Monatsh. Math. 141, 177–186 (2004) DOI 10.1007/s00605-002-0009-y
CR-Warped Products in Complex Projective Spaces with Compact Holomorphic Factor By
Bang-Yen Chen Michigan State University, East Lansing, MI, USA Dedicated to Professor G. D. Ludden on the occasion of his sixtieth birthday Received August 19, 2002 Published online July 15, 2003 # Springer-Verlag 2003 Abstract. A submanifold of a Kaehler manifold is called a CR-warped product if it is the warped product NT f N? of a complex submanifold NT and a totally real submanifold N?. There exist many CR-warped products NT f N? in CPhþp , h ¼ dimC NT and p ¼ dimR N? (see [5, 6]). In contrast, we prove in this article that the situation is quite different if the holomorphic factor NT is compact. For such CR-wraped products in CPm ð4Þ, we prove the following: (1) The complex dimension m of the ambient space is at least h þ p þ hp. (2) If m ¼ h þ p þ hp, then NT is CPh ð4Þ. We also obtain two geometric inequalities for CR-warped products in CPm with compact NT. 2000 Mathematics Subject Classification: 53C40, 53C42; 53B25 Key words: CR-submanifolds, warped product, CR-warped product, complex projective space, geometric inequality
1. Introduction Let B and F be two Riemannian manifolds with Riemannian metrics gB and gF, respectively, and f a positive differentiable function on B. The warped product B f F is the product manifold B F equipped with the Riemannian metric ð1:1Þ g ¼ gB þ f 2 gF : The function f is called the warping function. It is well-known that the notion of warped products plays some important roles in differential geometry as well as in physics (see [7, pp 364–367]). ~ be a Kaehler manifold with complex structure J and N a Riemannian Let M ~ . For each x 2 N, we denote by Hx the maxmanifold isometrically immersed in M imal holomorphic subspace of the tangent space TxN of N. If the dimension of Hx is the same for all x 2 N, the spaces Hx define a holomorphic distribution H on N, which is called the holomorphic distribution of N. A submanifold N in a Kaehler ~ is called a CR-submanifold if there exists a holomorphic distribution manifold M H on N whose orthogonal complement H? is a totally real distribution, i.e., JH? T ? N. A CR-submanifold is called a totally real submanifold if dim Hx ¼ 0. A submanifold of a Kaehler manifold is called a CR-product if it is the Riemannian product NT N? of a complex submanifold NT and a totally real
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submanifold N? . It was proved in [4] that (a) CR-products in complex hyperbolic spaces are either complex or totally real, (b) a submanifold in a complex Euclidean space is a CR-product if and only if it is the direct sum of a complex submanifold and a totally real submanifold of linear complex subspaces, and (c) CR-products in the complex projective space CPh þ p þ hp are obtained from the well-known Segre embedding in a natural way (see, [1, 4] for details). A submanifold of a Kaehler manifold is called a CR-warped product if it is the warped product NT f N? of a complex submanifold NT and a totally real submanifold N? (see [5]). The CR-warped product is called NT -totally geodesic if its second fundamental form satisfies ðX; YÞ ¼ 0 for X, Y tangent to NT . Similarly, it is called N? -totally geodesic if ðZ; WÞ ¼ 0 for Z, W tangent to N? . Throughout this paper we denote by h the complex dimension of NT and by p the real dimension of N?. It is known that there exist many CR-warped products NT f N? in complex projective m-space CPhþp (see [5, 6]). In contrast, for CR-warped products with compact holomorphic factor NT, we prove the following. Theorem 1. Let NT f N? be a CR-warped product in the complex projective m-space CPm ð4Þ of constant holomorphic sectional curvature 4. If NT is compact, we have m 5 h þ p þ hp. Theorem 2. If NT f N? is a CR-warped product in CPhþpþhp ð4Þ with compact NT, then NT is holomorphically isometric to CPh ð4Þ. Theorem 3. For any CR-warped product NT f N? in CPm ð4Þ with compact NT and any q 2 N?, we have ð kk2 dVT 5 4hp volðNT Þ; ð1:2Þ NT fqg
where kk is the norm of the second fundamental form. The equality sign of (1.2) holds identically if and only if we have: (i) The warping function f is constant. (ii) ðNT ; gNT Þ is holomorphically isometric to CPh ð4Þ and it is isometrically immersed in CPm ð4Þ as a totally goedesic complex submanifold. (iii) ðN? ; f 2 gN? Þ is isometric to an open portion of the real projective p-space p RP ð1Þ of constant sectional curvature one and it is isometrically immersed in CPm ð4Þ as a totally geodesic totally real submanifold. (iv) NT f N? is immersed linearly fully in a linear complex subspace CPhþpþhp ð4Þ of CPm ð4Þ; and moreover, the immersion is rigid. Theorem 4. Let NT f N? be a CR-warped product with compact NT in CPm ð4Þ. If the warping function f is non-constant, then, for each q 2 N? , we have ð ð 2 kk dVT 5 2p1 ðln f Þ2 dVT þ 4hp volðNT Þ; ð1:3Þ NT fqg
NT
where dVT, 1 and vol(NT) are the volume element, the first positive eigenvalue of the Laplacian and the volume of NT, respectively.
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Moreover, the equality sign of (1.3) holds identically if and only if we have (a) ln f ¼ 1 ln f . (b) The CR-warped products is both NT -totally geodesic and N? -totally geodesic. 2. Preliminaries We follow notations and terminology from [3, 5]. Let N be a Riemannian nmanifold with inner product h ; i and e1 ; . . . ; en an orthonormal frame field on N. For a differentiable function ’ on N, the gradient r’, the Hessian H ’ , and the Laplacian ’ of ’ are defined respectively by hr’; Xi ¼ X’;
ð2:1Þ
H ’ ðX; YÞ ¼ XY’ rX Y’ ;
ð2:2Þ
’ ¼
n X fðrej ej Þ’ ej ej ’g
ð2:3Þ
j¼1
for vector field X, Y tangent to N, where r is the Riemannian connection on N. Let N be a Riemannian manifold isometrically immersed in another Rieman~ . The formulas of Gauss and Weingarten are given respectively by nian manifold M ~ X Y ¼ rX Y þ ðX; YÞ; r
ð2:4Þ
~ X ¼ A X þ DX ð2:5Þ r ~ is the Riemannian connection on for X, Y tangent to N and normal to N, where r ~ M , the second fundamental form, D the normal connection and A the shape operator of the submanifold. The second fundamental form and the shape operator are related by hA X; Yi ¼ hðX; YÞ; i;
ð2:6Þ
~. where h ; i denotes the inner product on N as well as on M The equation of Gauss is given by ~ ðX; Y; Z; WÞ þ hðX; WÞ; ðY; ZÞi RðX; Y; Z; WÞ ¼ R hðX; ZÞ; ðY; WÞi;
ð2:7Þ
~ denote the curvature tensors of N and for X, Y, Z, W tangent to M, where R and R ~ M , respectively. with For the second fundamental form , we define its covariant derivative r respect to the connection on TM T ? M by X ÞðY; ZÞ ¼ DX ððY; ZÞÞ ðrX Y; ZÞ ðY; rX ZÞ: ð2:8Þ ðr The equation of Codazzi is given by X ÞðY; ZÞ ðr Y ÞðX; ZÞ; ~ ðX; YÞZÞ? ¼ ðr ðR
ð2:9Þ
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~ ðX; YÞZÞ? denotes the normal component of R ~ ðX; YÞZ. The equation of where ðR Ricci is given by ~ ðX; Y; ; Þ ¼ R? ðX; Y; ; Þ h½A ; A X; Yi: R
ð2:10Þ
?
where R is the curvature tensor of the normal connection D. ~ with complex structure J, we For a CR-submanifold N in a Kaehler manifold M denote by the complementary orthogonal subbundle of JH? in the normal bundle T ? N. Hence we have the following orthogonal direct sum decomposition: T ? N ¼ JH? ;
JH? ? :
ð2:11Þ
The Riemann curvature tensor of the complex projective m-space CPm ð4Þ with constant holomorphic sectional curvature 4 is given by ~ ðX; Y; Z; WÞ ¼ hX; WihY; Zi hX; ZihY; Wi R þ hJX; WihJY; Zi hJX; ZihJY; Wi þ 2hX; JYihJZ; Wi:
ð2:12Þ
3. Lemmas We need the following lemmas for later use. ~ . Then we Lemma 1. ([4]) Let M be a CR-submanifold in a Kaehler manifold M have (1) hrU Z; Xi ¼ hJAJZ U; Xi, (2) AJZ W ¼ AJW Z, and (3) AJ X ¼ A JX, for any vectors U tangent to M, X, Y in H, Z, W in H? , and in . For a vector field X tangent to N1, the lift of X on N1 f N2 is the tangent vector ~ on N1 f N2 whose value at each (p, q) is the lift Xp to ðp; qÞ. Thus, the lift field X of X is the unique vector field on N1 f N2 that is N1 -related to X and N2 -related to the zero vector field on N2. The set of all such lifts of vector fields on N1 is denoted by LðN1 Þ. Similarly, we denote by LðN2 Þ the lifts of vector fields from vector fields tangent to N2. For CR-warped products in Kaehler manifolds we have the following. ~ , then Lemma 2. If NT f N? is a CR-warped product in a Kaehler manifold M we have: (1) (2) (3) (4) (5) (6)
hhLðNT Þ; LðNT ÞÞ; JLðN? Þi ¼ 0; rX Z ¼ rZ X ¼ ðX ln f ÞZ; hðJX; ZÞ; JWi ¼ ðX ln f ÞhZ; Wi; rX JY ¼ JrX Y and ðX; JYÞ ¼ JðX; YÞ; ðJX; ZÞ ¼ ðX ln f ÞJZ þ J ðX; ZÞ; ðJX; ZÞ ¼ J ðX; ZÞ
for vector fields X, Y 2 LðNT Þ and Z, W 2 LðN? Þ, where ðX; ZÞ denotes the component of ðX; ZÞ.
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Proof. Statements (1), (2), and (3) are nothing but Lemma 4.1 of [5, part I]. For vector fields X, Y in LðNT Þ, we have ~ X ðJYÞ ¼ J r ~ X Y ¼ JðrX YÞ þ JðX; YÞ: rX JY þ ðX; JYÞ ¼ r ð3:1Þ Since NT is totally geodesic in NT f N? and NT is invariant under the action of J, rX JY and JðrX YÞ are tangent to the first factor NT and ðX; JYÞ and JðX; YÞ are normal to NT. Thus, we obtain statement (4) from (3.1). For vector fields X in LðNT Þ and Z in LðN? Þ, we obtain from statement (2) that ~ Z JX ¼ ðX ln f ÞJZ þ JðX; ZÞ rZ ðJXÞ þ ðZ; JXÞ ¼ r ð3:2Þ which implies statement (5). Statement (6) is a consequence of statement (5).
&
For CR-warped products in complex projective spaces, we have Lemma 3. Let NT f N? be a CR-warped product in CPm ð4Þ. Then hðX; YÞ; ðV; ZÞi ¼ h ðX; YÞ; ðV; ZÞi ¼ 0 for X, Y, V 2 LðNT Þ and Z 2 LðN? Þ. Proof. Let NT f N? be a CR-warped product in CPm ð4Þ and let X, Y, V be vector fields in LðNT Þ and Z in LðN? Þ. Since NT is totally geodesic in NT f N? and (2.12), we obtain from (2.12) that ~ ðX; Y; V; ZÞ ¼ 0: RðX; Y; V; ZÞ ¼ R ð3:3Þ Thus, by applying equation (2.7) of Gauss, we get hðX; VÞ; ðY; ZÞi ¼ hðX; ZÞ; ðY; VÞi:
ð3:4Þ
Also, from statement (1) of Lemma 2, we find ðX; VÞ ¼ ðX; VÞ
ð3:5Þ
for vector fields X, V 2 LðNT Þ. Using (3.4), (3.5), statement (4) of Lemma 2, and statement (1) of Lemma 2, we get hðX; VÞ; ðJX; ZÞi ¼ hðX; ZÞ; ðJX; VÞi
by ð3:4Þ
¼ hJðX; ZÞ; ðX; VÞi
by statement ð4Þ of Lemma 2
¼ hJðX; ZÞ; ðX; VÞi ¼ hðJX; ZÞ; ðX; VÞi
by ð3:5Þ by statement ð6Þ of Lemma 2
¼ hðJX; ZÞ; ðX; VÞi
by ð3:5Þ:
ð3:6Þ
Therefore, we obtain hðX; VÞ; ðJX; ZÞi ¼ 0 ¼ hðX; ZÞ; ðX; JVÞi
ð3:7Þ
for vector fields X, V in LðNT Þ and Z in LðN? Þ. Hence, for vector fields X, V in LðNT Þ and Z in LðN? Þ, we have hðX; ZÞ; ðX; VÞi ¼ 0. Thus, by applying polarization, we obtain hðX; ZÞ; ðY; VÞi þ hðY; ZÞ; ðX; VÞi ¼ 0:
ð3:8Þ
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Combining (3.4) and (3.8) yields hðX; ZÞ; ðY; VÞi ¼ 0
ð3:9Þ
for X, Y, V 2 LðNT Þ and Z 2 LðN? Þ. Now, equation h ðX; YÞ; ðV; ZÞi ¼ 0 follows from (3.9) and statement (1) of Lemma 2. This completes the proof of Lemma 3. & 4. Proofs of Theorems 1 and 2 The proofs of Theorems 1 and 2 are based on Lemma 2 and the following lemma. Lemma 4. Let NT f N? be a CR-warped product in CPm ð4Þ. Then, for X in LðNT Þ and Z in LðN? Þ, we have k ðX; ZÞk2 ¼ fhX; Xi þ HCln f ðX; XÞghZ; Zi;
ð4:1Þ
where 1 1 HCln f ðX; YÞ ¼ H ln f ðX; XÞ þ H ln f ðJX; JXÞ 2 2 is called the complex Hessian of ln f .
ð4:2Þ
Proof of Lemma 4. Assume that NT f N? is a CR-warped product in CPm ð4Þ. For vector fields X, Y 2 LðNT Þ and Z, W 2 LðN? Þ, equation (2.12) gives ~ ðX; JX; JZ; ZÞ ¼ 2hX; XihZ; Zi; R
ð4:3Þ
since we have X, Y, JX ? Z, W and JZ ? W. On the other hand, for X 2 LðNT Þ and Z 2 LðN? Þ, equation (2.9) of Codazzi implies that ~ ðX; JX; JZ; ZÞ ¼ hDJX ðX; ZÞ ðrJX X; ZÞ ðX; rJX ZÞ; JZi R hDX ðJX; ZÞ ðrX JX; ZÞ ðJX; rX ZÞ; JZi:
ð4:4Þ
Since NT is totally geodesic in NT f N? , rX Z and rJX Z lie in LðN? Þ and rX JX and rJX X lie in LðNT Þ. Hence, by applying (4.3) and statements (1), (2) and (3) of Lemma 2, we get 2hX; XihZ; Zi ¼ JXðhZ; ZiJX ln f Þ hðX; ZÞ; DJX JZi XðhZ; ZiX ln f Þ þ hðJX; ZÞ; DX JZi þ fðrJX JXÞ ln f þ ðrX XÞ ln f þ ðX ln f Þ2 þ ððJX ln f ÞÞ2 ghZ; Zi: ð4:5Þ Since XhZ; Zi ¼ 2ðX ln f Þ hZ; Zi by statement (2) of Lemma 2, we find JXðhZ; ZiJX ln f Þ þ XðhZ; ZiX ln f Þ ¼ fðJXÞ2 ln f þ X 2 ln f þ 2ðJX ln f Þ2 þ 2ðX ln f Þ2 ghZ; Zi:
ð4:6Þ
Because CPm ð4Þ is Kaehlerian, we have JrX Z þ JðX; ZÞ ¼ AJZ X þ DX JZ:
ð4:7Þ
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Applying (4.7) and statements (2) and (3) and Lemma 2, we find hðJX; ZÞ; DX JZi ¼ hðJX; ZÞ; JrX Zi þ hðJX; ZÞ; JðX; ZÞi ¼ ðX ln f Þ2 hZ; Zi þ hðJX; ZÞ; JðX; ZÞi
ð4:8Þ
for vector fields X in LðNT Þ and Z in LðN? Þ. On the other hand, if we denote by ðX; ZÞ the -component of ðX; ZÞ, then statement (3) of Lemma 1 implies that hðJX; ZÞ; JðX; ZÞi ¼ hðJX; ZÞ; J ðX; ZÞi ¼ hAJ ðX;ZÞ JX; Zi ¼ hA ðX;ZÞ X; Zi ¼ k ðX; ZÞk2 :
ð4:9Þ
Combining (4.8) and (4.9) yields hðJX; ZÞ; DX JZi ¼ ðX ln f Þ2 hZ; Zi þ k ðX; ZÞk2 :
ð4:10Þ
Similarly, we also have hðX; ZÞ; DJX JZi ¼ ðJX ln f Þ2 hZ; Zi k ðX; ZÞk2
ð4:11Þ
by statement (6) of Lemma 2. Because NT is a complex submanifold of CPm ð4Þ and NT is a totally geodesic submanifold of NT f N? , we get JrJX X ¼ rJX JX;
JrX JX ¼ rX X:
ð4:12Þ
Combining (4.5), (4.6), (4.10), (4.11) and (4.12) gives 2hX; XihZ; Zi ¼ fðrX X þ rJX JXÞ ln f X 2 ln f ðJXÞ2 ln f ghZ; Zi þ 2k ðX; ZÞk2 :
ð4:13Þ
Therefore, we have k ðX; ZÞk2 ¼ fhX; Xi þ HCln f ðX; XÞghZ; Zi;
ð4:14Þ
which is statement (i) of Lemma 4. Assume that fX1 ; . . . ; X2h g and fZ1 ; . . . ; Zp g are orthonormal bases of LðNT Þ and LðN? Þ, respectively. Since the complex Hessian HCln f , the inner product h ; i and the second fundamental form are bilinear, polarization and (4.14) yield h ðXi ; Zs Þ; ðXj ; Zs Þi ¼ fij þ HCln f ðXi ; Xj Þgst :
ð4:15Þ
Now, let us assume that NT is compact. Then the function ln f has an absolute minimum value at some point, say u 2 NT . At this critical point u, the complex Hessian HCln f is non-negative definite. Thus, each ðXi ; Zs Þ is nonzero at u by (4.14). Moreover, since HCln f is clearly self-adjoint, there exists an orthonormal basis X1 ; . . . ; X2h at u 2 NT such that HCln f at u is diagonalized with respect to X1 ; . . . ; X2h . Hence, from (4.15), we know that the vectors at u: ðXi ; Zt Þ; i ¼ 1; . . . ; 2h; t ¼ 1; . . . ; p;
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are mutually orthogonal nonzero vectors in the complex subspace u of the normal space at u. Therefore, the complex rank of the holomorphic vector bundle is at least hp. Hence, we get m 5 h þ p þ hp. This completes the proof of Theorem 1. For the proof of Theorem 2, we assume that NT f N? is a CR-warped product in CPh þ p þ hp (4) with compact NT . Let fZ1 ; . . . ; Zp g be a basis of LðN? Þ and let fX1 ; . . . ; X2h g be a basis of LðNT Þ which diagonalizes HCln f at the point u on which ln f attains its absolute minimum. Then we known from above that ðXi ; Zt Þ, i ¼ 1; . . . ; 2h; t ¼ 1; . . . ; p, are mutually orthogonal nonzero vectors in u . Hence, by continuity, we conclude that there exists an open neighborhood U of u so that, restricted to U f N? , the vector fields f ðXj ; Zs Þ; j ¼ 1; . . . ; 2h; p ¼ 1; . . . pg span the holomorphic vector bundle . On the other hand, from Lemma 3 and statement (1) of Lemma 2 we know that, for any X, Y 2 LðNT Þ, ðX; YÞ is a normal vector field in perpendicular to each ðXj ; Zs Þ. Thus, we have ðX; YÞ ¼ 0 identically on U f N? . Therefore, by using the fact that NT is totally geodesic in NT f N? , we conclude that, for any fixed q 2 N? , the nonempty open subset U fqg is immersed as a totally geodesic complex subspace CPh ð4Þ of CPh þ p þ hp ð4Þ. Hence, U fqg is immersed as an open portion of a linear complex submanifold CPp ð4Þ of CPh þ p þ hp ð4Þ. On the other hand, since NT fqg is a complex submanifold of the Kaehler manifold CPh þ p þ hp , NT fqg is real analytic. Hence, the whole complex submanifold NT fqg must be the linear complex subspace CPp ð4Þ. Consequently, NT is holomorphic isometric to CPh ð4Þ. This completes the proof of Theorem 2. & 5. Proofs of Theorems 3 and 4 Assume that NT f N? is a CR-warped product in CPm ð4Þ. Then statement (i) of Lemma 4 implies that 1 ln f 2 ln f ð5:1Þ k ðX; ZÞk ¼ hX; Xi þ ðH ðX; XÞ þ H ðJX; JXÞÞ hZ; Zi 2 for X in LðNT Þ and Z in LðN? Þ. On the other hand, from statement (3) of Lemma 2, we find kJH? ðJX; ZÞk2 ¼ hJH? ðJX; ZÞ; J 2 JH? ðJX; ZÞi ¼ ðX ln f ÞhJH? ðJX; ZÞ; JZi ¼ ðX ln f Þ2 hZ; Zi:
ð5:2Þ
kJH? ðX; ZÞk2 ¼ ððJXÞ ln f Þ2 hZ; Zi:
ð5:3Þ
Similarly, we also have
Combining (5.1), (5.2) and (5.3) gives kk2 5 2pfkr ln f k2 ðln f Þ þ 2hg;
ð5:4Þ
with equality sign holding if and only if we have ðLðNT Þ; LðNT ÞÞ ¼ ðLðN? Þ; LðN? ÞÞ ¼ 0; i.e., NT f N? is both NT and N? -totally geodesic.
ð5:5Þ
CR-Warped Products with Compact Holomorphic Factor
When NT is compact, (5.4) and Hopf’s lemma imply that ð ð kk2 dVT 5 2p kr ln f k2 dVT þ 4hp volðNT Þ NT fqg
185
ð5:6Þ
NT
for each q 2 N? . Obviously, inequality (5.6) implies inequality (1.2), with the equality sign holding if and only if (1) f is constant and (2) the equality kk2 ¼ 4hp holds identically. When f is constant, the warped product is the Riemannian product of ðNT ; gNT Þ and ðN? ; f 2 gN? Þ. So, NT f N? can be regarded as a CR-product in CPm ð4Þ. Thus, by applying Theorem 6.1 of [4, part I], we have: (1) NT is CPh ð4Þ which is immersed as a totally geodesic complex submanifold. (2) ðN? ; f 2 fN? Þ is an open portion of RPp ð1Þ which is immersed as a totally real totally geodesic submanifold. (3) NT f N? is immersed linearly fully in a linear complex subspace CPh þ p þ hp ð4Þ of CPm ð4Þ; moreover, the immersion is rigid. Conversely, if : NT f N? ! CPm ð4Þ is a CR-warped product such that conditions (i)–(iv) in Theorem 3 hold, then the CR-warped product is both NT and N? totally geodesic. On the other hand, since is a CR-product, Lemma 4 and statement (3) Lemma 2 imply that kðX; ZÞk2 ¼ hX; XihZ; Zi. Thus, we have kk2 ¼ 4hp identically. Therefore, the equality case of (1.2) holds identically. This completes the proof of Theorem 3. Next, let us assume that f is non-constant. Then the minimum principal on 1 yields (see [2, page 186]) ð ð 2 kr ln f k dVT 5 1 ðln f Þ2 dVT ð5:7Þ NT
NT
with equality holding if and only if ln f ¼ 1 ln f holds. So, by combining (5.6) and (5.7), we obtain inequality (1.3). Clearly, from above discussion, we know that the equality sign of (1.3) holds identically if and only if we have (a) ln f ¼ 1 ln f and (b) the warped product is both NT and N? -totally geodesic. & 6. Examples The following example shows that Theorems 1, 2 and 3 are sharp. Example 1. Let 1 be the identity map of CPh ð4Þ and 2 : RPp ð1Þ ! CPp ð4Þ a totally geodesic Lagrangian embedding of RPp ð1Þ into CPp ð4Þ. Denote by ¼ ð 1 ; 2 Þ : CPh ð4Þ RPp ð1Þ ! CPh ð4Þ CPp ð4Þ the product embedding of 1 and 2 . Let Sh;p be the Segre embedding of CPh ð4Þ CPp ð4Þ into CPhpþhþp ð4Þ. Then the composition ¼ Sh;p : ð 1 ; 2 Þ
CPh ð4Þ RPp ð1Þ ! CPh ð4Þ CPp ð4Þ totally geodesic Sh;p
! CPhp þ h þ p ð4Þ Segre embedding
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is a CR-warped product in CPh þ p þ hp whose holomorphic factor NT ¼ CPh ð4Þ is compact. Since the second fundamental from of satisfies kk2 ¼ 4hp, we have the equality case of (1.2) identically. The next example shows that the assumption of compactness cannot be removed. mþ1 Example 2. Let C ¼ C f0g, Cmþ1 f0g and fz0 ; . . . ; zh g is a nat mþ1¼ C ural complex coordinate system on C . Consider the action of C on Cmþ1 defined by ðz0 ; . . . ; zm Þ ¼ ðz0 ; . . . ; zm Þ for 2 C. Let ðzÞ denote the equivalence class containing z. Then the set of equivalence classes is the complex projective m-space CPm ð4Þ with the complex structure induced from the complex structure on Cmþ1 . hþpþ1 p For two natural numbers h and p, we define a map : Chþ1 S ð1Þ ! C by ðz0 ; . . . ; zh ; w0 ; . . . ; wp Þ ¼ ðw0 z0 ; P w1 z0 ; . . . ; wp z0 ; z1 ; . . . ; zh Þ for ðz0 ; . . . ; zh Þ in p p 2 Chþ1 and ðw0 ; . . . ; wp Þ in S ð1Þ with j¼0 wj ¼ 1. Since the image of is invariant under the action of C , the composition:
hþpþ1 p : Chþ1 ! CPhþp ð4Þ S ð1Þ ! C
induces a CR-immersion of the product manifold NT Sp ð1Þ into CPhþp ð4Þ, where NT ¼ fðz0 ; . . . ; zh Þ 2 CPh ð4Þ : z0 6¼ 0g is a proper open subset of CPh ð4Þ. Clearly, the induced metric on NT Sp ð1Þ is a warped product metric and the holomoprhic factor NT is non-compact. Notice that the complex dimension of the ambient space is h þ p; far less than h þ p þ hp. References [1] [2] [3] [4]
Bejancu A (1986) Geometry of CR-Submanifolds. Dordrecht: Reidel Berger M, Gauduchon P, Mazet E (1971) Le spectre d’une variete riemannienne. Berlin: Springer Chen BY (1973) Geometry of Submanifolds. New York: Dekker Chen BY (1981) CR-submanifolds of a Kaehler manifold I. J Differential Geometry 16: 305–322; II, ibid 16: 493–509 [5] Chen BY (2001) Geometry of warped product CR-submanifolds in Kaehler manifolds. Monatsh Math 133: 177–195; II, ibid 134: 103–119 [6] Chen BY (2002) Real hypersurfaces in complex space form which are warped products. Hokkaido Math J 31: 363–383 [7] O’Neill B (1983) Semi-Riemannian Geometry with Application to Relativity. New York: Academic Press Author’s address: Department of Mathematics, Michigan State University, East Lansing, MI 488241027, USA, e-mail:
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