Acta Mathematica Academiae Paedagogicae Nyíregyháziensis 21 (2005), 79 7 www.emis.de/journals ISSN 176-0091 WARPED PRODUCT SUBMANIFOLDS IN GENERALIZED COMPLEX SPACE FORMS ADELA MIHAI Abstract. B.Y. Chen [5] established a sharp inequality or the warping unction o a warped product submaniold in a Riemannian space orm in terms o the squared mean curvature. Later, in [], he studied warped product submaniolds in complex hyperbolic spaces. In the present paper, we establish an inequality between the warping unction (intrinsic structure) and the squared mean curvature H 2 and the holomorphic sectional curvature c (extrinsic structures) or warped product submaniolds M 1 M 2 in any generalized complex space orm M(c, α). Introduction The notion o warped product plays some important role in dierential geometry as well as in physics [3]. For instance, the best relativistic model o the Schwarzschild space-time that describes the out space around a massive star or a black hole is given as a warped product. One o the most undamental problems in the theory o submaniolds is the immersibility (or non-immersibility) o a Riemannian maniold in a Euclidean space (or, more generally, in a space orm). According to a well-known theorem on Nash, every Riemannian maniold can be isometrically immersed in some Euclidean spaces with suiciently high codimension. Nash s theorem implies, in particular, that every warped product M 1 M 2 can be immersed as a Riemannian submaniold in some Euclidean space. Moreover, many important submaniolds in real and complex space orms are expressed as a warped product submaniold. Every Riemannian maniold o constant curvature c can be locally expressed as a warped product whose warping unction satisies = c. For example, S n (1) is locally isometric to ( π 2, π 2 ) cos t S n 1 (1), E n is locally isometric to (0, ) x S n 1 (1) and H n ( 1) is locally isometric to R e x E n 1 (see [3]). 2000 Mathematics Subject Classiication. 53C0, 53C15, 53C2. Key words and phrases. Generalized complex space orms, warped products, CR-warped products, CR-products, warping unction. Supported by a JSPS postdoctoral ellowship. 79
0 ADELA MIHAI 1. Preliminaries Let M be an almost Hermitian maniold with almost complex structure J and Riemannian metric g. One denotes by the operator o covariant dierentiation with respect to g on M. Deinition 1.1. I the almost complex structure J satisies ( X J)Y + ( Y J)X = 0, or any vector ields X and Y on M, then the maniold M is called a nearly-kaehler maniold [10]. Remark 1.2. The above condition is equivalent to ( X J)X = 0, X ΓT M. For an almost complex structure J on the maniold M, the Nijenhuis tensor ield is deined by N J (X, Y ) = [JX, JY ] J[JX, Y ] J[X, JY ] [X, Y ], or any vector ields X, Y tangent to M, where [, ] is the Lie bracket. A necessary and suicient condition or a nearly-kaehler maniold to be Kaehler is the vanishing o the Nijenhuis tensor N J. Any -dimensional nearly-kaehler maniold is a Kaehler maniold. Example 1.3. Let S 6 be the 6-dimensional unit sphere deined as ollows. Let E 7 be the set o all purely imaginary Cayley numbers. Then E 7 is a 7-dimensional subspace o the Cayley algebra C. Let {1 = e 0, e 1,..., e 6 } be a basis o the Cayley algebra, 1 being the unit element o C. I X = 6 i=0 xi e i and Y = 6 i=0 yi e i are two elements o E 7, one deines the scalar product in E 7 by 6 < X, Y >= x i y i, and the vector product by X Y = i j i=0 x i y j e i e j, being the multiplication operation o C. Consider the 6-dimensional unit sphere S 6 in E 7 : S 6 = {X E 7 < X, X >= 1}. The scalar product in E 7 induces the natural metric tensor ield g on S 6. The tangent space T X S 6 at X S 6 can naturally be identiied with the subspace o E 7 orthogonal to X. Deine the endomorphism J X on T X S 6 by It is easy to see that J X Y = X Y, or Y T X S 6. g(j X Y, J X Z) = g(y, Z), Y, Z T X S 6. The correspondence X J X deines a tensor ield J such that J 2 = I. Consequently, S 6 admits an almost Hermitian structure (J, g). This structure is a non-kaehlerian nearly-kaehlerian structure (its Betti numbers o even order are 0).
WARPED PRODUCT SUBMANIFOLDS IN GENERALIZED COMPLEX SPACE FORMS 1 We will consider a class o almost Hermitian maniolds, called RK-maniolds, which contains nearly-kaehler maniolds. Deinition 1. ([9]). An RK-maniold ( M, J, g) is an almost Hermitian maniold or which the curvature tensor R is invariant by J, i.e. R(JX, JY, JZ, JW ) = R(X, Y, Z, W ), or any X, Y, Z, W ΓT M. An almost Hermitian maniold M is o pointwise constant type i or any p M and X T p M we have λ(x, Y ) = λ(x, Z), where λ(x, Y ) = R(X, Y, JX, JY ) R(X, Y, X, Y ) and Y and Z are unit tangent vectors on M at p, orthogonal to X and JX, i.e. g(x, X) = g(y, Y ) = 1, g(x, Y ) = g(jx, Y ) = g(x, Z) = g(jx, Z) = 0. The maniold M is said to be o constant type i or any unit X, Y ΓT M with g(x, Y ) = g(jx, Y ) = 0, λ(x, Y ) is a constant unction. Recall the ollowing result [9]. Theorem 1.5. Let M be an RK-maniold. Then M is o pointwise constant type i and only i there exists a unction α on M such that λ(x, Y ) = α[g(x, X)g(Y, Y ) (g(x, Y )) 2 (g(x, JY )) 2 ], or any X, Y ΓT M. Moreover, M is o constant type i and only i the above equality holds good or a constant α. In this case, α is the constant type o M. Deinition 1.6. A generalized complex space orm is an RK-maniold o constant holomorphic sectional curvature and o constant type. We will denote a generalized complex space orm by M(c, α), where c is the constant holomorphic sectional curvature and α the constant type, respectively. Each complex space orm is a generalized complex space orm. The converse statement is not true. The sphere S 6 endowed with the standard nearly-kaehler structure is an example o generalized complex space orm which is not a complex space orm. Let M(c, α) be a generalized complex space orm o constant holomorphic sectional curvature c and o constant type α. Then the curvature tensor R o M(c, α) has the ollowing expression [9]: R(X, Y )Z = c + 3α [g(y, Z)X g(x, Z)Y ] (1.1) + c α [g(x, JZ)JY g(y, JZ)JX + 2g(X, JY )JZ]. Let M be an n-dimensional submaniold o a 2m-dimensional generalized complex space orm M(c, α) o constant holomorphic sectional curvature c and constant type α. One denotes by K(π) the sectional curvature o M associated with a plane section π T p M, p M, and the Riemannian connection o M, respectively.
2 ADELA MIHAI Also, let h be the second undamental orm and R the Riemann curvature tensor o M. Then the equation o Gauss is given by (1.2) R(X, Y, Z, W ) = R(X, Y, Z, W ) + g(h(x, W ), h(y, Z)) g(h(x, Z), h(y, W )), or any vectors X, Y, Z, W tangent to M. Let p M and {e 1,..., e n,..., e 2m } an orthonormal basis o the tangent space T p M(c, α), such that e1,..., e n are tangent to M at p. We denote by H the mean curvature vector, that is (1.3) H(p) = 1 h(e i, e i ). n Also, we set (1.) h r ij = g(h(e i, e j ), e r ), i, j {1,..., n}, r {n + 1,..., 2m}. and (1.5) h 2 = g(h(e i, e j ), h(e i, e j )). i,j=1 For any tangent vector ield X to M, we put JX = P X + F X, where P X and F X are the tangential and normal components o JX, respectively. We denote by (1.6) P 2 = g 2 (P e i, e j ). i,j=1 Let M be a Riemannian n-maniold and {e 1,..., e n } be an orthonormal rame ield on M. For a dierentiable unction on M, the Laplacian o is deined by (1.7) = {( ej e j ) e j e j }. j=1 We recall the ollowing result o Chen or later use. Lemma 1.7 ([1]). Let n 2 and a 1,..., a n, b real numbers such that ( n ) 2 ( n ) a i = (n 1) a 2 i + b Then 2a 1 a 2 b, with equality holding i and only i a 1 + a 2 = a 3 =... = a n. 2. Warped product submaniolds Chen established a sharp relationship between the warping unction o a warped product M 1 M 2 isometrically immersed in a real space orm M(c) and the squared mean curvature H 2 (see [5]). In [7], we established a relationship between the warping unction o a warped product M 1 M 2 isometrically immersed in a complex space orm M(c) and the squared mean curvature H 2.
WARPED PRODUCT SUBMANIFOLDS IN GENERALIZED COMPLEX SPACE FORMS 3 Let (M 1, g 1 ) and (M 2, g 2 ) be two Riemannian maniolds and a positive dierentiable unction on M 1. The warped product o M 1 and M 2 is the Riemannian maniold M 1 M 2 = (M 1 M 2, g), where g = g 1 + 2 g 2 (see, or instance, [5]). Let x: M 1 M 2 M(c, α) be an isometric immersion o a warped product M 1 M 2 into a generalized complex space orm M(c, α). We denote by h the second undamental orm o x and H i = 1 n i trace h i, where trace h i is the trace o h restricted to M i and n i = dim M i (i = 1, 2). For a warped product M 1 M 2, we denote by D 1 and D 2 the distributions given by the vectors tangent to leaves and ibres, respectively. Thus, D 1 is obtained rom the tangent vectors o M 1 via the horizontal lit and D 2 by tangent vectors o M 2 via the vertical lit. Let M 1 M 2 be a warped product submaniold o a generalized complex space orm M(c, α) o constant holomorphic sectional curvature c and constant type α. Since M 1 M 2 is a warped product, it is known that (2.1) X Z = Z X = 1 (X)Z, or any vector ields X, Z tangent to M 1, M 2, respectively. I X and Z are unit vector ields, it ollows that the sectional curvature K(X Z) o the plane section spanned by X and Z is given by (2.2) K(X Z) = g( Z X X X Z X, Z) = 1 {( XX) X 2 }. We choose a local orthonormal rame {e 1,..., e n, e n+1,..., e 2m }, such that e 1,..., e n1 are tangent to M 1, e n1+1,..., e n are tangent to M 2, e n+1 is parallel to the mean curvature vector H. Then, using (2.2), we get n1 (2.3) = K(e j e s ), j=1 or each s {n 1 + 1,..., n}. From the equation o Gauss, we have (2.) n 2 H 2 = 2τ + h 2 n(n 1) c + 3α We set (2.5) δ = 2τ n(n 1) c + 3α 3 P 2 c α Then, (2.) can be written as (2.6) n 2 H 2 = 2(δ + h 2 ). 3 P 2 c α. n2 2 H 2. With respect to the above orthonormal rame, (2.6) takes the ollowing orm: ( n ) 2 n ii = 2 δ + ( ii ) 2 + 2m ( ij ) 2 + (h r ij) 2. i j r=n+2 i,j=1
ADELA MIHAI I we put a 1 = h11 n+1, a 2 = n 1 becomes ( 3 ) 2 3 a i = 2 δ + i=2 hn+1 ii a 2 i + and a 3 = n 1 i j n jj kk 2 j k n 1 ( t=n 1+1 hn+1 tt ij ) 2 + 2m, the above equation r=n+2 i,j=1 ss tt n 1+1 s t n Thus a 1, a 2, a 3 satisy the Lemma o Chen (or n = 3), i.e. ( 3 ) 2 ( ) 3 a i = 2 b +. a 2 i (h r ij) 2 Then 2a 1 a 2 b, with equality holding i and only i a 1 +a 2 = a 3. In the case under consideration, this means (2.7) jj kk + ss tt n 1+1 s<t n δ 2 + ( αβ )2 + 1 2m (h r 2 αβ) 2. Equality holds i and only i (2.) n 1 ii = Using again the Gauss equation, we have (2.9) n 2 = τ K(e j e k ) = τ n 1(n 1 1)(c + 3α) 3 c α 2m r=n+1 n 1+1 s<t n 1 α<β n t=n 1+1 2m tt. n 1 +1 s<t n. r=n+2 α,β=1 K(e s e t ) = (h r jjh r kk (h r jk) 2 ) r=n+1 g 2 (Je j, e k ) n 2(n 2 1)(c + 3α) (h r ssh r tt (h r st) 2 ) 3 c α n 1+1 s<t n Combining (2.7) and (2.9) and taking account o (2.3), we obtain (2.10) n 2 n(n 1)(c + 3α) c + 3α τ + n 1 n 2 g 2 (Je j, e k ) 3 c α 3 c α 1 j n 1 ;n 1 +1 t n ( jt ) 2 1 2 δ 2 n 1 +1 s<t n 2m (h r αβ) 2 r=n+2 α,β=1 g 2 (Je s, e t ) g 2 (Je s, e t ).
WARPED PRODUCT SUBMANIFOLDS IN GENERALIZED COMPLEX SPACE FORMS 5 + = τ 2m r=n+2 τ ((h r jk) 2 h r jjh r kk) + n(n 1)(c + 3α) 3 c α r=n+2 + n 1 n 2 c + 3α j=1 2m r=n+2 n 1 +1 s<t n δ 2 r=n+2 2m g 2 (Je j, e k ) 3 c α 2 ( 1 2m n 1 h r jj 1 2m n 2 2 n(n 1)(c + 3α) + n 1 n 2 c + 3α The equality sign o (2.10) holds i and only i ((h r st) 2 h r ssh r tt) r=n+1 1 j n 1;n 1+1 t n n 1+1 s<t n ) 2 t=n 1+1 δ 2 3c α 3 c α h r tt g 2 (Je s, e t ) (h r jt) 2 g 2 (Je j, e k ) g 2 (Je s, e t ). n 1 +1 s<t n (2.10.1) h r jt = 0, 1 j n 1, n 1 + 1 t n, n + 1 r 2m, and (2.10.2) n 1 h r ii = t=n 1 +1 h r tt = 0, n + 2 r 2m. Obviously (2.10.1) is equivalent to the mixed totally geodesicness o the warped product M 1 M 2 (i.e. h(x, Z) = 0, or any X in D 1 and Z in D 2 ) and (2.) and (2.10.2) imply n 1 H 1 = n 2 H 2. Using (2.5), we inally obtain Lemma 2.1. Let x: M 1 M 2 M(c, α) be an isometric immersion o an n- dimensional warped product into a 2m-dimensional generalized complex space orm M(c, α). Then: (2.11) + n 1 + 3 c α n 2 n 2 1 i n 1 n 1 +1 s n where n i = dim M i, i = 1, 2, and is the Laplacian operator o M 1. From the above Lemma, it ollows g 2 (Je i, e s ). Theorem 2.2. Let x: M 1 M 2 M(c, α) be an isometric immersion o an n- dimensional warped product into a 2m-dimensional generalized complex space orm M(c, α). Then: i) I c < α, then (2.12) + n 1. n 2 Moreover, the equality case o (2.12) holds identically i and only i x is a mixed totally geodesic immersion, n 1 H 1 = n 2 H 2, where H i, i = 1, 2, are the partial mean curvature vectors and JD 1 D 2.
6 ADELA MIHAI ii) I c = α, then (2.13) + n 1. n 2 Moreover, the equality case o (2.13) holds identically i and only i x is a mixed totally geodesic immersion and n 1 H 1 = n 2 H 2, where H i (i = 1, 2), are the partial mean curvature vectors. iii) I c > α, then (2.1) + n 1 n 2 + 3 c α P 2. Moreover, the equality case o (2.1) holds identically i and only i x is a mixed totally geodesic immersion, n 1 H 1 = n 2 H 2, where H i, i = 1, 2, are the partial mean curvature vectors and both M 1 and M 2 are totally real submaniolds. A submaniold N in a Kaehler maniold M is called a CR-submaniold i there exists on N a holomorphic distribution D whose orthogonal complementary distribution D is a totally real distribution, i.e., JD x T p N. A CR-submaniold o a Kaehler maniold M is called a CR-product i it is a Riemannian product o a Kaehler submaniold and a totally real submaniold. There do not exist warped product CR-submaniolds o the orm M M, with M a totally real submaniold and M a complex submaniold, other then CR-products. A CR-warped product is a warped product CR-submaniold o the orm M M, by reversing the two actors [2]. Obviously, any CR-warped product submaniold, in particular any CR-product, satisies JD 1 D 2. Corollary 2.3. Let M be an n-dimensional CR-warped product submaniold o a 2m-dimensional generalized complex space orm M(c, α). Then: (2.15) + n 1. n 2 Moreover, the equality case o (2.15) holds identically i and only i x is a mixed totally geodesic immersion, n 1 H 1 = n 2 H 2, where H i, i = 1, 2, are the partial mean curvature vectors. We derive the ollowing non-existence results. Corollary 2.. Let M(c, α) be a generalized complex space orm, M1 an n 1 - dimensional Riemannian maniold and a dierentiable unction on M 1. I there is a point p M 1 such that ()(p) > n 1 c+3α (p), then there do not exist any minimal CR-warped product submaniold M 1 M 2 in M(c, α). Corollary 2.5. Let M(c, α) be a generalized complex space orm, with c > α, M 1 an n 1 -dimensional totally real submaniold o M(c, α) and a dierentiable unction on M 1. I there is a point p M 1 such that ()(p) > n 1 c+3α (p), then there do not exist any totally real submaniold M 2 in M(c, α) such that M 1 M 2 be a minimal warped product submaniold in M(c, α).
WARPED PRODUCT SUBMANIFOLDS IN GENERALIZED COMPLEX SPACE FORMS 7 Reerences [1] B.-Y. Chen. Some pinching and classiication theorems or minimal submaniolds. Arch. Math., 60(6):56 57, 1993. [2] B.-Y. Chen. Geometry o warped product CR-submaniolds in Kaehler maniolds. Monatsh. Math., 133(3):177 195, 2001. [3] B.-Y. Chen. Geometry o warped products as Riemannian submaniolds and related problems. Soochow J. Math., 2(2):125 156, 2002. [] B.-Y. Chen. Non-immersion theorems or warped products in complex hyperbolic spaces. Proc. Japan Acad., Ser. A, 7(6):96 100, 2002. [5] B.-Y. Chen. On isometric minimal immersions rom warped products into real space orms. Proc. Edinb. Math. Soc., II. Ser., 5(3):579 57, 2002. [6] K. Matsumoto and I. Mihai. Warped product submaniolds in Sasakian space orms. SUT J. Math., 3(2):135 1, 2002. [7] A. Mihai. Warped product submaniolds in complex space orms. Acta Sci. Math., 70:19 27, 200. [] S. Nölker. Isometric immersions o warped products. Dier. Geom. Appl., 6(1):1 30, 1996. [9] F. Urbano. CR-Submaniolds o Nearly-Kaehler Maniolds. Doctoral thesis, Granada, 190. [10] K. Yano and M. Kon. Structures on maniolds, volume 3 o Series in Pure Mathematics. World Scientiic, Singapore, 19. Received October 26, 200. Faculty o Mathematics, University o Bucharest, Str. Academiei 1, 01001 Bucharest, Romania E-mail address: adela@math.math.unibuc.ro