On submanifolds with recurrent second fundamental form in spaces of constant curvature

On submanifolds with reurren t seond fundamen tal form in spaes of onstan t urv ature. On submanifolds with reurren t seond fundamen tal form in spaes of onstan t urv ature. Irina I. Bo drenk o 1 Abstrat The omplete lo al lassiation and geometri desription of n -dimensional submanifolds F n with reurren t nonparallel seond fundamen tal form in the spaes of onstan t urv ature M n + p ( c ) are obtained in this artile. In tro dution Let F n b e the n -dimensional ( n ≥ 2) smo oth submanifold in ( n + p ) -dimensional ( p ≥ 2) spae of onstan t urv ature M n + p () . W e denote b y b the seond fundamen tal form F n , b y ∇ the onnetedness of V an der V arden  Bortolotti. b is alled p ar al lel , if ∇ b ≡ 0 . Numerous artiles ha v e b een dev oted to the studying of submanifolds with ∇ b ≡ 0 . The syrv ey of those main results w e an see in [1, 2℄. A ording to the denition of the reurren t tensor eld [3, á. 279℄ the nonzero form b is alled the r e urr ent one , if there exists 1-form µ on F n su h that ∇ b = µ N b . W e will sa y that F n b elongs to the set R b if F n has reurren t the seond fundamen tal form b . The submanifolds F n from the set R b in Eulidean spae E n + p w ere b eing studied in [4℄. In this artile w e lassify the submanifolds F n from the set R b in spaes M n + p ( c ) . The set R b inludes all submanifolds with parallel the seond fundamen tal form but not only those ones. F or example, the pro dut of the plane urv e γ ⊂ E 2 with urv ature k 1 6 = 0 and ( n − 1) - dimensional spae E n − 1 is ylindrial h yp ersurfae F n = γ × E n − 1 ⊂ E n +1 b elongs to the set R b [4℄. F or ea h p oin t x ∈ F n w e denote b y T x F n and T ⊥ x F n tangen t and normal spaes of F n at x resp etiv ely . A ording to [5℄ w e will sa y that F n arries, in arbitrary domain U ⊂ F n the  onjugate system { L k 1 1 , . . . , L k m m } ( k 1 + . . . + k m = n , dim L k i i = k i , i = 1 , m ), if T y F n = L k 1 1 ( y ) ⊕ . . . ⊕ L k m m ( y ) at ea h p oin t y ∈ U , all distributions L k i i , i = 1 , m , are smo oth, in v olute and m utually onjugate. Moreo v er, if L k i i is m utually totally orthogonal then the onjugate system is alled the ortho gonal . Submanifold F ⊂ F n is alled the surfa e of urvatur e F n if, at ev ery p oin t x ∈ F , its tangen t subspaes T x F are prop er subspaes of matrixes B ξ of the seond fundsmen tal form b with resp et to arbitrary normal ξ ∈ T ⊥ x F n . 1   2005 Irina I. Bo drenk o, asso iate professor, Departmen t of Mathematis, V olgograd State Univ ersit y , Univ ersit y Prosp ekt 100, V olgograd, 400062, R USSIA. E.-mail: b o drenk omail.ru h ttp://www.b o drenk o.om h ttp://www.b o drenk o.org 1 Irina I. Bo drenk o. The main result of this artile is Theorem 1. L et F n in M n + p ( c ) b elongs to the set R b . If, at p oint x ∈ F n , ∇ b 6 = 0 then x b elongs to a  ertain domain U ⊂ F n , wher e 1) F n  arries the ortho gonal  onjugate system { L 1 1 , L n − 1 2 } ; 2) F n is dir e t R iemmanian pr o dut F 1 1 × F n − 1 2 of maximal inte gr al submanifolds F 1 1 , F n − 1 2 distributions L 1 1 , L n − 1 2 , r esp e tively; 3) F 1 i ( i = 1 , 2) is the line of urvatur e, F n − 1 2 , wher e n > 2 , is the surfa e of urvatur e of submanifold F n in M n + p ( c ) . F rom theorem 1 w e obtain the follo wing statemen t on geometri struture of submanifolds from the set R b . Theorem 2. L et F n b e n -dimensional submanifold in simply  onne te d sp a e M n + p ( c ) . If F n b elongs to the set R b and ∇ b 6 = 0 at e ah p oint x ∈ F n then F n is join of losur es of its domains whih 1) is op en p art of dir e t R iemmanian pr o dut F 1 1 × F n − 1 2 of urve F 1 1 with urvatur e k 1 6 =  onst and ( n − 1) -dimensional intrinsi al ly planar total ly ge o desi in F n submanifold F n − 1 2 ; 2) is  ontaine d in some ( n +1) -dimensional total ly ge o desi submanifold M n +1 ⊂ M n + p ( c ) . 1 The equations of submanifolds of the set R b . Denote b y e g Riemmanian metri on M n + p ( c ) , b y g indued Riemmanian metri on F n , b y f ∇ and ∇ Riemmanian onnetedness on M n + p ( c ) and F n o ordinated with e g and g resp etiv ely . The seond fundamen tal form b denes b y the equation [6, . 39℄: f ∇ X Y = ∇ X Y + b ( X, Y ) for an y v etor elds X , Y tangen tial to F n . Denote b y D normal onnetedness F n in M n + p ( c ) . F or an y v etor eld ξ noraml to F n , w e ha v e: f ∇ X ξ = − A ξ X + D X ξ , where A ξ is the seond fundsmen tal tensor orresp onding to ξ . F or an y v etor elds X , Y , Z tangen tial to F n , and v etor eld ξ normal to F n , w e ha v e: e g ( b ( X , Y ) , ξ ) = g ( A ξ X , Y ) , (1) ( ∇ X b )( Y , Z ) = D X ( b ( Y , Z )) − b ( ∇ X Y , Z ) − b ( Y , ∇ X Z ) , (2) ( ∇ X A ξ ) Y = ∇ X ( A ξ Y ) − A ξ ( ∇ X Y ) − A D X ξ Y , (3) e g (( ∇ X b )( Y , Z ) , ξ ) = g (( ∇ X A ξ ) Y , Z ) . (4) Denote b y R and R ⊥ the urv ature tensors of onnetednesses ∇ and D resp etiv ely . The equations of Gauss, P eterson  Co dai and Rii if F n is submanifold in M n + p ( c ) tak es the follo wing forms resp etiv ely [6℄: g ( R ( X, Y ) Z , W ) = c ( g ( X , W ) g ( Y , Z ) − g ( X , Z ) g ( Y , W ))+ 2 On submanifolds with reurren t seond fundamen tal form in spaes of onstan t urv ature. + e g ( b ( X , W ) , b ( Y , Z )) − e g ( b ( X , Z ) , b ( Y , W )) , (5) ( ∇ X b )( Y , Z ) = ( ∇ Y b )( X, Z ) , (6) e g ( R ⊥ ( X , Y ) ξ , η ) = g ([ A ξ , A η ] X , Y ) (7) for an y X , Y , Z , W ∈ T F n and ξ , η ∈ T ⊥ F n , where T F n and T ⊥ F n are tangen t and normal bundles on F n resp etiv ely . F n b elongs to the set R b if and only if, on F n , there exists 1-form µ su h that for an y X , Y , Z ∈ T F n ( ∇ X b )( Y , Z ) = µ ( X ) b ( Y , Z ) . (8) Equation system (5)  (8) determines submanifolds F n of the only set R b in M n + p ( c ) . 2 The struture of submanifolds from the set R b . Before pro of of teorems 1, 2 w e onsider some prop osals. Let { e i } n 1 b e the eld of basises in T F n . W e denote b y S and R 1 Rii form and Rii tensor in onnetedness ∇ resp etiv ely . F or an y v etor elds X , Y tangen t to F n form S is dened b y the follo wing equation S ( X , Y ) = P n i =1 g ( R ( e i , X ) Y , e i ) , op erator R 1 satises the follo wing equation S ( X , Y ) = g ( R 1 X , Y ) . Submanifold F n is alled Eisenstein , if R 1 = λI , where I is iden tial transformation, λ is Eisenstein onstan t. Lemma 1. L et F n in M n + p ( c ) b elongs to the set R b . If ∇ b 6 = 0 at e ah p oint x ∈ F n then F n is Eisenstein with Eisenstein  onstant λ = c ( n − 1) . Pro of. Let { n σ } p 1 b e the eld of orthonormalized basises in T ⊥ F n . Assume A σ = A n σ . W e denote b y H the v etor of mean urv ature of F n : H = H σ n σ , nH σ = trae A σ . F rom (5) using (1) w e obtain: S ( X , Y ) = c ( n − 1) g ( X , Y ) + n e g ( H , b ( X, Y )) − p X σ =1 g ( A σ X , A σ Y ) ∀ X, Y ∈ T F n . (9) W e will sho w that, on F n , holds the equation n e g ( H , b ( X , Y )) = p X σ =1 g ( A σ X , A σ Y ) ∀ X, Y ∈ T F n . (10) Notie that using (6) and (8) w e ha v e e g  b ( X, Y ) , ( ∇ Z b )( V , W )  is symmetri b y X , Y , Z , V , W ∈ T F n and therefore using (4), on F n , w e ha v e: n e g ( H , ( ∇ Z b )( X, Y )) = p X σ =1 g ( A σ X , ( ∇ Z A σ )( Y )) . ∀ X, Y , Z ∈ T F n . 3 Irina I. Bo drenk o. Then using (8) w e obtain the equation µ ( Z ) n e g ( H , b ( X, Y )) − p X σ =1 g ( A σ X , A σ Y ) ! = 0 ∀ X , Y , Z ∈ T F n . Sine µ 6 = 0 then from the ab o v e equation w e get (10). W e use (10) and w e obtain from (9) that for an y X , Y ∈ T F n S ( X , Y ) = c ( n − 1) g ( X, Y ) . Lemma is pro v ed. Denote, in T ⊥ x F n , linear subspae N 0 ( x ) = { ξ ( x ) ∈ T ⊥ x F n | A ξ ( x ) = 0 } . Denote b y N 1 ( x ) orthogonal omplemen t N 0 ( x ) in T ⊥ x F n : T ⊥ x F n = N 0 ( x ) ⊕ N 1 ( x ) . N 1 ( x ) is alled the rst normal sp a e of submanifold F n in x . Dimension N 1 ( x ) is alled the p oint  o dimension F n in x . Notie that at ea h p oin t x ∈ F n dim N 1 ( x ) ≤ min { p, n ( n + 1) 2 } . Lemma 2. L et F n b elongs to the set R b . If ∇ b 6 = 0 at p oint x ∈ F n then dim N 1 ( x ) = 1 . Pro of. Assume that dim N 1 ( x ) 6 = 1 at the giv en p oin t x . Sine b 6 = 0 then dim N 1 ( x ) > 1 and therefore there exist v etors t 1 , t 2 , t 3 , t 4 ∈ T x F n su h that v etors b ( t 1 , t 2 ) , b ( t 3 , t 4 ) ∈ T ⊥ x F n are not ollinear. Consider linear om bination of v etors: µ ( t 1 ) µ ( t 2 ) b ( t 3 , t 4 ) − µ ( t 3 ) µ ( t 4 ) b ( t 1 , t 2 ) . Using (6) and (8) w e ha v e: µ ( t 1 ) µ ( t 2 ) b ( t 3 , t 4 ) − µ ( t 3 ) µ ( t 4 ) b ( t 1 , t 2 ) = µ ( t 1 ) µ ( t 3 ) b ( t 2 , t 4 ) − µ ( t 3 ) µ ( t 4 ) b ( t 1 , t 2 ) = µ ( t 4 ) µ ( t 3 ) b ( t 2 , t 1 ) − µ ( t 3 ) µ ( t 4 ) b ( t 1 , t 2 ) = 0 . Using linear indep endene of v etors b ( t 1 , t 2 ) , b ( t 3 , t 4 ) w e ha v e µ ( t 1 ) µ ( t 2 ) = µ ( t 3 ) µ ( t 4 ) = 0 . Then for ∀ t ∈ T x F n µ ( t ) b ( t 1 , t 2 ) = µ ( t 1 ) b ( t, t 2 ) = µ ( t 2 ) b ( t, t 1 ) = 0 . Therefore, µ ( t ) = 0 , ∀ t ∈ T x F n on tradits to the follo wing ondition ∇ b 6 = 0 . Lemma is pro v ed. W e sa y that F n  arries planar normal  onne te dness , if R ⊥ ≡ 0 on F n . The last ondition is neessary and suien t for the follo wing: submanifold F n in M n + p ( c ) has n prinipal diretions at an y p oin t [6, . 99℄. Lemma 3. L et F n in M n + p ( c ) b elongs to the set R b . If ∇ b 6 = 0 at any p oint x ∈ F n then F n  arries planar normal  onne te dness. Pro of follo ws from lemma 2 and equations (7). Let N b e q -dimensional distribution putting the orresp ondene for ea h p oin t x ∈ F n some q -dimensional subspae N ( x ) ∈ T ⊥ F n . W e sa y that N is p ar al lel in normal  onne te dness , if for an y normal v etor eld ξ ∈ N the o v arian t deriv ativ e D X ξ ∈ N for all X ∈ T F n . Consider, on F n , distributions ∆ 0 and ∆ 1 su h that ∆ 0 ( x ) = N 0 ( x ) , ∆ 1 ( x ) = N 1 ( x ) . Lemma 4. L et F n in M n + p ( c ) b elongs to the set R b . If ∇ b 6 = 0 at any p oint x ∈ F n then ∆ 0 and ∆ 1 ar e p ar al lel in normal  onne te dness. 4 On submanifolds with reurren t seond fundamen tal form in spaes of onstan t urv ature. Pro of. Using lemma 2 w e get for an y p oin t x ∈ F n dim N 1 ( x ) = 1 . Let ξ b e the unit normal v etor eld generating the distribution ∆ 1 . Let { X i } n 1 b e the eld of basises of tangen tial v etors in T F n . Therefore N 1 ( x ) = span { b ( X i ( x ) , X j ( x ) } n i,j =1 ∀ x ∈ F n . F rom (2), using (8) w e ha v e: D X i ( b ( X j , X k )) = ( ∇ X i b )( X j , X k ) + b ( ∇ X i X j , X k ) + b ( X j , ∇ X i X k ) = = µ ( X i ) b ( X j , X k ) + b ( ∇ X i X j , X k ) + b ( X j , ∇ X i X k ) ∈ ∆ 1 . Therefore D X ( b ( Y , Z )) ∈ ∆ 1 ∀ X, Y , Z ∈ T F n and hene D X ξ ∈ ∆ 1 ∀ X ∈ T F n . Moreo v er, the eld ξ is parallel in normal onnetedness D . Sine e g ( ξ , ξ ) = 1 then e g ( D X ξ , ξ ) = 0 ∀ X ∈ T F n . Therefore, using the fat ∆ 1 is one-dimensional distribution w e obtain the equalit y D X ξ = 0 ∀ X ∈ T F n . W e add ξ in to the eld of orthonormalized basises { n σ } p 1 in T ⊥ F n assuming ξ = n 1 . Then the elds n 2 , . . . , n p generate distribution ∆ 0 . W e ha v e: e g ( ξ , n ρ ) = 0 , ρ = 2 , p . Hene e g ( ξ , D X n ρ ) = 0 ∀ X ∈ T F n , ρ = 2 , p and therefore D X n ρ ∈ ∆ 0 ∀ X ∈ T F n , ρ = 2 , p . Moreo v er without loss of generalit y w e assume that the normals n 2 , . . . , n p are parallel in normal onnetedness D . Lemma is pro v ed. Pro of theorem 1. F rom lemma 2 w e get that in some neigh b orho o d O ( x ) ⊂ F n of p oin t x there exists unit v etor eld ξ ∈ T ⊥ F n su h that v etor elds b ( X, Y ) are ollinear ξ for ¢á ¥ å X , Y ∈ T F n . W e add ξ in to the eld of orthonormalized basises { n σ } p 1 ¢ T ⊥ F n assuming ξ = n 1 . Then using the fat that A ξ is symmetri linear transformation, from (9) w e ha v e S ( X , Y ) = c ( n − 1) g ( X, Y ) + g ( A ξ X , Y ) trae A ξ − g ( A 2 ξ X , Y ) ∀ X, Y ∈ T F n . Then using lemma 1 w e obtain g ( trae A ξ A ξ X − A 2 ξ X , Y ) = 0 ∀ X, Y ∈ T F n . Hene ( trae A ξ A ξ − A 2 ξ ) X = 0 ∀ X ∈ T F n . (11) F rom lemma 3 w e ha v e the fat that F n has at ea h p oin t y ∈ O ( x ) n prinipal diretions. The prinipal diretions are m utually orthogonal and onjugate, and therefore are eigen v etors of op erator A ξ . Consider orthonoralized basis { t i } n 1 ∈ T y F n onsisting of eigen v etors of op erator A ξ ( y ) . Denote b y k i the eigen v alue of op erator A ξ ( y ) orresp onding to the eigen v etor t i : A ξ ( y ) t i = k i t i , i = 1 , n . Assume nh = P n j =1 k j . Then from (11) w e get: nhk i − k 2 i = 0 , i = 1 , n . That is ea h k i is ro ot of quadrati equation k 2 − nhk = 0 . Sine A ξ 6 = 0 then w e assume k 1 = . . . = k m = nh 6 = 0 , k m +1 = . . . = k n = 0 , where 1 ≤ m ≤ n . Sine nh = P n j =1 k j then nh = mnh and therefore m = 1 . Denote, in O ( x ) , t w o distributions L 1 and L 2 as: at an y p oin t y ∈ O ( x ) L 1 ( y ) = { t ∈ T y F n | A ξ ( y ) = k 1 t } , L 2 ( y ) = { t ∈ T y F n | A ξ ( y ) = 0 } . 5 Irina I. Bo drenk o. W e will sho w that L 1 and L 2 are dieren tiable distributions. Assume X 1 , X 2 , . . . , X n are dieren tiable v etor elds tangen t to F n su h that X 1 and X 2 , . . . , X n at the p oin t x are basises for L 1 ( x ) and L 2 ( x ) resp etiv ely . Denote v etor elds Y 1 , Y 2 , . . . , Y n as: Y 1 = A ξ X 1 , Y i = ( A ξ − k 1 I ) X i , i = 2 , n. Then using (11) w e obtain A ξ Y 1 = A 2 ξ X 1 = k 1 A ξ X 1 = k 1 Y 1 , A ξ Y i = A ξ (( A ξ − k 1 I ) X i ) = ( A 2 ξ − k 1 A ξ ) X i = 0 , i = 2 , n. Therefore Y 1 b elongs to L 1 , and Y 2 , . . . , Y n b elong to L 2 . Sine Y 1 , . . . , Y n are linearly indep enden t at p oin t x and hene in some domain U ⊂ O ( x ) then L 1 and L 2 ha v e, in U, lo al basises Y 1 and Y 2 . . . , Y n resp etiv ely . W e will sho w that L 1 and L 2 are in v olute. Using (4), w e write (6) as: ( ∇ X A ξ ) Y = ( ∇ Y A ξ ) X ∀ X, Y ∈ T F n . (12) Using (3) w e ha v e for an y X , Y ∈ T F n A ξ ([ X , Y ]) = A ξ ( ∇ X Y − ∇ Y X ) = A ξ ( ∇ X Y ) − A ξ ( ∇ Y X ) = = ( ∇ Y A ξ ) X − ( ∇ X A ξ ) Y + ∇ X ( A ξ Y ) − ∇ Y ( A ξ X ) + A D Y ξ X − A D X ξ Y . Then using (12) w e ha v e the follo wing equalit y A ξ ([ X , Y ]) = ∇ X ( A ξ Y ) − ∇ Y ( A ξ X ) + A D Y ξ X − A D X ξ Y ∀ X, Y ∈ T F n . (13) F rom lemma 4 w e ha v e that eld ξ is parallel in normal onnetedness, i.e. D X ξ = 0 ∀ X ∈ T F n . Therefore, from (13) for su h ξ w e obtain: A ξ ([ X , Y ]) = ∇ X ( A ξ Y ) − ∇ Y ( A ξ X ) ∀ X , Y ∈ T F n . (14) If v etor elds X , Y ∈ L 1 then using the fat L 1 is one-dimensional distribution w e assume X = ν Y 1 , Y = ρY 1 for some funtions ν, ρ. Then from (14) w e get A ξ ([ X , Y ]) = ∇ X ( A ξ Y ) − ∇ Y ( A ξ X ) = ∇ X ( k 1 Y ) − ∇ Y ( k 1 X ) = = k 1 ( ∇ X Y − ∇ Y X ) + X ( k 1 ) Y − Y ( k 1 ) X = = k 1 ( ∇ X Y − ∇ Y X ) + ν Y 1 ( k 1 ) ρY 1 − ρY 1 ( k 1 ) ν Y 1 = k 1 [ X , Y ] . Therefore [ X , Y ] ∈ L 1 for an y X , Y ∈ L 1 . If v etor elds X , Y ∈ L 2 then using the fat A ξ X = 0 ∀ X ∈ L 2 and (14) w e obtain the equalit y A ξ ([ X , Y ]) = 0 . Hene [ X , Y ] ∈ L 2 ∀ X, Y ∈ L 2 . It is lear that L 1 , L 2 are m utually orthogonal and onjugate, i.e. g ( X , Y ) = 0 , b ( X, Y ) = 0 ∀ X ∈ L 1 , ∀ Y ∈ L 2 . (15) 6 On submanifolds with reurren t seond fundamen tal form in spaes of onstan t urv ature. T y F n = L 1 ( y ) L L 2 ( y ) at an y p oin t y ∈ U . Hene, in domain U, distributions L 1 , L 2 form orthogonal onjugate system. Statemen t 1) of theorem is pro v ed. In order to omplete the pro of of statemen t 2) w e will sho w that distributions L 1 and L 2 are parallel in onnetedness ∇ . Notie Y ( k 1 ) = 0 , ∀ Y ∈ L 2 (16) Sine b ( X, X ) = k 1 g ( X , X ) ∀ X ∈ L 1 then ( ∇ Y b )( X, X ) = Y ( k 1 ) g ( X , X ) ∀ X ∈ L 1 , ∀ Y ∈ T F n . (17) On the other hand taking in to aoun t the fat that from lemma 3 w e ha v e R ⊥ ≡ 0 , using (2) and equations (6), (8) for an y X , Y ∈ T F n w e obtain: ( ∇ Y b )( X, X ) = ( ∇ Y b )( X, X ) = ( ∇ X b )( X, Y ) = µ ( X ) b ( X , Y ) . Hene, gran ting (15) w e get ( ∇ Y b )( X, X ) = 0 ∀ X ∈ L 1 , ∀ Y ∈ L 2 . DZਬ¥ïï ¯®á«¥¤¥¥ equalit y ¢ (17), ¯à¨å®¤¨¬ ª (16). Distributions L 1 and L 2 are parallel in onnetedness ∇ , if resp etiv ely ∇ Z X ∈ L 1 ∀ X ∈ L 1 , ∀ Z ∈ T F n ¨ ∇ Z Y ∈ L 2 ∀ Y ∈ L 2 , ∀ Z ∈ T F n . (18) Let X ∈ L 1 , Y ∈ L 2 . A t rst w e pro v e that ∇ Y X ∈ L 1 , ∇ X Y ∈ L 2 . F rom (14) w e ha v e: A ξ ( ∇ Y X − ∇ X Y ) = ∇ Y ( A ξ X ) − ∇ X ( A ξ Y ) = ∇ Y ( k 1 X ) . Hene using (16) w e obtain: A ξ ( ∇ Y X − ∇ X Y ) = k 1 ∇ Y X ∀ X ∈ L 1 , ∀ Y ∈ L 2 . (19) W e represen t v etor elds ∇ Y X and ∇ X Y as: ∇ Y X = Z 1 + Z 2 , ∇ X Y = V 1 + V 2 , where Z 1 , V 1 ∈ L 1 , Z 2 , V 2 ∈ L 2 . Then A ξ ( ∇ Y X − ∇ X Y ) = A ξ ( Z 1 − V 1 + Z 2 − V 2 ) = = A ξ ( Z 1 − V 1 ) + A ξ ( Z 2 − V 2 ) = k 1 ( Z 1 − V 1 ) . On the other hand from (19) w e get: A ξ ( ∇ Y X − ∇ X Y ) = k 1 ( Z 1 + Z 2 ) . Therefore, Z 2 = 0 , V 1 = 0 . Hene ∇ Y X ∈ L 1 , ∇ X Y ∈ L 2 ∀ X ∈ L 1 , ∀ Y ∈ L 2 . (20) Then from (15) for ∀ Z ∈ T F n the follo wing equalit y holds g ( ∇ Z X , Y ) + g ( X , ∇ Z Y ) = 0 . If Z ∈ L 1 then using (20) from the last equation w e obtain g ( ∇ Z X , Y ) = 0 . Similarily for Z ∈ 7 Irina I. Bo drenk o. L 2 w e ha v e g ( X , ∇ Z Y ) = 0 . Therefore the follo wing onditions hold: ∇ Z X ∈ L 1 ∀ Z ∈ L 1 and ∇ Z Y ∈ L 2 ∀ Z ∈ L 2 whi h with (20) bring to (18). Therefore, in domain U, w e an in tro due o ordinates ( u 1 , . . . , u n ) su h that v etor elds ∂ ∂ u 1 ¨ ∂ ∂ u 2 , . . . ∂ ∂ u n generate distributions L 1 and L 2 resp etiv ely . Moreo v er, g ∂ ∂ u 1 , ∂ ∂ u 1 ! = g 11 ( u 1 ) , k 1 = k 1 ( u 1 ) , g ∂ ∂ u i , ∂ ∂ u j ! = g ij ( u 2 , . . . , u n ) , g ∂ ∂ u 1 , ∂ ∂ u j ! = 0 , i, j = 2 , n. (21) Then, in domain U, F n is diret Riemmanian pro dut of maximal in tegral manifolds F 1 1 and F n − 1 2 distributions L 1 and L 2 resp etiv ely: F n = F 1 1 × F n − 1 2 . Statemen t 2) of theorem is pro v ed. The pro of of statemen t 3) is ompleted b y the follo wing b ∂ ∂ u 1 , ∂ ∂ u 1 ! = k 1 ( u 1 ) g 11 ( u 1 ) ξ , b ∂ ∂ u i , ∂ ∂ u j ! = 0 , b ∂ ∂ u 1 , ∂ ∂ u j ! = 0 , i, j = 2 , n. (22) Theorem is pro v ed. Note. In domain U determined b y the onditions of theorem 1 1-form µ = d ln | H | . A ording to [2, p. 33℄ w e will all submanifold F n lo  al ly r e duible , if in some neigh b ourho o d of an y its p oin t F n arries the orthogonal onjugate system { L k 1 1 , . . . , L k m m } ( m ≥ 2) su h that all distributions L k i i , i = 1 , m are parallel in onnetedness ∇ . Otherwise submanifold F n is alled nonr e duible . F rom theorem 1 w e ha v e Corollary . L et F n b e irr e duible submanifold in M n + p ( c ) . If F n b elongs to the set R b then F n has p ar al lel the se  ond fundamental form b . Lemma 5. In domain U, determine d by the  onditions of the or em 1 ther e exists the eld of orthonormalize d b asises { n σ } p 1 in T ⊥ F n p ar al lel in normal  onne te dness D and suh that n ρ =  onst , ρ = 2 , p . Pro of. Consider in domain U the eld of orthonormalized basises { n σ } p 1 in T ⊥ F n : normal v etor elds n 1 = ξ and n 2 , . . . , n p generate resp etiv ely distributions ∆ 1 and ∆ 0 . Using 8 On submanifolds with reurren t seond fundamen tal form in spaes of onstan t urv ature. lemma 4, without loss of generalit y w e assume that all normals n σ , σ = 1 , p are parallel in normal onnetedness D . Then in U using the fat A ρ = 0 , ρ = 2 , p , w e ha v e: f ∇ X n ρ = 0 , ∀ X ∈ T F n , ρ = 2 , p. (23) Lemma is pro v ed. Lemma 6. In domain U determine d by the  onditions of the or em 1, the exist lo  al  o or dinates ( v 1 , v 2 , . . . , v n ) suh that ve tor elds ∂ ∂ v i =  onst , i = 2 , n. Pro of. In tro due in domain U the lo al o ordinates ( u 1 , u 2 , . . . , u n ) su h that v etor elds ∂ ∂ u 1 and ∂ ∂ u 2 , . . . ∂ ∂ u n generate distributions L 1 1 and L n − 1 2 resp etiv ely . Then omp onen ts of fundamen tal quadrati forms of F n , in U, tak e the form (21), (22), and from equations (5) w e obtain R ( X , Y ) Z = 0 ∀ X , Y , Z ∈ T F n − 1 2 . (24) Therefore, in domain U w e an in tro due the lo al o ordinates ( v 1 , v 2 , . . . , v n ) : v 1 = u 1 , v i = v i ( u 2 , . . . , u n ) , i = 2 , n, in whi h g ∂ ∂ v 1 , ∂ ∂ v 1 ! = g 11 ( v 1 ) , g ∂ ∂ v 1 , ∂ ∂ v j ! = 0 , g ∂ ∂ v i , ∂ ∂ v j ! = δ ij , i, j = 2 , n, (25) where δ ij is Krone k er sym b ol, b ∂ ∂ v 1 , ∂ ∂ v 1 ! = k 1 ( v 1 ) g 11 ( v 1 ) ξ , b ∂ ∂ v i , ∂ ∂ v j ! = 0 , b ∂ ∂ v 1 , ∂ ∂ v j ! = 0 , i, j = 2 , n. (26) Using (25) and (26), in domain U, w e ha v e: f ∇ X ∂ ∂ v i = 0 ∀ X ∈ T F n , i = 2 , n. (27) Lemma is pro v ed. 9 Irina I. Bo drenk o. Pro of theorem 2. Let x b e arbitrary p oin t in F n . Then x is in some domain U ⊂ F n where the onditions of theorem 1 hold. Let, in domain U, the eld { n σ } p 1 is determined b y lemma 5, and the lo al o ordinates ( v 1 , v 2 , . . . , v n ) are determined b y lemma 6. In tro due, in M n + p ( c ) , in neigh b orho o d of p oin t x, the lo al o ordinates ( y 1 , . . . , y n + p ) . Then F n is giv en lo ally b y the follo wing equation system y a = y a ( v 1 , . . . , v n ) , a = 1 , n + p . F rom (27) w e ha v e that, in domain U, the follo wing onditions hold: ∂ y a ∂ v i = d a i = onst , i = 2 , n, a = 1 , n + p. Then F n , in U, an b e giv en b y the follo wing equations: y a = z a ( v 1 ) + n X i =2 d a i v i , d a i = onst , i = 2 , n, a = 1 , n + p. (28) Equations v 1 = v 1 0 = onst and v i = v i 0 = onst , i = 2 , n determine, in U, submanifolds F n − 1 2 and F 1 1 resp etiv ely . F rom the equalit y (24) w e get that F n − 1 2 is in ternally planar in F n . Sine b ( X, Y ) = 0 ∀ X , Y ∈ T F n − 1 2 then F n − 1 2 is ompletely geo desi in F n . The statemen t 1) of theorem is pro v ed. Pro of of statemen t 2) w e will do separately for ev ery ase: c = 0 , c > 0 , < 0 . Case 1. c = 0 , i.e. F n ⊂ E n + p . In tro due, in E n + p , the Cartesian o ordinates ( x 1 , . . . , x n + p ) , e g ab = δ ab , a, b = 1 , n + p . Let r = { x 1 ( v 1 , v 2 , . . . v n ) , . . . , x n + p ( v 1 , v 2 , . . . v n ) } b e radius v etor of arbitrary p oin t x ∈ U. Then using (23), in domain U, w e ha v e f ∇ X e g ( r, n ρ ) = e g ( X , n ρ ) + e g ( r , f ∇ X n ρ ) = 0 ∀ X ∈ T F n , ρ = 2 , p. Hene, e g ( r , n ρ ) = c ρ = onst , n ρ = onst , ρ = 2 , p. I.e. U is on tained in some ( n + 1) − dimensional plane E n +1 ⊂ E n + p normal to v etors n 2 , . . . , n p . Moreo v er, using (28) represen t radius v etor r as: r ( v 1 , v 2 , . . . , v n ) = R ( v 1 ) + n X i =2 v i d i , where R ( v 1 ) = { z 1 ( v 1 ) , . . . , z n + p ( v 1 ) } , d i = { d 1 i , . . . , d n + p i } = onst , e g ( d i , d j ) = δ ij , i, j = 2 , n. 10 On submanifolds with reurren t seond fundamen tal form in spaes of onstan t urv ature. Therefore, in domain U, using (25) for an y v etor X = X k ∂ ∂ v k w e ha v e: f ∇ X e g ( R, d i ) = e g ∂ r ∂ v 1 , d i ! X 1 = g ∂ ∂ v 1 , ∂ ∂ v i ! X 1 = 0 i = 2 , n. Hene e g ( R, d i ) = b i = onst , i = 2 , n. Therefore radius v etor R of urv e F 1 1 is on tained in some 2-dimensional plane E 2 ⊂ E n +1 normal to v etors d 2 , . . . , d n . Then without loss of generalit y w e ha v e: r ( v 1 , v 2 , . . . , v n ) = { x 1 ( v 1 ) , x 2 ( v 1 ) , v 2 , . . . , v n , 0 , . . . , 0 } . I.e. U is op en part of diret Riemmanian pro dut F 1 1 × E n − 1 ⊂ E n +1 ⊂ E n + p of urv e F 1 1 ⊂ E 2 and ( n − 1) -dimensional plane E n − 1 . Case 2. c > 0 . Consider M n + p ( c ) as h yp ersphere S n + p ( 1 √ c ) in E n + p +1 of radius 1 √ c with en ter at origin of o ordinates. Denote b y e b the seond fundamen tal form S n + p ( 1 √ c ) ¢ E n + p +1 : e b = √ c e g . Let ( x 1 , . . . , x n + p +1 ) b e the Cartesian o ordinates in E n + p +1 , g ∗ ab = δ ab , a, b = 1 , n + p + 1 b e Eulidean metri and ∇ ∗ b e Eulidean onnetedness in E n + p +1 . Let r b e radius v etor of p oin t x ∈ U in E n + p +1 . Denote b y n the unit normal at x on S n + p ( 1 √ c ) in E n + p +1 su h that n = − √ cr . In domain U, for an y X ∈ T F n w e get: ∇ ∗ X n σ = f ∇ X n σ + e b ( X, n σ ) = f ∇ X n σ + √ c e g ( X , n σ ) = f ∇ X n σ , σ = 1 , p. Then using (23) w e ha v e: ∇ ∗ X n ρ = 0 ∀ X ∈ T F n , ρ = 2 , p. (29) I.e. n ρ = onst , ρ = 2 , p, ¢ E n + p +1 . Using (29) for an y X ∈ T F n w e obtain: ∇ ∗ X g ∗ ( r , n ρ ) = g ∗ ( X , n ρ ) + g ∗ ( r , ∇ ∗ X n ρ ) = 0 , ρ = 2 , p. Therefore, g ∗ ( r , n ρ ) = c ρ = onst , n ρ = onst , ρ = 2 , p. (30) Consequen tly an y p oin t x ∈ U is on tained in ( n + 2) -dimensional plane E n +2 ⊂ E n + p +1 normal to v etors n 2 , . . . , n p and parallel to v etor n , and therefore passing through the origin of o ordinates. Hene U ⊂ S n +1 ( 1 √ c ) = S n + p ( 1 √ c ) ∩ E n +2 . Case 3. c < 0 . Let E n + p +1 1 b e ( n + p + 1 ) -dimensional Mink o wski spae with o ordinates ( x 0 , x 1 , . . . , x n + p ) . Pseudo-Eulidean metri g ∗ is determined as in [7, § 48℄: g ∗ = n + p X a,b =0 g ∗ ab dx a dx b = − dx 2 0 + n + p X a =1 dx 2 a . 11 Irina I. Bo drenk o. Let ∇ ∗ b e onnetedness in E n + p +1 1 o ordinated with g ∗ . Consider M n + p ( c ) as pseudosphere H n + p ( i √ − c ) ¢ E n + p +1 1 . Submanifold H n + p ( i √ − c ) ⊂ E n + p +1 1 is giv en as: − x 2 0 + n + p X a =1 x 2 a = 1 c , x 0 > 0 . Let x b e arbitrary p oin t of domain U ⊂ F n ⊂ H n + p ( i √ − c ) . Let r b e radius v etor of p oin t x in E n + p +1 1 . Without loss of generalit y w e an assume that r has the follo wing o ordinates: x 0 = 1 √ − c , x 1 = . . . = x n + p = 0 . Let n b e normal to H n + p ( i √ − c ) in x su h that n = √ − cr and g ∗ ( n, n ) = − 1 , i.e. n is imaginary unit v etor of axis O x 0 . Then ∇ ∗ X n = √ − cX, ∇ ∗ X n σ = f ∇ X n σ ∀ X ∈ T F n . Consequen tly as in the ase 2 w e obtain the onditions (30). Therefore an y p oin t x ∈ U is in ( n + 2) -dimensional Mink o wski spae E n +2 1 on taining the v etor n . Hene U ⊂ E n +2 1 ∩ H n + p i √ − c ! = H n +1 i √ − c ! . Theorem is pro v ed. 12 On submanifolds with reurren t seond fundamen tal form in spaes of onstan t urv ature. Referenes 1. Lumiste U.G. Semisymmetri submanifolds // Problems of geometry . 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