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Michigan State University, East Lansing, MI, USA. (Received 24 August 2000; ... manifold which satisfy the equality case of the inequality. Finally we classify ...
Monatsh. Math. 133, 177±195 (2001)

Geometry of Warped Product CR-Submanifolds in Kaehler Manifolds By

Bang-Yen Chen Michigan State University, East Lansing, MI, USA (Received 24 August 2000; in revised form 19 February 2001) Abstract. In this paper we study warped product CR-submanifolds in Kaehler manifolds and introduce the notion of CR-warped products. We prove several fundamental properties of CR-warped products in Kaehler manifolds and establish a general inequality for an arbitrary CR-warped product in an arbitrary Kaehler manifold. We then investigate CR-warped products in a general Kaehler manifold which satisfy the equality case of the inequality. Finally we classify CR-warped products in complex Euclidean space which satisfy the equality. 2000 Mathematics Subject Classi®cation: 53C40, 53C42, 53B25 Key words: CR-submanifolds, warped product, CR-product, CR-warped product, Kaehler manifold

1. Introduction ~ be a Kaehler manifold with complex structure J and let N be a Let M ~ For each x 2 N, denote by Dx Riemannian manifold isometrically immersed in M. the maximal holomorphic subspace of the tangent space Tx N of N. If the dimension of Dx is the same for all x in N; Dx gives a holomorphic distribution D on N. ~ is called a CR-submanifold if there A submanifold N in a Kaehler manifold M exists on N a differentiable holomorphic distribution D whose orthogonal com? plement D? is a totally real distribution, i.e., JD? x  Tx N (cf. [1], [3], [4]) (For a most recent survey on CR-submanifolds, see chapter 17 of [6]). A CR-submanifold is called a totally real submanifold if dim Dx ˆ 0 (cf. [8]). It is called proper if it is neither holomorphic nor totally real. A CR-submanifold N is called anti-holo? morphic if JD? x ˆ Tx N; x 2 N. Throughout this paper, we denote by h the complex rank of the holomorphic distribution D and by p the rank of the totally real distribution D? for CR-submanifolds. ~ is called a CR-product if it is a A CR-submanifold M of a Kaehler manifold M Riemannian product of a holomorphic submanifold NT and a totally real sub~ The notion of CR-products in Kaehler manifolds was introduced manifold N? of M. in [4]. It was proved in [4] that a CR-submanifold M in a complex Euclidean space is a CR-product if and only if it is a Riemannian product of a holomorphic submanifold of a linear complex subspace and a totally real submanifold of a linear complex subspace. Moreover, it was proved in [4] that there do not exist CR Dedicated to Professor Koichi Ogiue on the occasion of his 60th birthday.

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products in complex hyperbolic spaces other than holomorphic and totally real submanifolds. Furthermore, CR-products in complex projective space CPh‡p‡hp are obtained from the Segre embedding in a natural way. The latter was applied to establish in 1981 the ®rst converse theorem to C. Segre's embedding theorem originally published in 1891 (see [4], [5], [7], [13] for details). Let B and F be two Riemannian manifolds with Riemannian metrics gB and gF , respectively, and f > 0 a differentiable function on B. Consider the product manifold B  F with its projection  : B  F ! B and  : B  F ! F. The warped product M ˆ B f F is the manifold B  F equipped with the Riemannian structure such that …1:1† kXk2 ˆ k …X†k2 ‡ f 2 ……x††k …X†k2 2 for any tangent vector X 2 Tx M. Thus, we have g ˆ gB ‡ f gF . The function f is called the warping function of the warped product (cf. [2], [12]). In this paper we introduce a new class of CR-submanifolds in Kaehler manifolds; namely, the class of CR-warped products, and we establish the fundamental theory of such submanifolds. In Section 2 we provide the basic notations and basic formulas. In Section 3 we prove that if M ˆ N? f NT is a warped product CR~ such that N? is a totally real submanifold and submanifold of a Kaehler manifold M ~ then M is a CR-product. By contrast, we NT is a holomorphic submanifold of M, show that there exist many warped product CR-submanifolds NT f N? in Kaehler manifolds which are not CR-products by reversing the two factors NT and N? . For simplicity, we call a warped product CR-submanifold NT  f N? a CR-warped product, whenever N? is a totally real submanifold and NT is a holomorphic ~ (or, more generally, of an almost Hermitian submanifold of a Kaehler manifold M manifold). In Section 4 we prove a basic lemma for CR-warped products and obtain a simple characterization of CR-warped products. In Section 5 we establish a general inequality for an arbitrary CR-warped product in an arbitrary Kaehler manifold. In this section, we also establish the basic structure of CR-warped products which satisfy the equality case of the inequality. The last two sections are devoted to the complete identi®cation of the immersion of CR-warped products in complex Euclidean space satisfying the equality case of the inequality. This is done by a careful analysis of curvature constraints using the language of PDE's. In order to accomplish this aim, we determine in Section 6 the exact solutions of a special PDE system. By applying the exact solutions of such PDE system we are able to completely classify in the last section all CR-warped products in complex Euclidean space which satisfy the equality case of the inequality. 2. Preliminaries Let N be a Riemannian n-manifold with inner product h ; i and e1 ; . . . ; en be an orthonormal frame ®elds on N. For differentiable function ' on N, the gradient r' and the Laplacian ' of ' are de®ned respectively by hr'; Xi ˆ X'; …2:1† n X f…rej ej †' ÿ ej ej 'g …2:2† ' ˆ jˆ1

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for any vector ®eld X tangent to N, where r denotes the Riemannian connection on N. ~ If N is a Riemannian manifold isometrically immersed in a Kaehler manifold M ~m with complex structure J. Then the formulas of Gauss and Weingarten for N in M are given respectively by ~ X Y ˆ rX Y ‡ …X; Y†; …2:3† r ~ X  ˆ ÿA X ‡ DX  r

…2:4†

~ denotes the for any vector ®elds X; Y tangent to N and  normal to N, where r ~  the second fundamental form, D the normal Riemannian connection on M; ~ The second fundamental form and connection, and A the shape operator of N in M. the shape operator are related by hA X; Yi ˆ h…X; Y†; i, where h ; i denotes the ~ inner product on M as well as on M. ~ the equation of Gauss is given For a submanifold N of a Kaehler manifold M, by ~ R…X; Y; Z; W† ˆ R…X; Y; Z; W† ‡ h…X; Z†; …Y; W†i ÿ h…X; W†; …Y; Z†i;

…2:5†

~ denote the for X; Y; Z; W tangent to M and ;  normal to M, where R and R ~ m , respectively. curvature tensors of N and M  with For the second fundamental form , we de®ne its covariant derivative r respect to the connection on TM  T ? M by  X †…Y; Z† ˆ DX ……Y; Z†† ÿ …rX Y; Z† ÿ …Y; rX Z†: …2:6† …r The equation of Codazzi is  X h†…Y; Z† ÿ …r  Y h†…X; Z†; ~ …2:7† …R…X; Y†Z†? ˆ …r ? ~ ~ Y†Z. The equation of where …R…X; Y†Z† denotes the normal component of R…X; Ricci is given by ~ R…X; Y; ; † ˆ R? …X; Y; ; † ÿ h‰A ; A ŠX; Yi:

…2:8†

?

where R is the curvature tensor of the normal connection D. ~ we denote by  the comFor a CR-submanifold N in a Kaehler manifold M, plementary orthogonal subbundle of JD? in the normal bundle T ? N. Hence we have the following orthogonal direct sum decomposition: T ? N ˆ JD?  ;

JD? ? :

…2:9†

We recall the following general lemma from [4] for later use. ~ Then we Lemma 2.1. Let M be a CR-submanifold in a Kaehler manifold M. have (1) hrU Z; Xi ˆ hJAJZ U; Xi, (2) AJZ W ˆ AJW Z; and (3) AJ X ˆ ÿA JX, for any vectors U tangent to M, X, Y in D, Z, W in D? , and  in .

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3. Warped Products N? f NT in Kaehler Manifolds ~ which are In this section we study CR-submanifolds in a Kaehler manifold M warped products of the form N? f NT , where N? is a totally real submanifold and ~ NT is a holomorphic submanifold of M. Theorem 3.1. If M ˆ N? f NT be a warped product CR-submanifold of a ~ such that N? is a totally real submanifold and NT is a Kaehler manifold M ~ then M is a CR-product. holomorphic submanifold of M, Proof. Let M ˆ N? f NT be a warped product CR-submanifold in a Kaehler ~ such that N? is a totally real submanifold and NT is a holomorphic manifold M ~ Since the metric tensor of M is given by g ˆ gN? ‡ f 2 gNT ; N? is submanifold of M. a totally geodesic submanifold of M. Thus, for any vector ®elds Z; W on N? and X on NT , we have …3:1† hrZ W; Xi ˆ 0: ~ is Kaehlerian, the formulas of Gauss and Weingarten Since the ambient space M imply …3:2† ÿAJW Z ‡ DZ …JW† ˆ J…rZ W† ‡ J…Z; W†: Taking the inner product of (3.2) with JX gives hAJW Z; JXi ˆ ÿhrZ W; Xi:

…3:3†

By combining (3.1) and (3.3) we obtain h…D; D? †; JD? i ˆ 0:

…3:4†

On the other hand, from Lemma 7.3 of [2] we have rX Z ˆ rZ X ˆ …Z ln f †X; ?

…3:5† T

T

for any vector ®elds X in D and Z in D . Thus, if we denote by  and A the second fundamental form and the shape operator of NT in M, then we obtain from the formulas of Gauss and Weingarten that hT …X; Y†; Zi ˆ hATZ X; Yi ˆ ÿhrX Z; Yi ˆ ÿ…Z ln f †hX; Yi

…3:6†

?

for any X; Y in D and Z in D . Hence we ®nd T …X; Y† ˆ ÿr…ln †hX; Yi;

…3:7†

where r…ln † is the gradient r…ln † of ln . Equation (3.7) implies that NT is a totally umbilical submanifold of M. ~ Then Let ^ denote the second fundamental form of NT in the ambient space M. ^…X; Y† ˆ T …X; Y† ‡ …X; Y†;

…3:8†

for any X; Y tangent to NT . By applying (3.7) and (3.8) we ®nd h^ …X; X†; Zi ˆ ÿZ…ln †hX; Xi: …3:9† ~ we also have the following Since NT is a holomorphic submanifold of M, relations: ^…X; JY† ˆ ^…JX; Y† ˆ J ^…X; Y†:

…3:10†

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Hence, by combining (3.9) and (3.10), we obtain h^ …X; X†; Zi ˆ ÿh^ …JX; JX†; Zi ˆ Z…ln †hX; Xi:

…3:11†

(3.9) and (3.11) imply Z…ln † ˆ 0. Therefore, we obtain from (3.6) and (3.8) that h^ T …X; Y†; Zi ˆ hT …X; Y†; Zi ˆ 0

…3:12†

for any X; Y in D and Z in D? . Hence, by (3.8), (3.10) and (3.12), we obtain h…X; Y†; JZi ˆ h^ …X; Y†; JZi ˆ ÿh^ …X; JY†; Zi ˆ 0:

…3:13†

h…D; D†; JD? i ˆ 0:

…3:14†

Therefore Conditions (3.4) and (3.14) imply AJD? D ˆ 0. Therefore, by applying Lemma 4.2 & of [4], we conclude that M ˆ N?  NT is a CR-product. 4. A Simple Characterization of CR-Warped Products Theorem 3.1 shows that there do not exist warped product CR-submanifolds in the form N? f NT other than CR-products such that NT is a holomorphic sub~ So, from now on we consider manifold and N? is a totally real submanifold of M. warped product CR-submanifolds in the form: NT f N? , where NT is a holo~ by reversing the morphic submanifold and N? is a totally real submanifold of M two factors NT and N? . We simply call such warped product CR-submanifolds CRwarped products. A CR-warped product NT f N? is said to be trivial if its warping function f is constant. A trivial CR-warped product NT f N? is nothing but a CR-product NT  N?f , where N?f is the manifold with metric f 2 gN? which is homothetic to the original metric gN? on N? . For CR-warped products in Kaehler manifolds we have the following. Lemma 4.1. For a CR-warped product M ˆ NT f N? in any Kaehler manifold M, we have (1) h…D; D†; JD? i ˆ 0; (2) rX Z ˆ rZ X ˆ …X ln f †Z; (3) h…JX; Z†; JWi ˆ …X ln f †hZ; Wi; (4) DX …JZ† ˆ JrX Z; whenever …D; D? †  JD? ; (5) h…D; D? †; JD? i ˆ 0 if and only if M ˆ NT f N? is a trivial CR-warped ~ product in M, where X; Y are vector ®elds on NT and Z; W are on N? ~ is Kaehlerian, we have Proof. Since M JrX Z ‡ J…X; Z† ˆ ÿAJZ X ‡ DX JZ;

…4:1†

for any vector ®elds X; Y on NT and Z in N? . Thus, by taking the inner product of (4.1) and JY, we ®nd hrX Z; Yi ˆ ÿhAJZ X; JYi ˆ ÿh…X; JY†; JZi:

…4:2†

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On the other hand, since M ˆ NT f N? is a warped product, NT is a totally geodesic submanifold of M. Thus, we also have hrX Z; Yi ˆ 0. Combining this with (4.2), we obtain statement (1). Statement (2) is nothing but the statement (2) of Lemma 7.3 in [2]. By applying (3.5), Lemma 2.1 and statement (2), we get h…JX; Z†; JWi ˆ ÿhJAJW Z; Xi ˆ ÿhrZ W; Xi ˆ hrZ X; Wi ˆ …X ln f †hZ; Wi

…4:3†

for X on NT and Z; W on N? . This proves statement (3). Since NT is a totally geodesic submanifold in M; rX Z 2 D? . Thus JrX Z 2 JD? . On the other hand, condition …D; D? †  JD? implies J…X; Z† 2 TM. Therefore, by applying (4.1), we obtain statement (4). Finally, statement (5) follows from statement (3) and the de®nition of trivial CR-warped products. & We have the following simple characterization of CR-warped products. ~ is locally a Theorem 4.2. A proper CR-submanifold M of a Kaehler manifold M CR-warped product if and only if AJZ X ˆ ……JX††Z;

X 2 D;

Z 2 D? ;

…4:4†

?

for some function  on M satisfying W ˆ 0; W 2 D . ~ then Proof. If M is a CR-warped product NT f N? is a Kaehler manifold M, statements (1) and (2) of Lemma 4.1 imply AJZ X ˆ ÿ……JX† ln f †Z for each X 2 D and Z 2 D? . Since f is a function on NT , we also have W…ln f † ˆ 0 for all W 2 D? . Conversely, assume that M is a proper CR-submanifold of a Kaehler manifold ~ satisfying M AJZ X ˆ ……JX††Z;

X 2 D;

Z 2 D? ;

…4:5†

for some function  with W ˆ 0; W 2 D? . Then we have h…D; D†; JD? i ˆ 0;

…4:6†

h…JX; Z†; JWi ˆ ÿ…X†hZ; Wi;

…4:7†

?

for vectors X; Y in D and Z; W in D . Lemma 3.7 of [4,II] and (4.6) implies that the holomorphic distribution D is integrable and its leaves are totally geodesic in M. On the other hand, from Lemma 2.1 and (4.7) we have hZ X; Wi ˆ ÿhrZ W; Xi ˆ ÿhJAJW Z; Xi ˆ h…JX; Z†; JWi ˆ ÿ…X†hZ; Wi

…4:8†

for X in D and Z; W in D? . Since the totally real distribution D? of a CRsubmanifold of a Kaehler manifold is always integrable [4], (4.8) and the condition W ˆ 0; W 2 D? , imply that each leaf of D? is an extrinsic sphere in M, i.e., a totally umbilical submanifold with parallel mean curvature vector. Hence, by

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applying a result of [11] (cf. [10, Remark 2.1]), we know that M is locally the warped product NT f N? of a holomorphic submanifold and a totally real submanifold N? of M, where NT is a leaf of D and N? is a leaf of D? and f is a certain warping function. & 5. A General Inequality for CR-Warped Products ~ with a unit normal vector For a real hypersurface N of a Kaehler manifold M ®eld , the tangent vector ®eld J on N is called a characteristic vector ®eld of N. A unit tangent vector V on N is called a principal vector if V is an eigenvector of the shape operator A ; the corresponding eigenvalue is called the principal curvature at V. For CR-warped products in Kaehler manifolds we have the following. Theorem 5.1. Let M ˆ NT f N? be a CR-warped product in a Kaehler manifold M. We have (1) The squared norm of the second fundamental form of M satis®es kk2 5 2pkr…ln f †k2 ; …5:1† where r ln f is the gradient of ln f and p is the dimension of N? . (2) If the equality sign of (5.1) holds identically, then NT is a totally geodesic ~ Moreover, M is a submanifold and N? is a totally umbilical submanifold of M. ~ minimal submanifold in M. (3) When M is anti-holomorphic and p > 1. The equality sign of (5.1) holds ~ identically if and only if N? is a totally umbilical submanifold of M. (4) If M is anti-holomorphic and p ˆ 1, then the equality sign of (5.1) holds identically if and only if the characteristic vector ®eld J of M is a principal vector ®eld with zero as its principal curvature. (Notice that in this case, M is a real ~ Also, in this case, the equality sign of (5.1) holds identically if hypersurface in M.) ~ and only if M is a minimal hypersurface in M. Proof. From Lemma 4.1, we have h…Z; JX†; JZi ˆ h…JX; Z†; JZi ˆ X ln f …5:2† ? for any unit vector Z in D . Applying (5.2) we obtain inequality (5.1) immediately. For any vector ®elds X in D and Z; W in D? , Lemma 2.1 implies hrW Z; Xi ˆ hJAJZ W; Xi ˆ ÿh…JX; W†; JZi:

…5:3†

Hence, by using statement (2) of Lemma 4.1 and (5.3), we ®nd hrW Z; Xi ˆ ÿ…X ln f †hZ; Wi:

…5:4†

?

On the other hand, if we denote by  the second fundamental form of N ? in M ˆ NT f N? , we get h? …Z; W†; Xi ˆ hrW Z; Xi:

…5:5†

Combining (5.4) and (5.5) yields ? …Z; W† ˆ ÿhZ; Wir ln f

…5:6†

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Now, assume that the equality case of (5.1) holds identically. Then we obtain from (5.2) that …D; D† ˆ 0;

…D? ; D? † ˆ 0;

…D; D? †  JD? :

…5:7†

Since NT is a totally geodesic submanifold in M, the ®rst condition in (5.7) ~ implies that NT is totally geodesic in M. On the other hand, (5.6) shows that N? is totally umbilical in M. Therefore the ~ Moreover, second condition in (5.7) implies that N? is also totally umbilical in M. ~ This proves statement (2). from (5.7), we know that M is minimal in M. To prove statements (3) and (4) let us assume that M is an anti-holomorphic CR~ Then, from statement (1) of Lemma 4.1, we get warped product in M. …D; D† ˆ 0:

…5:8†

~ then there exists a normal vector ®eld H ^ of N? in If N? is totally umbilical in M, ~ satis®es ~ such that the second fundamental form ^ of N? in M M ^ ^…Z; W† ˆ hZ; WiH;

…5:9†

?

for Z; W tangent to N? . Since ^…Z; W† ˆ  …Z; W† ‡ …Z; W†, (5.9) implies that there is a normal vector ®eld  such that …Z; W† ˆ hZ; Wi: Therefore, for each unit vector W 2 D pendicular to W, we have

?

and each unit vector Z in D

h; JWi ˆ h…Z; Z†; JWi ˆ h…Z; W†; JZi ˆ hZ; Wih; JZi ˆ 0

…5:10† ?

per-

…5:11†

where we have applied Lemma 2.1. Since M is assumed to be anti-holomorphic, (5.11) implies either p ˆ 1 or …D? ; D? † ˆ 0:

…5:12†

Hence, (5.2), (5.8) and (5.12) implies the equality case of (5.1) whenever p > 1. ~ In this case, the characteristic When p ˆ 1; M is a real hypersurface of M. vector ®eld J is a principal vector ®eld with zero as its principal curvature if and only if (5.12) holds. So, in this case we also have equality case of (5.1) if the characteristic vector ®eld J is a principal vector ®eld with zero as its principal curvature. Also, from the ®rst condition in (5.7), we also know that condition (5.12) ~ holds if and only if M is minimal in M. By applying statement (2), the converse is easy to verify. & For CR-warped products in complex space forms, we have the following. Proposition 5.2. Let M ˆ NT f N? be a nontrivial CR-warped product satis~ of constant holomorphic fying kk2 ˆ 2pkr ln f k2 in a complex space form M…c† sectional curvature c. We have ~ (a) NT is a totally geodesic holomorphic submanifold of M…c†. Hence NT is a h complex space form N …c† of constant holomorphic sectional curvature c. ~ Hence, N? is a (b) N? is a totally umbilical totally real submanifold of M…c†. real space form of constant sectional curvature, say  > c=4.

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(c) When p ˆ dim N? > 1, the warping function f satis®es krf k2 ˆ  ÿ …c=4† f 2 . Proof. Under the hypothesis, we have (5.7). Statement (a) follows from the ®rst equation of (5.7) and the fact that NT is totally geodesic in M. From the second equation of (5.7) and that N? is totally umbilical in M, we ~ Hence, by (1.1) and the equation of know that N? is totally umbilical in M…4c†. Gauss, we know that N? is of constant curvature, say  5 c=4. From (5.6) we see that  ˆ c=4 occurs only when the warping function is constant. Thus, we have statement (b). Let RN? denote the Riemann curvature tensor of N? . Then we have (cf. [12, p. 210]) R…Z; W†V ˆ RN? …Z; W†V ÿ kr ln f k2 …hW; ViZ ÿ hZ; ViW†; …5:13† for vectors Z; W; V tangent to N? . By applying (1.1), (5.7), (5.13), the equation of Gauss, and statement (b), we obtain statement (c). & 6. Exact Solutions of a Special PDE System We need the exact solution of following PDE system for later use. Proposition 6.1. The solutions  ˆ …x1 ; y1 ; . . . ; xh ; yh † of the following system of partial differential equations: @2 @ @ @ @ ˆ ÿ ; …6:1† @xj @xk @yj @yk @xj @xk @2 @ @ @ @ ˆÿ ÿ ; @xj @yk @yj @xk @xj @yk

j; k ˆ 1; . . . ; h;

…6:2†

@2 @ @ @ @ ˆ ÿ @yj @yk @xj @xk @yj @yk are given by

o 1 n …6:4†  ˆ ln h ; zi2 ‡ hi ; zi2 ; 2 where z ˆ …x1 ‡ iy1 ; . . . ; xh ‡ iyh †; ˆ …a1 ; a3 ‡ ia4 ; . . . ; a2hÿ1 ‡ ia2h † is a vector in Ch and h ; zi denotes the Euclidean inner product of and z in Ch . Proof. From (6.2) we get

  @ @ @ ˆ ÿ2 ln : @x1 @y1 @x1

…6:5†

Solving (6.5) yields @ ˆ eÿ2 …y1 ; x2 ; y2 ; . . . ; xh ; yh †; @y1 ˆ …y1 ; x2 ; y2 ; . . . ; xh ; yh †. Therefore we obtain

for some function 1  ˆ ln……x1 ; x2 ; y2 ; . . . ; xh ; yh † ‡ …y1 ; x2 ; y2 ; . . . ; xh ; yh ††; 2

…6:6†

…6:7†

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for some functions  and  of 2h ÿ 1 variables. From (6.7) we get x1 y1 ; y1 ˆ ; x1 ˆ 2… ‡ † 2… ‡ † x1 x1 ˆ

x1 x1 … ‡ † ÿ x21

…6:8†

:

…6:9†

2… ‡ †x1 x1 ˆ x21 ‡ 2y1 :

…6:10†

2… ‡ †2

By (6.1), (6.8) and (6.9) we obtain Similarly, from (6.3) with j ˆ k ˆ 1 and (6.7), we also have 2… ‡ †y1 y1 ˆ x21 ‡ 2y1 :

…6:11†

By combining (6.10) and (6.11) we ®nd x1 x1 ˆ y1 y1 ˆ 2F…x2 ; y2 ; . . . ; xh ; yh †;

…6:12†

for some positive function F of 2h ÿ 2 variables. Therefore we obtain  ˆ F…x2 ; . . . ; yh †x21 ‡ G…x2 ; . . . ; yh †x1 ‡ H…x2 ; . . . ; yh †;  ˆ F…x2 ; . . . ; yh †y21 ‡ K…x2 ; . . . ; yh †y1 ‡ L…x2 ; . . . ; yh †;

…6:13†

for some functions G; H; K; L of 2h ÿ 2 variables. Substituting (6.13) into (6.10) gives 4F…H ‡ L† ˆ G2 ‡ K 2 . Hence we have o 1 n …2Fx1 ‡ G†2 ‡ …2Fy1 ‡ K†2 : …6:14† ‡ˆ 4F Hence, we get o 1 n …6:15†  ˆ ln …ax1 ‡ †2 ‡ …ay1 ‡ †2 ; 2 p where a…x2 ; . . . ; yh † ˆ F ; …x2 ; . . . ; yh † ˆ 2pGF, and …x2 ; . . . ; yh † ˆ 2pKF. From (6.15) we get a…ax1 ‡ † ; x1 ˆ …ax1 ‡ †2 ‡ …ay1 ‡ †2 a…ay1 ‡ † ; y1 ˆ …ax1 ‡ †2 ‡ …ay1 ‡ †2 …6:16† …ax1 ‡ †…axj x1 ‡ xj † ‡ …ay1 ‡ †…axj y1 ‡ xj † ; xj ˆ …ax1 ‡ †2 ‡ …ay1 ‡ †2 …ax1 ‡ †…ayj x1 ‡ yj † ‡ …ay1 ‡ †…ayj y1 ‡ yj † ; yj ˆ …ax1 ‡ †2 ‡ …ay1 ‡ †2 for j ˆ 2; . . . ; h. Hence, by applying (6.1) for x1 xj , we obtain …axj …2ax1 ‡ † ‡ a xj †……ax1 ‡ †2 ‡ …ay1 ‡ †2 † ÿ a…ax1 ‡ †‰…ax1 ‡ †…axj x1 ‡ xj † ‡ …ay1 ‡ †…axj y1 ‡ xj †Š ˆ a…ay1 ‡ †‰…ax1 ‡ †…ayj x1 ‡ yj † ‡ …ay1 ‡ †…ayj y1 ‡ yj †Š:

…6:17†

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By comparing the coef®cients of x31 and y31 in (6.17) we ®nd axj ˆ ayj ˆ 0 for j ˆ 2; . . . ; h; respectively. Hence a is a positive constant. Therefore (6.16) and (6.17) give xj ˆ yj ˆ

…ax1 ‡ † xj ‡ …ay1 ‡ †xj …ax1 ‡ †2 ‡ …ay1 ‡ †2 …ax1 ‡ † yj ‡ …ay1 ‡ †yj …ax1 ‡ †2 ‡ …ay1 ‡ †2

; ;

…6:18†

…ay1 ‡ † xj ÿ …ax1 ‡ †xj ˆ …ax1 ‡ † yj ‡ …ay1 ‡ †yj :

…6:19†

Since a is constant, (6.18) implies xj yk ˆ

xj yk ‡ xj yk ‡ …ax1 ‡ † xj yk ‡ …ay1 ‡ †xj yk ÿ2

…ax1 ‡ †2 ‡ …ay1 ‡ †2 ‰…ax1 ‡ † xj ‡ …ay1 ‡ †xj Š  ‰…ax1 ‡ † yk ‡ …ay1 ‡ †yk Š ……ax1 ‡ †2 ‡ …ay1 ‡ †2 †2

;

…6:20†

for 2 4 j; k 4 h. Hence, by applying (6.2) with j ˆ k, (6.18) and (6.20), we get ‰ xj yj ‡ xj yj ‡ …ax1 ‡ † xj yj ‡ …ay1 ‡ †xj yj Š  ‰…ax1 ‡ †2 ‡ …ay1 ‡ †2 Š ˆ 0

…6:21†

Therefore, we ®nd xj yj ˆ xj yj ˆ 0 for j ˆ 2; . . . ; h, by comparing the coef®cients of x31 and y31 in (6.21), respectively. Similarly, by using (6.18) and by comparing the coef®cients of x31 and y31 in other equations from (6.1)±(6.3), we may also obtain xj xj ˆ xk yj ˆ xj yk ˆ xj xj ˆ xk yj ˆ zj yk ˆ 0 for 2 4 j; k 4 h. Therefore, there exists constants a3 ; . . . ; a2h ; b3 ; . . . ; b2h such that

ˆ a3 x2 ‡ a4 y2 ‡    ‡ a2hÿ1 xh ‡ a2h yh ;  ˆ b3 x2 ‡ b4 y2 ‡    ‡ b2hÿ1 xh ‡ b2h yh :

…6:22†

Combining (6.15) and (6.22) yields o 1 n  ˆ ln …ax1 ‡ a3 x2 ‡ a4 y2 ‡    ‡ a2h yh †2 ‡ …ay1 ‡ b3 x2 ‡    ‡ b2h yh †2 : 2 …6:23† Finally, from (6.1) with j ˆ 1 and (6.23), we obtain a2kÿ1 ˆ b2k and a2k ˆ ÿb2kÿ1 . Thus, we obtain from (6.23) that 1 n  ˆ ln …ax1 ‡ a3 x2 ‡ a4 y2 ‡    ‡ a2hÿ1 xh ‡ a2h yh †2 2 o ‡ …ay1 ÿ a4 x2 ‡ a3 y2 ÿ    ÿ a2h xh ‡ a2hÿ1 yh †2 :

…6:24†

If we put ˆ …a; a3 ‡ ia4 ; . . . ; a2hÿ1 ‡ ia2h † and z ˆ …x1 ‡ iy1 ; . . . ; xh ‡ iyh †, & then (6.24) becomes (6.4) with a1 ˆ a.

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7. CR-Warped Products in Cm Satisfying kk2 ˆ 2pkr…ln f †k2 The purpose of this section is to prove the following classi®cation theorem. Theorem 7.1. A CR-warped product N ˆ NT f N? in a complex Euclidean mspace Cm satis®es kk2 ˆ 2pkr…ln f †k2 if and only if (1) NT is an open portion of a complex Euclidean h-space Ch , (2) N? is an open portion of the unit p-sphere S p , and (3) up to rigid motions of Cm , the immersion of N  Ch f S p into Cm is given by x…z; w† ˆ

z1 ‡ …w0 ÿ 1†a1

n X

aj zj ;    zh ‡ a…w0 ÿ 1†ah

jˆ1

w1

h X

aj zj ; . . . ; wp

jˆ1

h X

!

h X

aj zj ;

jˆ1

aj zj ; 0; . . . ; 0 ;

jˆ1

z ˆ …z1 ; . . . ; zh † 2 Ch ; where f ˆ

w ˆ …w0 ; . . . ; wh † 2 S p  E p‡1 ;

q ha; zi2 ‡ hia; zi2 ;

…7:1†

…7:2†

for some point a ˆ …a1 ; . . . ; ah † 2 Shÿ1  Eh . Proof. Assume that x : NT f N? ! Cm is a CR-warped product which satis®es kk2 ˆ 2pkr…ln f †k2 . Then, from Lemma 4.1 and Theorem 5.1, we have the following: (i) NT is a totally geodesic holmorphic submanifold of Cm ; (ii) N? is a totally umbilical totally real submanifold of Cm ; (iii) for any X 2 D and Z 2 D? , we have …JX; Z† ˆ …X ln f †JZ;

…7:3†

…D; D† ˆ …D? ; D? † ˆ 0:

…7:4†

and (iv) Applying statement (4) of Lemma 4.1 and (7.3), we ®nd DX JZ ˆ JrX Z

for any X 2 D;

Z 2 D? :

…7:5† m

Since NT is a totally geodesic holomorphic submanifold of C ; NT is an open portion of a linear complex subspace Ch of Cm . Also, since N? is a totally umbilical totally real submanifold of Cm ; N? is an open portion of an ordinary hypersphere Sp lying in a totally real …p ‡ 1†-subspace E p‡1 of Cm (cf. [9]). Without loss of generality, we may assume that the radius of Sp is one. Let z ˆ …z1 ; . . . ; zh † be a natural complex coordinate system on Ch . We put zj ˆ xj ‡ iyj ; j ˆ 1; . . . ; h. The standard ¯at metric on Ch is then given by g0 ˆ dx21 ‡ dy21 ‡    ‡ dx2h ‡ dy2h :

…7:6†

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189

On S p we consider a spherical coordinate system fu1 ; . . . ; up g so that the metric tensor g1 so S p is given by g1 ˆ du21 ‡ cos2 u1 du22 ‡    ‡ cos2 u1    cos2 upÿ1 du2p :

…7:7†

The warped product metric on NT f N? is then given by gˆ

h X …dx2k ‡ dy2k † ‡ f 2 fdu21 ‡ cos2 u1 du22 ‡    ‡ cos2 u1    cos2 upÿ1 du2p g: kˆ1

…7:8† From (7.8) and a straightforward computation we know that the Riemannian connection on NT f N? satis®es @ @ @ ˆ r@ ˆ r@ ˆ 0; j; k ˆ 1; . . . ; h; @xj @x @xj @y @yj @y k k k fx @ @ ˆ j ; j ˆ 1; . . . ; h; t ˆ 1; . . . ; p; r@ @xj @u f @ut t fy @ @ ˆ j ; j ˆ 1; . . . ; h; t ˆ 1; . . . ; p; r@ @yj @u f @ut t @ @ ˆ ÿtan us ; 1 4 s < t 4 p; r@ @us @u @ut t

r@

 tÿ1 h  Y X @ @ @ ˆÿ cos2 us ffxk ‡ ffyk t @ut @xk @yk sÿ1 kˆ1 ! tÿ1 tÿ1 X sin 2uq Y @ cos2 us ; t ˆ 1; . . . ; p: ‡ @uq 2 sˆq‡1 qˆ1

…7:9† …7:10† …7:11† …7:12†

r@u@

…7:13†

Using (7.9)±(7.13) we know that the Riemann curvature tensor of NT f N? satis®es    2  @ @ @ @  @ @ @ ; ˆ ‡ R @xj @ut @xk @xj @xk @xj @xk @ut    2  @ @ @ @  @ @ @ R ; ˆ ‡ @xj @ut @yk @xj @yk @xj @yk @ut    2  @ @ @ @  @ @ @ R ; ˆ ‡ : …7:14† @yj @ut @yk @yj @yk @yj @yk @ut From (7.3) we have



   @ @ @ @ ; J ˆÿ ;  @xj @ut @yj @ut     @ @ @ @ ; J ˆ ;  @yj @ut @xj @ut

…7:15†

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for j ˆ 1; . . . ; h; t ˆ 1; . . . ; p, where  ˆ ln f . Thus, by applying the equation of Gauss, (7.14), and (7.15), we obtain @2 @ @ @ @ ˆ ÿ ; @xj @xk @yj @yk @xj @xk @2 @ @ @ @ ˆÿ ÿ ; @xj @yk @yj @xk @xj @yk

j; k ˆ 1; . . . ; h;

…7:16†

@2 @ @ @ @ ˆ ÿ @yj @yk @xj @xk @yj @yk Therefore, after applying Proposition 6.1, we conclude that there exists a vector ˆ …a1 ; a3 ‡ ia4 ; . . . ; a2hÿ1 ‡ ia2h † in Ch such that o 1 n …7:17†  ˆ ln h ; zi2 ‡ hia; zi2 ; 2 where z ˆ …x1 ‡ iy1 ; . . . ; xh ‡ iyh †. Since  ˆ ln f , (7.17) implies q f ˆ h ; zi2 ‡ hi ; zi2 ; krf k ˆ k k: …7:18† After applying the following rotations: …a2jÿ1 xj ‡ a2j yj ; ÿa2j xj ‡ a2jÿ1 yj † 7!

q a22jÿ1 ‡ a22j …xj ; yj †;

for j ˆ 2; . . . ; h, we obtain from (7.18) that …7:19† f ˆ f…a1 x1 ‡    ‡ ah xh †2 ‡ …a1 y1 ‡    ‡ ah yh †2 g1=2 : Without loss of generality, we may assume a1 6ˆ 0. From the formula of Gauss, (7.4), (7.5), (7.9)±(7.13), and (7.19), we know that the immersion x of NT f N? in Cm satis®es xxj xk ˆ xxj yk ˆ xyj yk ˆ 0; j; k ˆ 1; . . . ; h; aj xxj ut ˆ 2 fa1 …x1 ÿ iy1 † ‡    ‡ ah …xh ÿ iyh †gxut f aj xyj ut ˆ 2 fa1 …y1 ‡ ix1 † ‡    ‡ ah …yh ‡ ixh †gxut f xus ut ˆ ÿ…tan us †xut ; 1 4 s < t 4 p; xut ut ˆ ÿ

tÿ1 Y sˆ1

cos2 us

…7:20† …7:21† …7:22†

h n X ak …a1 x1 ‡    ‡ ah xh †xxk kˆ1

‡ ak …a1 y1 ‡    ‡ ah yh †xyk

o

! tÿ1 tÿ1 X sin 2uq Y cos2 us xuq : ‡ 2 qˆ1 sˆq‡1

…7:24†

Solving (7.20) gives xˆ

h X kˆ1

^ k …u1 ; . . . ; up †xk ‡ A

h X kˆ1

^ k …u1 ; . . . ; up †yk ‡ C…u1 ; . . . ; up † B

…7:25†

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191

^1; . . . ; A ^h; B ^1; . . . ; B ^ h ; C. Substituting (7.25) into (7.21) gives for some functions A ^j ˆ ‰…a1 x1 ‡    ‡ ah xh †2 ‡ …a1 y1 ‡    ‡ ah yh †2 Š  A ut ( ) h h X X ^ k xk ‡ ^ k yk ‡ Cut ; B A aj fa1 …x1 ÿ iy1 † ‡    ‡ ah …xh ÿ iyh †g kˆ1

ut

kˆ1

ut

…7:26† for j ˆ 1; . . . ; h; t ˆ 1; . . . ; p. By comparing the coef®cients of x21 ; xk yl , and x1 in (7.26) respectively, we obtain ^ j ˆ a1 aj A ^1 ; a21 A ut ut

…7:27†

^k ; ^ lu ˆ iaj al A aj ak B ut t

…7:28†

Cu1 ˆ    ˆ Cup ˆ 0

…7:29†

for j; k; l ˆ 1; . . . ; h. Condition (7.29) implies that C is a constant vector in Cm . So, we may choose C ˆ 0 by applying a suitable translation on Cm if necessary. Since a1 6ˆ 0, (7.27) and (7.28) with j ˆ k ˆ 1 imply ^ j ; j ˆ 1; . . . ; h; t ˆ 1; . . . ; p: ^1 ; B ^ j ˆ aj A ^ uj ˆ iA …7:30† A ut ut t a1 ut Solving (7.30) gives

^ 1 ‡ ^j ; B ^ j ‡ ^j ; j ˆ 1; . . . ; h; ^ j ˆ aj A ^ j ˆ iA …7:31† A a1 for some constant vectors ^j ; ^j ; j ˆ 1; . . . ; h. (7.25) and (7.31) show that there exist a Cm -valued function A…u1 ; . . . ; uk † and some vectors j ; j ; j ˆ 1; . . . ; h, in Cm so that x ˆ A…u1 ; . . . ; up †

h X

ak …xk ‡ iyk † ‡

kˆ1

h X … k xk ‡ k yk †

…7:32†

kˆ1

Case (a): p ˆ 1. In this case, by substituting (7.32) into (7.24) with t ˆ 1, we ®nd h h n X X ak …xk ‡ iyk † ˆ ÿ ak …a1 x1 ‡    ‡ ah xh †…ak A ‡ k † Au1 u1 kˆ1

kˆ1

o ‡ ak …a1 y1 ‡    ‡ ah yh †…iak A ‡ k †

…7:33†

which implies Au1 u1 ‡ A ˆ 0; h X kˆ1

ak k ˆ

h X kˆ1

ak k ˆ 0;

…7:34† …7:35†

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B.-Y. Chen

since A is a function depending on u1 . Solving (7.34) gives A ˆ c1 cos u1 ‡ c2 sin u1 ;

…7:36†

m

for some vectors c1 ; c2 in C . Combining (7.32), (7.35) and (7.36) we obtain x…x1 ; y1 ; . . . ; xh ; yh ; u1 † ˆ …c1 cos u1 ‡ c2 sin u1 †

h X

ak …xk ‡ iyk † ‡

kˆ1

h X … k xk ‡ k yk †;

…7:37†

kˆ1

where k ; k ; k ˆ 1; . . . ; h, are vectors in Cm satisfying (7.35). If we choose the following initial conditions: xx1 …1; 0; . . . ; 0† ˆ …1; 0; . . . ; 0; 0; . . . ; 0†;    ; h-th z}|{ xxh …1; 0; . . . ; 0† ˆ …0; . . . ; 0; 1 ; 0; . . . ; 0†; xy1 …1; 0; . . . ; 0† ˆ …i; 0; . . . ; 0; 0; . . . ; 0†;    ; h-th z}|{ xyh …1; 0; . . . ; 0† ˆ …0; . . . ; 0; i ; 0; . . . ; 0†; …h‡1†-th z}|{ xu1 …1; 0; . . . ; 0† ˆ …0; . . . ; 0; 0; a1 ; 0; . . . ; 0†;

…7:38†

then we obtain from (7.37) and (7.38) that

h-th z}|{ a1 c1 ‡ 1 ˆ …1; 0; . . . ; 0†; . . . ; ah c1 ‡ h ˆ …0; . . . ; 0; 1 ; 0; . . . ; 0†; h-th z}|{ ia1 c1 ‡ 1 ˆ …i; 0; . . . ; 0†; . . . ; iah c1 ‡ h ˆ …0; . . . ; 0; i ; 0; . . . 0†; …h‡1†-th z}|{ c2 ˆ …0; . . . ; 0; 1 ; 0; . . .†:

Hence we ®nd c1

1

h 1

h

…h‡1†-th z}|{ 1 ˆ …a1 ; . . . ; ah ; 0; . . . ; 0†; c2 ˆ …0; . . . ; 0; 1 ; 0; . . .† b 1 ˆ …b ÿ a21 ; ÿa1 a2 ; . . . ; ÿa1 ah ; 0; . . . ; 0†; b  1 ˆ …ÿa1 ah ; . . . ; ÿahÿ1 ah ; b ÿ a2h ; 0; . . . ; 0†; b i ˆ …b ÿ a21 ; ÿa1 a2 ; . . . ; ÿa1 ah ; 0; . . . ; 0†; b  i ˆ …ÿa1 ah ; . . . ; ÿahÿ1 ah ; b ÿ a22 ; 0; . . . ; 0†: b

…7:39†

Geometry of Warped Product CR-Submanifolds in Kaehler Manifolds

where b ˆ

Ph

jˆ1

193

a2j . Therefore, we obtain from (7.37) that



x1 ‡ iy1 ‡

h X a1 …cos u1 ÿ 1† ak …xk ‡ iyk †; . . . ; b kˆ1

xh ‡ iyh ‡

h X ah …cos u1 ÿ 1† ak …xk ‡ iyk †; b kˆ1 !

sin u1

h X

ak …xk ‡ iyk †; 0; . . . ; 0 :

…7:40†

kˆ1

From (7.40) we ®nd

  1 hxx1 ; xx1 i ˆ 1 ‡ a21 1 ÿ sin2 u1 : b

…7:41†

Comparing (7.8) with (7.41) yields b ˆ 1. Therefore, we obtain (7.1) with p ˆ 1. Case (b): p > 1. In this case, we obtain krf k2 ˆ 1 by Proposition 5.2. Hence, from (7.18) we obtain a21 ‡ a22 ‡    ‡ a2h ˆ 1. Now, by substituting (7.32) into (7.23), we ®nd Aus ut ˆ ÿ…tan us †Aut ;

1 4 s < t 4 p:

…7:42†

Solving (7.42) for t ˆ p and s ˆ 1 yields A…u1 ; . . . ; up † ˆ D1 …u2 ; . . . ; up †cos u1 ‡ E1 …u1 ; . . . ; upÿ1 †

…7:43†

for some function D and E. Hence, we obtain from (7.32) that x ˆ …D1 …u2 ; . . . ; up †cos u1 ‡ E1 …u1 ; . . . upÿ1 ††

h X

ak …xk ‡ iyk †

kˆ1

‡

h X … k xk ‡ k yk †

…7:44†

kˆ1

Similarly, by substituting (7.44) into (7.23) with s ˆ 2; t ˆ p, we obtain D1 …u2 ; . . . ; up † ˆ D2 …u3 ; . . . up †cos u2 ‡ E2 …u2 ; . . . ; upÿ1 †: Continuing such procedure …p ÿ 1†-times we get n x ˆ Dpÿ1 …up †cos u1    cos upÿ1 ‡ E1 …u1 ; . . . ; upÿ1 † ‡ E2 …u2 ; . . . ; upÿ1 †cos u1 ‡    ‡ Epÿ1 …upÿ1 †cos u1    cos upÿ2

h oX

ak …xk ‡ iyk †

kˆ1

‡

h X … k xk ‡ k yk †: kˆ1

…7:45†

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By substituting (7.45) into (7.24) with t ˆ 1 and by applying a21 ‡ a22 ‡    ‡ a2h ˆ 1, we ®nd @ 2 E1 ‡ E1 ˆ 0; …7:46† @u21 h X

ak k ˆ

kˆ1

h X

ak k ˆ 0:

…7:47†

kˆ1

^ 1 …u2 ; . . . ; upÿ1 †cos u1 ‡ F2 …u2 ; . . . ; upÿ1 †sin u1 . Solving (7.46) gives E1 ˆ F Hence, we obtain from (7.45) that n x ˆ Dpÿ1 …up †cos u1    cos upÿ1 ‡ F1 …u2 ; . . . ; upÿ1 †cos u1 ‡ F2 …u2 ; . . . ; upÿ1 †sin u1 ‡ E3 …u3 ; . . . ; upÿ1 †cos u1 ‡    h oX ak …xk ‡ iyk † ‡ Epÿ1 …upÿ1 †cos u1    cos upÿ2 kˆ1 h X … k xk ‡ k yk †; ‡

…7:48†

kˆ1

^ 1 ‡ E2 . where F1 ˆ F Substituting (7.48) into (7.23) with s ˆ 1; 1 < t < p gives @F2 =@ut ˆ 0. Hence, F2 is a constant vector, say c2. Continuing such procedure suf®ciently many times, we may obtain n x…x1 ; y1 ; . . . ; xh ; yh ; u1 ; . . . ; up † ˆ c1 cos u1    cos up ‡ c2 sin u1 ‡ c3 sin u2 cos u1 ‡    ‡ cp‡1 sin up cos u1 . . . cos upÿ1

h oX

ak …xk ‡ iyk †

kˆ1

‡

h X

… k xk ‡ k yk †

…7:49†

kˆ1

for some vectors c1 ; . . . ; cp‡1 in Cm , where k ; k are vectors satisfying (7.47). Hence, after choosing suitable initial conditions, we obtain (7.1) from (7.49) and (7.50). The converse can be veri®ed by a straightforward long computation. & Remark 7.1. If we choose a ˆ …1; 0; . . . ; 0† 2 Shÿ1 , then the immersion x de®ned by (7.1) reduces to x…z; w† ˆ …z1 w0 ; . . . ; z1 wp ; z2 ; . . . ; zh †;

z 2 Ch ;

w 2 Sp  Ep‡1

…7:10 †

and the warping function f becomes f ˆ jz1 j. It is easy to see from (7.10 ) that, for each u 2 Ch and v 2 Sp , the immersion carries Ch 2 fvg onto a complex linear subspace and fug  Sp onto a hypersphere of radius jz1 j lying in a totally real …p ‡ 1†-plane of Cm . Remark 7.2. There exist many nontrivial CR-warped products NT f N? other ~ is a CR-warped than the ones de®ned by (7.1). For instance, if  : NT f N? ! M

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195

~ then, for any holomorphic submanifold BT of NT product in a Kaehler manifold M, and any submanifold B? of N? , the immersion 

~ BT ^f B?  NT f N? ! M is also a CR-warped product, where ~f is the restriction of f on BT . Remark 7.3. The classi®cation of CR-warped products satisfying the equality kk2 ˆ 2pkr…ln f †k2 in complex projective and complex hyperbolic spaces will be given in a subsequent paper. Acknowledgement. The author is grateful to the referee for several valuable suggestions to improve the presentation of the paper.

References [1] Bejancu A (1986) Geometry of CR-submanifolds. Dordrecht Boston, Mass: Reidel [2] Bishop RL, O'Neill B (1969) Manifolds of negative curvature. Trans Amer Math Soc 145: 1±49 [3] Blair DE, Chen BY (1979) On CR-submanifolds of Hermitian manifolds. Israel J Math 34: 353±363 [4] Chen BY (1981) CR-submanifolds of a Kaehler manifold. J Differential Geometry 16: 305±322; ibid 16: 493±509 [5] Chen BY (1981) Geometry of Submanifolds and Its Applications. Science University of Tokyo [6] Chen BY (2000) Riemannian Submanifolds. In: Dillen E, Verstraelen L (eds) Handbook of Differential Geometry, vol I, pp 187±418. Amsterdam: North Holland [7] Chen BY, Kuan WE (1981) The Segre embedding and its converse. Ann Fac Sci Toulouse Math 7: 1±28 [8] Chen BY, Ogiue K (1974) On totally real submanifolds. Trans Amer Math Soc 193: 257±266 [9] Chen BY, Ogiue K (1974) Two theorems on Kaehler manifolds. Michigan Math J 21: 225±229 [10] Dillen F, NoÈlker S (1993) Semi-parallelity, muti-rotation surfaces and the helix-property. J Reine Angew Math 435: 33±63 [11] Hiepko S (1979) Eine innere Kennzeichung der verzerrten Produkte. Math Ann 241: 209±215 [12] O'Neill B (1983) Semi-Riemannian Geometry with Applictions to Relativity. New York: Academic Press [13] Segre C (1891) Sulle varietaÁ che rappresentano le coppie di punti di due piani o spazi. Rend Cir Mat Palermo 5: 192±204 Author's address: Department of Mathematics, Michigan State University, East Lansing, MI 488241027, USA; e-mail: [email protected]