Mean Curvature and Shape Operator of Slant Immersions in a Sasakian Space Form

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1 Mean Cuvatue and Shape Opeato of Slant Immesions in a Sasakian Space Fom Muck Main Tipathi, Jean-Sic Kim and Son-Be Kim Abstact Fo submanifolds, in a Sasakian space fom, which ae tangential to the stuctue vecto field, we establish a basic inequality between squaed mean cuvatue and Ricci cuvatue. Equality cases ae also discussed. Some applications of these esults ae given fo slant, invaiant, anti-invaiant and CR-submanifolds. We also establish an inequality between the shape opeato and the sectional cuvatue fo slant submanifolds in a Sasakian space fom. In paticula, we give simila esults fo invaiant and anti-invaiant submanifolds. Mathematics Subject Classification: 53C0, 53C25. Key wods: Sasakian space fom, invaiant submanifold, anti-invaiant submanifold, slant submanifold, CR-submanifold, totally geodesic submanifold, Ricci cuvatue, sectional cuvatue and squaed mean cuvatue. 1 Intoduction Accoding to B.-Y. Chen, one of the basic poblems in submanifold theoy is to find simple elationships between the main extinsic invaiants and the main intinsic invaiants of a submanifold. In [5], he establishes a elationship between sectional cuvatue function K and the shape opeato fo submanifolds in eal space foms. In [6], he also gives a elationship between Ricci cuvatue and squaed mean cuvatue. A contact vesion of B.-Y. Chen s inequality and its applications to slant immesions in a Sasakian space fom M c ae given in []. In the pesent pape, we continue the study of submanifolds in a Sasakian space fom, which ae tangent to the stuctue vecto field. Necessay details about Sasakian space foms and slant submanifolds ae eviewed in section 2. In section 3, fo those submanifolds in Sasakian space foms which ae tangential to the stuctue vecto field, we establish a basic inequality between Ricci cuvatue and squaed mean cuvatue function. We also discuss equality cases. As applications, we state simila esults fo slant, invaiant, anti-invaiant and CR-submanifolds. In the last section, we establish an inequality between the shape opeato and the sectional cuvatue fo slant submanifolds in a Sasakian space fom. In paticula, we give simila esults fo invaiant and anti-invaiant submanifolds. Balkan Jounal of Geomety and Its Applications, Vol.7, No.1, 2002, pp c Balkan Society of Geometes, Geomety Balkan Pess 2002.

2 102 M.M. Tipathi, J.-S. Kim and S.-B. Kim 2 Peliminaies Let M be a 2m + 1-dimensional almost contact manifold endowed with an almost contact stuctue ϕ, ξ, η, that is, ϕ is a 1, 1 tenso field, ξ is a vecto field and η is 1-fom such that ϕ 2 = I + η ξ and η ξ = 1. Then, ϕ ξ = 0 and η ϕ = 0. The almost contact stuctue is said to be nomal if in the poduct manifold M R the induced almost complex stuctue J defined by J X, λd/dt = ϕx λξ, η X d/dt is integable, whee X is tangent to M, t is the coodinate of R and λ is a smooth function on M R. The condition fo almost contact stuctue to be nomal is equivalent to vanishing of the tosion tenso [ϕ, ϕ] + 2dη ξ, whee [ϕ, ϕ] is the Nijenhuis tenso of ϕ. Let g be a compatible Riemannian metic with the stuctue ϕ, ξ, η, that is, g ϕx, ϕy = g X, Y η X η Y o equivalently, g X, ϕy = g ϕx, Y and g X, ξ = η X fo all X, Y T M. Then, M becomes an almost contact metic manifold equipped with the almost contact metic stuctue ϕ, ξ, η, g. Moeove, if g X, ϕy = dη X, Y, then M is said to have a contact metic stuctue ϕ, ξ, η, g, and M is called a contact metic manifold. A nomal contact metic stuctue in M is a Sasakian stuctue and M is a Sasakian manifold. A necessay and sufficient condition fo an almost contact metic stuctue to be a Sasakian stuctue is 1 X ϕ Y = g X, Y ξ η Y X, X, Y T M, whee is the Levi-Civita connection of the Riemannian metic g. R and S ae equipped with standad Sasakian stuctues. Fo moe details, we efe to [2]. The sectional cuvatue K X ϕx of a plane section spanned by a unit vecto X othogonal to ξ is called a ϕ-sectional cuvatue. If M has constant ϕ-sectional cuvatue c then it is called a Sasakian space fom and is denoted by M c. The cuvatue tenso R of a Sasakian space fom M c is given by 2 R X, Y Z = c + 3 {g Y, Z X g X, Z Y } + + c 1 {g ϕy, Z ϕx g ϕx, Z ϕy 2g ϕx, Y ϕz + η X η Z Y η Y η Z X + g X, Z η Y ξ g Y, Z η X ξ}. Let M be an n + 1-dimensional submanifold immesed in an almost contact metic manifold Mϕ, ξ, η, g. Let g denote the induced metic on M also. We denote by the second fundamental fom of M and by A N the shape opeato associated to any vecto N in the nomal bundle T M. Then g X, Y, N = g A N X, Y fo all X, Y T M and N T M. The Gauss equation is 3 R X, Y, Z, W = R X, Y, Z, W g X, W, Y, Z + g X, Z, Y,W fo all X, Y, Z, W T M, whee R is the induced cuvatue tenso of M. The elative null space of M at a point p M is defined by N p = {X T p M X, Y = 0, fo all Y T p M}.

3 Mean Cuvatue and Shape Opeato of Slant Immesions 103 Let {e 1,..., e } be an othonomal basis of the tangent space T p M. The mean cuvatue vecto H p at p M is H p 1 e i, e i. n + 1 i=1 The submanifold M is totally geodesic in M if = 0; minimal if H = 0; and totally umbilical if X, Y = g X, Y H fo all X, Y T M. We put ij = g e i, e j, e and 2 = i,j=1 g e i, e j, e i, e j, whee e belongs to an othonomal basis {e n+2,..., e } of the nomal space T p M. The scala cuvatue τ p at p M is given by 5 τ p = i<j K e i e j, whee K e i e j is the sectional cuvatue of the plane section spanned by e i and e j. Fo a vecto field X in M, we put ϕx = P X + F X, P X T M, F X T M. Thus, P is an endomophism of the tangent bundle of M and satisfies g X, P Y = g P X, Y fo all X, Y T M. The squaed nom of P is given by P 2 = i,j=1 g e i, P e j 2 fo any local othonomal basis {e 1, e 2,..., e } fo T p M. A submanifold M of an almost contact metic manifold with ξ T M is called a semi-invaiant submanifold [1] o a contact CR submanifold [8] if thee exists two diffeentiable distibutions D and D on M such that i T M = D D E, ii the distibution D is invaiant by ϕ, i.e., ϕd = D, and iii the distibution D is anti-invaiant by ϕ, i.e., ϕd T M. The submanifold M tangent to ξ is said to be invaiant o anti-invaiant [8] accoding as F = 0 o P = 0. Thus, a CR-submanifold is invaiant o anti-invaiant accoding as D = {0} o D = {0}. A pope CR-submanifold is neithe invaiant no anti-invaiant. Fo each non zeo vecto X T p M, such that X is not popotional to ξ p, we denote the angle between ϕx and T p M by θ X. Then M is said to be slant [7],[3] if the angle θ X is constant, that is, it is independent of the choice of p M and X T p M {ξ}. The angle θ of a slant immesion is called the slant angle of the immesion. Invaiant and anti-invaiant immesions ae slant immesions with slant angle θ = 0 and θ = π/2 espectively. A pope slant immesion is neithe invaiant no anti-invaiant.

4 10 M.M. Tipathi, J.-S. Kim and S.-B. Kim 3 Mean cuvatue and Ricci cuvatue Let M be an n + 1-dimensional submanifold in a 2m + 1-dimensional Sasakian space fom Mc tangential to the stuctue vecto field ξ. In view of 2 and 3, it implies that 6 R X, Y, Z, W = c + 3 {g X, W g Y, Z g X, Z g Y, W } + + c 1 {g X, ϕw g Y, ϕz g X, ϕz g Y, ϕw 2g X, ϕy g Z, ϕw g X, W η Y η Z + g X, Z η Y η W g Y, Z η X η W + g Y, W η X η Z} + + g X, W, Y, Z g X, Z, Y, W fo all X, Y, Z, W T M, whee R is the induced cuvatue tenso of M. Thus, we have n H 2 = 2τ n n + 1 c P 2 2n c 1. In [6], B.-Y. Chen established a elationship between Ricci cuvatue and the squaed mean cuvatue fo a submanifold in a eal space fom as follows. Theoem 3.1 Let M be an n-dimensional submanifold in a eal space fom R m c. Then, 1. Fo each unit vecto X T p M, we have 8 H 2 {Ric X n 1 c}. n2 2. If Hp = 0, then a unit vecto X T p M satisfies the equality case of 8 if and only if X lies in the elative null space N p at p. 3. The equality case of 8 holds fo all unit vectos X T p M, if and only if eithe p is a totally geodesic point o n = 2 and p is a totally umbilical point. In this section, we find simila esults fo diffeent kind of submanifolds in a Sasakian space fom. Theoem 3.2 Let M be an n + 1-dimensional submanifold in a -dimensional Sasakian space fom Mc tangential to the stuctue vecto field ξ. Then, i Fo each unit vecto U T p M, we have 9 Ric U n H 2 +n c+3+{3 P U 2 n1 η U 2 1} c 1. ii If Hp = 0, a unit vecto U T p M satisfies the equality case of 9 if and only if U belongs to the elative null space N p. iii The equality case of 9 holds fo all unit vectos U T p M if and only if M is a totally geodesic submanifold.

5 Mean Cuvatue and Shape Opeato of Slant Immesions 105 Poof. We choose an othonomal basis {e 1,..., e, e n+2,..., e } such that e 1,..., e T p M. The squaed second fundamental fom and the squaed mean cuvatue vecto also satisfy 10 2 = 1 2 n + 12 H =n+2 j=2 1j 2 2 =n =n+2 2 i<j ii jj ij 2. Fom 7 and 10, we get 3 P 2 2n c n + 12 H 2 = τ 1 8 nn + 1 c =n+2 =n+2 2 i<j Fom the equation of Gauss we also have ii jj ij 2. =n+2 j=2 1j 2 which gives K e i e j = =n+2 + c 1 ii jj ij 2 + c g e i, P e j 2 η e i 2 η e j 2, 2 i<j Ke i e j = 2m =n+2 2 i<j ii jj ij nn 1 c {3 P 2 6 P e 1 2 2n11ηe 1 2 }c1. Fom 11 and 12, we get 1 n + 12 H 2 = τ 2 i<j 1 n c K e i e j 3 P e 1 2 n 1 η e c 1 + o Ric e 1 = =n { n H 2 + n c =n+2 j=2 1j 2.

6 106 M.M. Tipathi, J.-S. Kim and S.-B. Kim } 3 P e 1 2 n 1 η e c 1 =n =n+2 j=2 1j 2. Since e 1 = X can be chosen to be any abitay unit vecto in T p M, the above equation implies 9. In view of 13, the equality case of 9 is valid if and only if 1 11 = , 12 = = 1 = 0, {n + 2,..., 2m + 1}. If Hp = 0, 1 implies that e 1 = X belongs to the elative null space N p at p. Convesely, if e 1 = X lies in the elative null space, then 1 holds because Hp = 0 is assumed. This poves statement ii. Now, we pove iii. Assume that the equality case of 9 fo all unit tangent vectos to M at p M is tue. Then, in view of 13, fo each {n + 2,..., 2m + 1}, we have 15 2 ii = , i {1,..., n + 1}, ij = 0, i j. Thus, we have two cases, namely eithe n = 1 o n 1. In the fist case p is a totally umbilical point, while in the second case p is a totally geodesic point. Since ξ T M, theefoe each totally umbilical point is totally geodesic. Thus in both the cases, p is a totally geodesic point. The poof of convese pat is staightfowad. The above theoem implies the following thee esults fo slant, invaiant and anti-invaiant submanifolds isometically immesed in a Sasakian space fom. Theoem 3.3 Let M be an n + 1-dimensional θ-slant submanifold isometically immesed in a 2m + 1-dimensional Sasakian space fom Mc such that ξ T M. Then i Fo each unit vecto U T p M, we have 16 Ric U n H 2 + n c {3 cos 2 θ n cos 2 θ η U 2 1} c 1. ii If Hp = 0, a unit vecto U T p M satisfies the equality case of 16 if and only if U N p. iii The equality case of 16 holds fo all unit vectos U T p M if and only if M is a totally geodesic submanifold. Poof. A θ-slant submanifold M of an almost contact metic manifold satisfies 17 g P X, P Y = cos 2 θg ϕx, ϕy, g F X, F Y = sin 2 θg ϕx, ϕy fo all X, Y T M. In view of 17, fo a unit vecto U T p M, we get P U 2 = g P U, P U = cos 2 θ 1 η U 2. Using this in 9, we get 16. Rest of the poof is simila to that of Theoem 3.2.

7 Mean Cuvatue and Shape Opeato of Slant Immesions 107 Theoem 3. Let M be an n + 1-dimensional invaiant submanifold isometically immesed in a 2m + 1-dimensional Sasakian space fom Mc such that ξ T M. Then, i Fo each unit vecto U T p M, we have 18 Ric U n H 2 + n c {2 n + 2 η U 2 } c 1. ii If Hp = 0, a unit vecto U T p M satisfies the equality case of 18 if and only if U N p. iii The equality case of 18 holds fo all unit vectos U T p M if and only if M is a totally geodesic submanifold. Theoem 3.5 Let M be an n + 1-dimensional anti-invaiant submanifold isometically immesed in a 2m + 1-dimensional Sasakian space fom Mc such that ξ T M. Then, i Fo each unit vecto U T p M, we have 19 Ric U n H 2 + n c + 3 {n 1 η U 2 + 1} c 1. ii If Hp = 0, a unit vecto U T p M satisfies the equality case of 19 if and only if U N p. iii The equality case of 19 holds fo all unit vectos U T p M if and only if M is a totally geodesic submanifold. We also have the following Theoem 3.6 Let M be an n + 1-dimensional CR-submanifold in a Sasakian space fom M c. Then, the following statements ae tue. 1. Fo each unit vecto U D, we have 20 Ric U n H 2 + n + 2 c + 3n Fo each unit vecto U D, we have 21 Ric U n H 2 + n 1 c + 3n If Hp = 0, a unit vecto U D esp. D satisfies the equality case of 20 esp. 21 if and only if U N p. Shape opeato fo slant immesion Let M be an n + 1-dimensional θ-slant submanifold in a 2m + 1-dimensional Sasakian space fom Mc such that ξ T M. Let p M and a numbe b > c c 1 cos 2 θ such that the sectional cuvatue K b at p. We choose an othonomal basis {e 1,..., e = ξ, e n+2,..., e } at p such that e n+2 is paallel to the mean cuvatue vecto H, and e 1,..., e diagonalize the shape opeato A n+2. Then we have

8 108 M.M. Tipathi, J.-S. Kim and S.-B. Kim 22 A n+2 = a a , 0 0 a A = 23 ij, tace A = ii = 0, i, j = 1,..., n + 1; = n + 3,..., 2m + 1. i=1 Fo i j, we put 2 u ij a i a j = u ji. In view of Gauss equation 6, fo X = Z = e i, Y = W = e j, we have 25 u ij b c c 1 g e i, P e j 2 =n+3 ii jj ij 2. Now, we pove the following Lemma. Lemma.1 Fo u ij we have 1 Fo any fixed i {1,..., n + 1}, we find u ij n b c c 1 cos 2 θ. i j 2 Fo distinct i, j, k {1, [..., n + ] 1} it follows that ai 2 = u iju ik /u jk. n Fo a fixed k, 1 k and fo each B S k {B {1,..., n + 1} : B = 2 k}, we have u jt k n k + 1 b c c 1 cos 2 θ, j B t B whee B is the complement of B in {1,..., n + 1}. Fo distinct i, j {1,..., n + 1}, it follows that u ij > 0. Poof. 1 Fom 23, 2 and 25, we obtain i j u ij n b c c 1 cos 2 θ = n b c c 1 cos 2 θ = n b c c 1 n b c c 1 cos 2 θ =n+3 =n+3 cos 2 θ + > 0. =n+3 j=1 ii jj j i j i ii ii j =i 2 ij ij 2 = ij 2 =

9 Mean Cuvatue and Shape Opeato of Slant Immesions We have u ij u ik /u jk = a i a j a i a k /a j a k = a 2 i. 3 Let B = {1,..., k} and B = {k + 1,..., n + 1}. Then u jt k n k + 1 b c + 3 j B t B =n+3 k = k n k + 1 k + =n+3 k n k + 1 j=1 t=k+1 3 c 1 [ jj tt jt 2] b c c 1 j=1 t=k+1 cos 2 θ = cos 2 θ 2 k jt + jj j=1 b c c 1 cos 2 θ Fo i j, if u ij = 0 then a i = 0 o a j = 0. The statement a i = 0 implies that u il = a i a l = 0 fo all l {1,..., n + 1}, l i. Then, we get u ij = 0, j i which is a contadiction with 1. Thus, fo i j, it follows that u ij 0. We assume that u 1 < 0. Fom 2, fo 1 < i < n + 1, we get u 1i u i < 0. Without loss of geneality, we may assume u 12,..., u 1l,, u l+1,..., u n > 0, u 1 l+1,..., u 1, u 2,..., u l < 0, fo some [ n 2 + 1] l n. If l = n, then u 1 + u u n < 0, which contadicts to 1. Thus, l < n. Fom 2, we get: 27 a 2 = u i u t u i t > 0, whee 2 i l, l + 1 t n. By 26 and 27, we obtain u it < 0, which implies that l l n l u it = u it + u i + u 1t < 0, i=1 t=l+1 i=2 t=l+1 i=1 t=l+1 which is a contadiction to 3. Thus is poved. B.-Y. Chen establishes a shap elationship between the shape opeato and the sectional cuvatue fo submanifolds in eal space foms [5]. In the following theoem, we establish a simila inequality between the shape opeato and the sectional cuvatue fo slant submanifolds in a Sasakian space fom.

10 110 M.M. Tipathi, J.-S. Kim and S.-B. Kim Theoem.2 Let M be an n + 1-dimensional slant submanifold isometically immesed in a -dimensional Sasakian space fom Mc. If at a point p M thee exists a numbe b > c + 3 / + 3/ c 1 cos 2 θ such that the sectional cuvatue K b at p, then the shape opeato A H at the mean cuvatue vecto satisfies 28 A H > n n + 1 whee I n is the identity map. b c c 1 cos 2 θ I n, at p, Poof. Let p M and a numbe b > c + 3 / + 3/ c 1 cos 2 θ such that the sectional cuvatue K b at p. We choose an othonomal basis {e 1,..., e, e n+2,..., e } at p such that e n+2 is paallel to the mean cuvatue vecto H, and e 1,..., e diagonalize the shape opeato A n+2. Now, fom Lemma.1 it follows that a 1,..., a have the same sign. We assume that a j > 0 fo all j {1,..., n + 1}. Then u ij = a i a a a 2 i n b c c 1 29 cos 2 θ. j i Fom 29 and 22, we obtain a i n + 1 H n b c + 3 which implies that Hence, we get 28. a i H > > n 3 c 1 cos 2 θ b c c 1 cos 2 θ n b c c 1 cos 2 θ. n a 2 i In paticula, the above theoem implies the following two theoems. Theoem.3 Let M be an n + 1-dimensional anti-invaiant submanifold isometically immesed in a 2m + 1-dimensional Sasakian space fom Mc such that ξ T M. If at a point p M thee exists a numbe b > c + 3 / such that the sectional cuvatue K b at p, then the shape opeato A H at the mean cuvatue vecto satisfies A H > n b c I n, at p. n + 1 Theoem. Let M be an n + 1-dimensional invaiant submanifold isometically immesed in a 2m + 1-dimensional Sasakian space fom Mc such that ξ T M. If at a point p M thee exists a numbe b > c such that the sectional cuvatue K b at p, then the shape opeato A H at the mean cuvatue vecto satisfies 31 A H > n n + 1 b c I n, at p. The above equation is same as equation 5.1 of Theoem.1 in the pape of B.-Y. Chen [5]. Acknowledgement. Postdoctoal Reseache at Chonnam National Univesity, Koea in Bain Koea 21. Coodinato Poject: Pofesso Minkyu Kwak.,

11 Mean Cuvatue and Shape Opeato of Slant Immesions 111 Refeences [1] A. Bejancu, Geomety of CR-submanifolds, Mathematics and Its Applications East Euopean Seies, 23. D. Reidel Publishing Co., Dodecht, [2] D. E. Blai, Riemannian geomety of contact and symplectic manifolds, Pogess in Mathematics, 203. Bikhause Boston, Inc., Boston, MA, [3] J. L. Cabeizo, A. Caiazo, L. M. Fenandez and M. Fenandez, Slant submanifolds in Sasakian manifolds. Glasgow Math. J , no. 1, [] A. Caiazo, A contact vesion of B.-Y. Chen s inequality and its applications to slant immesions, Kyungpook Math. J , no. 2, [5] B.-Y. Chen, Mean cuvatue and shape opeato of isometic immesions in eal space fom, Glasgow Math. J , [6] B.-Y. Chen, Relations between Ricci cuvatue and shape opeato fo submanifolds with abitay codimensions, Glasgow Math. J , [7] A. Lotta, Slant submanifolds in contact geomety, Bull. Math. Soc. Roumanie , [8] K. Yano and M. Kon, Stuctues on manifolds, Seies in Pue Mathematics, 3. Wold Scientific Mukut Mani Tipathi Depatment of Mathematics and Astonomy, Lucknow Univesity, Lucknow , India Pesent Addess: Depatment of Mathematics, Chonnam National Univesity, Kwangju , Koea mm tipathi@hotmail.com Jeong-Sik Kim Depatment of Mathematics Education Sunchon National Univesity, Sunchon 50-72, Koea jskim01@hanmi.com Seon-Bu Kim Depatment of Mathematics, Chonnam National Univesity, Kwangju , Koea sbk@chonnam.chonnam.ac.k