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On the isolation phenomena of locally conformally flat manifolds with constant scalar curvature – Submanifolds versions
Accepted Manuscript On the isolation phenomena of locally conformally flat manifolds with constant scalar curvature – Submanifolds versions (revised version)
Xiuxiu Cheng, Zejun Hu
PII: DOI: Reference:
S0022-247X(18)30364-0 https://doi.org/10.1016/j.jmaa.2018.04.057 YJMAA 22211
To appear in:
Journal of Mathematical Analysis and Applications
Received date:
19 September 2017
Please cite this article in press as: X. Cheng, Z. Hu, On the isolation phenomena of locally conformally flat manifolds with constant scalar curvature – Submanifolds versions (revised version), J. Math. Anal. Appl. (2018), https://doi.org/10.1016/j.jmaa.2018.04.057
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ON THE ISOLATION PHENOMENA OF LOCALLY CONFORMALLY FLAT MANIFOLDS WITH CONSTANT SCALAR CURVATURE – SUBMANIFOLDS VERSIONS (REVISED VERSION) XIUXIU CHENG AND ZEJUN HU
Abstract. In this paper, from the viewpoint of submanifold theory, we study the isolation phenomena of Riemannian manifolds with constant scalar curvature and vanishing Weyl conformal curvature tensor. Firstly, for any locally strongly convex affine hyperspheres in an (n + 1)-dimensional equiaffine space Rn+1 with constant scalar curvature, we prove an inequality involving the traceless Ricci tensor, the Pick invariant and the scalar curvature. The inequality is optimal and we can further completely classify the affine hyperspheres which realize the equality case of the inequality. Secondly, and analogously, for Lagrangian minimal submanifolds of complex projective space CP n equipped with the Fubini-Study metric, under the condition that the Weyl conformal curvature tensor vanishes, we establish a similar but reverse inequality involving the traceless Ricci tensor, the scalar curvature and the squared norm of the second fundamental form. The inequality is also optimal and we can further completely classify the submanifolds which realize the equality case of the inequality.
1. Introduction The study of isolation phenomena of Riemannian manifolds with appropriate curvature properties is an interesting topic that has achieved a lot of results. In the present paper, we study such problem for Riemannian manifolds with both constant scalar curvature and vanishing Weyl conformal curvature tensor, under the additional condition that the Riemannian manifolds appear as hypersurfaces or submanifolds of some ambient spaces. As a matter of fact, we take this paper as a counterpart of Cheng-Hu-Li-Li’s article [5], where remarkable isolation phenomena of Einstein manifolds were shown from the viewpoint of submanifold theory. Recall that a Riemannian manifold (M n , g) of dimension n ≥ 4 has vanishing Weyl conformal curvature tensor if and only if it is locally conformally flat, which means that a neighborhood of each point of M n can be conformally immersed into the Euclidean n-space Rn . On the other hand, if n = 3, the Weyl conformal curvature tensor becomes automatically zero identically. To have a brief overview on the rigidity phenomena of locally conformally flat manifolds with constant scalar curvature, which have been studied extensively by many geometers, e.g. [2, 4, 7, 10–12, 21, 22, 24, 25], we recall that the earliest Key words and phrases. Isolation phenomenon, affine hypersphere, locally conformally flat manifold, constant scalar curvature, Lagrangian submanifold, minimal submanifold. 2010 Mathematics Subject Classification. Primary 53C24; Secondary 53A15, 53C25, 53D12. This project was supported by grants of NNSFC, No.11371330 and No.11771404. 1
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XIUXIU CHENG AND ZEJUN HU
result due to M. Tani [22] showed that the universal cover of a compact, orientable, n-dimensional, locally conformally flat Riemannian manifold (M n , g) with constant scalar curvature and positive Ricci curvature is a sphere. Then as a generalization, S.I. Goldberg [10] proved that an n-dimensional complete and locally conformally flat Riemannian manifold (M n , g) with positive constant scalar curvature R and Ricci curvature bounded from below is a space form for n ≥ 3, provided that the ˜ satisfies Ric ˜ 2 < R2 , where ·g denotes the tensorial traceless Ricci tensor Ric g n(n−1) norm with respect to g. Moreover, Pigola-Rigoli-Setti [21] further proved that the universal covering of an n-dimensional complete and locally conformally flat Riemannian manifold (M n , g) with positive constant scalar curvature is isometric ˜ 2 ≤ R2 and the strict inequality holds at to a sphere, provided that Ric g n(n−1) some point. In this respect, we would mention that Q.M. Cheng [4] have also improved Tani’s above result by proving that a compact, connected, oriented and locally conformally flat Riemannian n-manifold with constant scalar curvature is isometric to a space form or a Riemannian product Sn−1 (c) × S1 provided that its Ricci curvature is nonnegative. Furthermore, Cheng [4] also gave a topological classification of compact, connected, oriented, locally conformally flat Riemannian ˜ 2 ≤ R2 , stating that manifolds with nonnegative scalar curvature R and Ric g n(n−1) such manifolds either are isometric to Sn−1 (c) × S1 or have universal covering which is diffeomorphic to a sphere. There are also results on the global isolation phenomena of locally conformally flat Riemannian manifolds with constant scalar curvature. Indeed, Pigola-RigoliSetti [21] proved that a complete simply connected and locally conformally flat Riemannian n-manifold with scalar curvature R = 0 and n ≥ 3 is isometrically the ˜ n/2 < A(n), where · k denotes the Lk -norm, A(n) = Euclidean space if Ric 2/n 2n−5/2 (n − 1)1/2 (n − 2)3 ωn with ωn the volume of the unit sphere. Then, in [24], H.W. Xu and E.T. Zhao improved Pigola-Rigoli-Setti’s above pinching theorem by ˜ n/2 < B(n) with B(n) = 2n−5/2 (n − 1)1/2 (n − 2)(n2 − 2n + assuming that Ric 2/n 4)ωn . Moreover, Xu and Zhao [24] further proved that, for a complete locally conformally flat Riemannian n-manifold with constant scalar curvature R = 0, if ˜ n/2 < n(n − 1)ωn2/n , then it is isometrically a space form. n ≥ 6 and Ric In this paper, we concentrate on Riemannian manifolds with constant scalar curvature and vanishing Weyl conformal curvature tensor, which appear as hypersurfaces or generally the submanifolds of some ambient spaces. Our main purpose is to show that for such hypersurfaces or submanifolds, the norm of their traceless Ricci tensors satisfy an interesting optimal pointwise inequality, meaning that we can completely classify the hypersurfaces or submanifolds which realize the equality case of the inequality. Our proof makes use of two exciting achievements having been made in recent years in local differential geometry. The first one belongs to affine differential geometry concerning the complete classification of a large subclass of hyperbolic affine hyperspheres, namely the locally strongly convex affine hypersurfaces that have parallel cubic form (i.e. Fubini-Pick form) with respect to the Levi-Civita connection of the affine metric, due to Hu, Li and Vrancken (cf. [15], and [13, 14] for its preparatory stage). The second one is due to Dillen, Li, Vrancken and Wang [6] and [18], concerning the updated complete classification of Lagrangian submanifolds in complex projective space with parallel second fundamental form.
ISOLATION PHENOMENA OF LOCALLY CONFORMALLY FLAT MANIFOLDS
3
Firstly, we consider locally strongly convex affine hyperspheres in the (n + 1)dimensional equiaffine space Rn+1 such that their affine metrics are of constant scalar curvature. As a 2-dimensional Riemannian manifold with constant scalar curvature has constant sectional curvature, thus for n = 2 a locally strongly convex affine sphere with constant affine scalar curvature is affinely equivalent to either a quadric or the hyperbolic affine sphere x1 x2 x3 = 1 in R3 (cf. [17] or [23]). For the cases of n ≥ 3, we will state our main results as follows: Theorem 1.1. Let x : M n → Rn+1 (n ≥ 3) be a locally strongly convex affine hypersphere with affine mean curvature L1 such that its affine metric G has constant ˜ of scalar curvature R. Then we have JR ≤ 0, and the traceless Ricci tensor Ric M n satisfies ˜ 2 ≤ − (n+1)(n−2) JR, (1.1) Ric G
n+2
where J denotes the Pick invariant. Moreover, the equality sign in (1.1) holds identically if and only if (M n , G) has vanishing Weyl conformal curvature tensor W and, moreover, one of the following cases occurs: (i) R = 0, J = 0 and M n is affinely equivalent to the elliptic paraboloid xn+1 = 12 (x1 )2 + (x2 )2 + · · · + (xn )2 with L1 = 0; (ii) R > 0, J = 0 and M n is affinely equivalent to the ellipsoid (x1 )2 + (x2 )2 + · · · + (xn )2 + (xn+1 )2 = 1 with L1 = 1; (iii) R < 0, J = 0 and M n is affinely equivalent to the hyperboloid (x1 )2 + (x2 )2 + · · · + (xn )2 − (xn+1 )2 = −1 with L1 = −1; (iv) R = 0, J = 0 and M n is affinely equivalent to the flat and hyperbolic affine hypersphere Q(1, n) : x1 x2 · · · xn+1 = 1 with L1 = −(n + 1)−(n+1)/(n+2) ; (v) R < 0, J = 0 and M n is affinely equivalent to the hyperbolic affine hypersphere n (xn )2 − (x1 )2 − · · · − (xn−1 )2 (xn+1 )2 = 1 with L1 = −nn/(n+2) (n + 1)−(n+1)/(n+2) . Remark 1.1. Recall that a Riemannian manifold of dimension n ≥ 4 is locally conformally flat if and only if W vanishes; whereas W = 0 holds automatically if n = 3. Related to Theorems 1.1, we would also mention the well-known fact that, unlike in the case of the Euclidean hyperspheres, there are many different types of affine hyperspheres. In chapter 3 of the book [17], there are a comprehensive and systematic summary on the study of this large class of equiaffine hypersurfaces. To introduce the next result, we denote by CP n (4) the complex projective space equipped with the Fubuni-Study metric of constant holomorphic sectional curvature 4. We will consider Lagrangian minimal submanifolds of CP n (4) with constant scalar curvature R and vanishing Weyl conformal curvature tensor W . As the main result we will prove: Theorem 1.2. Let x : M n → CP n (4) (n ≥ 4) be a Lagrangian minimal submanifold such that the induced metric g of M n is locally conformally flat with constant ˜ of M n satisfies scalar curvature R. Then the traceless Ricci tensor Ric ˜ 2 ≥ (n−2)(n+1) SR, (1.2) Ric g
n(n−1)(n+2)
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XIUXIU CHENG AND ZEJUN HU
where S is the squared-norm of the second fundamental form of x : M n → CP n (4). Moreover, the equality sign in (1.2) holds identically if and only if one of the following cases occurs: (i) R = n(n−1), S = 0 and M n is congruent with the totally geodesic standard embedding of the real projective space RP n into CP n (4); (ii) R = 0, S = n(n − 1) and M n is congruent with the standard embedding of the Clifford torus T n into CP n (4); (iii) R > 0, S > 0, and, up to a reparametrization and a rigid motion of CP n (4), M n is the Calabi product of the Lagrangian totally geodesic submanifold ˜ with RP n−1 of CP n−1 (4) and a point, such that x = Π ◦ x x ˜ : R × Sn−1 → S2n+1 (1) being expressed by ti nti − n+1 n 1 n+1 (y1 , . . . , yn ), x ˜(t, y) = , e e n+1 n+1 and Π : S2n+1 (1) → CP n (4) is the Hopf fibration. From the fact that W vanishes automatically for n = 3 and the proof of Theorem 1.2 we obtain immediately: Theorem 1.3. Let x : M 3 → CP 3 (4) be a Lagrangian minimal submanifold such that the induced metric g of M 3 is of constant scalar curvature R. Then the squared˜ of M 3 satisfy norm S and the traceless Ricci tensor Ric ˜ 2 ≥ 2 SR. (1.3) Ric g
15
Moreover, the equality sign in (1.3) holds identically if and only if one of the cases (i), (ii) and (iii) in Theorem 1.2 occurs for n = 3. Remark 1.2. The standard embeddings of the real projective space RP n and the Clifford torus T n into CP n (4) are described in Li-Zhao [19] (cf. also [3, 9]). Moreover, according to [19] and [9], if x : M n → CP n (4) (n ≥ 2) is a Lagrangian minimal submanifold with constant sectional curvature, then it is the standard embedding of either the real projective space RP n or the Clifford torus T n into CP n (4). Acknowledgements. We would like to express our thanks to the editor and the referee for their valuable comments and helpful proposals on language issues in the original version of this paper. 2. Affine hyperspheres and proof of Theorem 1.1 2.1. Basic facts of equiaffine hypersurfaces. In this section, we briefly review the theory of local equiaffine hypersurfaces, for details we refer to Chapter 2 of [17] (cf. also [20]). Let Rn+1 be the equiaffine space equipped with its canonical flat connection and a parallel volume element, defined by the determinant det. Let M n be a connected and smooth n-dimensional manifold, and x : M n → Rn+1 be a locally strongly convex hypersurface immersion. We choose an equiaffine frame field {x; e1 , e2 , . . . , en , en+1 } on M n , such that det [e1 , e2 , . . . , en , en+1 ] = 1, e1 , e2 , . . . , en ∈ Tx M, Gij := G(ei , ej ) = δij , en+1 = Y,
ISOLATION PHENOMENA OF LOCALLY CONFORMALLY FLAT MANIFOLDS
5
where G and Y denote the (Blaschke-Berwald) affine metric and the equiaffine normal vector field of M n , respectively. Denote by B the equiaffine Weingarten form of x : M n → Rn+1 . The eigenvalues of B relative to G are called the affine principal curvatures of x(M ), and are denoted n by λ1 , . . . , λn . Then the equiaffine mean curvature is defined by L1 = n1 i=1 λi . If λ1 = · · · = λn = const. > 0 (resp. = 0, or < 0), the affine hypersurface is called an elliptic (resp. parabolic, or hyperbolic) affine hypersphere. Denote by Rijkl , Rij the components of Riemannian curvature tensor and Ricci tensor with respect to the affine metric, respectively, and by R the affine scalar curvature. Let Aijk and Aijk,l be the components of the Fubini-Pick form A and its covariant derivative with respect to the Levi-Civita connection of the affine metric. Then, we have the following structural equations (cf. Section 2.5 of [17]): (2.1) (2.2) Rijkl
Aijk,l − Aijl,k = 12 (δik Bjl + δjk Bil − δil Bjk − δjl Bik ), = (Aiml Ajmk − Aimk Ajml ) + 12 (δik Bjl + δjl Bik − δil Bjk − δjk Bil ), m
(2.3)
Rij =
Aikl Ajkl +
k,l
(2.4)
n−2 2 Bij
+ n2 L1 δij ,
Aiij = 0, 1 ≤ j ≤ n,
i
(2.5)
J=
1 n(n−1)
(Aijk )2 , χ = J + L1 , χ =
1 n(n−1) R,
where J and χ are called the Pick invariant and normalized affine scalar curvature, respectively. Next, in order to help the understanding of Theorems 1.1, we briefly review the following canonical locally strongly convex affine hyperspheres and their equiaffine invariants, for more details we refer to [17]. Example 1. Hyperquadrics in the equiaffine space Rn+1 are characterized by having vanishing Pick invariant, namely J = 0. They consist of the following three families of hypersurfaces corresponding to the value of L1 : 1) The elliptic paraboloid, defined by xn+1 = 12 (x1 )2 + (x2 )2 + · · · + (xn )2 , has the properties that J = λ1 = · · · = λn = χ = 0 and G is a flat metric. 2) The ellipsoid, defined by (x1 )2 + (x2 )2 + · · · + (xn )2 + (xn+1 )2 = r2 , r > 0, has the properties that J = 0, λ1 = · · · = λn = χ = r−(2n+2)/(n+2) and G is of constant sectional curvature r−(2n+2)/(n+2) . 3) The hyperboloid, defined by (x1 )2 + (x2 )2 + · · · + (xn )2 − (xn+1 )2 = −r2 , r > 0, xn+1 ≥ r, has the properties that J = 0, λ1 = · · · = λn = χ = −r−(2n+2)/(n+2) and G is of constant sectional curvature −r−(2n+2)/(n+2) . Example 2. The hypersurface Q(c, n) in Rn+1 , defined by x1 x2 · · · xn+1 = c, c > 0,
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XIUXIU CHENG AND ZEJUN HU
has the properties that λ1 = · · · = λn = −J = −(n+1)−(n+1)/(n+2) c−2/(n+2 , R = 0 and G is a flat metric. Example 3. The Calabi composition (of a hyperbolic affine hypersphere M and a point) defined by (cf. [1, 14], or pages 101-102 of [17])
√ x(t, u) = c exp √atn x (u), c exp{− n at} , u ∈ M , t ∈ R, where c , c , a are positive constants; and x : M → Rn is an (n − 1)-dimensional hyperbolic affine hypersphere (centered at the origin) with affine mean curvature L1 and affine metric G . A direct computation shows that the composition x is a new locally strongly convex hyperbolic affine hypersphere with affine metric G and affine mean curvature L1 being given by
n+1 nL1 2 2 , L1 = (n+1)Λ , G = Λ n(−L ) a dt ⊕ G 1
2
n where Λ is a positive constant such that Λn+2 = − n+1 L1 (c )2n (c )2 .
2.2. Affine hyperspheres with constant scalar curvature. In this section, we consider n (n ≥ 3)-dimensional locally strongly convex affine hyperspheres with constant affine mean curvature L1 and constant affine scalar curvature R. We begin with the following result, which modifies a statement in [16]. Lemma 2.1. Let x : M n → Rn+1 be a locally strongly convex affine hypersphere with constant affine scalar curvature R, then we have n+2 ˜ ij )2 + (n + 1)JR = 0, (2.6) (Aijk,l )2 + (Wijkl )2 + n−2 (R ˜ ij denote the components of the Weyl conformal curvature tensor where Wijkl and R and the traceless Ricci tensor, respectively. Proof. Choose as in section 2.1 a local equiaffine frame field {x; e1 , . . . , en+1 } along M n . Since x(M ) is an affine hypersphere, (2.1) - (2.3) yield immediately that Aijk,l = Aijl,k ,
(2.7) (2.8)
Rijkl =
(Aiml Ajmk − Aimk Ajml ) + L1 (δik δjl − δil δjk ),
m
(2.9)
Rij =
Aikl Ajkl + (n − 1)L1 δij .
k,l
From (2.4) and (2.7), and applying the well-known Ricci identity, we calculate the Laplacian of the Fubini-Pick form A to get ΔAijk = Aijk,ll = Aijl,kl l
(2.10)
=
l
=
l
l
Aijl,lk +
l
Aijr Rrlkl +
Aijr Rrlkl +
l
l
Airl Rrjkl +
Airl Rrjkl +
l
l
Arjl Rrikl .
Arjl Rrikl
ISOLATION PHENOMENA OF LOCALLY CONFORMALLY FLAT MANIFOLDS
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It follows that the Laplacian of the Pick invariant J can be calculated by 1 Δ (Aijk )2 ΔJ = n(n−1)
2 Aijk Aijk,ll = n(n−1) (Aijk,l )2 + (2.11) 2 Aijk Aijr Rrlkl = n(n−1) (Aijk,l )2 +
+ (Aijk Airl − Aijl Airk )Rrjkl . Substituting (2.8) and (2.9) into (2.11), we get 1 (2.12) (Aijk,l )2 + (Rij )2 + (Rijkl )2 − (n + 1)RL1 . 2 n(n − 1)ΔJ = ˜ ij = Rij − 1 Rδij , we have Using that R n ˜ ij )2 + 1 R2 . (R (2.13) (Rij )2 = n On the other hand, from the decomposition (2.14)
Rijkl = Wijkl +
1 n−2 (δik Rjl
+ δjl Rik − δil Rjk − δjk Ril ) −
we have the computation that (2.15) (Rijkl )2 = (Wijkl )2 +
R (n−1)(n−2) (δik δjl
4 n−2
(Rij )2 −
− δil δjk ),
2R2 (n−1)(n−2) .
Combining (2.13), (2.15) with R = n(n − 1)(J + L1 ), we can rewrite (2.12) as 1 n+2 ˜ ij )2 (Aijk,l )2 + (Wijkl )2 + n−2 (R 2 n(n − 1)ΔJ = (2.16) + (n + 1)JR. Then, noting that R = const. implies that J = const., the assertion (2.6) follows immediately. As direct consequence of Lemma 2.1, we have: Corollary 2.1. Let x : M n → Rn+1 be a locally strongly convex affine hypersphere with constant affine scalar curvature R, then we have ˜ ij )2 ≤ − (n−2)(n+1) JR. (2.17) (R n+2 Moreover, the equality sign in (2.17) holds identically if and only if Aijk,l = Wijkl = 0 for any i, j, k, l, namely that M n has parallel cubic form and that (M n , G) has vanishing Weyl conformal curvature tensor. 2.3. Proof of Theorem 1.1. Due to Lemma 2.1 and Corollary 2.1, we are left to discuss the situation that the equality sign in (2.17) holds identically. If it is the case, then we can apply the Classification Theorem of [15] to see that only the following three cases can occur: (1) J = 0 and M n is affinely equivalent to a hyperquadric, as stated by (i), (ii) and (iii). (2) M n is obtained either as the Calabi composition of an (n − 1)-dimensional hyperbolic affine hypersphere M1 with parallel cubic form and a point, or the Calabi composition of two lower dimensional hyperbolic affine hyperspheres, M1 and M2 , both with parallel cubic form. In this case, according to the results of [14] (cf. also [1, 8]), we can assume that (M n , G) takes the form either as
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XIUXIU CHENG AND ZEJUN HU
(2-1) (I × M1 , c dt2 ⊕ c1 G1 ); or (2-2) (I × M1 × M2 , c dt2 ⊕ c1 G1 ⊕ c2 G2 ), where c1 , c2 , c3 are constants, G1 and G2 are the affine metrics of M1 and M2 , respectively. For both of the above two subcases, we take an orthonormal basis {e1 , . . . , en } ∂ . Then we have R1ijk = 0, ∀ i, j, k. This, together with of M n such that e1 = √1c ∂t (2.14) and Wijkl = 0, implies that (2.18)
R11 = 0, Rij =
R n−1 δij ,
i, j ≥ 2.
Then using (2.14), (2.18) and Wijkl = 0 again, we obtain that (2.19)
Rijkl =
R (n−1)(n−2) (δik δjl
− δjk δil ), if i, j, k, l ≥ 2.
If case (2-1) occurs, then (2.19) implies that M1 is an (n − 1)-dimensional hyperbolic affine hypersphere with constant sectional curvature. From [23], we see that M1 is affinely equivalent to either Q(1, n − 1) : x1 x2 · · · xn = 1 in Rn , or the hyperboloid (x1 )2 +(x2 )2 +· · ·+(xn−1 )2 −(xn )2 = −1 in Rn . In the former case, M n is affinely equivalent to Q(1, n) : x1 x2 · ·· xn+1 = 1 in Rn+1 , as statedin (iv). In the n latter case, M n is affinely equivalent to (xn )2 − (x1 )2 − · · · − (xn−1 )2 (xn+1 )2 = 1 n+1 in R , as stated in (v). If case (2-2) occurs, then it is easily seen that R(v, w, v, w) = 0 for v ∈ T M1 and w ∈ T M2 . This implies by (2.19) that R = 0 and thus G is a flat metric. From [23] (cf. [16] or Theorem 3.8 of [17]), we obtain that M n is locally affinely equivalent to Q(1, n), as stated in (iv). (3) M n is affinely equivalent to one of the standard embedding of the noncompact symmetric spaces: SL(m, R)/SO(m), SL(m, C)/SU(m), SU∗ (2m)/Sp(m) for each m ≥ 3 and E6(−26) /F4 , with dimensions 21 m(m+1)−1, m2 −1, 2m2 −m−1 and 26, respectively. On the other hand, it was shown in [5] thatall these hyperbolic affine hyperspheres are actually of Einstein affine metrics with (Wijkl )2 = −(n+1)JR = 0. This contradicts the assertion of Corollary 2.1 that W = 0. We have completed the proof of Theorem 1.1. 3. Lagrangian minimal submanifolds and proof of Theorem 1.2 3.1. Lagrangian minimal submanifolds of CP n (4). In this section, we briefly review the theory of Lagrangian submanifolds of the n-dimensional complex projective space CP n (4) with the Fubini-Study metric g of constant holomorphic sectional curvature 4 and the canonical almost complex structure J. For details we refer to [3] and [19]. Suppose that M n is a Lagrangian submanifold of CP n (4), that means that J carries each tangent space of M n to be its corresponding normal space. We choose a local orthonormal frame field {e1 , e2 , . . . , e2n } of CP n (4) such that, restricted to M n , the vectors e1 , e2 , . . . , en are tangent to M n and en+i = Jei for 1 ≤ i ≤ n. In what follows we shall make use of the indices convention: 1 ≤ i, j, k, . . . ≤ n. Let hkij denote the components of the second fundamental form h of M n → CP n (4) with respect kto the frame field {e1 , . . . , en , Je1 , . . . , Jen }, namely we assume hij Jek . Then we have the totally symmetry relation that h(ei , ej ) = (3.1)
hkij = hjik = hikj .
ISOLATION PHENOMENA OF LOCALLY CONFORMALLY FLAT MANIFOLDS
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From now on we further suppose that M n is minimal, then it holds (3.2) hkii = 0. i
Rijkl , Rij , hlij,k
respectively the components of Riemannian curvature Denote by tensor, Ricci tensor and the covariant derivative ∇h of the second fundamental form of M n . Then we have the following equations: (3.3) (3.4)
hlij,k = hlik,j , Rijkl = δik δjl − δil δjk +
m m m (hm ik hjl − hil hjk ),
(3.5)
Rij = (n − 1)δij −
(3.6)
R = n(n − 1) − S, S =
hlik hljk ,
(hkij )2 ,
where S and R denote the squared-norm of the second fundamental form and the scalar curvature of M n , respectively. 3.2. Proof of Theorem 1.2. We begin with the following result, which modifies a similar statement in [5]. Lemma 3.1. Let x : M n → CP n (4) (n ≥ 4) be a Lagrangian minimal submanifold with constant scalar curvature, then we have n+2 n+1 ˜ ij )2 − (3.7) (hlij,k )2 − n−2 (R (Wijkl )2 + n(n−1) SR = 0. Proof. Using (3.1) – (3.3) and the Ricci identity, we have hkij,ll = hkil,jl Δhkij = = hkil,lj + hkir Rrljl + hkrl Rrijl + hril Rrkjl (3.8) = hkir Rrljl + hkrl Rrijl + hril Rrkjl . From the computation of the Laplacian of 12 S, using (2.13), (2.15), (3.1), (3.4)(3.6) and (3.8) , we obtain 1 (hkij,l )2 + (hirj hikl − hkij hril )Rkrjl + hkij hkir Rrj 2 ΔS = = (hkij,l )2 − (Rij )2 − (Rijkl )2 + (n + 1)R (3.9) n+2 n+1 ˜ ij )2 − = (hkij,l )2 − n−2 (R (Wijkl )2 + n(n−1) SR. Noting that R = const. implies that S = const., and the assertion (3.7) follows.
Next, we assume that Wijkl = 0. Then from (3.7) we have n+2 n+1 ˜ ij )2 = n+1 SR + (R (hkij,l )2 ≥ n(n−1) SR. (3.10) n−2 n(n−1) It follows that (1.2) is proven. Moreover, from (3.10) we see that the equality sign in (1.2) holds identically if and only if (3.11)
hkij,l = 0, 1 ≤ i, j, k, l ≤ n,
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XIUXIU CHENG AND ZEJUN HU
i.e., M n has parallel second fundamental form. If it is the case, we can apply the Classification Theorem of [6] and Theorem 1.5 of [18] to obtain that only one of the following three cases should occur: (1) M n is totally geodesic. Then, according to [19] (cf. also [3] and [9]), M n is congruent with the standard embedding of the real projective space RP n into CP n (4), and we have S = 0 and R = n(n − 1), as stated in (i) of Theorem 1.2. (2) M n is obtained as the Calabi product of an (n − 1)-dimensional Lagrangian minimal submanifold with parallel second fundamental form and a point, or the Calabi product of two lower dimensional Lagrangian minimal submanifolds with parallel second fundamental form. Then, similar arguments and computations to case (2) in our proof of Theorem 1.1, with using Theorem B of [19] and Theorem 1.5 of [18], we obtain that either (ii) or (iii) occurs, as stated in Theorem 1.2. (3) M n is congruent to one of the standard embedding of the following compact symmetric spaces: SU(m)/SO(m), SU(m), SU(2m)/Sp(m) for each m ≥ 3 and E6 /F4 , with dimensions 12 m(m+1)−1, m2 −1, 2m2 −m−1 and 26, respectively. On the other hand, it was shown in [5] that all theseLagrangian minimal submanifolds n+1 SR = 0. This are actually of Einstein induced metrics with (Wijkl )2 = n(n−1) n contradicts the assumption that the induced metric on M is locally conformally flat. We have completed the proof of Theorem 1.2. References [1] E. Calabi, Complete affine hypersurfaces I, Symposia Math. 10 (1972), 19-38. [2] G. Catino, On conformally flat manifolds with constant positive scalar curvature, Proc. Amer. Math. Soc. 144 (2016), 2627-2634. [3] B.Y. Chen and K. Ogiue, On totally real submanifold, Trans. Amer. Math. Soc. 193 (1974), 257-266. [4] Q.M. Cheng, Compact locally conformally flat Riemannian manifolds, Bull. London Math. Soc. 33 (2001), 459-465. [5] X. Cheng, Z. Hu, A.-M. Li and H. Li, On the isolation phenomena of Einstein manifolds – Submanifolds versions, Proc. Amer. Math. Soc. 146 (2018), 17311740. [6] F. Dillen, H. Li, L. Vrancken and X. Wang, Lagrangian submanifolds in complex projective space with parallel second fundamental form, Pacific J. Math. 255 (2012), 79-115. [7] F. Dillen, M. Petrovic and L. Verstraelen, Einstein, conformally flat and semi-symmetric submanifolds satisfying Chen’s equality, Israel J. Math. 100 (1997), 163-169. [8] F. Dillen and L. Vrancken, Calabi-type composition of affine spheres, Differ. Geom. Appl. 4 (1994), 303-328. [9] N. Ejiri, Totally real minimal immersions of n-dimensional real space forms into n-dimensional complex space forms, Proc. Amer. Math. Soc. 81 (1982), 213-216. [10] S.I. Goldberg, An application of Yau’s maximum principle to conformally flat spaces, Proc. Amer. Math. Soc. 79 (1980), 268–270. [11] T. Hasanis, Conformally flat spaces and a pinching problem on the Ricci tensor, Proc. Amer. Math. Soc. 86 (1982), 312-315.
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[12] Z. Hu, H. Li, and U. Simon, Schouten curvature functions on locally conformally flat Riemannian manifolds, J. Geom. 88 (2008), 75-100. [13] Z. Hu, H. Li, U. Simon and L. Vrancken, On locally strongly convex affine hypersurfaces with parallel cubic form. Part I, Differ. Geom. Appl. 27 (2009), 188-205. [14] Z. Hu, H. Li and L. Vrancken, Characterization of the Calabi product of hyperbolic affine hyperspheres, Results Math. 52 (2008), 299-314. [15] Z. Hu, H. Li and L. Vrancken, Locally strongly convex affine hypersurfaces with parallel cubic form, J. Diff. Geom. 87 (2011), 239-307. [16] A.-M. Li, Some theorems in affine differential geometry, Acta Math. Sinica (N.S.), 5 (1989), 345-354. [17] A.-M. Li, U. Simon, G.S. Zhao and Z. Hu, Global Affine Differential Geometry of Hypersurfaces, 2nd edition, Walter de Gruyter, Berlin/Boston, 2015. [18] H. Li and X. Wang, Calabi product Lagrangian immersions in complex projective space and complex hyperbolic space, Results Math. 59 (2011), 453-470. [19] A.-M. Li and G.S. Zhao, Totally real minimal submanifolds in CP n , Arch. Math. 62 (1994), 562-568. [20] K. Nomizu and T. Sasaki, Affine Differential Geometry. Geometry of Affine Immersions, Cambridge University Press, Cambridge, 1994. [21] S. Pigola, M. Rigoli and A.G. Setti, Some characterizations of space-forms, Trans. Amer. Math. Soc. 359 (2007), 1817-1828 [22] M. Tani, On a conformally flat Riemannian space with positive Ricci curvature, Tˆohoku Math. J. (2) 19 (1967), 227-231. [23] L. Vrancken, A.-M. Li and U. Simon, Affine spheres with constant affine sectional curvature, Math. Z. 206 (1991), 651-658. [24] H.W. Xu and E.T. Zhao, Lp Ricci curvature pinching theorems for conformally flat Riemannian manifolds, Pacific J. Math. 245 (2010), 381-396. [25] S.H. Zhu, The classification of complete locally conformally flat manifolds of nonnegative Ricci curvature, Pacific J. Math. 163 (1994), 189-199.
School of Mathematics and Statistics, Zhengzhou University, Zhengzhou 450001, People’s Republic of China. E-mail addresses: [email protected]; [email protected]
Xiuxiu Cheng, Zejun Hu
PII: DOI: Reference:
S0022-247X(18)30364-0 https://doi.org/10.1016/j.jmaa.2018.04.057 YJMAA 22211
To appear in:
Journal of Mathematical Analysis and Applications
Received date:
19 September 2017
Please cite this article in press as: X. Cheng, Z. Hu, On the isolation phenomena of locally conformally flat manifolds with constant scalar curvature – Submanifolds versions (revised version), J. Math. Anal. Appl. (2018), https://doi.org/10.1016/j.jmaa.2018.04.057
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ON THE ISOLATION PHENOMENA OF LOCALLY CONFORMALLY FLAT MANIFOLDS WITH CONSTANT SCALAR CURVATURE – SUBMANIFOLDS VERSIONS (REVISED VERSION) XIUXIU CHENG AND ZEJUN HU
Abstract. In this paper, from the viewpoint of submanifold theory, we study the isolation phenomena of Riemannian manifolds with constant scalar curvature and vanishing Weyl conformal curvature tensor. Firstly, for any locally strongly convex affine hyperspheres in an (n + 1)-dimensional equiaffine space Rn+1 with constant scalar curvature, we prove an inequality involving the traceless Ricci tensor, the Pick invariant and the scalar curvature. The inequality is optimal and we can further completely classify the affine hyperspheres which realize the equality case of the inequality. Secondly, and analogously, for Lagrangian minimal submanifolds of complex projective space CP n equipped with the Fubini-Study metric, under the condition that the Weyl conformal curvature tensor vanishes, we establish a similar but reverse inequality involving the traceless Ricci tensor, the scalar curvature and the squared norm of the second fundamental form. The inequality is also optimal and we can further completely classify the submanifolds which realize the equality case of the inequality.
1. Introduction The study of isolation phenomena of Riemannian manifolds with appropriate curvature properties is an interesting topic that has achieved a lot of results. In the present paper, we study such problem for Riemannian manifolds with both constant scalar curvature and vanishing Weyl conformal curvature tensor, under the additional condition that the Riemannian manifolds appear as hypersurfaces or submanifolds of some ambient spaces. As a matter of fact, we take this paper as a counterpart of Cheng-Hu-Li-Li’s article [5], where remarkable isolation phenomena of Einstein manifolds were shown from the viewpoint of submanifold theory. Recall that a Riemannian manifold (M n , g) of dimension n ≥ 4 has vanishing Weyl conformal curvature tensor if and only if it is locally conformally flat, which means that a neighborhood of each point of M n can be conformally immersed into the Euclidean n-space Rn . On the other hand, if n = 3, the Weyl conformal curvature tensor becomes automatically zero identically. To have a brief overview on the rigidity phenomena of locally conformally flat manifolds with constant scalar curvature, which have been studied extensively by many geometers, e.g. [2, 4, 7, 10–12, 21, 22, 24, 25], we recall that the earliest Key words and phrases. Isolation phenomenon, affine hypersphere, locally conformally flat manifold, constant scalar curvature, Lagrangian submanifold, minimal submanifold. 2010 Mathematics Subject Classification. Primary 53C24; Secondary 53A15, 53C25, 53D12. This project was supported by grants of NNSFC, No.11371330 and No.11771404. 1
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XIUXIU CHENG AND ZEJUN HU
result due to M. Tani [22] showed that the universal cover of a compact, orientable, n-dimensional, locally conformally flat Riemannian manifold (M n , g) with constant scalar curvature and positive Ricci curvature is a sphere. Then as a generalization, S.I. Goldberg [10] proved that an n-dimensional complete and locally conformally flat Riemannian manifold (M n , g) with positive constant scalar curvature R and Ricci curvature bounded from below is a space form for n ≥ 3, provided that the ˜ satisfies Ric ˜ 2 < R2 , where ·g denotes the tensorial traceless Ricci tensor Ric g n(n−1) norm with respect to g. Moreover, Pigola-Rigoli-Setti [21] further proved that the universal covering of an n-dimensional complete and locally conformally flat Riemannian manifold (M n , g) with positive constant scalar curvature is isometric ˜ 2 ≤ R2 and the strict inequality holds at to a sphere, provided that Ric g n(n−1) some point. In this respect, we would mention that Q.M. Cheng [4] have also improved Tani’s above result by proving that a compact, connected, oriented and locally conformally flat Riemannian n-manifold with constant scalar curvature is isometric to a space form or a Riemannian product Sn−1 (c) × S1 provided that its Ricci curvature is nonnegative. Furthermore, Cheng [4] also gave a topological classification of compact, connected, oriented, locally conformally flat Riemannian ˜ 2 ≤ R2 , stating that manifolds with nonnegative scalar curvature R and Ric g n(n−1) such manifolds either are isometric to Sn−1 (c) × S1 or have universal covering which is diffeomorphic to a sphere. There are also results on the global isolation phenomena of locally conformally flat Riemannian manifolds with constant scalar curvature. Indeed, Pigola-RigoliSetti [21] proved that a complete simply connected and locally conformally flat Riemannian n-manifold with scalar curvature R = 0 and n ≥ 3 is isometrically the ˜ n/2 < A(n), where · k denotes the Lk -norm, A(n) = Euclidean space if Ric 2/n 2n−5/2 (n − 1)1/2 (n − 2)3 ωn with ωn the volume of the unit sphere. Then, in [24], H.W. Xu and E.T. Zhao improved Pigola-Rigoli-Setti’s above pinching theorem by ˜ n/2 < B(n) with B(n) = 2n−5/2 (n − 1)1/2 (n − 2)(n2 − 2n + assuming that Ric 2/n 4)ωn . Moreover, Xu and Zhao [24] further proved that, for a complete locally conformally flat Riemannian n-manifold with constant scalar curvature R = 0, if ˜ n/2 < n(n − 1)ωn2/n , then it is isometrically a space form. n ≥ 6 and Ric In this paper, we concentrate on Riemannian manifolds with constant scalar curvature and vanishing Weyl conformal curvature tensor, which appear as hypersurfaces or generally the submanifolds of some ambient spaces. Our main purpose is to show that for such hypersurfaces or submanifolds, the norm of their traceless Ricci tensors satisfy an interesting optimal pointwise inequality, meaning that we can completely classify the hypersurfaces or submanifolds which realize the equality case of the inequality. Our proof makes use of two exciting achievements having been made in recent years in local differential geometry. The first one belongs to affine differential geometry concerning the complete classification of a large subclass of hyperbolic affine hyperspheres, namely the locally strongly convex affine hypersurfaces that have parallel cubic form (i.e. Fubini-Pick form) with respect to the Levi-Civita connection of the affine metric, due to Hu, Li and Vrancken (cf. [15], and [13, 14] for its preparatory stage). The second one is due to Dillen, Li, Vrancken and Wang [6] and [18], concerning the updated complete classification of Lagrangian submanifolds in complex projective space with parallel second fundamental form.
ISOLATION PHENOMENA OF LOCALLY CONFORMALLY FLAT MANIFOLDS
3
Firstly, we consider locally strongly convex affine hyperspheres in the (n + 1)dimensional equiaffine space Rn+1 such that their affine metrics are of constant scalar curvature. As a 2-dimensional Riemannian manifold with constant scalar curvature has constant sectional curvature, thus for n = 2 a locally strongly convex affine sphere with constant affine scalar curvature is affinely equivalent to either a quadric or the hyperbolic affine sphere x1 x2 x3 = 1 in R3 (cf. [17] or [23]). For the cases of n ≥ 3, we will state our main results as follows: Theorem 1.1. Let x : M n → Rn+1 (n ≥ 3) be a locally strongly convex affine hypersphere with affine mean curvature L1 such that its affine metric G has constant ˜ of scalar curvature R. Then we have JR ≤ 0, and the traceless Ricci tensor Ric M n satisfies ˜ 2 ≤ − (n+1)(n−2) JR, (1.1) Ric G
n+2
where J denotes the Pick invariant. Moreover, the equality sign in (1.1) holds identically if and only if (M n , G) has vanishing Weyl conformal curvature tensor W and, moreover, one of the following cases occurs: (i) R = 0, J = 0 and M n is affinely equivalent to the elliptic paraboloid xn+1 = 12 (x1 )2 + (x2 )2 + · · · + (xn )2 with L1 = 0; (ii) R > 0, J = 0 and M n is affinely equivalent to the ellipsoid (x1 )2 + (x2 )2 + · · · + (xn )2 + (xn+1 )2 = 1 with L1 = 1; (iii) R < 0, J = 0 and M n is affinely equivalent to the hyperboloid (x1 )2 + (x2 )2 + · · · + (xn )2 − (xn+1 )2 = −1 with L1 = −1; (iv) R = 0, J = 0 and M n is affinely equivalent to the flat and hyperbolic affine hypersphere Q(1, n) : x1 x2 · · · xn+1 = 1 with L1 = −(n + 1)−(n+1)/(n+2) ; (v) R < 0, J = 0 and M n is affinely equivalent to the hyperbolic affine hypersphere n (xn )2 − (x1 )2 − · · · − (xn−1 )2 (xn+1 )2 = 1 with L1 = −nn/(n+2) (n + 1)−(n+1)/(n+2) . Remark 1.1. Recall that a Riemannian manifold of dimension n ≥ 4 is locally conformally flat if and only if W vanishes; whereas W = 0 holds automatically if n = 3. Related to Theorems 1.1, we would also mention the well-known fact that, unlike in the case of the Euclidean hyperspheres, there are many different types of affine hyperspheres. In chapter 3 of the book [17], there are a comprehensive and systematic summary on the study of this large class of equiaffine hypersurfaces. To introduce the next result, we denote by CP n (4) the complex projective space equipped with the Fubuni-Study metric of constant holomorphic sectional curvature 4. We will consider Lagrangian minimal submanifolds of CP n (4) with constant scalar curvature R and vanishing Weyl conformal curvature tensor W . As the main result we will prove: Theorem 1.2. Let x : M n → CP n (4) (n ≥ 4) be a Lagrangian minimal submanifold such that the induced metric g of M n is locally conformally flat with constant ˜ of M n satisfies scalar curvature R. Then the traceless Ricci tensor Ric ˜ 2 ≥ (n−2)(n+1) SR, (1.2) Ric g
n(n−1)(n+2)
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XIUXIU CHENG AND ZEJUN HU
where S is the squared-norm of the second fundamental form of x : M n → CP n (4). Moreover, the equality sign in (1.2) holds identically if and only if one of the following cases occurs: (i) R = n(n−1), S = 0 and M n is congruent with the totally geodesic standard embedding of the real projective space RP n into CP n (4); (ii) R = 0, S = n(n − 1) and M n is congruent with the standard embedding of the Clifford torus T n into CP n (4); (iii) R > 0, S > 0, and, up to a reparametrization and a rigid motion of CP n (4), M n is the Calabi product of the Lagrangian totally geodesic submanifold ˜ with RP n−1 of CP n−1 (4) and a point, such that x = Π ◦ x x ˜ : R × Sn−1 → S2n+1 (1) being expressed by ti nti − n+1 n 1 n+1 (y1 , . . . , yn ), x ˜(t, y) = , e e n+1 n+1 and Π : S2n+1 (1) → CP n (4) is the Hopf fibration. From the fact that W vanishes automatically for n = 3 and the proof of Theorem 1.2 we obtain immediately: Theorem 1.3. Let x : M 3 → CP 3 (4) be a Lagrangian minimal submanifold such that the induced metric g of M 3 is of constant scalar curvature R. Then the squared˜ of M 3 satisfy norm S and the traceless Ricci tensor Ric ˜ 2 ≥ 2 SR. (1.3) Ric g
15
Moreover, the equality sign in (1.3) holds identically if and only if one of the cases (i), (ii) and (iii) in Theorem 1.2 occurs for n = 3. Remark 1.2. The standard embeddings of the real projective space RP n and the Clifford torus T n into CP n (4) are described in Li-Zhao [19] (cf. also [3, 9]). Moreover, according to [19] and [9], if x : M n → CP n (4) (n ≥ 2) is a Lagrangian minimal submanifold with constant sectional curvature, then it is the standard embedding of either the real projective space RP n or the Clifford torus T n into CP n (4). Acknowledgements. We would like to express our thanks to the editor and the referee for their valuable comments and helpful proposals on language issues in the original version of this paper. 2. Affine hyperspheres and proof of Theorem 1.1 2.1. Basic facts of equiaffine hypersurfaces. In this section, we briefly review the theory of local equiaffine hypersurfaces, for details we refer to Chapter 2 of [17] (cf. also [20]). Let Rn+1 be the equiaffine space equipped with its canonical flat connection and a parallel volume element, defined by the determinant det. Let M n be a connected and smooth n-dimensional manifold, and x : M n → Rn+1 be a locally strongly convex hypersurface immersion. We choose an equiaffine frame field {x; e1 , e2 , . . . , en , en+1 } on M n , such that det [e1 , e2 , . . . , en , en+1 ] = 1, e1 , e2 , . . . , en ∈ Tx M, Gij := G(ei , ej ) = δij , en+1 = Y,
ISOLATION PHENOMENA OF LOCALLY CONFORMALLY FLAT MANIFOLDS
5
where G and Y denote the (Blaschke-Berwald) affine metric and the equiaffine normal vector field of M n , respectively. Denote by B the equiaffine Weingarten form of x : M n → Rn+1 . The eigenvalues of B relative to G are called the affine principal curvatures of x(M ), and are denoted n by λ1 , . . . , λn . Then the equiaffine mean curvature is defined by L1 = n1 i=1 λi . If λ1 = · · · = λn = const. > 0 (resp. = 0, or < 0), the affine hypersurface is called an elliptic (resp. parabolic, or hyperbolic) affine hypersphere. Denote by Rijkl , Rij the components of Riemannian curvature tensor and Ricci tensor with respect to the affine metric, respectively, and by R the affine scalar curvature. Let Aijk and Aijk,l be the components of the Fubini-Pick form A and its covariant derivative with respect to the Levi-Civita connection of the affine metric. Then, we have the following structural equations (cf. Section 2.5 of [17]): (2.1) (2.2) Rijkl
Aijk,l − Aijl,k = 12 (δik Bjl + δjk Bil − δil Bjk − δjl Bik ), = (Aiml Ajmk − Aimk Ajml ) + 12 (δik Bjl + δjl Bik − δil Bjk − δjk Bil ), m
(2.3)
Rij =
Aikl Ajkl +
k,l
(2.4)
n−2 2 Bij
+ n2 L1 δij ,
Aiij = 0, 1 ≤ j ≤ n,
i
(2.5)
J=
1 n(n−1)
(Aijk )2 , χ = J + L1 , χ =
1 n(n−1) R,
where J and χ are called the Pick invariant and normalized affine scalar curvature, respectively. Next, in order to help the understanding of Theorems 1.1, we briefly review the following canonical locally strongly convex affine hyperspheres and their equiaffine invariants, for more details we refer to [17]. Example 1. Hyperquadrics in the equiaffine space Rn+1 are characterized by having vanishing Pick invariant, namely J = 0. They consist of the following three families of hypersurfaces corresponding to the value of L1 : 1) The elliptic paraboloid, defined by xn+1 = 12 (x1 )2 + (x2 )2 + · · · + (xn )2 , has the properties that J = λ1 = · · · = λn = χ = 0 and G is a flat metric. 2) The ellipsoid, defined by (x1 )2 + (x2 )2 + · · · + (xn )2 + (xn+1 )2 = r2 , r > 0, has the properties that J = 0, λ1 = · · · = λn = χ = r−(2n+2)/(n+2) and G is of constant sectional curvature r−(2n+2)/(n+2) . 3) The hyperboloid, defined by (x1 )2 + (x2 )2 + · · · + (xn )2 − (xn+1 )2 = −r2 , r > 0, xn+1 ≥ r, has the properties that J = 0, λ1 = · · · = λn = χ = −r−(2n+2)/(n+2) and G is of constant sectional curvature −r−(2n+2)/(n+2) . Example 2. The hypersurface Q(c, n) in Rn+1 , defined by x1 x2 · · · xn+1 = c, c > 0,
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XIUXIU CHENG AND ZEJUN HU
has the properties that λ1 = · · · = λn = −J = −(n+1)−(n+1)/(n+2) c−2/(n+2 , R = 0 and G is a flat metric. Example 3. The Calabi composition (of a hyperbolic affine hypersphere M and a point) defined by (cf. [1, 14], or pages 101-102 of [17])
√ x(t, u) = c exp √atn x (u), c exp{− n at} , u ∈ M , t ∈ R, where c , c , a are positive constants; and x : M → Rn is an (n − 1)-dimensional hyperbolic affine hypersphere (centered at the origin) with affine mean curvature L1 and affine metric G . A direct computation shows that the composition x is a new locally strongly convex hyperbolic affine hypersphere with affine metric G and affine mean curvature L1 being given by
n+1 nL1 2 2 , L1 = (n+1)Λ , G = Λ n(−L ) a dt ⊕ G 1
2
n where Λ is a positive constant such that Λn+2 = − n+1 L1 (c )2n (c )2 .
2.2. Affine hyperspheres with constant scalar curvature. In this section, we consider n (n ≥ 3)-dimensional locally strongly convex affine hyperspheres with constant affine mean curvature L1 and constant affine scalar curvature R. We begin with the following result, which modifies a statement in [16]. Lemma 2.1. Let x : M n → Rn+1 be a locally strongly convex affine hypersphere with constant affine scalar curvature R, then we have n+2 ˜ ij )2 + (n + 1)JR = 0, (2.6) (Aijk,l )2 + (Wijkl )2 + n−2 (R ˜ ij denote the components of the Weyl conformal curvature tensor where Wijkl and R and the traceless Ricci tensor, respectively. Proof. Choose as in section 2.1 a local equiaffine frame field {x; e1 , . . . , en+1 } along M n . Since x(M ) is an affine hypersphere, (2.1) - (2.3) yield immediately that Aijk,l = Aijl,k ,
(2.7) (2.8)
Rijkl =
(Aiml Ajmk − Aimk Ajml ) + L1 (δik δjl − δil δjk ),
m
(2.9)
Rij =
Aikl Ajkl + (n − 1)L1 δij .
k,l
From (2.4) and (2.7), and applying the well-known Ricci identity, we calculate the Laplacian of the Fubini-Pick form A to get ΔAijk = Aijk,ll = Aijl,kl l
(2.10)
=
l
=
l
l
Aijl,lk +
l
Aijr Rrlkl +
Aijr Rrlkl +
l
l
Airl Rrjkl +
Airl Rrjkl +
l
l
Arjl Rrikl .
Arjl Rrikl
ISOLATION PHENOMENA OF LOCALLY CONFORMALLY FLAT MANIFOLDS
7
It follows that the Laplacian of the Pick invariant J can be calculated by 1 Δ (Aijk )2 ΔJ = n(n−1)
2 Aijk Aijk,ll = n(n−1) (Aijk,l )2 + (2.11) 2 Aijk Aijr Rrlkl = n(n−1) (Aijk,l )2 +
+ (Aijk Airl − Aijl Airk )Rrjkl . Substituting (2.8) and (2.9) into (2.11), we get 1 (2.12) (Aijk,l )2 + (Rij )2 + (Rijkl )2 − (n + 1)RL1 . 2 n(n − 1)ΔJ = ˜ ij = Rij − 1 Rδij , we have Using that R n ˜ ij )2 + 1 R2 . (R (2.13) (Rij )2 = n On the other hand, from the decomposition (2.14)
Rijkl = Wijkl +
1 n−2 (δik Rjl
+ δjl Rik − δil Rjk − δjk Ril ) −
we have the computation that (2.15) (Rijkl )2 = (Wijkl )2 +
R (n−1)(n−2) (δik δjl
4 n−2
(Rij )2 −
− δil δjk ),
2R2 (n−1)(n−2) .
Combining (2.13), (2.15) with R = n(n − 1)(J + L1 ), we can rewrite (2.12) as 1 n+2 ˜ ij )2 (Aijk,l )2 + (Wijkl )2 + n−2 (R 2 n(n − 1)ΔJ = (2.16) + (n + 1)JR. Then, noting that R = const. implies that J = const., the assertion (2.6) follows immediately. As direct consequence of Lemma 2.1, we have: Corollary 2.1. Let x : M n → Rn+1 be a locally strongly convex affine hypersphere with constant affine scalar curvature R, then we have ˜ ij )2 ≤ − (n−2)(n+1) JR. (2.17) (R n+2 Moreover, the equality sign in (2.17) holds identically if and only if Aijk,l = Wijkl = 0 for any i, j, k, l, namely that M n has parallel cubic form and that (M n , G) has vanishing Weyl conformal curvature tensor. 2.3. Proof of Theorem 1.1. Due to Lemma 2.1 and Corollary 2.1, we are left to discuss the situation that the equality sign in (2.17) holds identically. If it is the case, then we can apply the Classification Theorem of [15] to see that only the following three cases can occur: (1) J = 0 and M n is affinely equivalent to a hyperquadric, as stated by (i), (ii) and (iii). (2) M n is obtained either as the Calabi composition of an (n − 1)-dimensional hyperbolic affine hypersphere M1 with parallel cubic form and a point, or the Calabi composition of two lower dimensional hyperbolic affine hyperspheres, M1 and M2 , both with parallel cubic form. In this case, according to the results of [14] (cf. also [1, 8]), we can assume that (M n , G) takes the form either as
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XIUXIU CHENG AND ZEJUN HU
(2-1) (I × M1 , c dt2 ⊕ c1 G1 ); or (2-2) (I × M1 × M2 , c dt2 ⊕ c1 G1 ⊕ c2 G2 ), where c1 , c2 , c3 are constants, G1 and G2 are the affine metrics of M1 and M2 , respectively. For both of the above two subcases, we take an orthonormal basis {e1 , . . . , en } ∂ . Then we have R1ijk = 0, ∀ i, j, k. This, together with of M n such that e1 = √1c ∂t (2.14) and Wijkl = 0, implies that (2.18)
R11 = 0, Rij =
R n−1 δij ,
i, j ≥ 2.
Then using (2.14), (2.18) and Wijkl = 0 again, we obtain that (2.19)
Rijkl =
R (n−1)(n−2) (δik δjl
− δjk δil ), if i, j, k, l ≥ 2.
If case (2-1) occurs, then (2.19) implies that M1 is an (n − 1)-dimensional hyperbolic affine hypersphere with constant sectional curvature. From [23], we see that M1 is affinely equivalent to either Q(1, n − 1) : x1 x2 · · · xn = 1 in Rn , or the hyperboloid (x1 )2 +(x2 )2 +· · ·+(xn−1 )2 −(xn )2 = −1 in Rn . In the former case, M n is affinely equivalent to Q(1, n) : x1 x2 · ·· xn+1 = 1 in Rn+1 , as statedin (iv). In the n latter case, M n is affinely equivalent to (xn )2 − (x1 )2 − · · · − (xn−1 )2 (xn+1 )2 = 1 n+1 in R , as stated in (v). If case (2-2) occurs, then it is easily seen that R(v, w, v, w) = 0 for v ∈ T M1 and w ∈ T M2 . This implies by (2.19) that R = 0 and thus G is a flat metric. From [23] (cf. [16] or Theorem 3.8 of [17]), we obtain that M n is locally affinely equivalent to Q(1, n), as stated in (iv). (3) M n is affinely equivalent to one of the standard embedding of the noncompact symmetric spaces: SL(m, R)/SO(m), SL(m, C)/SU(m), SU∗ (2m)/Sp(m) for each m ≥ 3 and E6(−26) /F4 , with dimensions 21 m(m+1)−1, m2 −1, 2m2 −m−1 and 26, respectively. On the other hand, it was shown in [5] thatall these hyperbolic affine hyperspheres are actually of Einstein affine metrics with (Wijkl )2 = −(n+1)JR = 0. This contradicts the assertion of Corollary 2.1 that W = 0. We have completed the proof of Theorem 1.1. 3. Lagrangian minimal submanifolds and proof of Theorem 1.2 3.1. Lagrangian minimal submanifolds of CP n (4). In this section, we briefly review the theory of Lagrangian submanifolds of the n-dimensional complex projective space CP n (4) with the Fubini-Study metric g of constant holomorphic sectional curvature 4 and the canonical almost complex structure J. For details we refer to [3] and [19]. Suppose that M n is a Lagrangian submanifold of CP n (4), that means that J carries each tangent space of M n to be its corresponding normal space. We choose a local orthonormal frame field {e1 , e2 , . . . , e2n } of CP n (4) such that, restricted to M n , the vectors e1 , e2 , . . . , en are tangent to M n and en+i = Jei for 1 ≤ i ≤ n. In what follows we shall make use of the indices convention: 1 ≤ i, j, k, . . . ≤ n. Let hkij denote the components of the second fundamental form h of M n → CP n (4) with respect kto the frame field {e1 , . . . , en , Je1 , . . . , Jen }, namely we assume hij Jek . Then we have the totally symmetry relation that h(ei , ej ) = (3.1)
hkij = hjik = hikj .
ISOLATION PHENOMENA OF LOCALLY CONFORMALLY FLAT MANIFOLDS
9
From now on we further suppose that M n is minimal, then it holds (3.2) hkii = 0. i
Rijkl , Rij , hlij,k
respectively the components of Riemannian curvature Denote by tensor, Ricci tensor and the covariant derivative ∇h of the second fundamental form of M n . Then we have the following equations: (3.3) (3.4)
hlij,k = hlik,j , Rijkl = δik δjl − δil δjk +
m m m (hm ik hjl − hil hjk ),
(3.5)
Rij = (n − 1)δij −
(3.6)
R = n(n − 1) − S, S =
hlik hljk ,
(hkij )2 ,
where S and R denote the squared-norm of the second fundamental form and the scalar curvature of M n , respectively. 3.2. Proof of Theorem 1.2. We begin with the following result, which modifies a similar statement in [5]. Lemma 3.1. Let x : M n → CP n (4) (n ≥ 4) be a Lagrangian minimal submanifold with constant scalar curvature, then we have n+2 n+1 ˜ ij )2 − (3.7) (hlij,k )2 − n−2 (R (Wijkl )2 + n(n−1) SR = 0. Proof. Using (3.1) – (3.3) and the Ricci identity, we have hkij,ll = hkil,jl Δhkij = = hkil,lj + hkir Rrljl + hkrl Rrijl + hril Rrkjl (3.8) = hkir Rrljl + hkrl Rrijl + hril Rrkjl . From the computation of the Laplacian of 12 S, using (2.13), (2.15), (3.1), (3.4)(3.6) and (3.8) , we obtain 1 (hkij,l )2 + (hirj hikl − hkij hril )Rkrjl + hkij hkir Rrj 2 ΔS = = (hkij,l )2 − (Rij )2 − (Rijkl )2 + (n + 1)R (3.9) n+2 n+1 ˜ ij )2 − = (hkij,l )2 − n−2 (R (Wijkl )2 + n(n−1) SR. Noting that R = const. implies that S = const., and the assertion (3.7) follows.
Next, we assume that Wijkl = 0. Then from (3.7) we have n+2 n+1 ˜ ij )2 = n+1 SR + (R (hkij,l )2 ≥ n(n−1) SR. (3.10) n−2 n(n−1) It follows that (1.2) is proven. Moreover, from (3.10) we see that the equality sign in (1.2) holds identically if and only if (3.11)
hkij,l = 0, 1 ≤ i, j, k, l ≤ n,
10
XIUXIU CHENG AND ZEJUN HU
i.e., M n has parallel second fundamental form. If it is the case, we can apply the Classification Theorem of [6] and Theorem 1.5 of [18] to obtain that only one of the following three cases should occur: (1) M n is totally geodesic. Then, according to [19] (cf. also [3] and [9]), M n is congruent with the standard embedding of the real projective space RP n into CP n (4), and we have S = 0 and R = n(n − 1), as stated in (i) of Theorem 1.2. (2) M n is obtained as the Calabi product of an (n − 1)-dimensional Lagrangian minimal submanifold with parallel second fundamental form and a point, or the Calabi product of two lower dimensional Lagrangian minimal submanifolds with parallel second fundamental form. Then, similar arguments and computations to case (2) in our proof of Theorem 1.1, with using Theorem B of [19] and Theorem 1.5 of [18], we obtain that either (ii) or (iii) occurs, as stated in Theorem 1.2. (3) M n is congruent to one of the standard embedding of the following compact symmetric spaces: SU(m)/SO(m), SU(m), SU(2m)/Sp(m) for each m ≥ 3 and E6 /F4 , with dimensions 12 m(m+1)−1, m2 −1, 2m2 −m−1 and 26, respectively. On the other hand, it was shown in [5] that all theseLagrangian minimal submanifolds n+1 SR = 0. This are actually of Einstein induced metrics with (Wijkl )2 = n(n−1) n contradicts the assumption that the induced metric on M is locally conformally flat. We have completed the proof of Theorem 1.2. References [1] E. Calabi, Complete affine hypersurfaces I, Symposia Math. 10 (1972), 19-38. [2] G. Catino, On conformally flat manifolds with constant positive scalar curvature, Proc. Amer. Math. Soc. 144 (2016), 2627-2634. [3] B.Y. Chen and K. Ogiue, On totally real submanifold, Trans. Amer. Math. Soc. 193 (1974), 257-266. [4] Q.M. Cheng, Compact locally conformally flat Riemannian manifolds, Bull. London Math. Soc. 33 (2001), 459-465. [5] X. Cheng, Z. Hu, A.-M. Li and H. Li, On the isolation phenomena of Einstein manifolds – Submanifolds versions, Proc. Amer. Math. Soc. 146 (2018), 17311740. [6] F. Dillen, H. Li, L. Vrancken and X. Wang, Lagrangian submanifolds in complex projective space with parallel second fundamental form, Pacific J. Math. 255 (2012), 79-115. [7] F. Dillen, M. Petrovic and L. Verstraelen, Einstein, conformally flat and semi-symmetric submanifolds satisfying Chen’s equality, Israel J. Math. 100 (1997), 163-169. [8] F. Dillen and L. Vrancken, Calabi-type composition of affine spheres, Differ. Geom. Appl. 4 (1994), 303-328. [9] N. Ejiri, Totally real minimal immersions of n-dimensional real space forms into n-dimensional complex space forms, Proc. Amer. Math. Soc. 81 (1982), 213-216. [10] S.I. Goldberg, An application of Yau’s maximum principle to conformally flat spaces, Proc. Amer. Math. Soc. 79 (1980), 268–270. [11] T. Hasanis, Conformally flat spaces and a pinching problem on the Ricci tensor, Proc. Amer. Math. Soc. 86 (1982), 312-315.
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School of Mathematics and Statistics, Zhengzhou University, Zhengzhou 450001, People’s Republic of China. E-mail addresses: [email protected]; [email protected]