Screen transversal conformal half-lightlike submanifolds

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1 Annals of the Unversty of Craova, Mathematcs and Computer Scence Seres Volume 40(2), 2013, Pages ISSN: Screen transversal conformal half-lghtlke submanfolds Wenje Wang, Yanng Wang, and Xmn Lu Abstract. In ths paper, we study half-lghtlke submanfolds of a sem-remannan manfold such that the shape operator of screen dstrbuton s conformal to the shape operator of screen transversal dstrbuton. We manly obtan some results concernng the nduced Rcc curvature tensor and the null sectonal curvature of screen transversal conformal half-lghtlke submanfolds Mathematcs Subject Classfcaton. Prmary 53C50; Secondary 53C40, 53C42. Key words and phrases. Half-lghtlke submanfold, screen transversal conformal, sem-remannan manfold, nduced Rcc tensor. 1. Introducton Snce the ntersecton of the normal bundle and the tangent bundle of a submanfold of a sem-remannan manfold may be not trval, t s more dffcult and nterestng to study the geometry of lghtlke submanfolds than non-degenerate submanfolds. Two standard methods to deal wth the above dffcultes were developed by Kupel [11] and Duggal-Bejancu [4] (see also Duggal-Jn [6] and Duggal-Sahn [8]), respectvely. It s obvous that there are two cases of codmenson 2 lghtlke submanfolds M snce for ths type the dmenson of ther radcal dstrbuton s ether 1 or 2, a codmenson 2 lghtlke submanfold s called a half-lghtlke submanfold (see [3]) f dm(rad(t M)) = 1. For more results about half-lghtlke submanfolds, we refer the readers to [5] and [9]. There exsts a shape operator of the screen dstrbuton n geometry of lghtlke submanfolds whch s very dfferent from the non-degenerate submanfold case, therefore, the conformalty of the operator of screen dstrbuton and the operator of lghtlke transversal bundle of a lghtlke hypersurface of sem-remannan manfolds was ntroduced by Atndogbe-Duggal [1]. Recently, lots of works (see [2] and [7]) have been done to nvestgate lghtlke submanfolds wth screen conformal condton. Moreover, we also refer the reader to [13] by the present authors for some related results on half-lghtlke submanfolds satsfyng the co-screen conformal condton. In ths paper, we manly dscuss the propertes of half-lghtlke submanfolds of a sem-remannan manfold such that the shape operator of screen dstrbuton and the shape operator of screen transversal dstrbuton are conformal. Under such a geometrc condton we prove that the Rcc tensor of a half-lghtlke submanfold nduced from ambent space s symmetrc. Moreover, we obtan some suffcent geometrc condtons to characterze screen transversal conformal half-lghtlke submanfolds. Receved May 28, Revsed September 17, Ths work was supported by Natural Scence Foundaton of Chna (No ). 140

2 SCREEN TRANSVERSAL CONFORMAL HALF-LIGHTLIKE SUBMANIFOLDS Prelmnares In ths secton, accordng to [5] and [10] developed by Duggal-Jn, we collect some notatons and fundamental equatons of half-lghtlke submanfolds of sem- Remannan manfolds. A submanfold (M, g) of dmenson m mmersed n a sem-remannan manfold (M, g) of dmenson (m + n) s called a lghtlke submanfold f the metrc g nduced from ambent space s degenerate and radcal dstrbuton Rad(T M) s of rank r, where m 2 and 1 r mn {m, n}. In partcular, (M, g) s called a half-lghtlke submanfold f n = 2 and r = 1. It s well known that the radcal dstrbuton s gven by Rad(T M) = T M T M, where T M s called the normal bundle of M n M. Thus there exst two non-degenerate complementary dstrbutons S(T M) and S(T M ) of Rad(T M) n T M and T M respectvely, whch are called the screen and the screen transversal dstrbuton on M respectvely. Thus we have T M = Rad(T M) orth S(T M), (1) and T M = Rad(T M) orth S(T M ), (2) where orth denotes the orthogonal drect sum. Consder the orthogonal complementary dstrbuton S(T M) to S(T M) n T M, t s easy to see that T M s a subbundle of S(T M). As S(T M ) s a non-degenerate subbundle of S(T M), the orthogonal complementary dstrbuton S(T M ) to S(T M ) n S(T M) s also a non-degenerate dstrbuton. Clearly, Rad(T M) s a subbundle of S(T M ). Choose L Γ(S(T M )) as a unt vector feld wth g(l, L) = ϵ = ±1. For any null secton ξ Rad(T M), there exsts a unquely defned null vector feld N Γ(S(T M ) ) satsfyng g(ξ, N) = 1, g(n, N) = g(n, X) = g(n, L) = 0, X Γ(S(T M)). Denote by ltr(t M) the vector subbundle of S(T M ) locally spanned by N. Then we show that S(T M ) = Rad(T M) ltr(t M). Let tr(t M) = S(T M ) orth ltr(t M). We call N, ltr(t M) and tr(t M) the lghtlke transversal vector feld, lghtlke transversal vector bundle and transversal vector bundle of M wth respect to the chosen screen dstrbuton S(T M) respectvely. Then T M s decomposed as follows: T M = T M tr(t M) = Rad(T M) tr(t M) orth S(T M) = Rad(T M) ltr(t M) orth S(T M) orth S(T M ). Let P be the projecton morphsm of T M on S(T M) wth respect to the decomposton (1). For any X, Y Γ(T M), N Γ(ltr(T M)), ξ Γ(Rad(T M)) and L Γ(S(T M) ), the Gauss and Wengarten formulas of M and S(T M) are gven by X Y = X Y + D 1 (X, Y )N + D 2 (X, Y )L, (4) X N = A N X + ρ 1 (X)N + ρ 2 (X)L, (5) X L = A L X + ε 1 (X)N + ε 2 (X)L, (6) X P Y = XP Y + E(X, P Y )ξ, (7) X ξ = A ξx + u 1 (X)ξ, (8) respectvely, where and are nduced connecton on T M and S(T M) respectvely, D 1 and D 2 are called locally second fundamental forms of M, E s called the locally second fundamental form on S(T M). A N, A ξ and A L are lnear operators on T M and ρ 1, ρ 2, ε 1, ε 2 and u 1 are 1-forms on T M. Note the connecton s torson free (3)

3 142 W. J. WANG, Y. N. WANG, AND X. M. LIU but s not metrc tensor, the connecton s metrc. D 1 and D 2 are both symmetrc tensors on Γ(T M). It s easy to check ε 2 = 0 and the followng: D 1 (X, ξ) = 0, D 1 (X, P Y ) = g(a ξx, P Y ), g(a ξx, N) = 0, (9) ϵd 2 (X, Y ) = g(a L X, Y ) ε 1 (X)η(Y ), g(a L X, N) = ϵρ 2 (X), (10) E(X, P Y ) = g(a N X, P Y ), g(a N X, N) = 0, u 1 (X) = ρ 1 (X) (11) for any X Γ(T M) and N Γ(ltr(T M)), where η(x) = g(x, N). From the above equatons we see that A ξ and A N are Γ(T M)-valued shape operators related to D 1 and D 2 respectvely, and A ξ s self-adjont on T M and satsfes A ξξ = 0. (12) Denote by R and R the curvature tensor of sem-remannan connecton of M and the nduced connecton on M respectvely, we obtan the followng Gauss- Codazz equatons for M and S(T M). g(r(x, Y )Z, P W ) = g(r(x, Y )Z, P W ) + D 1 (X, Z)E(Y, P W ) D 1 (Y, Z)E(X, P W ) + ϵd 2 (X, Z)D 2 (Y, P W ) ϵd 2 (Y, Z)D 2 (X, P W ), (13) g(r(x, Y )Z, N) = g(r(x, Y )Z, N) + ϵρ 2 (Y )D 2 (X, Z) ϵρ 2 (X)D 2 (Y, Z), (14) g(r(x, Y )Z, ξ) = g(r(x, Y )Z, ξ) + ε 1 (X)D 2 (Y, P Z) ε 1 (Y )D 2 (X, P Z), (15) for any X, Y, Z, W Γ(T M), N Γ(ltr(T M)) and ξ Γ(Rad(T M)). The Rcc curvature tensor of M denoted by Rc s defned by Rc(X, Y ) = trace{z R(X, Z)Y }, (16) where R(X, Y, Z, W ) = g(r(x, Y )Z, W ). Locally, Rc(X, Y ) s gven by Rc(X, Y ) = ϵ g(r(x, E )Y, E ), (17) where {E 1, E 2,, E m+2 } s an orthogonal frame feld of T M and g(e, E ) = ϵ s the sgn of E for 1 m + 2. In partcular, f Rc(X, Y ) = kg(x, Y ) for any X, Y Γ(T M), then M s called an Ensten manfold, where k s a constant on M. 3. Screen transversal conformal half-lghtlke submanfolds In ths secton, we consder a class of half-lghtlke submanfolds wth screen transversal conformal geometrc condton defned as follows. Defnton 3.1. Let M be a half-lghtlke submanfold of a sem-remannan manfold, M s called screen transversal locally (resp. globally) conformal f on any coordnate neghborhood U (resp. U = M) there exsts a non-zero smooth functon θ such that for any null vector feld ξ Γ(Rad(T M)) the relaton A L X = θa ξx, X Γ(T M) (18) holds, where L s a unt vector feld of screen transversal bundle of M.

4 SCREEN TRANSVERSAL CONFORMAL HALF-LIGHTLIKE SUBMANIFOLDS 143 In the sequel, by a screen transversal conformal we shall mean globally screen transversal conformal unless otherwse specfed. From (12) we know the shape operator A ξ s S(T M)-valued, thus for a screen transversal conformal half-lghtlke submanfold we obtan ρ 2 = 0 followng from (10). Also, we have the followng condtons to characterze screen transversal conformal half-lghtlke submanfolds. Theorem 3.1. Let M be a half-lghtlke submanfold of a sem-remannan manfold, then M s screen transversal conformal f and only f D 2 (X, P Y ) = ϵθd 1 (X, P Y ), ρ 2 (X) = 0, X, Y Γ(T M), where θ s a non-zero smooth functon on M. Proof. If M s a screen transversal conformal half-lghtlke submanfold, then t follows from (9) and (10) that ϵd 2 (X, P Y ) = g(a L X, P Y ) = θg(a ξx, P Y ) = θd 1 (X, P Y ). Conversely, we know from (10) that f ρ 2 (X) = ϵg(a L X, N) = 0, then the shape operator A L s S(T M)-valued. As S(T M) s a non-degenerate dstrbuton, then D 2 (X, P Y ) = ϵθd 1 (X, P Y ) mples that A L = ϵθa ξ. Ths completes the proof. Defnton 3.2 (see [8]). A lghtlke submanfold M n sem-remannan manfold (M, g) s sad to be rrotatonal f X ξ Γ(T M) for any X Γ(T M), where ξ Γ(Rad(T M)). For a half-lghtlke submanfold M, t follows from (9) that the above defnton s equvalent to D 2 (X, ξ) = ϵε 1 (X) = 0 for any X Γ(T M). If M s a screen transversal conformal half-lghtlke submanfold, then we have D 2 (ξ, P Y ) = 0 for any X Γ(T M) and ξ Γ(Rad(T M)). From (9) and Theorem 3.1, we have the followng proposton on screen transversal conformal half-lghtlke submanfolds. Proposton 3.2. Let M be a screen transversal conformal half-lghtlke submanfold, then M s sad to be rrotatonal provded ε 1 (ξ) = 0. Theorem 3.3. Let M be a screen transversal conformal half-lghtlke submanfold of a sem-remannan space form (M(c), g), then where ξ Γ(Rad(T M)). ρ 1 (ξ) + ϵθε 1 (ξ) = 0, Proof. It follows from (4) (6) and (9) (11) that g(r(x, Y )P Z, ξ) = ( X D 1 )(Y, P Z) ( Y D 1 )(X, P Z) + ρ 1 (X)D 1 (Y, P Z) ρ 1 (Y )D 1 (X, P Z) + ε 1 (X)D 2 (Y, P Z) ε 1 (Y )D 2 (X, P Z), (19) and g(r(x, Y )P Z, u) = ϵ ( ( X D 2 )(Y, P Z) ( Y D 2 )(X, P Z) ) + ρ 2 (X)D 1 (Y, P Z) ρ 2 (Y )D 1 (X, P Z) respectvely. By dfferentatng D 2 (Y, P Z) = ϵθd 1 (Y, P Z), we have (20) ( X D 2 )(Y, P Z) = ϵx(θ)d 1 (Y, P Z) + ϵθ( X D 1 )(Y, P Z). (21) From Theorem 3.1 t turns out that f M s screen transversal conformal then ρ 2 = 0, thus, equaton (20) becomes ( X D 2 )(Y, P Z) ( Y D 2 )(X, P Z) = 0. (22)

5 144 W. J. WANG, Y. N. WANG, AND X. M. LIU Substtutng (22) nto (21) and usng D 2 (Y, P Z) = ϵθd 1 (Y, P Z) and (19), we have D 1 (X, P Z) ( ρ 1 (Y ) + ϵθε 1 (Y ) ) D 1 (Y, P Z) ( ρ 1 (X) + ϵθε 1 (X) ) = 0. (23) Replacng Y by ξ n the above equaton we have D 1 (X, P Z) ( ρ 1 (ξ) + ϵθε 1 (ξ) ) = 0 (24) for any X, Z Γ(T M). Ths completes the proof. Corollary 3.4. Let M be an rrotatonal screen transversal conformal half-lghtlke submanfold of a sem-remannan space form (M(c), g), then where ξ Γ(Rad(T M)). ρ 1 (ξ) = 0, Let M be an m-dmensonal screen transversal conformal half-lghtlke submanfold of sem-remannan space form M(c) and S(T M) = span{e 1, e 2,, e m }, where {e } s an orthogonal frame felds of Γ(S(T M)) and g(e, e ) = ϵ. Then from the Gauss- Codazz equatons we have g(r(x, e )Y, e ) = g(r(x, e )Y, e ) + D 1 (X, Y ) ( E(e, e ) + ϵθ 2 D 1 (e, e ) ) and D 1 (e, Y ) ( E(X, e ) + ϵθ 2 D 1 (X, e ) ), (25) g(r(x, ξ)y, N) = g(r(x, ξ)y, N). (26) Makng use of the above equatons and the defnton of Rcc curvature tensor, we have Rc(X, Y ) = ϵ g(r(x, e )Y, e ) + g(r(x, ξ)y, N) = ϵ g(r(x, e )Y, e ) + g(r(x, ξ)y, N) D 1 (X, Y ) E(e, e ) D 1 (X, Y ) ϵθ 2 D 1 (e, e ) + D 1 (e, Y ) ( E(X, e ) + ϵθ 2 D 1 (X, e ) ) (27) = (1 m)cg(x, Y ) D 1 (X, Y ) ( E(e, e ) + ϵθ 2 D 1 (e, e ) ) + ϵθ 2 D 1 (e, X)D 1 (e, Y ) + D 1 (Y, e )E(X, e ). If there exst an orthogonal frame felds {e } of S(T M) such that the assocated matrces of shape operators A N and A ξ, that s, ( E(e, e j ) ) ( and (m 1) (m 1) D1 (e, e j ) ), can be dagonalzed, then we have (m 1) (m 1) D 1 (P Y, e )E(P X, e ) = D 1 (P X, e )E(P Y, e ), X, Y Γ(T M). It follows from (11) that E(ξ, P X) = 0 s equvalent to A N ξ = 0, therefore, we obtan the followng theorem. Theorem 3.5. Let M be an m-dmensonal screen transversal conformal half-lghtlke submanfold of sem-remannan space form M(c) wth A N ξ = 0, f the shape operator of screen bundle and the shape operator of lghtlke transversal bundle are symmetrc, then the nduced Rcc curvature tensor s symmetrc. In partcular, we have the followng corollary to characterze Ensten half-lghtlke surface.

6 SCREEN TRANSVERSAL CONFORMAL HALF-LIGHTLIKE SUBMANIFOLDS 145 Corollary 3.6. Let M be a screen transversal conformal half-lghtlke surface of 4- dmensonal sem-remannan space form M(c) wth A N ξ = 0, then the nduced Rcc curvature tensor s symmetrc. Moreover, the surface M s an Ensten surface. Proof. We obtan from (27) that Rc(X, Y ) = cg(x, Y ) D 1 (X, Y ) ( E(e, e) + ϵθ 2 D 1 (e, e) ) + D 1 (e, Y ) ( E(X, e) + ϵθ 2 D 1 (X, e) ) = cg(x, Y ), where e s a unt vector feld of S(T M). Ths proves the corollary. (28) Recall the followng noton of null sectonal curvature shown n [4]. Let x M and ξ be a null vector of T x M. A plane H of T x M s called a null plane drected by ξ f t contans ξ, g x (ξ, W ) = 0 for any W H and there exsts W o H such that g x (W o, W o ) 0. Thus the null secton curvature of H wth respectve ξ and the nduced connecton of M, s defned as a real number K ξ (H) = g x(r(w, ξ)ξ, W ), g x (W, W ) where W 0 s any vector n H lnearly ndependent wth respect to ξ. From [12] we see that an n(n 3)-dmensonal Lorentzan manfold s of constant curvature f and only f ts null sectonal curvatures are everywhere zero. Theorem 3.7. Let M be a screen transversal conformal half-lghtlke submanfold of sem-remannan space form M(c), then the null sectonal curvature of M s gven by K ξ (H) = ϵθε 1 (ξ)d 1 (W, W ). Proof. From (13) we have K ξ (H) = g(r(w, ξ)ξ, W ) + D 1 (ξ, ξ)e(w, P W ) D 1 (W, ξ)e(ξ, P W ) + ϵd 2 (ξ, ξ)d 2 (W, P W ) ϵd 2 (W, ξ)d 2 (ξ, P W ). By (9) and Theorem 3.1, we obtan K ξ (H) = ϵθε 1 (ξ)d 1 (W, W ). Ths completes the proof. If M s a screen transversal conformal half-lghtlke submanfold, t follows from Theorem 3.1 that D 2 (ξ, P X) = 0. Makng use of Defnton 3.2, we have the followng corollary. Corollary 3.8. Let M be a screen transversal conformal half-lghtlke submanfold of sem-remannan space form M(c), then the null sectonal curvature of M vanshes f and only f D 1 (W, W ) = 0 or M s rrotatonal. Defnton 3.3. A half-lghtlke submanfold M of sem-remannan manfold s sad to be totally D 1 (resp. D 2 )-umblcal f on each coordnate neghborhood U there exsts a smooth functon H 1 (resp. H 2 ) such that D 1 (X, Y ) = H 1 g(x, Y ) (resp. D 2 (X, Y ) = H 2 g(x, Y )) for any X, Y Γ(T M). In partcular, f H 1 = 0 (resp. H 2 = 0), then M s sad to be D 1 (resp. D 2 )-geodesc. Note that M s totally umblcal (resp. geodesc) (see [5]) f and only f M are both D 1 and D 2 umblcal (reps. geodesc). Together wth Theorem 3.1 and the above defnton, we have the followng theorem. (29)

7 146 W. J. WANG, Y. N. WANG, AND X. M. LIU Theorem 3.9. Let M be a nowhere geodesc half-lghtlke submanfold of sem- Remannan manfold M, then the the followng assertons are equvalent: (1) M s totally umblcal half-lghtlke submanfold of M wth ρ 2 (X) = 0, X Γ(T M); (2) M s screen transversal conformal and totally D 1 -umblcal half-lghtlke submanfold of M wth ε 1 (X) = 0, X Γ(T M); (3) M s screen transversal conformal and totally D 2 -umblcal half-lghtlke submanfold of M wth ε 1 (X) = 0, X Γ(T M). Proof. (1) (2): f M s a totally umblcal half-lghtlke submanfold of M, by Defnton 3.3 we have D 2 (X, Y ) = θd 1 (X, Y ), where θ = H2 H 1. Then t follows from Theorem 3.1 that M s screen transversal conformal. Also, t follows from (9) that ε 1 (X) = 0. Conversely, f M s screen transversal conformal then we obtan D 2 (X, P Y ) = ϵθd 1 (X, P Y ). Notcng that M s totally D 1 -umblcal f and only f D 1 (X, Y ) = H 1 g(x, Y ), thus, by Theorem 3.1, we prove the equvalence of (1) and (2). The proof of equvalence between (1) and (3) s the same wth that of (1) and (2). Theorem Suppose that M s a screen transversal conformal half-lghtlke submanfold of sem-remannan manfold, then M s totally umblcal f and only f A ξ X = H 1P X and ε 1 (X) = 0 for any X Γ(T M). Proof. If M s a totally umblcal half-lghtlke submanfold, t follows from (9) that D 1 (X, P Y ) = H 1 g(x, P Y ) = g(a ξx, P Y ). Snce the screen dstrbuton s nondegenerate, we have A ξ X = H 1P X. Moreover, t follows from (10) that D 2 (ξ, X) = H 2 g(ξ, P X) = ϵε 1 (X) = 0. Conversely, f A ξ X = H 1P X, t follows from (9) that D 1 (X, P Y ) = g(a ξ X, P Y ) = H 1 g(x, P Y ). In vew of D 2 (X, P Y ) = ϵθd 1 (X, P Y ) = ϵθh 1 g(x, P Y ) and ϵ 1 (X) = 0, we see that M s umblcal. Ths completes the proof. Acknowledgement The authors would lke to thank the referees for the valuable suggestons and comments n the mprovement of ths paper. References [1] C. Atndogbe and K. L. Duggal, Conformal screen on lghtlke hypersurfaces, Int. J. Pure Appl. Math. 11 (2004), no. 4, [2] K. L. Duggal, On canoncal screen for lghtlke submanfolds of codmensonal two, Cent. Eur. J. Math. 5 (2007), no. 4, [3] K. L. Duggal and A. Bejancu, Lghtlke submanfolds of codmensonal two, Math. J. Toyama Unv. 15 (1992), [4] K. L. Duggal and A. Bejancu, Lghtlke submanfolds of sem-remannan manfolds and applcatons, Kluwer Academc Publshers, Dordrecht, [5] K. L. Duggal and D. H. Jn, Half lghtlke submanfolds of codmensonal 2, Math. J. Toyama Unv. 22 (1999), [6] K. L. Duggal and D. H. Jn, Null curves and hypersurfaces of sem-remannan manfolds, World Scentfc, Sngapore, [7] K. L. Duggal and B. Sahn, Screen conformal half-lghtlke submanfolds, Int. J. Math. Math. Sc. 68 (2004), [8] K. L. Duggal and B. Sahn, Dfferental geometry of lghtlke submanfolds, Brkhauser Veralag AG, Basel, Boston, Berln, 2010.

8 SCREEN TRANSVERSAL CONFORMAL HALF-LIGHTLIKE SUBMANIFOLDS 147 [9] D. H. Jn, Ensten half lghtlke submanfold wth a Kllng co-screen dstrbuton, Honam Math. J. 30 (2008), no. 3, [10] D. H. Jn, Geometry of half lghtlke submanfolds of a sem-remannan space form wth a sem-symmetrc metrc connecton, J. Chungcheong Math. Soc. 24 (2011), [11] D. N. Kupel, Sngular sem-remnnan Geometry, Mathematcs and Its Applcatons, Kluwer Academc Publshers, Dordrecht, [12] B. O Nell, Sem-Remannan Geometry wth Applcaton to Relatvty, Academc Press, New York, [13] Y. N. Wang and X. M. Lu, Co-screen conformal half lghtlke submanfolds, Tamkang J. Math. 44 (2013), no. 4, [14] W. J. Wang and X. M. Lu, On parallelsm of ascreen half lghtlke submanfolds of ndefnte cosymplectc manfolds, Int. Electron. J. Geom. 6 (2013), no. 2, (Wenje Wang, Yanng Wang) Department of Mathematcs, South Chna Unversty of Technology, Guangzhou , Guangdong, P. R. Chna E-mal address: wangwj072@163.com, wyn051@163.com (Xmn Lu) School of Mathematcal Scences, Dalan Unversty of Technology, Dalan , Laonng, P. R. Chna E-mal address: xmnlu@dlut.edu.cn