y, φx, X, Y Γ D. 4.2 Thus, our assertion follows from 4.1 and 4.2. Corollary 4.2. Let M be a generalized transversal lightlike submanifold of indefinite Sasakian manifold M, g.thend is not integrable. Proof. It follows from Lemma 3.8 that g X, Y,V 2g Y, φx for any X, Y Γ D. Suppose that D is integrable, then we have g Y, φx 0, which is a contradiction to the fact that D is a nondegenerate distribution of M. Theorem 4.3. Let M be a generalized transversal lightlike submanifold of indefinite Sasakian manifold M, g. ThenD is integrable if and only if D l X, FY D l Y, FX and A FX Y A FY X for any X, Y Γ D. Proof. For X, Y Γ D, we have φy FY and TY QY 0. Thus, it follows from 3.8 that T X Y A FY X Qh l X, Y Sh s X, Y g X, Y V.Ash and g are symmetric, by interchanging the roles of X and Y in the above equation and subtracting the resulting equations, we have T X, Y A FX Y A FY X, X, Y Γ D. 4.3 Similarly, it follows from 3.9 that Q X Y D l X, FY. Interchanging the roles of X and Y in the above equation and subtracting the resulting equations, we have Q X, Y D l X, FY D l Y, FX, X, Y Γ D. 4.4 Noticing Lemma 3.8, we have g X, Y,V 2g Y, φx 0 for any X, Y Γ D. Thus, our assertion follows from 4.3 and 4.4. Remark 4.4. Let M be a generalized transversal lightlike submanifold of indefinite Sasakian manifold M, g. IfD {0}, then M is a transversal lightlike submanifolds 9. It follows from Theorem 4.3 that S TM is integrable if and only if D l X, FY D l Y, FX for all X, Y Γ S TM, which is just one of the conclusions shown in 10.
11 International Journal of Mathematics and Mathematical Sciences 11 Theorem 4.5. Let M be a generalized transversal lightlike submanifold of indefinite Sasakian manifold M, g. Then Rad TM {V } is integrable if and only if A φx Y A φy X and D s X, φy D s Y, φx has no components in φ D for any X, Y Γ Rad TM. Proof. Together with 2.14, 2.15, and the fact that is a metric connection, we have g X, Y,Z g X φy Y φx, φz g A φx Y A φy X, φz, 4.5 where Z Γ D. Similarly, for W Γ D, we have g X, Y,W g X φy Y φx, φw g D s X, φy D s Y, φx,φw, 4.6 Thus, our assertion follows from 4.5 and 4.6. Corollary 4.6. Let M be a generalized transversal lightlike submanifold of indefinite Sasakian manifold M, g.then Rad TM is not integrable. Proof. It follows from Lemma 3.8 that g X, Y,V 2g Y, φx holds for any X, Y Γ Rad TM. Suppose that Rad TM is integrable, then we have g Y, φx 0. On the other hand, for any X Γ Rad TM, there must exists Y Γ Rad TM such that g X, φy / 0as φ Rad TM ltr TM, a contradiction. Then we complete the proof. Using the similar method in the proof of Theorems and Corollaries 4.2 and 4.6, we have the following results. Theorem 4.7. Let M be a generalized transversal lightlike submanifold of indefinite Sasakian manifold M, g. Then, 1 Rad TM D {V } is integrable if and only if h s X, φy h s Y, φx has no components in φ D for any X, Y Γ Rad TM D. 2 Rad TM D {V } is integrable if and only if A φy X A φx Y has no components in D for any X, Y Γ Rad TM D. 3 Rad TM D is not integrable. 4 Rad TM D is not integrable. We know from 3 that a distribution on M is said to define a totally geodesic foliation if any leaf of D is geodesic. we focus on this property of generalized transversal lightlike submanifolds in the following of this section. Theorem 4.8. Let M be a generalized transversal lightlike submanifold of indefinite Sasakian manifold M, g. Then the screen distribution defines a totally geodesic foliation if and only if g TY,A φn X g FY,hs X, φn for any X, Y Γ S TM.
12 12 International Journal of Mathematics and Mathematical Sciences Proof. It is known that S TM defines a totally geodesic foliation if and only if X Y Γ S TM for any X, Y Γ S TM. For X Γ S TM, we have QX 0. Also, we have g X Y, N g X φy, φn g φy, X φn g φy, A φn X h s X, φn g TY,A φn X g FY,h s X, φn. 4.7 Thus we complete the proof. Remark 4.9. Let M be a generalized transversal lightlike submanifold of indefinite Sasakian manifold M, g. IfD {0}, then M is radical transversal lightlike submanifolds. For X Γ S TM, we have FX 0. Thus, it follows from Theorem 4.8 that S TM defines a totally geodesic foliation if and only if A X has no components in S TM, which is just the φn Theorem 3.6 proved in 10. Theorem Let M be a generalized transversal lightlike submanifold of indefinite Sasakian manifold M, g. Then, The following assertions are equivalent. 1 D defines a totally geodesic foliation. 2 D l X, φy 0 and A φy X has no components in D. 3 h s X, φz and h s X, ξ has no components φ D,whereX, Y, Z Γ D and ξ Γ Rad TM. Proof. It is easy to see that D defines a totally geodesic foliation if and only if X Y Γ D for any X, Y Γ D From 2.14 and 2.15, we have that g X Y, N g φ X Y, φn g X φy, φn g D l X, φy,φn, 4.8 where X, Y Γ D and N Γ ltr TM.Also,forZ Γ D, we have g X Y, Z g φ X Y, φz g X φy, φz g A φy X, φz. 4.9 Noticing Lemma 3.8, we have g X, Y,V 2g Y, φx 0 for all X, Y Γ D. Then the equivalence between 1 and 2 follows from 4.8 and 4.9. Noticing that is a metric connection, then from 4.8 and 4.9 we have that g X Y, N g φy, h s X, φn, g X Y, Z g h s X, φz,φy Thus, the equivalence between 1 and 3 follows from From 4.9, we know that g X Y, Z g A φy X, φz for any X, Y Γ D Rad TM and Z Γ D, then the following corollary holds.
13 International Journal of Mathematics and Mathematical Sciences 13 Corollary Let M be a generalized transversal lightlike submanifold of indefinite Sasakian manifold M, g. ThenD Rad TM {V } defines a totally geodesic foliation if and only if A φy X has no components in D for any X, Y Γ D Rad TM {V }. Theorem Let M be a generalized transversal lightlike submanifold of indefinite Sasakian manifold M, g. ThenD {V } defines a totally geodesic foliation if and only if A φn X and A φwx has no components in D for any X Γ D and W Γ D and N Γ ltr TM. Proof. Noticing that is a metric connection and 2.14 and 2.15, we have g X Y, N g φ X Y, φn g X φy, φn g A φn X, φy Similarly, for W Γ D, we have g X Y, W g φ X Y, φw g X φy, φ g A φw X, φy Then the proof follows from 4.11 and 4.12 and Lemma 3.8. It follows from 2.13 that φv 0, then 4.11 and 4.12 hold for any X, Y Γ D {V }. Thus it is easy to get the following corollary. Corollary Let M be a generalized transversal lightlike submanifold of indefinite Sasakian manifold M, g.thend cannot define a totally geodesic foliation. Proof. It follows from Lemma 3.8 that g X Y, V g Y, φx for any X, Y Γ D. Suppose that D defines a totally geodesic foliation, then we have g Y, φx 0. Which is a contradiction to the fact that D is a nondegenerate distribution of M. We say that M is a contact generalized transversal lightlike product manifold if D {V } and D Rad TM define totally geodesic foliations in M. Theorem Let M be a generalized transversal lightlike submanifold of indefinite Sasakian manifold M, g. Then, M is a contact generalized transversal lightlike product manifold if and only if 1 h X, φy Lh s X, Y for any X, Y Γ D {V } ; 2 A φx Y has no components in D for any X, Y Γ D Rad TM. Proof. For X, Y Γ D {V }, we have φx TX and QX FX 0asφV 0. It follows from 3.9 and 3.10, respectively, that Q X Y h l X, TY, F X Y h s X, TY Lh s X, Y On the other hand, for X, Y Γ D Rad TM, we have TX 0. Then it follows from 3.8 that A φy X T X Y φh l X, Y Sh s X, Y g X, Y V. 4.14
14 14 International Journal of Mathematics and Mathematical Sciences Noticing that X Y belongs to Γ D {V } or Γ D Rad TM if and only if Q X Y F X Y 0orT X Y 0, respectively, then our assertion follows from 4.13 and Totally Contact Umbilical Lightlike Submanifolds In this section, we study totally contact umbilical generalized transversal lightlike submanifolds of indefinite Sasakian manifolds defined by Duggal and Sahin 9. We mainly obtain a classification theorem for such lightlike submanifolds. A plane section of a Sasakian manifold M, φ, η, V, g is called a φ-section if it is spanned by a unit vector X orthogonal to V and φx, where X is a non-null vector field on M. The sectional curvature K X, φx of a φ-section is called a φ-sectional curvature. If M has a φ-sectional curvature c which is not depend on the φ-section at each point, then c is a constant and M is called a Sasakian space form, denoted by M c. The curvature tensor R of a Sasakian space form M c is given in 13 as follows: R X, Y Z c 3 { } g Y, Z X g X, Z Y 4 c 1 { ɛη X η Z Y ɛη Y η Z X g X, Z η Y V 4 g Y, Z η X V g φy, Z φx g φz, X φy 2g φx, Y φz }, 5.1 where X, Y, Z Γ TM. Definition 5.1 see 9. A lightlike submanifold of an indefinite Sasakian manifold is contact totally umbilical if h l X, Y { g X, Y η X η Y } α L η X h l Y, V η Y h l X, V, h s X, Y { g X, Y η X η Y } α S η X h s Y, V η Y h s X, V, where X, Y Γ TM, α L Γ ltr TM,andα S Γ S TM. Theorem 5.2. Let M be a totally contact umbilical proper generalized transversal lightlike submanifold of indefinite Sasakian manifold M, g.thenα L 0 if and only if D is integrable. Proof. Let X, Y Γ D, it follows from 2.15 and 5.2 that g X, Y,N g φ X Y φ Y X, φn g X φy Y φx, φn g h l X, φy h l Y, φx,φn g X, φy g Y, φx g α L,φN. 5.4
15 International Journal of Mathematics and Mathematical Sciences 15 Noticing that 2.15 implies g X, φy g Y, φx 0 for any X, Y Γ TM, then it follows from 5.4 that g X, Y,N 2g X, φy g α L,φN. On the other hand, for any Z Γ D it follows from 2.14, 2.15, and 5.3 that g X, Y,Z g φ X Y φ Y X, φz g X φy Y φx, φz g h s X, φy h s Y, φx,φz 0, 5.5 which means that X, Y have components in D for any X, Y Γ D. Noticing Lemma 3.8, we know that X, Y has no components in {V } for any X, Y Γ D. Thus, the proof is complete. Theorem 5.3. Let M be a totally contact umbilical proper generalized transversal lightlike submanifold of indefinite Sasakian manifold M, g.thenα S 0. Proof. Let X Γ D, from 5.3, we have h s X, Y g X, Y α S, which implies h s X, φy h s Y, φx 2g X, φx α S. Also, it follows from 3.10 that h s X, φy F X Y Lh s X, Y. 5.6 Interchanging the role of X and Y in the above equation and subtracting, we have h s X, φy h s Y, φx F X, Y. Using the same method as shown in the proof of Theorem 5.2, we have α S 0. Which proves the theorem. Lemma 5.4. Let M be a totally contact umbilical proper generalized transversal lightlike submanifold of indefinite Sasakian manifold M, g. Then, for any X, Y Γ D, we have g X φx, V g X, X and g φx X, V g X, X. Proof. Since is a metric connection, we have X g φx, V g X φx, V g φx, X V Noticing 2.16 and the fact that X Γ D implies g φx, V 0andφX Γ D, then from 5.7 and 2.14, we have g X φx, V g X, X. Interchanging the role of X and φx in 5.7 we get g φx X, V g X, X. Lemma 5.5. Let M be a totally contact umbilical proper generalized transversal lightlike submanifold of indefinite Sasakian manifold M, g. Then for any X, Y Γ D, one have g X ξ, φx 0 and g φx ξ, X 0. Proof. Since is a metric connection, we have φx g X, ξ g φx X, ξ g X, φx ξ
16 16 International Journal of Mathematics and Mathematical Sciences For X Γ D, itiseasytoseethatg φx, ξ 0andφX Γ D, then g φx ξ, X g h l φx, X,ξ. Using 5.2 we have g X ξ, φx 0. Interchanging the role of X and φx in 5.8 we get g φx ξ, X 0. At last, we complete this paper by a classification theorem for generalized transversal lightlike submanifolds. Theorem 5.6. Let M be a totally contact umbilical proper generalized transversal lightlike submanifold of indefinite Sasakian space form M c, g.thenc 3. Proof. For any X, Y, Z Γ TM, it follows from that R X, Y Z R X, Y Z A h l X,Z Y A h l Y,Z X A h s X,Z Y A h s Y,Z X X h l Y, Z Y h l X, Z D l X, h s Y, Z D l Y, h s X, Z X h s Y, Z Y h s X, Z D s X, h l Y, Z D l Y, h l X, Z, 5.9 where X h l Y, Z l X hl Y, Z h l X Y, Z h l X Z, Y and X h s Y, Z s X hs Y, Z h s X Y, Z h s X Z, Y. Noticing Theorem 5.3 then we have h s X, Y 0 for any X, Y Γ TM {V }. Let X, Y Γ D and ξ, ξ Γ Rad TM, replacing Z by ξ in 5.9 and using 5.2, we have g R X, Y ξ, ξ g X h l Y, ξ Y h l X, ξ,ξ g h l Y X, ξ h l X Y, ξ h l X, X ξ h l Y, X ξ,ξ g η Y X h l ξ, V η X Y h l ξ, V g X, Y ξ α L g Y, X ξ α L,ξ Replacing Y by φx in 5.10 and using Proposition 3.11 and Lemmas 5.4 and 5.5, we have g R X, Y ξ, ξ 2g X, X g φξ, ξ On the other hand, it follows from 5.1 that g R X, Y ξ, ξ 1 c 2 g φx, Y g φξ, ξ Replacing Y by φx in 5.12 and using 5.11, we have that c 3 g X, X g φξ, ξ 0 for all X Γ D. AsD is nondegenerate, so we may choose X such that g X, X / 0inthe above equation. Thus, we complete the proof.
17 International Journal of Mathematics and Mathematical Sciences 17 Acknowledgments The authors would like to thank the referee for his or her valuable suggestions and comments on the improvement of this paper. This work is supported by NSFC no and Natural Science Foundation of Guangdong Province of China no. S References 1 B. O Neill, Semi-Riemannian Geometry with Application to Relativity, Academic Press, London, UK, D. N. Kupeli, Singular Semi-Riemannian Geometry, vol. 366, Kluwer Academic Publishers, Dordrecht, The Netherlands, K. L. Duggal and B. Bejancu, Lightlike Submainfolds of Semi-Riemannian Mainfolds and Applications, Kluer Academic Publishers, Dordrecht, The Netherlands, K. L. Duggal and B. Sahin, Generalized Cauchy-Riemann lightlike submanifolds of indefinite Sasakian manifolds, Acta Mathematica Hungarica, vol. 122, no. 1-2, pp , K. L. Duggal and B. Sahin, Screen Cauchy Riemann lightlike submanifolds, Acta Mathematica Hungarica, vol. 106, no. 1-2, pp , B. Sahin and R. Gunes, Geodesic CR-lightlike submanifolds, Beiträge zur Algebra und Geometrie, vol. 42, no. 2, pp , K. L. Duggal and B. Sahin, Generalized Cauchy-Riemann lightlike submanifolds of Kaehler manifolds, Acta Mathematica Hungarica, vol. 112, no. 1-2, pp , B. Sahin, Transversal lightlike submanifolds of indefinite Kaehler manifolds, Analele Universitaii de Vest, vol. 44, no. 1, pp , K. L. Duggal and B. Sahin, Lightlike submanifolds of indefinite Sasakian manifolds, International Journal of Mathematics and Mathematical Sciences, vol. 2007, Article ID 57585, 21 pages, C. Yıldırım andb.şahin, Transversal lightlike submanifolds of indefinite Sasakian manifolds, Turkish Journal of Mathematics, vol. 34, no. 4, pp , K. L. Duggal, Space time manifolds and contact structures, International Journal of Mathematics & Mathematical Sciences, vol. 13, no. 3, pp , S. Sasakian, Almost Contact Manifolds, Lecture notes, Mathematical Institute, Tohoku University, Volumes I, II, III, K. Ogiue, On almost contact manifolds admitting axiom of planes or axiom of free mobility, Kōdai Mathematical Seminar Reports, vol. 16, pp , 1964.
18 Advances in Operations Research Advances in Decision Sciences Journal of Applied Mathematics Algebra Journal of Probability and Statistics The Scientific World Journal International Journal of Differential Equations Submit your manuscripts at International Journal of Advances in Combinatorics Mathematical Physics Journal of Complex Analysis International Journal of Mathematics and Mathematical Sciences Mathematical Problems in Engineering Journal of Mathematics Discrete Mathematics Journal of Discrete Dynamics in Nature and Society Journal of Function Spaces Abstract and Applied Analysis International Journal of Journal of Stochastic Analysis Optimization
Research Article GCR-Lightlike Product of Indefinite Sasakian Manifolds
Advances in Mathematical Physics Volume 2011, Article ID 983069, 13 pages doi:10.1155/2011/983069 Research Article GCR-Lightlike Product of Indefinite Sasakian Manifolds Rakesh Kumar, 1 Varun Jain, 2 andr.k.nagaich
SCREEN TRANSVERSAL LIGHTLIKE SUBMANIFOLDS OF INDEFINITE KENMOTSU MANIFOLDS
SARAJEVO JOURNAL OF MATHEMATICS Vol.7 (19) (2011), 103 113 SCREEN TRANSVERSAL LIGHTLIKE SUBMANIFOLDS OF INDEFINITE KENMOTSU MANIFOLDS RAM SHANKAR GUPTA AND A. SHARFUDDIN Abstract. In this paper, we introduce
SCREEN TRANSVERSAL LIGHTLIKE SUBMANIFOLDS OF INDEFINITE SASAKIAN MANIFOLDS
An. Şt. Univ. Ovidius Constanţa Vol. 18(2), 2010, 315 336 SCREEN TRANSVERSAL LIGHTLIKE SUBMANIFOLDS OF INDEFINITE SASAKIAN MANIFOLDS Cumali Yıldırım, Bayram Ṣahin Abstract We introduce screen transversal
Research Article Some Results on Warped Product Submanifolds of a Sasakian Manifold
International Mathematics and Mathematical Sciences Volume 2010, Article ID 743074, 9 pages doi:10.1155/2010/743074 Research Article Some Results on Warped Product Submanifolds of a Sasakian Manifold Siraj
A CHARACTERIZATION OF WARPED PRODUCT PSEUDO-SLANT SUBMANIFOLDS IN NEARLY COSYMPLECTIC MANIFOLDS
Journal of Mathematical Sciences: Advances and Applications Volume 46, 017, Pages 1-15 Available at http://scientificadvances.co.in DOI: http://dx.doi.org/10.1864/jmsaa_71001188 A CHARACTERIATION OF WARPED
ON SEMI-INVARIANT SUBMANIFOLDS OF A NEARLY KENMOTSU MANIFOLD WITH THE CANONICAL SEMI-SYMMETRIC SEMI-METRIC CONNECTION. Mobin Ahmad. 1.
MATEMATIQKI VESNIK 62, 3 (2010), 189 198 September 2010 originalni nauqni rad research paper ON SEMI-INVARIANT SUBMANIFOLDS OF A NEARLY KENMOTSU MANIFOLD WITH THE CANONICAL SEMI-SYMMETRIC SEMI-METRIC CONNECTION
SCREEN SLANT RADICAL TRANSVERSAL NULL SUBMANIFOLDS OF PARA-SASAKIAN MANIFOLDS. Bilal Eftal Acet, Selcen Yüksel Perktaş. 1.
FACTA UNIVERSITATIS (NIŠ) Ser. Math. Inform. Vol. 31, No 2 (2016), 543 557 SCREEN SLANT RADICAL TRANSVERSAL NULL SUBMANIFOLDS OF PARA-SASAKIAN MANIFOLDS Bilal Eftal Acet, Selcen Yüksel Perktaş Abstract.
An Inequality for Warped Product Semi-Invariant Submanifolds of a Normal Paracontact Metric Manifold
Filomat 31:19 (2017), 6233 620 https://doi.org/10.2298/fil1719233a Published by Faculty of Sciences Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat An Inequality for
A Semi-Riemannian Manifold of Quasi-Constant Curvature Admits Lightlike Submanifolds
International Journal of Mathematical Analysis Vol. 9, 2015, no. 25, 1215-1229 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ijma.2015.5255 A Semi-Riemannian Manifold of Quasi-Constant Curvature
CHAPTER 1 PRELIMINARIES
CHAPTER 1 PRELIMINARIES 1.1 Introduction The aim of this chapter is to give basic concepts, preliminary notions and some results which we shall use in the subsequent chapters of the thesis. 1.2 Differentiable
Warped Product Bi-Slant Submanifolds of Cosymplectic Manifolds
Filomat 31:16 (2017) 5065 5071 https://doi.org/10.2298/fil1716065a Published by Faculty of Sciences and Mathematics University of Niš Serbia Available at: http://www.pmf.ni.ac.rs/filomat Warped Product
Legendre surfaces whose mean curvature vectors are eigenvectors of the Laplace operator
Note di Matematica 22, n. 1, 2003, 9 58. Legendre surfaces whose mean curvature vectors are eigenvectors of the Laplace operator Tooru Sasahara Department of Mathematics, Hokkaido University, Sapporo 060-0810,
GENERALIZED NULL SCROLLS IN THE n-dimensional LORENTZIAN SPACE. 1. Introduction
ACTA MATHEMATICA VIETNAMICA 205 Volume 29, Number 2, 2004, pp. 205-216 GENERALIZED NULL SCROLLS IN THE n-dimensional LORENTZIAN SPACE HANDAN BALGETIR AND MAHMUT ERGÜT Abstract. In this paper, we define
K. A. Khan, V. A. Khan and Sirajuddin. Abstract. B.Y. Chen [4] showed that there exists no proper warped CRsubmanifolds
Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.yu/filomat Filomat 21:2 (2007), 55 62 WARPED PRODUCT CONTACT CR-SUBMANIFOLDS OF TRANS-SASAKIAN MANIFOLDS
An inequality for warped product pseudo-slant submanifolds of nearly cosymplectic manifolds
Al-Solamy Journal of Inequalities and Applications (2015) 2015:306 DOI 10.1186/s13660-015-0825-y R E S E A R C H Open Access An inequality for warped product pseudo-slant submanifolds of nearly cosymplectic
On Indefinite Almost Paracontact Metric Manifold
International Mathematical Forum, Vol. 6, 2011, no. 22, 1071-1078 On Indefinite Almost Paracontact Metric Manifold K. P. Pandey Department of Applied Mathematics Madhav Proudyogiki Mahavidyalaya Bhopal,
MEHMET AKIF AKYOL, LUIS M. FERNÁNDEZ, AND ALICIA PRIETO-MARTÍN
Konuralp Journal of Mathematics Volume No. 1 pp. 6 53 (016) c KJM THE L-SECTIONAL CURVATURE OF S-MANIFOLDS MEHMET AKIF AKYOL, LUIS M. FERNÁNDEZ, AND ALICIA PRIETO-MARTÍN Abstract. We investigate L-sectional
A CONSTRUCTION OF TRANSVERSE SUBMANIFOLDS
UNIVERSITATIS IAGELLONICAE ACTA MATHEMATICA, FASCICULUS XLI 2003 A CONSTRUCTION OF TRANSVERSE SUBMANIFOLDS by J. Szenthe Abstract. In case of Riemannian manifolds isometric actions admitting submanifolds
Complex and real hypersurfaces of locally conformal Kähler manifolds
Complex and real hypersurfaces of locally conformal Kähler manifolds Odessa National Economic University Varna 2016 Topics 1 Preliminaries 2 Complex surfaces of LCK-manifolds 3 Real surfaces of LCK-manifolds
Research Article On Submersion of CR-Submanifolds of l.c.q.k. Manifold
International Scholarly Research Network ISRN Geometry Volume 01, Article ID 309145, 13 pages doi:10.540/01/309145 Research Article On Submersion of CR-Submanifolds of l.c.q.k. Manifold Majid Ali Choudhary,
Research Article New Examples of Einstein Metrics in Dimension Four
International Mathematics and Mathematical Sciences Volume 2010, Article ID 716035, 9 pages doi:10.1155/2010/716035 Research Article New Examples of Einstein Metrics in Dimension Four Ryad Ghanam Department
CΛ-SUBMANIFOLDS OF A COMPLEX SPACE FORM
J. DIFFERENTIAL GEOMETRY 16 (1981) 137-145 CΛ-SUBMANIFOLDS OF A COMPLEX SPACE FORM AUREL BEJANCU, MASAHIRO KON & KENTARO YANO Dedicated to Professor Buchin Su on his Wth birthday 0. Introduction The CΉ-submanifolds
arxiv:math/ v2 [math.dg] 25 May 2007
arxiv:math/0604008v2 [math.dg] 25 May 2007 A Note on Doubly Warped Product Contact CR-Submanifolds in trans-sasakian Manifolds Marian-Ioan Munteanu Abstract Warped product CR-submanifolds in Kählerian
Some Properties of a Semi-symmetric Non-metric Connection on a Sasakian Manifold
Int. J. Contemp. Math. Sciences, Vol. 8, 2013, no. 16, 789-799 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ijcms.2013.28172 Some Properties of a Semi-symmetric Non-metric Connection on a Sasakian
SCREEN CONFORMAL EINSTEIN LIGHTLIKE HYPERSURFACES OF A LORENTZIAN SPACE FORM
Commun. Korean Math. Soc. 25 (2010), No. 2, pp. 225 234 DOI 10.4134/CKMS.2010.25.2.225 SCREEN CONFORMAL EINSTEIN LIGHTLIKE HYPERSURFACES OF A LORENTZIAN SPACE FORM Dae Ho Jin Abstract. In this paper, we
Research Article Warped Product Semi-Invariant Submanifolds in Almost Paracontact Riemannian Manifolds
Mathematical Problems in Engineering Volume 2009, Article ID 621625, 16 pages doi:10.1155/2009/621625 Research Article Warped Product Semi-Invariant Submanifolds in Almost Paracontact Riemannian Manifolds
Symmetries in Lightlike Hypersufaces of Indefinite Kenmotsu Manifolds
International Journal of Contemporary Mathematical Sciences Vol. 12, 2017, no. 3, 117-132 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ijcms.2017.611 Symmetries in Lightlike Hypersufaces of Indefinite
ON SOME INVARIANT SUBMANIFOLDS IN A RIEMANNIAN MANIFOLD WITH GOLDEN STRUCTURE
ANALELE ŞTIINŢIFICE ALE UNIVERSITĂŢII AL.I. CUZA DIN IAŞI (S.N.) MATEMATICĂ, Tomul LIII, 2007, Supliment ON SOME INVARIANT SUBMANIFOLDS IN A RIEMANNIAN MANIFOLD WITH GOLDEN STRUCTURE BY C.-E. HREŢCANU
ON SOME SUBMANIFOLDS OF A LOCALLY PRODUCT MANIFOLD
G. PITIS KODAI MATH. J. 9 (1986), 327 333 ON SOME SUBMANIFOLDS OF A LOCALLY PRODUCT MANIFOLD BY GHEORGHE PITIS An investigation of properties of submanifolds of the almost product or locally product Riemannian
Differential Geometry of Warped Product. and Submanifolds. Bang-Yen Chen. Differential Geometry of Warped Product Manifolds. and Submanifolds.
Differential Geometry of Warped Product Manifolds and Submanifolds A warped product manifold is a Riemannian or pseudo- Riemannian manifold whose metric tensor can be decomposes into a Cartesian product
Real Hypersurfaces with Pseudo-parallel Normal Jacobi Operator in Complex Two-Plane Grassmannians
Filomat 31:12 (2017), 3917 3923 https://doi.org/10.2298/fil1712917d Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Real Hypersurfaces
CR-submanifolds of Kaehlerian product manifolds
CR-submanifolds of Kaehlerian product manifolds Mehmet Atçeken Abstract. In this paper, the geometry of F -invariant CR-submanifolds of a Kaehlerian product manifold is studied. Fundamental properties
The geometry of secants in embedded polar spaces
The geometry of secants in embedded polar spaces Hans Cuypers Department of Mathematics Eindhoven University of Technology P.O. Box 513 5600 MB Eindhoven The Netherlands June 1, 2006 Abstract Consider
Distributions of Codimension 2 in Kenmotsu Geometry
Distributions of Codimension 2 in Kenmotsu Geometry Constantin Călin & Mircea Crasmareanu Bulletin of the Malaysian Mathematical Sciences Society ISSN 0126-6705 Bull. Malays. Math. Sci. Soc. DOI 10.1007/s40840-015-0173-6
LECTURE: KOBORDISMENTHEORIE, WINTER TERM 2011/12; SUMMARY AND LITERATURE
LECTURE: KOBORDISMENTHEORIE, WINTER TERM 2011/12; SUMMARY AND LITERATURE JOHANNES EBERT 1.1. October 11th. 1. Recapitulation from differential topology Definition 1.1. Let M m, N n, be two smooth manifolds
Null Bertrand curves in Minkowski 3-space and their characterizations
Note di Matematica 23, n. 1, 2004, 7 13. Null Bertrand curves in Minkowski 3-space and their characterizations Handan Balgetir Department of Mathematics, Firat University, 23119 Elazig, TURKEY hbalgetir@firat.edu.tr
Differential-Geometrical Conditions Between Geodesic Curves and Ruled Surfaces in the Lorentz Space
Differential-Geometrical Conditions Between Geodesic Curves and Ruled Surfaces in the Lorentz Space Nihat Ayyildiz, A. Ceylan Çöken, Ahmet Yücesan Abstract In this paper, a system of differential equations
Totally quasi-umbilic timelike surfaces in R 1,2
Totally quasi-umbilic timelike surfaces in R 1,2 Jeanne N. Clelland, University of Colorado AMS Central Section Meeting Macalester College April 11, 2010 Definition: Three-dimensional Minkowski space R
is constant [3]. In a recent work, T. IKAWA proved the following theorem for helices on a Lorentzian submanifold [1].
ANALELE ŞTIINŢIFICE ALE UNIVERSITĂŢII AL.I.CUZA IAŞI Tomul XLVI, s.i a, Matematică, 2000, f.2. ON GENERAL HELICES AND SUBMANIFOLDS OF AN INDEFINITE RIEMANNIAN MANIFOLD BY N. EKMEKCI Introduction. A regular
RICCI CURVATURE OF SUBMANIFOLDS IN SASAKIAN SPACE FORMS
J. Austral. Math. Soc. 72 (2002), 27 256 RICCI CURVATURE OF SUBMANIFOLDS IN SASAKIAN SPACE FORMS ION MIHAI (Received 5 June 2000; revised 19 February 2001) Communicated by K. Wysocki Abstract Recently,
LECTURE 8: THE SECTIONAL AND RICCI CURVATURES
LECTURE 8: THE SECTIONAL AND RICCI CURVATURES 1. The Sectional Curvature We start with some simple linear algebra. As usual we denote by ( V ) the set of 4-tensors that is anti-symmetric with respect to
On constant isotropic submanifold by generalized null cubic
On constant isotropic submanifold by generalized null cubic Leyla Onat Abstract. In this paper we shall be concerned with curves in an Lorentzian submanifold M 1, and give a characterization of each constant
On the 5-dimensional Sasaki-Einstein manifold
Proceedings of The Fourteenth International Workshop on Diff. Geom. 14(2010) 171-175 On the 5-dimensional Sasaki-Einstein manifold Byung Hak Kim Department of Applied Mathematics, Kyung Hee University,
Research Article Remarks on Asymptotic Centers and Fixed Points
Abstract and Applied Analysis Volume 2010, Article ID 247402, 5 pages doi:10.1155/2010/247402 Research Article Remarks on Asymptotic Centers and Fixed Points A. Kaewkhao 1 and K. Sokhuma 2 1 Department
Qing-Ming Cheng and Young Jin Suh
J. Korean Math. Soc. 43 (2006), No. 1, pp. 147 157 MAXIMAL SPACE-LIKE HYPERSURFACES IN H 4 1 ( 1) WITH ZERO GAUSS-KRONECKER CURVATURE Qing-Ming Cheng and Young Jin Suh Abstract. In this paper, we study
Scalar curvature on lightlike hypersurfaces
Scalar curvature on lightlike hypersurfaces Cyriaque Atindogbé Abstract. Recently, the concept of induced scalar curvature of lightlike hypersurfaces is introduced, restricting on a special class of the
Differential Geometry of Lightlike Submanifolds
Frontiers in Mathematics Differential Geometry of Lightlike Submanifolds Bearbeitet von Krishan L. Duggal, Bayram Sahin 1st Edition. 2010. Taschenbuch. 488 S. Paperback ISBN 978 3 0346 0250 1 Format (B
The Schouten-van Kampen affine connections adapted to almost (para)contact metric structures (the work in progress) Zbigniew Olszak
The Schouten-van Kampen affine connections adapted to almost (para)contact metric structures (the work in progress) Zbigniew Olszak Wroc law University of Technology, Wroc law, Poland XVII Geometrical
Bulletin of the Malaysian Mathematical Sciences Society Warped product submanifolds of nearly cosymplectic manifolds with slant immersion
Bulletin of the Malaysian Mathematical Sciences Society Warped product submanifolds of nearly cosymplectic manifolds with slant immersion --Manuscript Draft-- Manuscript Number: Full Title: Article Type:
Generalized Contact Structures
Generalized Contact Structures Y. S. Poon, UC Riverside June 17, 2009 Kähler and Sasakian Geometry in Rome in collaboration with Aissa Wade Table of Contents 1 Lie bialgebroids and Deformations 2 Generalized
LECTURE 1: LINEAR SYMPLECTIC GEOMETRY
LECTURE 1: LINEAR SYMPLECTIC GEOMETRY Contents 1. Linear symplectic structure 3 2. Distinguished subspaces 5 3. Linear complex structure 7 4. The symplectic group 10 *********************************************************************************
On a Type of Para-Kenmotsu Manifold
Pure Mathematical Sciences, Vol. 2, 2013, no. 4, 165-170 HIKARI Ltd, www.m-hikari.com On a Type of Para-Kenmotsu Manifold T. Satyanarayana Department of Mathematics Pragati Engineering College, Surampalem,
A Note About the Torsion of Null Curves in the 3-Dimensional Minkowski Spacetime and the Schwarzian Derivative
Filomat 9:3 05), 553 56 DOI 0.98/FIL503553O Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat A Note About the Torsion of Null Curves
Research Article Some New Fixed-Point Theorems for a (ψ, φ)-pair Meir-Keeler-Type Set-Valued Contraction Map in Complete Metric Spaces
Applied Mathematics Volume 2011, Article ID 327941, 11 pages doi:10.1155/2011/327941 Research Article Some New Fixed-Point Theorems for a (ψ, φ)-pair Meir-Keeler-Type Set-Valued Contraction Map in Complete
A brief introduction to Semi-Riemannian geometry and general relativity. Hans Ringström
A brief introduction to Semi-Riemannian geometry and general relativity Hans Ringström May 5, 2015 2 Contents 1 Scalar product spaces 1 1.1 Scalar products...................................... 1 1.2 Orthonormal
Non-existence of Screen Homothetic Lightlike Hypersurfaces of an Indefinite Kenmotsu Manifold
Applied Mathematical Sciences, Vol. 9, 2015, no. 18, 853-865 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2015.4121001 Non-existence of Screen Homothetic Lightlike Hypersurfaces of an Indefinite
Research Article Modulus of Convexity, the Coeffcient R 1,X, and Normal Structure in Banach Spaces
Abstract and Applied Analysis Volume 2008, Article ID 135873, 5 pages doi:10.1155/2008/135873 Research Article Modulus of Convexity, the Coeffcient R 1,X, and Normal Structure in Banach Spaces Hongwei
THE TANAKA WEBSTER CONNECTION FOR ALMOST S-MANIFOLDS AND CARTAN GEOMETRY
ARCHIVUM MATHEMATICUM (BRNO) Tomus 40 (2004), 47 61 THE TANAKA WEBSTER CONNECTION FOR ALMOST S-MANIFOLDS AND CARTAN GEOMETRY ANTONIO LOTTA AND ANNA MARIA PASTORE Abstract. We prove that a CR-integrable
1-TYPE AND BIHARMONIC FRENET CURVES IN LORENTZIAN 3-SPACE *
Iranian Journal of Science & Technology, Transaction A, ol., No. A Printed in the Islamic Republic of Iran, 009 Shiraz University -TYPE AND BIHARMONIC FRENET CURES IN LORENTZIAN -SPACE * H. KOCAYIGIT **
Hard Lefschetz Theorem for Vaisman manifolds
Hard Lefschetz Theorem for Vaisman manifolds Antonio De Nicola CMUC, University of Coimbra, Portugal joint work with B. Cappelletti-Montano (Univ. Cagliari), J.C. Marrero (Univ. La Laguna) and I. Yudin
Contact pairs (bicontact manifolds)
Contact pairs (bicontact manifolds) Gianluca Bande Università degli Studi di Cagliari XVII Geometrical Seminar, Zlatibor 6 September 2012 G. Bande (Università di Cagliari) Contact pairs (bicontact manifolds)
Research Article Global Existence and Boundedness of Solutions to a Second-Order Nonlinear Differential System
Applied Mathematics Volume 212, Article ID 63783, 12 pages doi:1.1155/212/63783 Research Article Global Existence and Boundedness of Solutions to a Second-Order Nonlinear Differential System Changjian
1 First and second variational formulas for area
1 First and second variational formulas for area In this chapter, we will derive the first and second variational formulas for the area of a submanifold. This will be useful in our later discussion on
A Joint Adventure in Sasakian and Kähler Geometry
A Joint Adventure in Sasakian and Kähler Geometry Charles Boyer and Christina Tønnesen-Friedman Geometry Seminar, University of Bath March, 2015 2 Kähler Geometry Let N be a smooth compact manifold of
CALCULUS ON MANIFOLDS. 1. Riemannian manifolds Recall that for any smooth manifold M, dim M = n, the union T M =
CALCULUS ON MANIFOLDS 1. Riemannian manifolds Recall that for any smooth manifold M, dim M = n, the union T M = a M T am, called the tangent bundle, is itself a smooth manifold, dim T M = 2n. Example 1.
ON KENMOTSU MANIFOLDS
J. Korean Math. Soc. 42 (2005), No. 3, pp. 435 445 ON KENMOTSU MANIFOLDS Jae-Bok Jun, Uday Chand De, and Goutam Pathak Abstract. The purpose of this paper is to study a Kenmotsu manifold which is derived
SOME ALMOST COMPLEX STRUCTURES WITH NORDEN METRIC ON THE TANGENT BUNDLE OF A SPACE FORM
ANALELE ŞTIINŢIFICE ALE UNIVERSITĂŢII AL.I.CUZA IAŞI Tomul XLVI, s.i a, Matematică, 2000, f.1. SOME ALMOST COMPLEX STRUCTURES WITH NORDEN METRIC ON THE TANGENT BUNDLE OF A SPACE FORM BY N. PAPAGHIUC 1
ESSENTIAL KILLING FIELDS OF PARABOLIC GEOMETRIES: PROJECTIVE AND CONFORMAL STRUCTURES. 1. Introduction
ESSENTIAL KILLING FIELDS OF PARABOLIC GEOMETRIES: PROJECTIVE AND CONFORMAL STRUCTURES ANDREAS ČAP AND KARIN MELNICK Abstract. We use the general theory developed in our article [1] in the setting of parabolic
A note on submanifolds of generalized Kähler manifolds
arxiv:1708.01409v1 [math.dg] 4 Aug 2017 A note on submanifolds of generalized Kähler manifolds by Izu Vaisman ABSTRACT. In this note, we consider submanifolds of a generalized Kähler manifold that are
GENERALIZED RAYCHAUDHURI S EQUATION FOR NULL HYPERSURFACES. Fortuné Massamba and Samuel Ssekajja. 1. Introduction
MATEMATIČKI VESNIK MATEMATIQKI VESNIK 70, 1 (2018), 79 88 March 2018 research paper originalni nauqni rad GENERALIZED RAYCHAUDHURI S EQUATION FOR NULL HYPERSURFACES Fortuné Massamba and Samuel Ssekajja
ON OSCULATING, NORMAL AND RECTIFYING BI-NULL CURVES IN R 5 2
Novi Sad J. Math. Vol. 48, No. 1, 2018, 9-20 https://doi.org/10.30755/nsjom.05268 ON OSCULATING, NORMAL AND RECTIFYING BI-NULL CURVES IN R 5 2 Kazım İlarslan 1, Makoto Sakaki 2 and Ali Uçum 34 Abstract.
A GEODESIC EQUATION IN THE SPACE OF SASAKIAN METRICS. Dedicated to Professor S.T. Yau on the occasion of his 60th birthday
A GEODESIC EQUATION IN THE SPACE OF SASAKIAN ETRICS PENGFEI GUAN AND XI ZHANG Dedicated to Professor S.T. Yau on the occasion of his 60th birthday This paper is to draw attention to a geodesic equation
HOLONOMY GROUPOIDS OF SINGULAR FOLIATIONS
j. differential geometry 58 (2001) 467-500 HOLONOMY GROUPOIDS OF SINGULAR FOLIATIONS CLAIRE DEBORD Abstract We give a new construction of Lie groupoids which is particularly well adapted to the generalization
LECTURE 5: COMPLEX AND KÄHLER MANIFOLDS
LECTURE 5: COMPLEX AND KÄHLER MANIFOLDS Contents 1. Almost complex manifolds 1. Complex manifolds 5 3. Kähler manifolds 9 4. Dolbeault cohomology 11 1. Almost complex manifolds Almost complex structures.
REGULAR TRIPLETS IN COMPACT SYMMETRIC SPACES
REGULAR TRIPLETS IN COMPACT SYMMETRIC SPACES MAKIKO SUMI TANAKA 1. Introduction This article is based on the collaboration with Tadashi Nagano. In the first part of this article we briefly review basic
On the exponential map on Riemannian polyhedra by Monica Alice Aprodu. Abstract
Bull. Math. Soc. Sci. Math. Roumanie Tome 60 (108) No. 3, 2017, 233 238 On the exponential map on Riemannian polyhedra by Monica Alice Aprodu Abstract We prove that Riemannian polyhedra admit explicit
Research Article Minor Prime Factorization for n-d Polynomial Matrices over Arbitrary Coefficient Field
Complexity, Article ID 6235649, 9 pages https://doi.org/10.1155/2018/6235649 Research Article Minor Prime Factorization for n-d Polynomial Matrices over Arbitrary Coefficient Field Jinwang Liu, Dongmei
Research Article Constrained Solutions of a System of Matrix Equations
Journal of Applied Mathematics Volume 2012, Article ID 471573, 19 pages doi:10.1155/2012/471573 Research Article Constrained Solutions of a System of Matrix Equations Qing-Wen Wang 1 and Juan Yu 1, 2 1
Warped product submanifolds of Kaehler manifolds with a slant factor
ANNALES POLONICI MATHEMATICI 95.3 (2009) Warped product submanifolds of Kaehler manifolds with a slant factor by Bayram Sahin (Malatya) Abstract. Recently, we showed that there exist no warped product
Acta Mathematica Academiae Paedagogicae Nyíregyháziensis 21 (2005), ISSN
Acta Mathematica Academiae Paedagogicae Nyíregyháziensis 21 (2005), 79 7 www.emis.de/journals ISSN 176-0091 WARPED PRODUCT SUBMANIFOLDS IN GENERALIZED COMPLEX SPACE FORMS ADELA MIHAI Abstract. B.Y. Chen
CR-SUBMANIFOLDS OF A NEARLY TRANS-SASAKIAN MANIFOLD
IJMMS 31:3 2002) 167 175 PII. S0161171202109288 http://ijmms.hindawi.com Hindawi Publishing Corp. CR-SUBMANIFOLDS OF A NEARLY TRANS-SASAKIAN MANIFOLD FALLEH R. AL-SOLAMY Received 26 September 2001 This
A local characterization for constant curvature metrics in 2-dimensional Lorentz manifolds
A local characterization for constant curvature metrics in -dimensional Lorentz manifolds Ivo Terek Couto Alexandre Lymberopoulos August 9, 8 arxiv:65.7573v [math.dg] 4 May 6 Abstract In this paper we
Real Hypersurfaces in Complex Two-Plane Grassmannians with Vanishing Lie Derivative
Canad. Math. Bull. Vol. 49 (1), 2006 pp. 134 143 Real Hypersurfaces in Complex Two-Plane Grassmannians with Vanishing Lie Derivative Young Jin Suh Abstract. In this paper we give a characterization of
Proper curvature collineations in Bianchi type-ii space-times( )
IL NUOVO CIMENTO Vol. 121 B, N. 3 Marzo 2006 DOI 10.1393/ncb/i2006-10044-7 Proper curvature collineations in Bianchi type-ii space-times( G. Shabbir( Faculty of Engineering Sciences, GIK Institute of Engineering
The parallelism of shape operator related to the generalized Tanaka-Webster connection on real hypersurfaces in complex two-plane Grassmannians
Proceedings of The Fifteenth International Workshop on Diff. Geom. 15(2011) 183-196 The parallelism of shape operator related to the generalized Tanaka-Webster connection on real hypersurfaces in complex
CR SINGULAR IMAGES OF GENERIC SUBMANIFOLDS UNDER HOLOMORPHIC MAPS
CR SINGULAR IMAGES OF GENERIC SUBMANIFOLDS UNDER HOLOMORPHIC MAPS JIŘÍ LEBL, ANDRÉ MINOR, RAVI SHROFF, DUONG SON, AND YUAN ZHANG Abstract. The purpose of this paper is to organize some results on the local
SEMI-RIEMANNIAN SUBMANIFOLDS OF A SEMI-RIEMANNIAN MANIFOLD WITH A SEMI-SYMMETRIC NON-METRIC CONNECTION
Commun. Korean Math. Soc. 27 (2012), No. 4, pp. 781 793 http://dx.doi.org/10.4134/ckms.2012.27.4.781 SEMI-RIEMANNIAN SUBMANIFOLDS OF A SEMI-RIEMANNIAN MANIFOLD WITH A SEMI-SYMMETRIC NON-METRIC CONNECTION
LINEAR CONNECTIONS ON NORMAL ALMOST CONTACT MANIFOLDS WITH NORDEN METRIC 1
LINEAR CONNECTIONS ON NORMAL ALMOST CONTACT MANIFOLDS WITH NORDEN METRIC 1 Marta Teofilova Abstract. Families of linear connections are constructed on almost contact manifolds with Norden metric. An analogous
Complete spacelike hypersurfaces with positive r-th mean curvature in a semi-riemannian warped product
DOI: 10.1515/auom-2015-0041 An. Şt. Univ. Ovidius Constanţa Vol. 23(2),2015, 259 277 Complete spacelike hypersurfaces with positive r-th mean curvature in a semi-riemannian warped product Yaning Wang and
Covariant Derivatives on Null Submanifolds
Covariant Derivatives on Null Submanifolds arxiv:1108.2332v1 [gr-qc] 11 Aug 2011 Don Hickethier Department of Mathematics, Flathead Valley Community College, Kalispell, MT 59901 dhicketh@fvcc.edu Tevian
ISOMETRIC IMMERSIONS IN CODIMENSION TWO OF WARPED PRODUCTS INTO SPACE FORMS
Illinois Journal of Mathematics Volume 48, Number 3, Fall 2004, Pages 711 746 S 0019-2082 ISOMETRIC IMMERSIONS IN CODIMENSION TWO OF WARPED PRODUCTS INTO SPACE FORMS MARCOS DAJCZER AND RUY TOJEIRO Abstract.
30 Surfaces and nondegenerate symmetric bilinear forms
80 CHAPTER 3. COHOMOLOGY AND DUALITY This calculation is useful! Corollary 29.4. Let p, q > 0. Any map S p+q S p S q induces the zero map in H p+q ( ). Proof. Let f : S p+q S p S q be such a map. It induces
Homogeneous para-kähler Einstein manifolds. Dmitri V. Alekseevsky
Homogeneous para-kähler Einstein manifolds Dmitri V. Alekseevsky Hamburg,14-18 July 2008 1 The talk is based on a joint work with C.Medori and A.Tomassini (Parma) See ArXiv 0806.2272, where also a survey
Research Article A Two-Grid Method for Finite Element Solutions of Nonlinear Parabolic Equations
Abstract and Applied Analysis Volume 212, Article ID 391918, 11 pages doi:1.1155/212/391918 Research Article A Two-Grid Method for Finite Element Solutions of Nonlinear Parabolic Equations Chuanjun Chen
Research Article On Prime Near-Rings with Generalized Derivation
International Mathematics and Mathematical Sciences Volume 2008, Article ID 490316, 5 pages doi:10.1155/2008/490316 Research Article On Prime Near-Rings with Generalized Derivation Howard E. Bell Department
CHARACTERIZATION OF TOTALLY GEODESIC SUBMANIFOLDS IN TERMS OF FRENET CURVES HIROMASA TANABE. Received October 4, 2005; revised October 26, 2005
Scientiae Mathematicae Japonicae Online, e-2005, 557 562 557 CHARACTERIZATION OF TOTALLY GEODESIC SUBMANIFOLDS IN TERMS OF FRENET CURVES HIROMASA TANABE Received October 4, 2005; revised October 26, 2005
Reduction of Homogeneous Riemannian structures
Geometric Structures in Mathematical Physics, 2011 Reduction of Homogeneous Riemannian structures M. Castrillón López 1 Ignacio Luján 2 1 ICMAT (CSIC-UAM-UC3M-UCM) Universidad Complutense de Madrid 2 Universidad
GEOMETRY OF WARPED PRODUCTS AS RIEMANNIAN SUBMANIFOLDS AND RELATED PROBLEMS
SOOCHOW JOURNAL OF MATHEMATICS Volume 28, No. 2, pp. 125-156, April 2002 GEOMETRY OF WARPED PRODUCTS AS RIEMANNIAN SUBMANIFOLDS AND RELATED PROBLEMS BY BANG-YEN CHEN Abstract. The warped product N 1 f
THE GAUSS MAP OF TIMELIKE SURFACES IN R n Introduction
Chin. Ann. of Math. 16B: 3(1995),361-370. THE GAUSS MAP OF TIMELIKE SURFACES IN R n 1 Hong Jianqiao* Abstract Gauss maps of oriented timelike 2-surfaces in R1 n are characterized, and it is shown that
3.2 Frobenius Theorem
62 CHAPTER 3. POINCARÉ, INTEGRABILITY, DEGREE 3.2 Frobenius Theorem 3.2.1 Distributions Definition 3.2.1 Let M be a n-dimensional manifold. A k-dimensional distribution (or a tangent subbundle) Δ : M Δ