Geodesic Lightlike Submanifolds of Indefinite Sasakian Manifolds *

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1 Advance in Pure Mahemaic doi:104236/apm Pubihed Onine November 2011 (hp://wwwscirporg/journa/apm) Geodeic Lighike Submanifod of Indefinie Saakian Manifod * Abrac Junhong Dong 1 imin Liu 2 1 Deparmen of Mahemaic Souh China Univeriy of Technoogy Guangzhou China 2 Schoo of Mahemaica Science Daian Univeriy of Technoogy Daian China E-mai: dongjunhong-run@163com ximiniu@dueducn Received Juy ; revied Augu ; acceped Augu In hi paper we udy geodeic conac CR-ighike ubmanifod geodeic creen CR-ighike (SCR) ubmanifod of indefinie Saakian manifod Some neceary ufficien condiion for oay geodeic mixed geodeic D -geodeic D -geodeic conac CR-ighike ubmanifod SCR ubmanifod are obained Keyword: CR-Lighike Submanifod Saakian Manifod Toay Geodeic Submanifod 1 Inroducion A ubmanifod M of a emi-riemannian manifod M i caed ighike ubmanifod if he induced meric on M i degenerae The genera heory of a ighike ubmanifod ha been deveoped by Kupei [1] Bejancu- Dugga [2] The geomery of CR-ighike ubmanifod of indefinie Kaeher manifod wa udied by Gugga Bejancu [2] The geodeic CR-ighike ubmanifod in indefinie Kaeher manifod were udied by Sahin Güne [34] Lighike ubmanifod of indefinie Saakian manifod can be defined according o he behavior of he amo conac rucure conac CR-ighike ubmanifod creen CR-ighike (SCR) ubmanifod of indefinie Saakian manifod were udied by Dugga Sahin in [5] The udy of he geomery of ubmanifod of indefinie Saakian manifod ha been deveoped by [6] oher In hi paper geodeic conac CR-ighike ubmanifod geodeic creen CR-ighike (SCR) ubmanifod of indefinie Saakian manifod are conidered Some neceary ufficien condiion for oay geodeic mixed geodeic D -geodeic D -geodeic conac CR-ighike ubmanifod SCR ubmanifod are obained * Thi work i uppored by NSFC ( ) 2 Preiminarie m A ubmanifod M immered in a emi-riemannian m manifod ( M n g) i caed a ighike ubmanifod if i admi a degenerae meric g induced from g whoe radica diribuion RadTM i of rank r where 1 r m RadTM = TM TM where TM = u TxM g u v= 0 v TxM Le STM xm be a creen diribuion which i emi- Riemannian compemenary diribuion of RadTM in TM ie TM = RadTM S( TM A STM i a nondegenerae vecor ubbunde of TM M we pu TM M = STM STM e conider a nondegenerae vecor ubbunde of STM which i a compemenary vecor bunde of RadTM in TM Since for any oca bai { i } of RadTM here exi a oca frame { N i } of ecion wih vaue in he orhogona compemen of S TM uch ha gi Ni = ij g NiN j =0 here exi a ighike ranvera vecor bunde r TM ocay panned by { N i } Le r TM be he compemenary (bu no orhogona) vecor bunde o TM in TM M Then r TM = r TM S TM T M = S TM RadTM r TM S TM Copyrigh 2011 SciRe

2 J H DONG ET AL 379 Now e be he evi-civia connecion on M we have = TM g Z g Z g Z Z (21) = h TM (22) V = AV V TM (23) V rtm where AV h V beong o TM r TM repecivey Uing he projecor : rtm STM : r TM r TM from [1] we have = h h TM (24) N = A N D N N r TM N (25) = A D S TM P = P h P (27) = A (28) (26) Denoe he projecion of TM o S TM by P we have he decompoiion for any TM RadTM N r TM From he above equaion we have g h = g A = g h P N g A P N (29) (210) g h =0 A =0 (211) Definiion 21 A (2n + 1)-dimeniona Semi-Riemannian manifod M g i caed a conac meric manifod if here i a (11) enor fied a vecor fied V caed he characeriic vecor fied i dua 1-form uch ha g = g g V V = 2 (212) = V g V = (213) = d g TM (214) where = 1 From he above definion i foow ha V =0 =0 V =1 (215) The ( V g) i caed a conac meric rucure of M If N d V =0 we ay ha M ha a norma conac rucure where N i he Nijenhui enor fied of A norma conac meric manifod i caed a Saakian manifod for which we have V = (216) = Le g V (217) M g S TM S TM be a ighike ubmanifod of M g For any vecor fied angen o M we pu = P Q (218) where P Q are he angenia he ranvera par of repecivey Le uppoe V i a paceike vecor fied o ha =1 i imiar when V i a imeike vecor fied 3 Geodeic Invarian Lighike Submanifod Definiion 31 Le M g S TM S TM be a ighike ubmanifod angen o he rucure vecor fied V V STM immered in an indefinie Saakian manifod M g we ay ha M i an invarian ubmanifod of M if he foowing condiion are aified RadTM = RadTM S TM = S TM (31) From (216) (217) (218) (24) we have h V = h V = 0 V = V = P (32) = = h h h TM (33) From (31) (212) we have = = r TM r TM S TM S TM Theorem 31 Le (34) M g S TM S TM be an invarian ighike ubmanifod of an indefinie Saakian manifod M hen M i oay geodeic if ony if h h of M are parae Proof Suppoe h i parae for any Z TM we have h V= h Vh V h V= 0 By (32) we have h V = h V =0 Copyrigh 2011 SciRe

3 380 o h Tha i o ay = 0 h P = 0 In a imiar way we can ge h P= 0 Thu M i oay geodeic Converey if h = h = 0 ince o h h Z= h Zh Z h Z= 0 h Z= h Zh Z h Z= 0 h are parae which compee he proof 4 Geodeic Conac CR-Lighike Submanifod Definiion 41 Le M g STM STM be a ighike ubmanifod angen o he rucure vecor fied V immered in an indefinie Saakian manifod M g e ay ha M i a conac CR-ighike ubmanifod of M if he foowing condiion are aified [(A)] RadTM i a diribuion on M uch ha RadTM ( RadTM )={0} [(B)] There exi vecor bunde D over M uch ha D 0 0 S TM = RadTM D D V D = D D = L L L = 1 where i non-degenerae 0 r TM L i 2 a vecor ubbunde of ST M So we have he decompoiion TM = D D V D = RadTM RadTM D D 0 If we denoe D = D V hen we have TM = D D D = D Definiion 42 A conac CR-ighike ubmanifod of an indefinie Saakian manifod i caed D -geodeic conac CR-ighike ubmanifod if i econd fundamena form h aified h =0 for any D Definiion 43 A conac CR-ighike ubmanifod of an indefinie Saakian manifod i caed mixed geodeic conac CR-ighike ubmanifod if i econd fundamena form h aified h Z = 0 for any D Z D Definiion 44 A conac CR-ighike ubmanifod of an indefinie Saakian manifod i caed D -geodeic conac CR-ighike ubmanifod if i econd fundamena form h aified hz U = 0 for any Z U D Theorem 41 Le M be a conac CR-ighike ubmanifod of an indefinie Saakian manifod M J H DONG ET AL Then M i oay geodeic if ony if g A = w g D ha no componen in L 1 TM panv or ha no componen in L1 Proof e know ha M i oay geodeic if ony if h = 0 for any TM By he definiion of he econd fundamena form h = 0 i equivaen o gh =0gh =0 for any RadTM STM From (24) (27) we have = g h g = g = g = g g g V = g g = g g g g A g D g h g = = = g A D = (41) (42) Thu from (41) (42) he proof i compeed Theorem 42 Le M be a conac CR-ighike ubmanifod of an indefinie Saakian manifod M Then M i mixed geodeic if ony if A ha no componen in RadTM L2 Proof By he definiion M i mixed geodeic if ony if g h = 0 g h x D D Then we have = g h g ga g ga = g = g = 0 = g g g V = g ( ) g = ( ) = Copyrigh 2011 SciRe

4 J H DONG ET AL 381 = g h g ga = g = g = g g g V = g = Thu he proof of he heorem i compee Theorem 43 Le M be a conac CR-ighike ubmanifod of an indefinie Saakian manifod M Then M i D -geodeic if ony if RadTM L2 ha no componen in L 2 D Proof M i D -geodeic if ony if gh =0 gh =0 for any D RadTM STM Then we have = g h g g g = g = g = g g g V = g = = = g h g = g = g = g g g V = g = g Thu he aerion of he heorem foow Theorem 44 Le M be a conac CR-ighike ubmanifod of an indefinie Saakian manifod M Then M i D -geodeic if ony if A A have no componen in L2 RadTM D Proof M i D -geodeic if ony if =0 =0 for any STM g h g h D RadTM So we have = = g h g g = g A = = = ga g h g g Thu he aerion of he heorem foow 5 Geodeic Conac SCR-Lighike Submanifod Definiion 51 Le M g S TM S TM be a ighike ubmanifod angen o he rucure vecor fied V immered in an indefinie Saakian manifod M g e ay ha M i a conac SCR-ighike ubmanifod of M if he foowing condiion are aified [(A)] There exi rea non-nu diribuion D D uch ha = S TM D D V D S TM D D = 0 where D i he orhogona compemenary o D V in ST M [(B)] D = D RadTM = RadTM r TM = r TM Hence we have he decompoiion TM = D D V D = D RadTM Le u denoe D = D V Definiion 52 A conac SCR-ighike ubmanifod of an indefinie Saakian manifod i caed mixed geodeic conac SCR-ighike ubmanifod if i econd fundamena form h aified h =0 for any D D Theorem 51 Le M be a conac SCR-ighike ubmanifod of an indefinie Saakian manifod M Then M i oay geodeic if ony if Lg L g TM = = 0 RadTM S TM Proof e know M i oay geodeic if ony if g h =0 g h =0 D D Copyrigh 2011 SciRe

5 382 J H DONG ET AL From (21) Lie derivaive we obain = g g g g g g g g h g = = = Lg g Lg gh 2 g h = L g = g g g g = = Hence we have In a imiar way we can ge 2 g h = L g hu he proof i compeed Theorem 52 Le M be a conac SCR-ighike ubmanifod of an indefinie Saakian manifod M Then M i mixed geodeic if ony if D A D for any D D Proof For any D D RadTM S TM denoe by = P Q = B C where P D Q D B D C STM D If M i mixed geodeic hen h = = 0 From he definiion here exi STM uch ha = Thu we have 0= = = A = PA QA B C From he definiion of he Q C we know ha QA = C = 0 So we have D A D From = (213) we have = hu he proof i compeed Theorem 53 Le M be a conac SCR-ighike ubmanifod of an indefinie Saakian manifod M Then D define a oay geodeic foiaion if ony if h Z h N ha no componen in D D ZD Proof From he definiion we have ha D i a oay geodeic foiaion if ony if D D which i equivaen o for any Then we have g Z = g N = 0 Z D N r TM = = g ZZ g Z g Z g ZV Z g Z g h Z g Z g Z g Z = = = = = = g NN g N g N g N g h N g N g N = = = g g V N = = = Thu he aerion i proved 6 Reference [1] D N Kupei Singuar Semi-Riemannian Geomery Kuwer Dordrech 1996 [2] K L Dugga A Bejancu Lighike Submanifod of Semi-Riemannian Manifod Appicaion Kuwer Academic Dordrech 1996 [3] B Sahin Tranvera Lighike Submanifod of Indefinie Kaeher Manifod Anaee Univeriaii de Ve Timioara Seria Maemaica Informaica Vo 44 No pp [4] B Sahin R Güne Geodeic CR-Lighike Submanifod Conribuion o Agebra Geomery Vo 42 No pp Copyrigh 2011 SciRe

6 J H DONG ET AL 383 [5] K L Dugga B Sahin Lighike Submanifod of Indefinie Saakian Manifod Inernaiona Journa of Mahemaic Mahemaica Science Arice ID Page [6] K L Dugga B Sahin Generaized Cauchy-Rieman Lighike Submanifod of Indefinie Saakian Manifod Aca Mahemaica Hungarica Vo 122 No pp doi:101007/ Copyrigh 2011 SciRe