Totally Umbilical Screen Transversal Lightlike Submanifolds of Semi-Riemannian Product Manifolds

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1 Advance in Pure Mahemaic, 01,, hp://dx.doi.org/10.436/apm Pubihed Onine Juy 01 (hp:// Toay Umbiica Screen Tranvera Lighike Submanifod of Semi-Riemannian Produc Manifod S. M. Khurheed Haider, Advin, Mama Thakur Deparmen of Mahemaic, Jamia Miia Iamia, New Dehi, India Emai: {advin.maeih, Received February 17, 01; revied March 5, 01; acceped March 1, 01 ABSTRACT We udy oay umbiica creen ranvera ighike ubmanifod immered in a emi-riemannian produc manifod obain neceary ufficien condiion for induced connecion on a oay umbiica radica creen ranvera ighike ubmanifod o be meric connecion. We prove a heorem which caifie oay umbiica ST-ani-invarian ighike ubmanifod immered in a emi-riemannian produc manifod. Keyword: Semi-Riemannian Produc Manifod; Lighike Submanifod; Toay Umbiica Radica ST-Lighike Submanifod; Toay Umbiica ST-Ani-Invarian Lighike Submanifod 1. Inroducion I i we known ha he geomery of ighike ubmanifod of emi-riemannian manifod i differen from he geomery of ubmanifod immered in a Riemannian manifod ince he norma vecor bunde of ighike ubmanifod inerec wih angen bunde making i more inereing o udy. The genera heory of ighike ubmanifod of a emi-riemannian manifod ha been deveoped by Dugga-Bejancu [1] Kupei []. Toay umbiica CR-ubmanifod of a Kaeher manifod wih Riemannian meric were udied by Bejancu [3], Dehmukh Huain [4] many more wherea, oay umbiica ighike ubmanifod of emi-riemannian manifod of conan curvaure wa inveigaed by Dugga-Jin [5] oay umbiica CR-ighike ubmanifod of an indefinie Kaeher manifod were udied by Dugga-Bejancu [1] Gogna e a. [6]. In [7], B. Sahin iniiaed he udy of ranvera ighike ubmanifod of an indefinie Kaeher manifod inveigaed he exience of uch ighike ubmanifod in an indefinie pace form. Thee ubmanifod in Saakian eing were udied by Yidirim Sahin [8]. A a generaizaion of rea nu curve of indefinie Kaeher manifod, B. Sahin [9] inroduced he noion of creen ranvera ighike ubmanifod obained many inereing reu. In hi paper, we udy oay umbiica creen ranvera ighike ubmanifod of emi- Riemannian produc manifod. Thi paper i arranged a foow. In Secion 3, we give he baic concep on ighike ubmanifod emi-riemannian produc manifod needed for hi paper. In Secion 4, we udy he inegrabiiy of diribuion invoved in he definiion of oay umbiica radica creen ranvera ighike ubmanifod obain neceary ufficien condiion for induced connecion on oay umbiica radica creen ranvera ighike ubmanifod o be meric connecion. In Secion 5, we prove a heorem which how ha he induced connecion on a oay umbiica ST-ani-invarian ighike ubmanifod i a meric connecion under ome condiion. We ao prove a heorem which caifie oay umbiica ST-ani-invarian ighike ubmanifod immered in a emi-riemannian produc manifod.. Preiminarie We foow [1] for he noaion fundamena equaion for ighike ubmanifod ued in hi paper. A ubmanifod M m immered in a emi-riemannian manifod m M n, g i caed a ighike ubmanifod if i i a ighike manifod wih repec o he meric g induced from g radica diribuion RadTM i of rank r, be a creen diribuion which i a emi-riemannian compemenary diribuion of RadTM in TM, i.e., where 1 r m. Le STM TM RadTM S TM Conider a creen ranvera vecor bunde S TM, which i a emi-riemannian compemenary vecor bun- Copyrigh 01 SciRe.

2 86 S. M. K. HAIDER ET AL. de of RadTM in TM. Since for any oca bai of RadTM, here exi a oca nu frame Ni of ecion wih vaue in he orhogona compemen of S TM in STM uch ha g i, N j ij, i foow ha here exi a ighike ranvera vecor bunde r(tm) ocay panned by N i [[1]; pg-144]. Le r(tm) be compemenary (bu no orhogona) vecor bunde o TM in TM. Then M M r TM r TM S TM, RadT r TM TM S TM M S TM. Foowing are four ubcae of a ighike ubmanifod M, STM M, g, S T. Cae 1: r-ighike if r < min{m, n}. Cae : Co-ioropic if r = n < m; STM = 0. Cae 3: Ioropic if r = m < n; STM = 0. Cae 4: Toay ighike if r = n = m; STM = 0 = S TM. The Gau Weingaren formuae are, Y, Y TM Y Y h (.1) U A U U TM, U r TM where Y, AU h, Y, U TM r TM, i on STM. Le,,, (.) beong o, repecivey, are inear connecion on M on he vecor bunde r TM, repecivey. Moreover, we have Y Y h, Yh,Y (.3) N N AN D, N (.4) W A D, W (.5) W W, Y TM, N r TM TM Denoe he projecion of TM on STM W S. by P. Then, by uing (.1), (.3)-(.5) he fac ha i a meric connecion, we obain W,,, W. g h, Y, W g Y, D, W g A, Y, g D N W g N A PY PY h, PY, A, (.6) From he decompoiion of he angen bunde of a ighike ubmanifod, we have TM RadTM (.7) for, Y. In genera, he induced connecion on M i no a meric connecion wherea i a meric connecion M g S TM S TM be a ighike ubmanifod of M, g. For any vecor fied angen o M, we pu F f (.8) where f are he angenia ranvera par of F repecivey. For V rtm FV BV CV (.9) where BV CV are he angenia ranvera par of FV repecivey. 3. Semi-Riemannian Produc Manifod Le (M 1, g 1 ) (M, g ) be wo m 1 m -dimeniona emi-riemannian manifod wih conan indice q 1 > 0 q > 0 repecivey. Le π : M1M M1, : M1 M M be he projecion which are given by π x, y x x, y y for any x, ym1 M. We denoe he produc manifod by M M M g, where, π, π, 1, g Y g Y g Y 1 for any,y TM, where denoe he differen- ia mapping. Then we have π π,, π π 0 π I where I i he ideniy map of M1 M. Thu M, g i a (m 1 + m )- dimeniona emi-riemannian manifod wih conan index (q 1 + q ). The Riemannian produc manifod M M, 1 M g i characerized by M 1 M which are oay geodeic ubmanifod of M. Now, if we pu F π hen we can eaiy ee ha F = I, g, FY g F Y (3.1) for any, Y TM, where F i caed amo Riemannian produc rucure on M1 M. If we denoe he Levi-Civia connecion on M by, hen for any o., Y 0 TM F Y (3.), ha i, F i parae wih repec 4. Toay Umbiica Radica ST-Lighike Submanifod In hi ecion, we udy oay umbiica radica STighike ubmanifod of a emi-riemannian produc manifod. We fir reca he foowing definiion from [9]. Definiion 4.1. A r-ighike ubmanifod M of a emi- Riemannian produc manifod M i aid o be a creen ranvera (ST) ighike ubmanifod of M if here Copyrigh 01 SciRe.

3 S. M. K. HAIDER ET AL. 87 exi a creen ranvera bunde S TM FRadTM STM. uch ha Definiion 4.. A ST-ighike ubmanifod M of a emi-riemannian produc manifod M i aid o be a radica ST-ighike ubmanifod if STM i invarian wih repec o F. We ao need he foowing definiion of oay umbiica ighike ubmanifod of a emi-riemannian manifod. Definiion 4.3. [5] A ighike ubmanifod (M, g) of a emi-riemannian manifod M, g i caed oay umbiica in M, if here i a mooh ranvera vecor fied H r TM of M, caed he ranvera curvaure vecor of M, uch ha for a, Y TM,, g, h Y Y H I i known ha M i oay umbiica if ony if on each co-ordinae neighborhood U, here exi mooh vecor fied H r TM H STM uch ha,, YH, h,, D W 0, TM W S TM h Y g Y, g Y H. (4.1) for any Y In repec of he inegrabiiy of he diribuion invoved in he definiion of oay umbiica radica ST-ighike ubmanifod immered in a emi-riemannian produc manifod, we have: Theorem 4.4. Le M be a oay umbiica radica ST-ighike ubmanifod of a Semi-Riemannian produc manifod. Then he creen diribuion S(TM) i away inegrabe. Proof. From (.3) (3.), a direc cacuaion how ha,, g h, FY, YS N r TM g Y N h Y, F, FN for in (4.), we ge (4.) TM. Uing (4.1),, N 0 g Y, from which our aerion foow. Theorem 4.5. Le M be a oay umbiica radica ST-ighike ubmanifod of a emi-riemannian produc manifod. Then he diribuion RadTM i away inegrabe. Proof. For Z, W RadTM STM, from (.3) (3.) we ge,,,, g h W g ZW g h Z F FW, F, FZ. Taking accoun of (4.1) in (4.3), we obain (4.3) gz, W, 0, which prove our aerion. The neceary ufficien condiion under which H 0 i given by he foowing reu. Theorem 4.6. Le M be a oay umbiica radica ST-ighike ubmanifod of a emi-riemannian produc manifod M. Then h, Y 0 if ony if H ha no componen in F RadTM for any Y, STM. Proof. Uing (.3) (3.), for any,y STM, we obain,, FY h FY h FY F Y Fh, Y Fh, Y. (4.4) Taking inner produc of (4.4) wih FN for any N r TM uing he fac ha F I, we ge g h FY FN g Y N,,,. (4.5) From (.7), (4.1) (4.5), we have g, FY g H, FN g h, Y, N. (4.6) Thu, our aerion foow from (4.6). I i known ha he induced connecion on a ighike ubmanifod immered in a emi-riemannian manifod i no a meric connecion. In view of hi, i i inereing o ee under wha condiion he induced connecion on a oay umbiica radica ST-ighike ubmanifod i a meric connecion. The foowing heorem give he geomeric condiion for he induced connecion o be a meric connecion. Theorem 4.7. Le M be a oay umbiica radica ST-ighike ubmanifod of a emi-riemannian produc manifod M. Then he induced connecion on M i a meric connecion if ony if AF 0 for TM, RadTM. Proof. For TM, RadTM, from (3.) we have F F. F,,,. A F f Fh Bh Ch (4.7) Uing (.3), (.5), (.8), (.9) (4.1) in (4.7), we obain Taking angenia componen of he above equaion hen uing (4.1), we arrive a f A F, which prove our aerion. Coroary 4.8. Le M be a oay umbiica radica ST-ighike ubmanifod of a emi-riemannian produc manifod M. Then he diribuion RadTM i parae if Copyrigh 01 SciRe.

4 88 S. M. K. HAIDER ET AL. ony if F 1 0 for any 1, RadTM. Proof. From (3.), for any 1, RadTM obain A F F. 1 1 Uing (.3), (.5), (.8), (.9) (4.1) in he above equaion, we ge AF F f Bh Fh 1 1, Ch, 1 AF f, 1 1, 1 we (4.8) Conidering he angenia componen of (4.8) uing (4.1), we arrive a from which our aerion foow. Lemma 4.9. Le M be a oay umbiica ST-ighike ubmanifod of a emi-riemannian produc manifod M. Then A W g H, W S TM TM for any Proof. For, YTM have W S., from (.6) (4.1), we W, g, YgH W g A Y,. (4.9) If RadTM, hen from (4.9) we infer ha AW 0. Moreover, if STM, hen due o non-degeneracy of S TM, we have A g H, W, W which prove he aerion. For he induced connecion of a oay umbiica radica ST-ighike ubmanifod in emi-riemannian produc manifod o be a meric connecion on r TM, we have: Theorem Le M be a oay umbiica radica ST-ighike ubmanifod of a emi-riemannian produc manifod M. Then i a meric connecion on r TM if ony if N ha no componen in for any TM N TM. Proof. For TM,WS rtm NrTM, uing (.), (.5), (.9) (3.), we ge, g NW g A FN FN D FN BW C1W CW, BW RadTM C W r TM. Uing (4.1) (4.10), we obain,,, (4.10) where, 1 CW g, NW, g A CW, g FNCW,. FN 1 Conidering emma 4.9, we ge g, NW, g FNCW,. (4.11) Thu our aerion foow from (4.11) Theorem.3 page 159 of [1]. Theorem Le M be a oay umbiica radica ST-ighike ubmanifod of a emi-riemannian produc manifod M. Then A A F1 F 1 for a 1, RadTM. Proof. For any 1, RadTM., uing produc rucure on M, we ge from which we have F F, 1 1 F 1 Fh 1, Fh 1, AF 1 F 1, (4.1) where we have ued (.3), (.5) (3.). Inerchanging 1 in (4.1) hen ubracing he reuing equaion from (4.1), we obain F. 1 F 1 AF 1 AF1 F 1 F 1 (4.13) Taking inner produc of (4.13) wih STM, we ge 1 F F 1 g, F g, F g A A,. 1 1 (4.14) Now, from (.3) (4.1), a direc cacuaion how ha 1 g, F 0, g, F 0. (4.15) 1 Uing (4.15) in (4.14), we ge F1 F g A, 0. A 1 (4.16) Thu our aerion foow from (4.16) ogeher wih non-degeneracy of STM. 5. Toay Umbiica ST-Ani-Invarian Lighike Submanifod In hi ecion, we udy oay umbiica ST-ani-invarian ighike ubmanifod immered in a emi-riemannian produc manifod. Fir we reca he foowing definiion from [9]. Definiion 5.1. [9] A ST-ighike ubmanifod M of a emi-riemannian produc Manifod M i aid o be a ST-ani-invarian ighike ubmanifod of M if STM i creen ranvera wih repec o F, i.e., STM F S TM. Copyrigh 01 SciRe.

5 S. M. K. HAIDER ET AL. 89 The neceary ufficien condiion for he induced connecion on a oay umbiica ST-aniinvarian ighike ubmanifod M o be a meric connecion i given by he foowing reu. Theorem 5.. Le M be a oay umbiica ST-aniinvarian ighike ubmanifod of a emi-riemannian produc manifod M. Then he induced connecion on M i a meric connecion if ony if F ha no componen in F S TM for a TM, RadTM. Proof. Uing (.3), (.5), (.8), (.9), (3.) (4.1), we arrive a AF F Ch, Bh, Ch,. Taking inner produc of (5.1) wih FY for YSTM hen uing (4.1), we obain g F, FY g, Y, (5.1) which prove he aerion. Theorem 5.3. Le M be a oay umbiica ST-aniinvarian ighike ubmanifod of a emi-riemannian produc manifod M. Then RadTM i parae if ony if F ha no componen in F S TM for RadTM 1 1, a. Proof. From (.3), (.5), (.8), (.9), (3.) (4.1), we have A F Ch F Ch 1,, Bh 1, for any 1, RadTM. Uing (4.1) in he above equaion, we ge A F. (5.) F Taking inner produc of (5.) wih FY for Y S TM, we obain g F, FY g, Y, 1 1 from which our aerion foow. Theorem 5.4. Le M be a oay umbiica ST-aniinvarian ighike ubmanifod of a emi-riemannian produc manifod M. Then H ha no componen in F r TM. Proof. For, Y STM, uing (.3), (.5) (3.) we ge AFY FY D, FY F Y Fh, Y Fh, Y Taking inner produc of (5.3) wih RadTM hen uing (4.1) we obain g, Y g H, F 0. (5.3) from which we have our aerion. Theorem 5.5. Le M be a oay umbiica ST-aniinvarian ighike ubmanifod of a emi-riemannian produc manifod M. Then H 0 if ony if F ha no componen in F S TM for a S TM. Proof. Uing (.3), (.5), (.8), (.9), (3.) (4.1) we ge F,,, A F Ch for any S TM of (5.4), we arrive a Bh Ch F Ch, Ch,. (5.4). From creen ranvera par Taking inner produc of he above equaion wih RadTM uing (.8), (4.1) we ge for g F, F g, g H,, F which prove our aerion. The foowing heorem caifie oay umbiica STani-invarian ighike ubmanifod immered in a emi- Riemannian produc manifod. Theorem 5.6. Le M be a oay umbiica ST-aniinvarian ighike ubmanifod of a emi-riemannian produc manifod M. Then eiher H ha no componen in F STM or dimstm 1. Proof. Taking inner produc of he angenia componen of (5.4) wih Z STM uing (3.1) (.9), we ge for any (.6) we have F STM g A, Z g h,, FZ (5.5). On he oher h, by virue of,,, F g A Z g h Z F (5.6) Combining (5.5) (5.6), we ge g h,, FZ g h, Z, F Uing (4.1) in he above equaion, we obain g, g H, FZ g, Z g H, F. (5.7) Inerchanging Z in (5.7) rearranging he erm, we ge g, Z g H, F gh, FZ. (5.8) g Z, Z From (5.7) (5.8), we concude ha g, Z,, g H, F g H, F. (5.9) g g Z Z Thu our aerion foow from (5.9). Copyrigh 01 SciRe.

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