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Michigan State University, East Lansing, MI, USA. (Received 13 March ... complex hyperbolic spaces which satisfy the equality of the inequality. In contrast to ...
Monatsh. Math. 134, 103±119 (2001)

Geometry of Warped Product CR-Submanifolds in Kaehler Manifolds, II By

Bang-Yen Chen Michigan State University, East Lansing, MI, USA (Received 13 March 2001; in revised form 10 August 2001) ~ is called a CR-warped product if N is the Abstract. A CR-submanifold N of a Kaehler manifold M ~ warped product NT f N? of a holomorphic submanifold NT and a totally real submanifold N? of M. This notion of CR-warped products was introduced in part I of this series. It was proved in part I that every CR-warped product NT f N? in a Kaehler manifold satis®es a basic inequality: kk2 52p kr…ln f †k2 . The classi®cation of CR-warped products in complex Euclidean space satisfying the equality case of the inequality is archived in part I. The main purpose of this second part of this series is to classify CR-warped products in complex projective and complex hyperbolic spaces 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 ~ 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 comple? mentary distribution D? is a totally real distribution, i.e., JD? x  Tx N. A CRsubmanifold is called proper if it is neither holomorphic nor totally real. It is called anti-holomorphic if JD? ˆ T ? N. It was proved in [2] that the totally real distribution of a CR-submanifold is completely integrable and its holomorphic distribution is a minimal distribution. ~ 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 submani~ The notion of CR-products in Kaehler manifolds was introduced in fold N? of M. [2]. It was proved in [2] 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 [2] that there do not exist CR-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 imbedding in a natural way. The latter was applied to establish in 1981 the ®rst converse theorem to C. Segre's embedding theorem published in 1891 (cf. [10]). (For the most recent surveys on CR-submanifolds and CR-products, see [3, 5].)  Dedicated to Professor Koichi Ogiue on the occasion of his sixtieth birthday.

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In the ®rst part of this series [4], the author introduced and investigated the fundamental properties of a much larger class of CR-submanifolds; namely, the class of CR-warped products which are de®ned as follows: A CR-submanifold N is called a CR-warped product if it is the warped product NT f N? of a holomorphic submanifold NT and a totally real submanifold N? , where f is the warping function. The author proved in [4] that every CR-warped product NT f N? in a Kaehler manifold satis®es the basic inequality: kk2 52pkr…ln f †k2 , where p denotes the dimension of N? , kk2 the squared norm of the second fundamental form, and r…ln f † the gradient of ln f . The author has classi®ed all CR-warped products in complex Euclidean space satisfying the equality of the inequality. The main purpose of the present part is devoted to the complete identi®cation of the immersions of all CR-warped products in complex projective spaces and in complex hyperbolic spaces which satisfy the equality of the inequality. In contrast to CR-products in complex projective spaces, our results imply surprisingly that there exist many CR-warped products in a complex projective space with the smallest possible codimension. Moreover, our results give rise to many explicit new examples of CR-submanifolds in complex hyperbolic spaces. Since there do not exist proper CR-products in complex hyperbolic spaces according to [2], our examples provide the ``simplest'' proper CR-submanifolds in complex hyperbolic spaces given in explicit form. 2. Preliminaries If N is a Riemannian manifold isometrically immersed in a Kaehler manifold ~ with complex structure J. Then the formulas of Gauss and Weingarten are given M respectively by (cf. [1]) ~ X Y ˆ rX Y ‡ …X; Y†; r ~ X ˆ r

A  X ‡ DX 

…2:1† …2:2†

~ denotes the for vector ®elds X; Y tangent to N and  normal to N, where r ~  the second fundamental form, D the normal Riemannian connection on M, connection, and A the shape operator. The equation of Gauss is given by ~ R…X; Y; Z; W† ˆ R…X; Y; Z; W† ‡ h…X; Z†; …Y; W†i h…X; W†; …Y; Z†i

…2:3†

~ denote the curvature tensors of N and for X; Y; Z; W tangent to N, where R and R ~ respectively. M,  with For the second fundamental form , we de®ne its covariant derivative r respect to the connection on TN  T ? N by  X †…Y; Z† ˆ DX ……Y; Z†† …rX Y; Z† …Y; rX Z†: …r …2:4† The equation of Codazzi is  X †…Y; Z† ~ Y†Z†? ˆ …r …R…X;

 Y †…X; Z†; …r ~ Y†Z. ~ Y†Z† denotes the normal component of R…X; where …R…X; ?

…2:5†

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105

~ we denote by  the compleFor a CR-submanifold N in a Kaehler manifold M, mentary orthogonal subbundle of JD? in the normal bundle T ? N. Hence we have the following orthogonal direct sum decomposition: T ? N ˆ JD?  ;

JD? ? :

…2:6†

We recall the following results for later use. ~ Then we Lemma 2.1 ‰2Š: Let N 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 A JX for U tangent to N, X; Y in D, Z; W in D? , and  in . Lemma 2.2 [4]. For a CR-warped product N ˆ NT f N? in any Kaehler mani~ we have fold M, (1) (2) (3) (4)

h…D; D†; JD? i ˆ 0; rX Z ˆ rZ X ˆ …X ln f †Z; h…JX; Z†; JWi ˆ …X ln f †hZ; Wi; DX …JZ† ˆ JrX Z, whenever …D; D? †  JD?,

where X; Y are vector ®elds on NT and Z; W are on N? . For a CR-warped product N ˆ NT f N? , we put h ˆ dimC NT

and p ˆ dimR N?

…2:7†

through out this paper. Theorem 2.3 [4]. Let N ˆ NT f N? be a CR-warped product in a Kaehler ~ We have manifold M. (1) The squared norm of the second fundamental form of N satis®es kk2 52pkr…ln f †k2 ;

…2:8†

where r ln f is the gradient of ln f and p is the dimension of N? . (2) If the equality sign of (2.8) holds identically, then NT is a totally geodesic ~ submanifold and N? is a totally umbilical submanifold of M. (3) When N is anti-holomorphic and p > 1. Then the equality sign of (2.8) ~ holds identically if and only if N? is a totally umbilical submanifold of M. (4) If N is anti-holomorphic and p ˆ 1, then the equality sign of (2.8) holds identically if and only if the characteristic vector ®eld J of N is a principal vector ®eld ~ with zero as its principal curvature, where  is a unit normal vector ®eld of N in M. 3. Anti-Holomorphic CR-Warped Products ~ m …c† of constant holomorphic sectional curvature c is A Kaehler manifold M called a complex space form. Let CPm …c† be a complex projective N-space endowed with the Fubini-Study metric of constant holomorphic sectional

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curvature c > 0 and let Cm denote the complex Euclidean m-space endowed with the usual Hermitian metric. And let CH m …c† be the complex hyperbolic m-space with constant holomorphic sectional curvature c < 0. The following proposition implies that we only need to consider antiholomorphic submanifolds for the classi®cation of CR-warped products in complex space forms satisfying the equality case of the inequality (2.8). Proposition 3.1. Let N ˆ NT f N? be a CR-warped product in a complete ~ m …c†. If N satis®es kk2 ˆ 2pkr…ln f †k2 , simply-connected complex space form M ~ h‡p …c† of then N is contained in a totally geodesic holomorphic submanifold M m ~ …c† as an anti-holomorphic submanifold. M Proof. From Lemma 2.2 we have h…Z; JX†; JZi ˆ h…JX; Z†; JZi ˆ X ln f

…3:1†

?

for any X 2 D and any unit vector Z in D . Thus we have …D; D† ˆ 0;

…D? ; D? † ˆ 0;

2

…D; D? †  JD? ;

…3:2†

2

whenever kk ˆ 2pkr…ln f †k holds. Condition (3.2) implies that the ®rst normal space x ˆ Im x at each point x 2 M satis®es x  JD? x . From statement (4) of Lemma 2.2 and (3.2), we have DX JD?  JD? :

…3:3† ?

On the other hand, for any vector ®elds Z; W in D , we obtain from (3.2) that DZ JW ˆ JrZ W ‡ AJW Z ?

?

…3:4† ?

which implies DZ JD  JD . Combining this with (3.3) we know that JD is a parallel subbundle of the normal bundle T ? M (with respect to the normal connection). Since each ®rst normal space x ˆ Im x of N is contained in JD? and JD? is a parallel normal subbundle, N is thus contained in an …h ‡ p†-di~ h‡p …c† of the complex mensional totally geodesic holomorphic submanifold M m ~ …c† by a reduction theorem. Because the codimension of N in space form M ~ h‡p …c† is equal to the dimension of N? , N is an anti-holomorphic submanifold of M ~ h‡p …c†. & M 4. A General Construction Procedure We ®x notations and at the same time present a procedure to determine the desired submanifolds in a complex projective or complex hyperbolic space. m‡1 Case (1): CPm …4†. Let C ˆ C f0g and Cm‡1 f0g. Consider the  ˆC  m‡1 action of C on C de®ned by   …z0 ; . . . ; zm † ˆ …z0 ; . . . ; zm †. The set of equivalent classes obtained from this action is denoted by CPm. Let …z† denote the m equivalent class contains z. Then  : Cm‡1  ! CP is a surjection. It is known that m the CP admits a complex structure induced from the complex structure on Cm‡1 and a Kaehler metric g with constant holomorphic sectional curvature 4.  ˆ  1 …M† is a Assume : M ! CPm …4† is an isometric immersion. Then M  m‡1 1  C -bundle over M and the lift :  …M† ! C of is an isometric immersion  is invariant under the C -action. satisfying    ˆ  . M

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Conversely, if  : N ! Cm‡1  is an isometric immersion invariant under the   ! CPm …4† satisC -action, then there is a unique isometric immersion : …N† fying    ˆ  . There is an alternate way to view the lift  :  1 …N† ! Cm‡1  via the Hopf ®bration as follows: Let S2m‡1 denote the unit hypersphere of Cm‡1 centered at the  ˆ 1g: Then we have a U…1†-action on S2m‡1 origin and let U…1† ˆ f 2 C :  2m‡1 de®ned by z 7! z. At z 2 S  Cm‡1 , the vector V ˆ iz is tangent to the ¯ow of the action. The quotient space S2m‡1 = , obtained from the U…1†-action, is CPm …4†. The almost complex structure J on CPm …4† is induced from the complex structure J on Cm‡1 via the Hopf ®bration: ' : S2m‡1 ! CPm …4†. ^ ˆ ' 1 …M† is a principal If : M ! CPm …4† is an isometric immersion, then M ^ ! S2m‡1 of is an circle bundle over M with totally geodesic ®bers. The lift ^ : M ^ isometric immersion satisfying '  ˆ  '. ^ ! S2m‡1 is an isometric immersion invariant under the Conversely, if : M ^ ! CPm …4† U…1†-action, then there is a unique isometric immersion ' : '…M† ^ satisfying '  ' ˆ '  '. Since V ˆ iz generates the vertical subspaces of the Hopf ®bration, we have an orthogonal decomposition: T S2m‡1 ˆ …T CPm †  SpanfVg: …4:1† z

'…z†

For each vector X tangent to CPm …4†, we denote by X  a horizontal lift of X via the Hopf ®bration '. The horizontal lift X  and X have the same length, since the Hopf ®bration is a Riemannian submersion.  ˆ  1 …M† is diffeomorphic For an isometric immersion : M ! CPm …4†, M ! ^ where R ˆ R f0g and M ^ ˆ ' 1 …M†. The immersion  : M to R  M m‡1 2m‡1 ^ ^ C is related to the immersion : M ! S by  …t; q† ˆ t ^…q†; t 2 R ; q 2 M: ^ …4:2†  is the cone over M ^ with the origin of Cm‡1 as its vertex. The metric tensor g of M  ^ are related by M and the metric tensor ^g of M g ˆ t2 ^g ‡ dt2 :

…4:3†

Case (2): CH m … 4†. Consider the complex number …m ‡P1†-space C1m‡1 endowed with the pseudo-Euclidean metric g0 ˆ dz0 dz0 ‡ m dzj dzj . Put m‡1  -action on Cm‡1 by   jˆ1 Cm‡1 ˆ C f0g. Consider the C …z ; 0 . . . ; zm † ˆ 1 1 1 …z0 ; . . . ; zm †. The set of equivalent classes obtained from this action is denoted by CH m. Just like CPm , there is an alternate way to view CH m as follows: Let : hz; zi ˆ H12m‡1 ˆ fz ˆ …z1 ; z2 ; . . . ; zm‡1 † 2 Cm‡1 1

1g;

…4:4†

induced from the pseudo-Euclidean metric where h ; i is the inner product on Cm‡1 1 g0 . H12m‡1 is known as the anti-de Sitter space-time. We put Tz0 ˆ fz 2 Cm‡1 : Rehu; zi ˆ Rehu; izi ˆ 0g: 1 Then we have an U…1†-action on H12m‡1 de®ned by z 7! z. At each point z 2 H12m‡1 , the vector iz is tangent to the ¯ow of the action. The orbit lies in the

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negative de®nite plane spanned by z and iz. The quotient space H12m‡1 =  under the U…1†-action is exactly the complex hyperbolic space CH m with constant holomorphic sectional curvature 4. The complex structure J on CH m is induced from the canonical complex structure J on Cm‡1 via the Riemannian submersion: 1 ': H12m‡1 ! CH m … 4†;

…4:5†

which has totally geodesic ®bers. The submersion (4.5) is called the hyperbolic Hopf ®bration.  ˆ ' 1 …M† is Assume : M ! CH m … 4† is an isometric immersion. Then M  m‡1 m‡1  ! C1  C a C -bundle over M and the lift  : M of is the isometric  is invariant under the action of C . immersion satisfying    ˆ  . M  ! Cm‡1 Conversely, if  : M under the 1 is an isometric immersion invariant   ! CH m … 4† satisfying C -action, then there is an isometric immersion : …M†    ˆ  .  ˆ  1 …M† is diffeoFor an isometric immersion : M ! CH m … 4†, M  1 ^ where M ^ ˆ ' …M†. The immersion  : M  ! Cm‡1 morphic to R  M, 1 is 2m‡1 ^ ^ ! H1 related to : M by …t; q† ˆ t ^…q†; t 2 R ; q 2 M: ^ …4:6† 5. CR-Warped Products in CPm In this and the next sections we determine all CR-warped products NT f N? in complex projective and complex hyperbolic spaces which satisfy the basic equality kk2 ˆ 2pkr…ln f †k2. If the warping function f is constant, such a CR-warped product is totally geodesic. Hence, we assume that the CR-warped products are non-trivial, i.e., the warping function is non-constant. ^ ! S2m‡1 and Assume : M ! CPm …4† is an isometric immersion. Let ^ : M m‡1 m‡1 2m‡1 m :M  ! C be the lifts of via ' : S ! CP …4† and  : C ! CPm …4†,  r ^ and r denote the Riemannian respectively, as described in Section 4. Let r;  M ^ and M respectively, and let ^ be the second fundamental connections on M; ^ in S2m‡1 . Then we have form of M ^  Y  ˆ …rX Y† hPX; YiV; …5:1† r X  ^  V ˆ …PX† ; ^ V X ˆ r r …5:2† X

^…X  ; Y  † ˆ ……X; Y†† ;

^ V V ˆ 0; r ^…X  ; V† ˆ …FX† ;

…5:3† ^…V; V† ˆ 0

…5:4†

for vector ®elds X; Y tangent to M, where PX and FX are the tangential and the normal components of JX, respectively. ^  S2m‡1  Cm‡1 For a vector U tangent to M  , we extend U to a vector ®eld in m‡1  over C , also denoted by U, by parallel translation along the rays of the cone M ^ We obtain from (4.2) that M. 1 ^ …5:5† …U; W†…t; q† ˆ ^…U; W†…q†; t 2 R ; q 2 M; t

Geometry of Warped Product CR-Submanifolds in Kaehler Manifolds, II

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    @ @ @ ˆ  ; ˆ0  U; @t @t @t ^ where  is the second fundamental form of . for U; W tangent to M,

…5:6†

Proposition 5.1. Let N ˆ NT f N? be a CR-warped product in CPm …4†. Then we have (1) N ˆ  1 …N† is isometric to NT tf N? , where N T ˆ  1 …NT † and f ˆ f  ,  : Cm‡1 ! CPm …4†.  (2) N is isometrically immersed in Cm‡1  as a CR-warped product such that N T is as a holomorphic submanifold and N? as a totally real submanifold. (3) NT f N? satis®es kk2 ˆ 2pkr…ln f †k2 if and only if N T tf N? satis®es  tf †k2. the corresponding equality k k2 ˆ 2pkr…ln Proof. Assume NT f N? is a CR-warped product in CPm …4†. Denote by D and ^ be D the holomorphic and the totally real distributions of N, respectively. Let D the distribution on N^ spanned by D and V ˆ iz, where D ˆ fX  ; X 2 Dg and X  a horizontal lift of X. Since D is integrable, (5.1)±(5.3) implies that the dis^ is also integrable. From (5.1)±(5.3), we also know that each leaf of D ^ tribution D ^ is a totally? geodesic submanifold of N. ^ ˆ fZ  2 T N^ : Z 2 D? g which is the orthogonal complementary disLet D ^ in T N. ^ For any Z; W 2 D?, (5.1) gives tribution of D ^  W  ˆ …rZ W† : r …5:7† ?

Z

^ ? is an integrable distribution. Since D? is integrable, (5.7) implies that D On the other hand, it is known from [4] that, for any vector ®eld X in D and Z; W in D? , we have

…5:8† hrZ W; Xi ˆ …X ln f †hZ; Wi:  Thus, by (5.7), (5.8), h…rZ W† ; Vi ˆ 0, and the fact that the Hopf ®bration is a Riemannian submersion, we obtain ^  W  ; X  i ˆ …X ln f †hZ  ; W  i; hr Z

^  W  ; Vi ˆ 0: hr Z

…5:9† ? ^ ^ Thus, each leaf of D is an extrinsic sphere in N, i.e., a totally umbilical submanifold with parallel mean curvature vector. Therefore, according to a result ^ N a of Hiepko [8], N^ is a warped product N^ T ^f N? , where N^ T is a leaf of D, ? ^ ^ horizontal lift of N? and f the warping function. From the de®nitions of D, N^ T and ', we may choose N^ T to be ' 1 …NT †. Because the Hopf ®bration ' : S2m‡1 ! CPm …4† is a Riemannian submersion, d' preserves the length of vectors normal to ®bres. In particular, d' preserves the inner product of any two vectors in F? . Therefore, the warping function f^ of N^T ^f N? is given by f  '. Consequently, we obtain N^ ˆ N^T f ' N? , where N? is a horizontal lift of N ? via '. Since N is the punctured cone over N^ with 0 as its vertex, N is nothing but N T tf N? , where N T ˆ  1 …NT †, f ˆ f  , and N ? is a horizontal lift of N? via . Because N ? is isometric to N? , N is thus isometric to N T tf N? . This proves statement (1).

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Since NT is a holomorphic submanifold of CPm …4†, we know from the de®nition of  that N T ˆ  1 …NT † is a holomorphic submanifold of Cm‡1  . Because N? is a totally real submanifold of CPm …4† and N ? is a horizontal lift of N? via , N ? is a totally real submanifold of Cm‡1  . Therefore, N is isometrically immersed in Cm‡1 as a CR-warped product. Thus, we have statement (2).  To prove statement (3), let us assume that N ˆ NT f N? is a CR-warped product in CPm …4† satisfying the basic equality kk2 ˆ 2pkr…ln f †k2. Then, according to Proposition 3.1, N is contained in a totally geodesic holomorphic linear subspace CP h‡p …4† of CPm …4† as an anti-holomorphic submanifold. Without loss of generality, we may put m ˆ h ‡ p. From the proof of Proposition 3.1, we have …D; D† ˆ 0;

…D? ; D? † ˆ 0;

…5:10†

…X; Z† ˆ ……JX† ln f †JZ …5:11† ? for any X 2 D and any unit vector Z in D .  denote the distribution on N spanned by D ^ and @=@t and let D  ? denote Let D  in T N.  ? is spanned by vectors in Cm‡1  Then D the orthogonal distribution of D  ? ^ by parallel translation along rays of the cone N over N. ^ obtained from D From (5.4), (5.5) and the second condition in (5.10) we obtain  ? † ˆ 0:  ?; D …D Also, from (5.4)±(5.6) and the ®rst condition in (5.10), we have  D†  ˆ 0: …D;

…5:12† …5:13†

Applying (5.4)±(5.68) yields 1 ……X; Z†† ; t 1 …V; Z  † ˆ JZ  ; t   @  ;Z ˆ0  @t

…X  ; Z  † ˆ

…5:14† …5:15† …5:16†

 and Z in D? . Hence, by applying (4.6) and (5.11)±(5.16), we ®nd for X in D   1 ‡ kr…ln f †k2 k k2 ˆ 2p : …5:17† t2 On the other hand, from V…tf † ˆ t…V ^f † ˆ 0 and @t@ …tf † ˆ 1=t, it is easy to see  that the gradient of ln…tf † on N satis®es kr…ln tf †k2 ˆ …1 ‡ kr…ln f †k2 †=t2 .  k2 ˆ 2pkr…ln tf †k2 . Hence, N T tf N? satis®es the corresponding equality: k m Conversely, if N ˆ NT f N? is a CR-warped product in CP …4† such that  f †k2 , then  k2 ˆ 2pkrln…t N T tf N? satis®es the corresponding equality: k  D†  ˆ …D  ?; D  ? † ˆ 0; …D; …U; Z  † ˆ

……JU†ln…tf ††JZ 

…5:18† …5:19†

Geometry of Warped Product CR-Submanifolds in Kaehler Manifolds, II

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 and Z  in D ? . Hence we obtain (5.10) from (5.4), (5.5) and (5.18). for any U in D Moreover, from (5.4), (5.5), (5.19), and f ˆ f  , we obtain (5.11). Therefore, the equality kk2 ˆ 2pkr…ln f †k2 must holds for N in CPm …4†. & The following result determines all CR-warped products in CPm …4† satisfying the basic equality. Theorem 5.2. A non-trivial CR-warped product NT f N? in CPm …4† satis®es the basic equality kk2 ˆ 2pkr…ln f †k2 if and only if we have (1) NT is an open portion of complex Euclidean h-space Ch , (2) N? is an open portion of a unit p-sphere Sp , and (3) up to rigid motions, the immersion x of NT f N? into CPm …4† is the composition    x, where  h h X X  x…z; w† ˆ z0 ‡ …w0 1†a0 aj zj ; . . . ; zh ‡ …w0 1†ah aj zj ; jˆ0

w1

h X

aj zj ; . . . ; wp

jˆ0

h X

 aj zj ; 0; . . . ; 0 ;

jˆ0

…5:20†

jˆ0

m  is the projection  : Cm‡1 . . . ; ah are real numbers satisfying a20 ‡  ! CP …4†, a0 ;h‡1 2 2 a1 ‡    ‡ ah ˆ 1, z ˆ …z0 ; z1 ; . . . ; zh † 2 C and w ˆ …w0 ; . . . ; wp † 2 S p  Ep‡1 .

Proof. Follows from Theorem 7.1 of [4] and Propositions 3.1 and 5.1

&

Remark 5.1. It is very interesting to compare Theorem 5.2 with the following result on CR-products from [2]. Theorem 5.3. We have (1) If NT  N? is a CR-product in CPm …4†, then m5h ‡ p ‡ hp. (2) Every CR-product NT  N? in CPh‡p‡hp …4† is a standard CR-product. Remark 5.2. Notice that the essential codimension for the CR-warped products given in Theorem 5.2 is merely p, but the essential codimension for the CRproducts given in Theorem 5.3 is p ‡ 2hp. 6. CR-Warped Products in Ch‡p and in CH m … 4† 1 The main purpose of this section is to determine CR-warped products in complex hyperbolic space satisfying the basic equality. Since complex hyperbolic spaces do not admit proper CR-products according to [2], every CR-warped product in a complex hyperbolic space is non-trivial. ^ ! H12m‡1 and Let : M ! CH m … 4† be an isometric immersion. Let ^ : M 2m‡1 m :M  ! Cm‡1 via ' : H1 ! CH … 4† and  : Cm‡1 1 be the lifts of 1 ! m CH … 4†, respectively (cf. Section 4). Since V ˆ iz generates the time-like vertical subspace of the Riemannian submersion ' : H12m‡1 ! CH m … 4†, we have an orthogonal decomposition: Tz H12m‡1 ˆ …T'…z† CH m †  SpanfVg:

…6:1†

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For each X tangent to CH m … 4† we denote by X  a horizontal lift of X via '.  r ^ and r denote the Riemannian Then X  and X have the same length. Let r;  ^ connections on M; M and M respectively, and let ^ denote the second fundamental ^ in H12m‡1 . Then we have (cf. [7]) form of M ^ V X ˆ r ^  V ˆ …PX† ; ^  Y  ˆ …rX Y† ‡ hPX; YiV; r r X X ^ V V ˆ ^…V; V† ˆ 0; ^…X  ; Y  † ˆ ……X; Y†† ; ^…X  ; V† ˆ …FX† …6:2† r ^  H12m‡1  for X; Y tangent to M. As in Section 5, for each vector U tangent to M m‡1 m‡1 C1 , we extend U to a vector ®eld in C1 by parallel translation along rays of  over M. ^ We have the cone M 1 ^ …6:3† …U; W†…t; q† ˆ ^…U; W†…q†; t 2 R ; q 2 M; t     @ @ @ ˆ  ; ˆ0 …6:4†  U; @t @t @t ~ be the connection of Cm‡1 . From (4.6), (6.2)±(6.4) ^ Let r for U; W tangent to M.  ^ is contained in H12m‡1 , we obtain and the fact that M ~  Y  ˆ 1 f…rX Y† ‡ …X; Y† ‡ hJX; Yiiz ‡ hX; Yizg; r X t 1 ~ ~VV ˆ z ; rX V ˆ …JX† ; r t t @   ~@X ˆ X ; r ~@ @ ˆ 0 ~ V @ ˆ V; r ~  ˆr …6:5† r X @t @t @t @t @t  on M  satis®es for X; Y tangent to M. (6.5) implies that the connection r     Y  ˆ 1 …rX Y† ‡ hJX; YiV ‡ hX; Yi @ ; r X t @t 1 1 @ VV ˆ  V ˆ r  V X  ˆ …PX† ; r ; r X t t @t   @ ˆr  @ X ˆ X; r  @ V ˆ V; r V @ ˆ r  @ @ ˆ 0: r …6:6† X @t @t @t @t @t @t We need the following. Proposition 6.1. Let N ˆ NT f N? be a CR-warped product in CH m … 4†. Then we have (1) N ˆ  1 …N† is isometric to N T tf N? , where NT ˆ  1 …NT † and f ˆ f  . (2) N is isometrically immersed in Cm‡1 N T is as a holomorphic 1 such thatm‡1 submanifold and N? as a totally real submanifold of C1 . (3) NT f N? satis®es kk2 ˆ 2pkr…ln f †k2 if and only if N T tf N? is immersed as an anti-holomorphic submanifold in a holomorphic linear subspace Ch‡p‡1 of Cm‡1 1 1 whose second fundamental form  satis®es  ? † ˆ 0;  D†  ˆ …D  ?; D …6:7† …D;  and D  ? are the holomorphic and the totally real distributions. where D (4) dim N T ˆ 2 ‡ dim NT and N T is a semi-Riemannian manifold of index 2.

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Proof. Since statements (1) and (2) can be proved in the same way as statements (1) and (2) of Proposition 5.1 with only minor modi®cation, so we omit their proof. For statement (3), let us assume that the CR-warped product NT f N? satis®es the equality kk2 ˆ 2pkr…ln f †k2. Then Proposition 3.1 implies that NT f N? is contained in a totally geodesic holomorphic submanifold, say CH h‡p … 4† of CH m … 4†. Since N T tf N? is the preimage of NT f N? via the projection m  : Cm‡1 … 4†, N T tf N? is immersed into a holomorphic linear subspace  ! CH h‡p‡1 m‡1 C1 of C1 . From the proof of Proposition 3.1, we have …D; D† ˆ …D? ; D? † ˆ 0: …6:8† Thus, by applying (6.2), (6.3), (6.4) and (6.8), we obtain (6.7).  T   N? is contained in a holomorphic linear subConversely, assume that N tf h‡p‡1 space C1 as an anti-holomorphic submanifold satisfying (6.7). Then NT f N? ˆ …N T tf N? † is an anti-holomorphic submanifold in CPh‡p … 4†. From (6.2), (6.3), and (6.7), we have (6.8). On the other hand, since N is an anti-holomorphic CR-warped product in CPh‡p … 4†, we also have …X; Z† ˆ ……JX†ln f †JZ …6:9† for X in D and Z in D? . Hence, from (6.8) and (6.9) we obtain kk2 ˆ 2p kr…ln f †k2 . This proves statement (3). Statement (4) follows from the de®nition of N T . & In view of Proposition 6.1, we investigate anti-holomorphic CR-warped products BT f B? in Ch‡p which satisfy the following three conditions: 1 (i) B? is a p-dimensional Riemannian manifold and BT is a (real) 2hdimensional semi-Riemannian manifold of index 2. (ii) BT is immersed as a holomorphic submanifold and B? as a totally real submanifold of Ch‡p 1 . (iii) The second fundamental form  of BT f B? in Ch‡p satis®es 1 ? ? …D; D† ˆ …D ; D † ˆ 0; …6:10† ? where D and D are the holomorphic and totally real distributions. The following result classi®es all anti-holomorphic submanifolds in Ch‡p 1 which satisfy conditions (i), (ii) and (iii). Proposition 6.2. Let B ˆ BT f B? be an anti-holomorphic CR-warped product in Ch‡p with f being a non-constant function. Then B satis®es conditions 1 (i), (ii) and (iii) if and only if one of the following two cases occurs: (1) BT is an open portion of Ch1 , B? is an open portion of a unit p-sphere S  Ep‡1 and, up to rigid motions of Ch‡p 1 , the immersion is given by  h h X X aj zj ; z2 ‡ a2 …w0 1† aj zj ; . . . ; x…z; w† ˆ z1 ‡ a1 …1 w0 † p

zh ‡ ah …w0

1†

jˆ1 h X jˆ1

aj zj ; w1

h X jˆ1

jˆ1

aj zj ; . . . ; wp

h X jˆ1

 aj zj

…6:11†

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for some real numbers a1 ; . . . ; ah satisfying a21 a22    a2h ˆ 1, where z ˆ …z1 ; . . . ; zh † 2 Ch1 and w ˆ …w0 ; . . . ; wp † 2 Sp  Ep‡1 . (2) p ˆ 1, BT is an open portion of Ch1 and, up to rigid motions of Ch‡p 1 , the immersion is given by  h h X X x…z; t† ˆ z1 ‡ a1 …cosh t 1† aj zj ; z2 ‡ a2 …1 cosh t† aj zj ; . . . ; zh jˆ1

‡ ah …1

cosh u1 †

h X

aj zj ; sinh t

jˆ1

h X

jˆ1

 aj zj ;

…6:12†

jˆ1

for some real numbers a1 ; . . . ; ah satisfying a21

a22



a2h ˆ 1.

Proof. Assume  : B ˆ BT f B? ! Ch‡p is a non-trivial anti-holomorphic 1 submanifold satisfying conditions (i), (ii) and (iii). Then the normal bundle of B is space-like. Let D and D? denote the holomorphic and the totally real distributions on B. For any vector ®elds X in D and Z; W in D? , we have JrZ W ‡ J…Z; W† ˆ

AJW Z ‡ DZ JW:

…6:13†

By taking the inner product of (6.13) with JX, we ®nd hrZ W; Xi ˆ

hAJW Z; JXi ˆ

h…JX; Z†; JWi:

…6:14†

Since B ˆ BT f B? is a warped product, we have [9, p.206] rX Z ˆ rZ X ˆ …X ln f †Z:

…6:15†

By applying (6.14) and (6.15), we get h…JX; Z†; JWi ˆ

hrZ W; Xi ˆ hrZ X; Wi ˆ …X ln f †hZ; Wi:

…6:16†

Since BT f B? is anti-holomorphic, (6.16) implies …X; Z† ˆ

……JX†ln f †JZ

…6:17†

for any vector ®elds X in D and Z in D? . Because BT is totally geodesic in BT f B? and BT is holomorphic in Cm‡1 , 1 (6.8) implies that BT is a holomorphic totally geodesic submanifold of Cm‡1 . 1 Hence, BT is an open portion of a linear complex subspace Ch1 of Ch‡p . Also, since 1 B? is totally umbilical in BT f B? and B? is totally real in Ch‡p 1 , (6.8) implies that B? is totally real and totally umbilical in Ch‡p 1 . Hence, B? is an open portion of an ordinary hypersphere Sp . Without loss of generality, we may assume the radius of Sp is one. Let z ˆ …z1 ; . . . ; zh † be a natural complex coordinate system on Ch1 . We put zj ˆ xj ‡ iyj ; j ˆ 1; . . . ; h. The standard ¯at metric on Ch1 is given by g0 ˆ

dx21

dy21 ‡

h X …dx2j ‡ dy2j †:

…6:18†

jˆ2

By choosing a spherical coordinate system fu1 ; . . . ; up g on Sp , we have g ˆ g0 ‡ f 2 fdu21 ‡ cos2 u1 du22 ‡    ‡ cos2 u1    cos2 up 1 du2p g:

…6:19†

Geometry of Warped Product CR-Submanifolds in Kaehler Manifolds, II

From (6.19) we obtain @ @ @ ˆ r@x@ ˆ r@@y ˆ 0; r@x@ j @xk j @ yk j @ yk

115

j; k ˆ 1; . . . ; h;

…6:20†

r@

fx @ @ ˆ j ; @ut f @ut

j ˆ 1; . . . ; h;

t ˆ 1; . . . ; p;

…6:21†

r@

fy @ @ ˆ j ; f @ut @ut

j ˆ 1; . . . ; h;

t ˆ 1; . . . ; p;

…6:22†

@xj

@ yj

@ @ ˆ tan us ; 14s < t4p; @ut @ut   t 1 h  Y X @ @ @ @ @ 2 ˆf cos us fx1 ‡ fy 1 ‡ fyk r@ fx k @ut @u @x1 @y1 kˆ2 @xk @yk t sˆ1   t 1 t 1 X sin 2uq Y @ ‡ cos2 us ; t ˆ 1; . . . ; p @u 2 q qˆ1 sˆq‡1 r@

@us

…6:23†

…6:24†

which are exactly (7.9)±(7.13) of [4]. Also, from (6.17)±(6.19), we have         @ @ @ @ @ @ @ @ ˆ ;  ˆ ; …6:25† ; J ; J  @xj @ut @yj @ut @yj @ut @xj @ut for j ˆ 1; . . . ; h; t ˆ 1; . . . ; p, where  ˆ ln f . Thus we may obtain @2 @ @ ˆ @xj @xk @yj @yk @2 ˆ @xj @yk

@ @ @yj @xk

@2 @ @ ˆ @yj @yk @xj @xk

@ @ ; @xj @xk @ @ ; @xj @yk

j; k ˆ 1; . . . ; h;

@ @ @yj @yk

…6:26†

as we did in [4]. Therefore, we may apply Proposition 6.1 of [4] to obtain q …6:27† f ˆ h ; zi2 ‡ hi ; zi2 : for some ˆ …a1 ; . . . ; ah † 2 Eh where z ˆ …z1 ; . . . ; zh † 2 Ch . By applying the formula of Gauss and (6.18)±(6.27), we know that the immersion x of NT f N? in Ch‡p satis®es 1 xxj xk ˆ xxj yk ˆ xyj yk ˆ 0;

j; k ˆ 1; . . . ; h;

aj fa1 …x1 iy1 † ‡    ‡ ah …xh iyh †gxut ; f2 aj ˆ 2 fa1 …y1 ‡ ix1 † ‡    ‡ ah …yh ‡ ixh †gxut ; f

…6:28†

xxj ut ˆ

…6:29†

xyj ut

…6:30†

xus ut ˆ

…tan us †xut ;

14s < t4p;

…6:31†

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xut ut ˆ

t 1 Y sˆ1

‡

  cos us …a1 x1 ‡    ‡ ah xh † a1 xx1 2

 ‡ …a1 y1 ‡    ‡ ah yh † a1 xy1

t 1  t 1 X sin 2uq Y

2

qˆ1

2

sˆq‡1



h X kˆ2

 ak xxk

h X kˆ2

 ak xyk

cos us xuq :

…6:32†

After solving (6.28) and (6.29) as we did in [4], we ®nd h h X X x…z; u1 ; . . . ; up † ˆ A…u1 ; . . . ; up † ak zk ‡ … k xk ‡ k yk †; kˆ1

kˆ1

z ˆ …z1 ; . . . ; zh † 2 Ch1 ;

…6:33†

for some vector-valued function A and vectors j ; j ; j ˆ 1; . . . ; h, in Ch‡p 1 . Case (a): p ˆ 1. Substituting (6.33) into (6.32) with t ˆ 1 yields   h h X X Au1 u1 ak zk ˆ …a1 x1 ‡‡ah xh † a1 …a1 A‡ 1 † ak …ak A‡ k † kˆ1

kˆ2

  h X ak …iak A‡k † ‡…a1 y1 ‡‡ah yh † a1 …ia1 A‡1 †

…6:34†

kˆ2

which implies Au1 u1 ‡ A ˆ 0; a1 1

h X

ˆ

a21 ‡ a22 ‡    ‡ a2h ;

ak k ˆ a1 1

kˆ2

h X

ak k ˆ 0:

…6:35† …6:36†

kˆ2

CASE (i): > 0. In this case, we put ˆ b2 , b > 0. We may assume a2 6ˆ 0. By solving (6.35), we get A ˆ c1 cos…bu1 † ‡ c2 sin…bu1 †; for some vectors c1 ; c2 in

Ch‡1 1 .

…6:37†

Combining (6.33), (6.36) and (6.37) we ®nd

x…z; u1 † ˆ …c1 cos…bu1 † ‡ c2 sin…bu1 ††

h X

ak zk ‡

kˆ1

h X … k xk ‡ k yk †;

…6:38†

kˆ1

satisfying (6.36). where k ; k ; k ˆ 1; . . . ; h; are vectors in Ch‡1 1 After choosing the following initial conditions: j-th z}|{ xxj …0; 1; 0; . . . ; 0† ˆ …0; . . . ; 0; 1 ; 0; . . . ; 0; 0†; xyj …0; 1; 0; . . . ; 0† ˆ ixx1 …0; 1; 0; . . . ; 0†; j ˆ 1; . . . ; h; xu1 …0; 1; 0; . . . ; 0† ˆ …0; . . . ; 0; 0; a2 †;

…6:39†

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we obtain from (6.38) and (6.39) that  h X a1 x ˆ z1 ‡ 2 …1 cos…bu1 †† ak zk ; . . . ; zh b kˆ1 ah ‡ 2 …cos…bu1 † b

 h X 1 1† ak zk ; sin…bu1 † ak zk : b kˆ1 kˆ1 h X

…6:40†

It is straightforward to verify that hxu1 ; xu1 i ˆ b2 f 2 . Hence, by comparing this with (6.34), we get b ˆ 1. Therefore, we obtain (6.11) with p ˆ 1 from (6.40). CASE (ii): ˆ 0. In this case, A ˆ c1 u1 ‡ c2 for some vectors c1 ; c2 . Hence, (6.33) reduces to h h X X x…z1 ; . . . ; zh ; u1 † ˆ …c1 u1 ‡ c2 † ak zk ‡ … k xk ‡ k yk †: …6:41† kˆ1

kˆ1

Substituting (6.41) into (6.31) with t ˆ 1 gives   X  h h h X X 0ˆ ak xk a1 xx1 ak xxk ‡ ak yk a1 xy1 kˆ1

kˆ2

kˆ1

h X kˆ2

 ak xyk

which is impossible, since xx1 ; . . . ; xxh ; xy1 ; . . . ; xyh are linearly independent and f 6ˆ 0. CASE (iii): < 0. In this case, a1 6ˆ 0. We may put ˆ b2 . Then we obtain from (6.35) that A ˆ c1 cosh…bu1 † ‡ c2 sinh…bu1 †. Thus, (6.33) reduces to x ˆ …cosh…bu1 † ‡ c2 sinh…bu1 ††

h X

ak zk ‡

kˆ1

h X … k xk ‡ k yk †:

…6:42†

kˆ1

After choosing some suitable initial conditions, we obtain  h X a1 x ˆ z1 ‡ 2 …cosh…bu1 † 1† ak zk ; . . . ; b kˆ1 ah zh ‡ 2 …1 b

 h X 1 cosh…bu1 †† ak zk ; sinh…bu1 † ak zk : b kˆ1 kˆ1 h X

…6:43†

It is easy to verify that hxu1 ; xu1 i ˆ b2 f 2 . Hence, by (6.19) and (6.27) we obtain b ˆ 1. Consequently, we have (6.12). Case (b): p > 1. In this case, we obtain (6.11) in the same way as case (b) in the proof of Theorem 7.1 of [4]. The converse is easy to verify. & Theorem 6.3. A CR-warped product NT f N? in CH m … 4† satis®es the basic equality kk2 ˆ 2pkr…ln f †k2 if and only if one of the following two cases occurs: (1) NT is an open portion of complex Euclidean h-space Ch , N? is an open portion of a unit p-sphere Sp and, up to rigid motions, the immersion is the

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m composition    x, where  is the projection  : Cm‡1 1 ! CH … 4† and  h h X X  x…z; w† ˆ z0 ‡ a0 …1 w0 † aj zj ; z1 ‡ a1 …w0 1† aj zj ; . . . ; jˆ0

zh ‡ ah …w0

1†

jˆ0

h X

aj zj ; w1

jˆ0

h X

aj zj ; . . . ; wp

jˆ0

h X

 aj zj ; 0; . . . ; 0

jˆ0

…6:44† a20 p

a21

a2h

for some real numbers a0 ; . . . ; ah satisfying  ˆ 1, where z ˆ p‡1 …z0 ; . . . ; zh † 2 Ch‡1 and w ˆ …w ; . . . ; w † 2 S  E . 0 p 1 (2) p ˆ 1, NT is an open portion of Ch and, up to rigid motions, the immersion is the composition   x, where  h h X X  x…z; t† ˆ z0 ‡ a0 …cosh t 1† aj zj ; z1 ‡ a1 …1 cosh t† aj zj ; . . . ; zh jˆ0

‡ ah …1

cosh t†

h X jˆ0

jˆ0

aj zj ; sinh t

h X



aj zj ; 0; . . . ; 0

…6:45†

jˆ0

for some real numbers a0 ; a1 . . . ; ah‡1 satisfying a20

a21



a2h ˆ 1.

Proof. Recall that every CR-warped product in a complex hyperbolic space is non-trivial according to [2] and every CR-warped product NT f N? in CH m … 4† satisfying the basic equality kk2 ˆ 2pkr…ln f †k2 is anti-holomorphic in some totally geodesic holomorphic submanifold CH h‡p … 4† according to Proposition 3.1. Hence, by Proposition 6.1, the preimage  1 …NT f N? † is an antiholomorphic CR-warped product in Ch‡p‡1 which satis®es conditions (i), (ii) 1 and (iii). Hence, by Proposition 6.2, the CR-warped product in CH h‡p … 4† is the image under the projection  either of (6.44) or of (6.45). Conversely, since (6.44) and (6.45) are invariant under the C -action, both are m projectable under the projection  : Cm‡1 … 4†. Since (6.44) and (6.45) 1 ! CH h‡p‡1 satisfying conditions (i), de®ne anti-holomorphic CR-warped products in C1 (ii) and (iii), Proposition 6.1 implies that their projections under  are CR-warped products in CH m … 4† satisfying the basic equality kk2 ˆ 2pkr…ln f †k2. & m ~ …4c† Remark 6.1. CR-warped products NT f N? in a complex space form M also satisfy another general inequality: kk2 52pfkr…ln f †k2 ‡ …ln f † ‡ 2hcg. For the complete classi®cation of CR-warped products satisfying the equality case of this second general inequality, see [6]. References [1] Chen BY (1973) Geometry of Submanifolds. New York: Dekker [2] Chen BY (1981) CR-submanifolds of a Kaehler manifold, I. J Differential Geometry 16: 305±322; II, ibid 16: 493±509 [3] Chen BY (2000) Riemannian Submanifolds. In: Dillen F, Verstraelen L (eds) Handbook of Differential Geometry, vol I, pp 187±418. Amsterdam: North Holland [4] Chen BY (2001) Geometry of warped product CR-submanifolds in Kaehler manifolds. Monatsh Math 133: 177±195

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[5] Chen BY (2001) Riemannian geometry of Lagrangian submanifolds. Taiwanese J Math 5: 681±720. [6] Chen BY (2001) CR-warped products in complex space forms (preprint) [7] Chen BY, Ludden GD, Montiel S (1984) Real submanifolds of a Kaehler manifold. Algebras Groups Geom 1: 176±212 [8] Hiepko S (1979) Eine innere Kennzeichung der verzerrten Produkte. Math Ann 241: 209±215 [9] O'Neill B (1983) Semi-Riemannian Geometry with Applications to Relativity. New York: Academic Press [10] Segre C (1891) Sulle varieta 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]