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 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 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 CPhphp 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 B f F is the manifold B F equipped with the Riemannian structure such that
1:1 kXk2 k
Xk2 f 2
xk
Xk2 2 for any tangent vector X 2 Tx M. Thus, we have g 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 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 X';
2:1 n X f
rej ej ' ÿ ej ej 'g
2:2 ' j1
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179
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 rX Y
X; Y;
2:3 r ~ X 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 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 R
X; Y; Z; W h
X; Z;
Y; Wi ÿ h
X; W;
Y; Zi;
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 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; YZ?
r ? ~ ~ YZ. The equation of where
R
X; YZ denotes the normal component of R
X; Ricci is given by ~ R
X; Y; ; R?
X; Y; ; ÿ hA ; A X; 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 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 hJAJZ U; Xi, (2) AJZ W AJW Z; and (3) AJ X ÿA JX, for any vectors U tangent to M, X, Y in D, Z, W in D? , and in .
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B.-Y. Chen
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 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 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 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 0: ~ is Kaehlerian, the formulas of Gauss and Weingarten Since the ambient space M imply
3:2 ÿAJW Z DZ
JW J
rZ W J
Z; W: Taking the inner product of (3.2) with JX gives hAJW Z; JXi ÿhrZ W; Xi:
3:3
By combining (3.1) and (3.3) we obtain h
D; D? ; JD? i 0:
3:4
On the other hand, from Lemma 7.3 of [2] we have rX Z rZ X
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 hATZ X; Yi ÿhrX Z; Yi ÿ
Z ln f hX; Yi
3:6
?
for any X; Y in D and Z in D . Hence we ®nd T
X; Y 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 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 ÿZ
ln hX; Xi:
3:9 ~ we also have the following Since NT is a holomorphic submanifold of M, relations: ^
X; JY ^
JX; Y J ^
X; Y:
3:10
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181
Hence, by combining (3.9) and (3.10), we obtain h^
X; X; Zi ÿh^
JX; JX; Zi Z
ln hX; Xi:
3:11
(3.9) and (3.11) imply Z
ln 0. Therefore, we obtain from (3.6) and (3.8) that h^ T
X; Y; Zi hT
X; Y; Zi 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 h^
X; Y; JZi ÿh^
X; JY; Zi 0:
3:13
h
D; D; JD? i 0:
3:14
Therefore Conditions (3.4) and (3.14) imply AJD? D 0. Therefore, by applying Lemma 4.2 & of [4], we conclude that M 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 NT f N? in any Kaehler manifold M, we have (1) h
D; D; JD? i 0; (2) rX Z rZ X
X ln f Z; (3) h
JX; Z; JWi
X ln f hZ; Wi; (4) DX
JZ JrX Z; whenever
D; D? JD? ; (5) h
D; D? ; JD? i 0 if and only if M 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 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 ÿhAJZ X; JYi ÿh
X; JY; JZi:
4:2
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B.-Y. Chen
On the other hand, since M NT f N? is a warped product, NT is a totally geodesic submanifold of M. Thus, we also have hrX Z; Yi 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 ÿhJAJW Z; Xi ÿhrZ W; Xi hrZ X; Wi
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
JXZ;
X 2 D;
Z 2 D? ;
4:4
?
for some function on M satisfying W 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 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 0 for all W 2 D? . Conversely, assume that M is a proper CR-submanifold of a Kaehler manifold ~ satisfying M AJZ X
JXZ;
X 2 D;
Z 2 D? ;
4:5
for some function with W 0; W 2 D? . Then we have h
D; D; JD? i 0;
4:6
h
JX; Z; JWi ÿ
XhZ; 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 ÿhrZ W; Xi ÿhJAJW Z; Xi h
JX; Z; JWi ÿ
XhZ; 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 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
Geometry of Warped Product CR-Submanifolds in Kaehler Manifolds
183
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 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 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 h
JX; Z; JZi 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 hJAJZ W; Xi ÿh
JX; W; JZi:
5:3
Hence, by using statement (2) of Lemma 4.1 and (5.3), we ®nd hrW Z; Xi ÿ
X ln f hZ; Wi:
5:4
?
On the other hand, if we denote by the second fundamental form of N ? in M NT f N? , we get h?
Z; W; Xi hrW Z; Xi:
5:5
Combining (5.4) and (5.5) yields ?
Z; W ÿhZ; Wir ln f
5:6
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B.-Y. Chen
Now, assume that the equality case of (5.1) holds identically. Then we obtain from (5.2) that
D; D 0;
D? ; D? 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 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 hZ; WiH;
5:9
?
for Z; W tangent to N? . Since ^
Z; W
Z; W
Z; W, (5.9) implies that there is a normal vector ®eld such that
Z; W hZ; Wi: Therefore, for each unit vector W 2 D pendicular to W, we have
?
and each unit vector Z in D
h; JWi h
Z; Z; JWi h
Z; W; JZi hZ; Wih; JZi 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 1 or
D? ; D? 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 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 NT f N? be a nontrivial CR-warped product satis~ of constant holomorphic fying kk2 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.
Geometry of Warped Product CR-Submanifolds in Kaehler Manifolds
185
(c) When p dim N? > 1, the warping function f satis®es krf k2 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 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; WV RN?
Z; WV 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
x1 ; y1 ; . . . ; xh ; yh of the following system of partial differential equations: @2 @ @ @ @ ÿ ;
6:1 @xj @xk @yj @yk @xj @xk @2 @ @ @ @ ÿ ÿ ; @xj @yk @yj @xk @xj @yk
j; k 1; . . . ; h;
6:2
@2 @ @ @ @ ÿ @yj @yk @xj @xk @yj @yk are given by
o 1 n
6:4 ln h; zi2 hi; zi2 ; 2 where z
x1 iy1 ; . . . ; xh iyh ;
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
@ @ @ ÿ2 ln : @x1 @y1 @x1
6:5
Solving (6.5) yields @ eÿ2
y1 ; x2 ; y2 ; . . . ; xh ; yh ; @y1
y1 ; x2 ; y2 ; . . . ; xh ; yh . Therefore we obtain
for some function 1 ln
x1 ; x2 ; y2 ; . . . ; xh ; yh
y1 ; x2 ; y2 ; . . . ; xh ; yh ; 2
6:6
6:7
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B.-Y. Chen
for some functions and of 2h ÿ 1 variables. From (6.7) we get x1 y1 ; y1 ; x1 2
2
x1 x1
x1 x1
ÿ x21
6:8
:
6:9
2
x1 x1 x21 2y1 :
6:10
2
2
By (6.1), (6.8) and (6.9) we obtain Similarly, from (6.3) with j k 1 and (6.7), we also have 2
y1 y1 x21 2y1 :
6:11
By combining (6.10) and (6.11) we ®nd x1 x1 y1 y1 2F
x2 ; y2 ; . . . ; xh ; yh ;
6:12
for some positive function F of 2h ÿ 2 variables. Therefore we obtain F
x2 ; . . . ; yh x21 G
x2 ; . . . ; yh x1 H
x2 ; . . . ; yh ; 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 G2 K 2 . Hence we have o 1 n
2Fx1 G2
2Fy1 K2 :
6:14 4F Hence, we get o 1 n
6:15 ln
ax1 2
ay1 2 ; 2 p where a
x2 ; . . . ; yh F ;
x2 ; . . . ; yh 2pGF, and
x2 ; . . . ; yh 2pKF. From (6.15) we get a
ax1 ; x1
ax1 2
ay1 2 a
ay1 ; y1
ax1 2
ay1 2
6:16
ax1
axj x1 xj
ay1
axj y1 xj ; xj
ax1 2
ay1 2
ax1
ayj x1 yj
ay1
ayj y1 yj ; yj
ax1 2
ay1 2 for j 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 a
ay1
ax1
ayj x1 yj
ay1
ayj y1 yj :
6:17
Geometry of Warped Product CR-Submanifolds in Kaehler Manifolds
187
By comparing the coef®cients of x31 and y31 in (6.17) we ®nd axj ayj 0 for j 2; . . . ; h; respectively. Hence a is a positive constant. Therefore (6.16) and (6.17) give xj yj
ax1 xj
ay1 xj
ax1 2
ay1 2
ax1 yj
ay1 yj
ax1 2
ay1 2
; ;
6:18
ay1 xj ÿ
ax1 xj
ax1 yj
ay1 yj :
6:19
Since a is constant, (6.18) implies xj yk
xj yk xj yk
ax1 xj yk
ay1 xj yk ÿ2
ax1 2
ay1 2
ax1 xj
ay1 xj
ax1 yk
ay1 yk
ax1 2
ay1 2 2
;
6:20
for 2 4 j; k 4 h. Hence, by applying (6.2) with j k, (6.18) and (6.20), we get xj yj xj yj
ax1 xj yj
ay1 xj yj
ax1 2
ay1 2 0
6:21
Therefore, we ®nd xj yj xj yj 0 for j 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 xk yj xj yk xj xj xk yj zj yk 0 for 2 4 j; k 4 h. Therefore, there exists constants a3 ; . . . ; a2h ; b3 ; . . . ; b2h such that
a3 x2 a4 y2 a2hÿ1 xh a2h yh ; b3 x2 b4 y2 b2hÿ1 xh b2h yh :
6:22
Combining (6.15) and (6.22) yields o 1 n ln
ax1 a3 x2 a4 y2 a2h yh 2
ay1 b3 x2 b2h yh 2 : 2
6:23 Finally, from (6.1) with j 1 and (6.23), we obtain a2kÿ1 b2k and a2k ÿb2kÿ1 . Thus, we obtain from (6.23) that 1 n 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
a; a3 ia4 ; . . . ; a2hÿ1 ia2h and z
x1 iy1 ; . . . ; xh iyh , & then (6.24) becomes (6.4) with a1 a.
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B.-Y. Chen
7. CR-Warped Products in Cm Satisfying kk2 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 NT f N? in a complex Euclidean mspace Cm satis®es kk2 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
z1
w0 ÿ 1a1
n X
aj zj ; zh a
w0 ÿ 1ah
j1
w1
h X
aj zj ; . . . ; wp
j1
h X
!
h X
aj zj ;
j1
aj zj ; 0; . . . ; 0 ;
j1
z
z1 ; . . . ; zh 2 Ch ; where f
w
w0 ; . . . ; wh 2 S p E p1 ;
q ha; zi2 hia; zi2 ;
7:1
7:2
for some point a
a1 ; . . . ; ah 2 Shÿ1 Eh . Proof. Assume that x : NT f N? ! Cm is a CR-warped product which satis®es kk2 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
X ln f JZ;
7:3
D; D
D? ; D? 0:
7:4
and (iv) Applying statement (4) of Lemma 4.1 and (7.3), we ®nd DX JZ 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 p1 of Cm (cf. [9]). Without loss of generality, we may assume that the radius of Sp is one. Let z
z1 ; . . . ; zh be a natural complex coordinate system on Ch . We put zj xj iyj ; j 1; . . . ; h. The standard ¯at metric on Ch is then given by g0 dx21 dy21 dx2h dy2h :
7:6
Geometry of Warped Product CR-Submanifolds in Kaehler Manifolds
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 du21 cos2 u1 du22 cos2 u1 cos2 upÿ1 du2p :
7:7
The warped product metric on NT f N? is then given by g
h X
dx2k dy2k f 2 fdu21 cos2 u1 du22 cos2 u1 cos2 upÿ1 du2p g: k1
7:8 From (7.8) and a straightforward computation we know that the Riemannian connection on NT f N? satis®es @ @ @ r@ r@ 0; j; k 1; . . . ; h; @xj @x @xj @y @yj @y k k k fx @ @ j ; j 1; . . . ; h; t 1; . . . ; p; r@ @xj @u f @ut t fy @ @ j ; j 1; . . . ; h; t 1; . . . ; p; r@ @yj @u f @ut t @ @ ÿtan us ; 1 4 s < t 4 p; r@ @us @u @ut t
r@
tÿ1 h Y X @ @ @ ÿ cos2 us ffxk ffyk t @ut @xk @yk sÿ1 k1 ! tÿ1 tÿ1 X sin 2uq Y @ cos2 us ; t 1; . . . ; p: @uq 2 sq1 q1
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 @ @ @ @ @ @ @ ; R @xj @ut @xk @xj @xk @xj @xk @ut 2 @ @ @ @ @ @ @ R ; @xj @ut @yk @xj @yk @xj @yk @ut 2 @ @ @ @ @ @ @ R ; :
7:14 @yj @ut @yk @yj @yk @yj @yk @ut From (7.3) we have
@ @ @ @ ; J ÿ ; @xj @ut @yj @ut @ @ @ @ ; J ; @yj @ut @xj @ut
7:15
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B.-Y. Chen
for j 1; . . . ; h; t 1; . . . ; p, where ln f . Thus, by applying the equation of Gauss, (7.14), and (7.15), we obtain @2 @ @ @ @ ÿ ; @xj @xk @yj @yk @xj @xk @2 @ @ @ @ ÿ ÿ ; @xj @yk @yj @xk @xj @yk
j; k 1; . . . ; h;
7:16
@2 @ @ @ @ ÿ @yj @yk @xj @xk @yj @yk Therefore, after applying Proposition 6.1, we conclude that there exists a vector
a1 ; a3 ia4 ; . . . ; a2hÿ1 ia2h in Ch such that o 1 n
7:17 ln h; zi2 hia; zi2 ; 2 where z
x1 iy1 ; . . . ; xh iyh . Since ln f , (7.17) implies q f h; zi2 hi; zi2 ; krf k kk:
7:18 After applying the following rotations:
a2jÿ1 xj a2j yj ; ÿa2j xj a2jÿ1 yj 7!
q a22jÿ1 a22j
xj ; yj ;
for j 2; . . . ; h, we obtain from (7.18) that
7:19 f f
a1 x1 ah xh 2
a1 y1 ah yh 2 g1=2 : Without loss of generality, we may assume a1 6 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 xxj yk xyj yk 0; j; k 1; . . . ; h; aj xxj ut 2 fa1
x1 ÿ iy1 ah
xh ÿ iyh gxut f aj xyj ut 2 fa1
y1 ix1 ah
yh ixh gxut f xus ut ÿ
tan us xut ; 1 4 s < t 4 p; xut ut ÿ
tÿ1 Y s1
cos2 us
7:20
7:21
7:22
h n X ak
a1 x1 ah xh xxk k1
ak
a1 y1 ah yh xyk
o
! tÿ1 tÿ1 X sin 2uq Y cos2 us xuq : 2 q1 sq1
7:24
Solving (7.20) gives x
h X k1
^ k
u1 ; . . . ; up xk A
h X k1
^ k
u1 ; . . . ; up yk C
u1 ; . . . ; up B
7:25
Geometry of Warped Product CR-Submanifolds in Kaehler Manifolds
191
^1; . . . ; A ^h; B ^1; . . . ; B ^ h ; C. Substituting (7.25) into (7.21) gives for some functions A ^j
a1 x1 ah xh 2
a1 y1 ah yh 2 A ut ( ) h h X X ^ k xk ^ k yk Cut ; B A aj fa1
x1 ÿ iy1 ah
xh ÿ iyh g k1
ut
k1
ut
7:26 for j 1; . . . ; h; t 1; . . . ; p. By comparing the coef®cients of x21 ; xk yl , and x1 in (7.26) respectively, we obtain ^ j a1 aj A ^1 ; a21 A ut ut
7:27
^k ; ^ lu iaj al A aj ak B ut t
7:28
Cu1 Cup 0
7:29
for j; k; l 1; . . . ; h. Condition (7.29) implies that C is a constant vector in Cm . So, we may choose C 0 by applying a suitable translation on Cm if necessary. Since a1 6 0, (7.27) and (7.28) with j k 1 imply ^ j ; j 1; . . . ; h; t 1; . . . ; p: ^1 ; B ^ j aj A ^ uj iA
7:30 A ut ut t a1 ut Solving (7.30) gives
^ 1 ^j ; B ^ j ^j ; j 1; . . . ; h; ^ j aj A ^ j iA
7:31 A a1 for some constant vectors ^j ; ^j ; j 1; . . . ; h. (7.25) and (7.31) show that there exist a Cm -valued function A
u1 ; . . . ; uk and some vectors j ; j ; j 1; . . . ; h, in Cm so that x A
u1 ; . . . ; up
h X
ak
xk iyk
k1
h X
k xk k yk
7:32
k1
Case (a): p 1. In this case, by substituting (7.32) into (7.24) with t 1, we ®nd h h n X X ak
xk iyk ÿ ak
a1 x1 ah xh
ak A k Au1 u1 k1
k1
o ak
a1 y1 ah yh
iak A k
7:33
which implies Au1 u1 A 0; h X k1
ak k
h X k1
ak k 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 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
c1 cos u1 c2 sin u1
h X
ak
xk iyk
k1
h X
k xk k yk ;
7:37
k1
where k ; k ; k 1; . . . ; h, are vectors in Cm satisfying (7.35). If we choose the following initial conditions: xx1
1; 0; . . . ; 0
1; 0; . . . ; 0; 0; . . . ; 0; ; h-th z}|{ xxh
1; 0; . . . ; 0
0; . . . ; 0; 1 ; 0; . . . ; 0; xy1
1; 0; . . . ; 0
i; 0; . . . ; 0; 0; . . . ; 0; ; h-th z}|{ xyh
1; 0; . . . ; 0
0; . . . ; 0; i ; 0; . . . ; 0;
h1-th z}|{ xu1
1; 0; . . . ; 0
0; . . . ; 0; 0; a1 ; 0; . . . ; 0;
7:38
then we obtain from (7.37) and (7.38) that
h-th z}|{ a1 c1 1
1; 0; . . . ; 0; . . . ; ah c1 h
0; . . . ; 0; 1 ; 0; . . . ; 0; h-th z}|{ ia1 c1 1
i; 0; . . . ; 0; . . . ; iah c1 h
0; . . . ; 0; i ; 0; . . . 0;
h1-th z}|{ c2
0; . . . ; 0; 1 ; 0; . . .:
Hence we ®nd c1
1
h 1
h
h1-th z}|{ 1
a1 ; . . . ; ah ; 0; . . . ; 0; c2
0; . . . ; 0; 1 ; 0; . . . b 1
b ÿ a21 ; ÿa1 a2 ; . . . ; ÿa1 ah ; 0; . . . ; 0; b 1
ÿa1 ah ; . . . ; ÿahÿ1 ah ; b ÿ a2h ; 0; . . . ; 0; b i
b ÿ a21 ; ÿa1 a2 ; . . . ; ÿa1 ah ; 0; . . . ; 0; b i
ÿa1 ah ; . . . ; ÿahÿ1 ah ; b ÿ a22 ; 0; . . . ; 0: b
7:39
Geometry of Warped Product CR-Submanifolds in Kaehler Manifolds
where b
Ph
j1
193
a2j . Therefore, we obtain from (7.37) that
x
x1 iy1
h X a1
cos u1 ÿ 1 ak
xk iyk ; . . . ; b k1
xh iyh
h X ah
cos u1 ÿ 1 ak
xk iyk ; b k1 !
sin u1
h X
ak
xk iyk ; 0; . . . ; 0 :
7:40
k1
From (7.40) we ®nd
1 hxx1 ; xx1 i 1 a21 1 ÿ sin2 u1 : b
7:41
Comparing (7.8) with (7.41) yields b 1. Therefore, we obtain (7.1) with p 1. Case (b): p > 1. In this case, we obtain krf k2 1 by Proposition 5.2. Hence, from (7.18) we obtain a21 a22 a2h 1. Now, by substituting (7.32) into (7.23), we ®nd Aus ut ÿ
tan us Aut ;
1 4 s < t 4 p:
7:42
Solving (7.42) for t p and s 1 yields A
u1 ; . . . ; up D1
u2 ; . . . ; up cos u1 E1
u1 ; . . . ; upÿ1
7:43
for some function D and E. Hence, we obtain from (7.32) that x
D1
u2 ; . . . ; up cos u1 E1
u1 ; . . . upÿ1
h X
ak
xk iyk
k1
h X
k xk k yk
7:44
k1
Similarly, by substituting (7.44) into (7.23) with s 2; t p, we obtain D1
u2 ; . . . ; up D2
u3 ; . . . up cos u2 E2
u2 ; . . . ; upÿ1 : Continuing such procedure
p ÿ 1-times we get n x 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
k1
h X
k xk k yk : k1
7:45
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B.-Y. Chen
By substituting (7.45) into (7.24) with t 1 and by applying a21 a22 a2h 1, we ®nd @ 2 E1 E1 0;
7:46 @u21 h X
ak k
k1
h X
ak k 0:
7:47
k1
^ 1
u2 ; . . . ; upÿ1 cos u1 F2
u2 ; . . . ; upÿ1 sin u1 . Solving (7.46) gives E1 F Hence, we obtain from (7.45) that n x 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 k1 h X
k xk k yk ;
7:48
k1
^ 1 E2 . where F1 F Substituting (7.48) into (7.23) with s 1; 1 < t < p gives @F2 =@ut 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 c1 cos u1 cos up c2 sin u1 c3 sin u2 cos u1 cp1 sin up cos u1 . . . cos upÿ1
h oX
ak
xk iyk
k1
h X
k xk k yk
7:49
k1
for some vectors c1 ; . . . ; cp1 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
1; 0; . . . ; 0 2 Shÿ1 , then the immersion x de®ned by (7.1) reduces to x
z; w
z1 w0 ; . . . ; z1 wp ; z2 ; . . . ; zh ;
z 2 Ch ;
w 2 Sp Ep1
7:10
and the warping function f becomes f 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
Geometry of Warped Product CR-Submanifolds in Kaehler Manifolds
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 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]