The Differentiable Sphere Theorem for Manifolds with Positive Ricci Curvature

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 140, Number 3, March 2012, Pages 1011–1021 S 0002-9939(2011)10952-3 Article electronically published on July 21, 2011

THE DIFFERENTIABLE SPHERE THEOREM FOR MANIFOLDS WITH POSITIVE RICCI CURVATURE

HONG-WEI XU AND JUAN-RU GU

(Communicated by Jianguo Cao)

Abstract. We prove that if M n is a compact Riemannian n-manifold and if − Ricmin > (n 1)τnKmax,whereKmax(x):=maxπ⊂TxM K(π), Ricmin(x):= · · minu∈UxM Ric(u), K( )andRic( ) are the sectional curvature and Ricci cur- − 6 vature of M respectively, and τn =1 5(n−1) ,thenM is diffeomorphic to a spherical space form. In particular, if M is a compact simply connected manifold with K ≤ 1andRicM > (n − 1)τn,thenM is diffeomorphic to the standard n-sphere Sn. We also extend the differentiable sphere theorem above to submanifolds in Riemannian manifolds with codimension p.

1. Introduction and main results In 1989, Shen [14] obtained the following important topological sphere theorem for manifolds of positive Ricci curvature (see also [16]). Theorem A. Let M n be an n-dimensional complete and simply connected manifold. If the sectional curvature satisfies KM ≤ 1, and the Ricci curvature satisfies RicM ≥ (n − 1)δn,where 5 − 3 8 8(n−1) for even n, δn = 5 − 3 8 4(n−1) for odd n, then M is homeomorphic to the n-sphere Sn. A natural question related to Shen’s sphere theorems is as follows. Question. Can one prove a differentiable sphere theorem for manifolds satisfying similar pinching condition? In 1966, Calabi (unpublished) and Gromoll [8] first investigated the differentiable pinching problem for positive pinched compact manifolds. During the past four decades, there has been much progress on differentiable pinching problems for Riemannian manifolds and submanifolds [1, 2, 5, 16]. In 2007, Brendle and Schoen

Received by the editors November 6, 2010 and, in revised form, December 11, 2010. 2010 Mathematics Subject Classification. Primary 53C20; Secondary 53C40. Key words and phrases. Compact manifolds, differentiable sphere theorem, Ricci curvature, Ricci flow, second fundamental form. Research supported by the NSFC, grant No. 10771187, 11071211, and the Trans-Century Training Programme Foundation for Talents by the Ministry of Education of China.

c 2011 American Mathematical Society Reverts to public domain 28 years from publication 1011

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proved a remarkable diﬀerentiable sphere theorem [6, 7] for manifolds with quarter pinched curvature in the pointwise sense. Recently Brendle [3] obtained the following striking convergence theorem for the Ricci ﬂow.

Theorem B. Let (M,g0) be a compact Riemannian manifold of dimension n(≥ 4). Assume that

2 2 R1313 + λ R1414 + R2323 + λ R2424 − 2λR1234 > 0

for all orthonormal four-frames {e1,e2,e3,e4} and all λ ∈ [0, 1]. Then the normalized Ricci flow with initial metric g0 ∂ 2 g(t)=−2Ric + r g(t) ∂t g(t) n g(t) exists for all time and converges to a constant curvature metric as t →∞.Here rg(t) denotes the mean value of the scalar curvature of g(t). Let M n be an n-dimensional submanifold in an (n + p)-dimensional Riemann- ian manifolds N n+p.DenotebyH and S the mean curvature and the squared length of the second fundamental form of M, respectively. Let K(π) be the sectional curvature of N for the tangent 2-plane π(⊂ TxN) at the point x ∈ N.Set (k) Kmax(x):=maxπ⊂TxN K(π). Denote by Ricmin(x) the minimum of the k-th Ricci curvature of N at the point x ∈ N. The geometry and topology of the k-th Ricci curvature were initiated by Hartman [10] in 1979 and developed by Wu [18], Shen [14, 15] and others. In this paper, we investigate the differentiable pinching problem for compact submanifolds in Riemannian manifolds with codimension p(≥ 0). Using Brendle’s convergence theorem for the Ricci flow, we prove a differentiable sphere theorem for manifolds with lower bound for Ricci curvature and upper bound for sectional curvature, which is an answer to our question. We first prove the following differentiable sphere theorem for general submanifolds.

Theorem 1.1. Let M n be an n-dimensional, n ≥ 3, complete submanifold in an (n + p)-dimensional Riemannian manifold N n+p with codimension p (≥ 0).If

2 2 − n H − 5 (k) − − 6 sup S Ricmin (k )Kmax < 0, M n − 1 3 5 for some integer k ∈ [2,n+ p − 1],thenM is diﬀeomorphic to a spherical space form. In particular, if M is simply connected, then M is diﬀeomorphic to Sn.

Moreover, we get the following diﬀerentiable sphere theorem.

Theorem 1.2. Let M n be an n-dimensional, n ≥ 3, complete submanifold in an (n + p)-dimensional Riemannian manifold N n+p with codimension p (≥ 0).If √ − 5 2 (k) − − 6 sup S Ricmin (k )Kmax < 0, M 3 5 for some integer k ∈ [2,n+ p − 1],thenM is diﬀeomorphic to a spherical space form. In particular, if M is simply connected, then M is diﬀeomorphic to Sn.

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Let K(π) be the sectional curvature of M for the tangent 2-plane π ⊂ TxM at the point x ∈ M, and let Ric(u) be the Ricci curvature of M for the unit tan- ∈ ∈ gent vector u UxM at x M.SetKmax(x):=maxπ⊂TxM K(π), Ricmin(x):=

minu∈UxM Ric(u). We obtain the following diﬀerentiable sphere theorem for Rie- mannian manifolds with pinched curvatures in the pointwise sense.

Theorem 1.3. Let M n be an n-dimensional, n ≥ 3, compact Riemannian mani- − − 6 fold. If Ricmin > (n 1)τnKmax,whereτn =1 5(n−1) ,thenM is diﬀeomorphic to a spherical space form. In particular, if M is simply connected, then M is diffeomorphic to Sn .

2. Notation and formulas Let M n be an n-dimensional submanifold in an (n + p)-dimensional Riemannian manifolds N n+p. We shall make use of the following convention on the range of indices: 1 ≤ A,B,C,...≤ n + p;1≤ i,j,k,...≤ n; if p ≥ 1,n+1≤ α,β,γ,...≤ n + p. For an arbitrary fixed point x ∈ M ⊂ N, we choose an orthonormal local frame n+p field {eA} in N such that the ei’s are tangent to M.Denoteby{ωA} the dual frame field of {eA}.Let Rm = Rijklωi ⊗ ωj ⊗ ωk ⊗ ωl, i,j,k,l Rm = RABCDωA ⊗ ωB ⊗ ωC ⊗ ωD A,B,C,D be the Riemannian curvature tensors of M and N, respectively. Denote by h the second fundamental form of M.Whenp =0,h is identically equal to zero. When ≥ α ⊗ ⊗ p 1, we set h = α,i,j hijωi ωj eα. The squared norm S of the second fundamental form and the mean curvature H of M are given by 1 S = (hα )2,H= hα e . ij n ii α α,i,j α,i Then we have the Gauss equation

(2.1) Rijkl = Rijkl + h(ei,ek),h(ej,el) − h(ei,el),h(ej,ek) . Denote by Ric(·)andRic(·) the Ricci curvature of M and N, respectively. Set Ric(ei)= Rijij,Ricmin(x)= min Ric(u); u∈UxM j Ric(eA)= RABAB, Ricmin(x)= min Ric(u). u∈UxN B ∈ ∈ k For any unit tangent vector u UxM at the point x M, let Vx be a k-dimensional ⊥ k { } subspace of TxM satisfying u Vx . Choose an orthonormal basis ei in TxM { } k ≤ ≤ such that ej0 = u, span ej1 ,...,ejk = Vx , where the indices 1 j0,j1,...,jk n

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are distinct from each other. We set k (k) k (k) (2.2) Ric (u; Vx )=Ric (ej0 ; ej1 ,...,ejk )= Rj0jsj0js , s=1 (k) (k) k (2.3) Ric (u)= min Ric (u; Vx ), ⊥ k⊂ u Vx TxM (k) (k) (k) k (2.4) Ric (x)= min Ric (u)= min min Ric (u; Vx ). min ∈ ∈ ⊥ k⊂ u UxM u UxM u Vx TxM

(k) k (k) Deﬁnition 2.1. We call Ric (u; Vx )thek-th Ricci curvature of M,andRicmin(x) is called the minimum of the k-th Ricci curvature of M at the point x ∈ M. ∈ ∈ k For any unit tangent vector u UxN at the point x N, let Vx be a k- ⊥ k dimensional subspace of TxN satisfying u Vx . Choose an orthonormal basis { } { } k eA in TxN such that eA0 = u, span eA1 ,...,eAk = Vx , where the indices 1 ≤ A0,A1,...,Ak ≤ n + p are distinct from each other. We deﬁne the k-th Ricci curvature and the minimum of the k-th Ricci curvature of N at the point x ∈ N as follows: k (k) k (2.5) Ric (u; Vx )= RA0AsA0As , s=1 (k) (k) k (2.6) Ric (u)= min Ric (u; Vx ), ⊥ k⊂ u Vx TxN (k) (k) k (2.7) Ricmin(x)= min min Ric (u; Vx ). ∈ ⊥ k⊂ u UxN u Vx TxN

Denote by K(π) the sectional curvature of M for the tangent 2-plane π(⊂ TxM) at the point x ∈ M and by K(π) the sectional curvature of N for the tangent ⊂ ∈ 2-plane π( TxN)atthepointx N.SetKmin(x)=minπ⊂TxM K(π), Kmax(x)=

maxπ⊂TxM K(π), Kmin(x)=minπ⊂TxN K(π), Kmax(x)=maxπ⊂TxN K(π). Then by Berger’s inequality, we have 2 (2.8) |R |≤ (K − K ) ijkl 3 max min for all distinct indices i, j, k, l,and 2 (2.9) |R |≤ (K − K ) ABCD 3 max min for all distinct indices A, B, C, D.

3. Proof of the theorems Theorem 3.1. Let M n be an n-dimensional, n ≥ 4, compact submanifold in an (n + p)-dimensional Riemannian manifold N n+p with codimension p(≥ 0).If 2 2 5 (k) 6 n H S< Ric − k − K + , 3 min 5 max n − 1 for some integer k ∈ [2,n+ p − 1],thenM is diﬀeomorphic to a spherical space form. In particular, if M is simply connected, then M is diﬀeomorphic to Sn. Proof. When p =0,itiseasytoseefrom(2.2)that (k) ≤ − Ricmin Kmin +(k 1)Kmax.

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Then we have ≥ (k) − − (3.1) Kmin Ricmin (k 1)Kmax.

Suppose {e1,e2,e3,e4} is an orthonormal four-frame and λ ∈ R. From (2.2), (2.8) and (3.1) we obtain

R1313 + R2323 −|R1234| k+1 2 ≥ Ric(k) − R − (K − K ) min i3i3 3 max min (3.2) i=3 ≥ (k) − − − 2 − (k) Ricmin (k 2)Kmax (kKmax Ricmin) 3 5 6 ≥ Ric(k) − k − K . 3 min 5 max Similarly we get 5 6 (3.3) R + R −|R |≥ Ric(k) − k − K . 1414 2424 1234 3 min 5 max From (3.2), (3.3) and the assumption we obtain 2 2 R1313 + λ R1414 + R2323 + λ R2424 − 2λR1234 2 ≥ R1313 + R2323 −|R1234| + λ (R1414 + R2424 −|R1234|) (3.4) 5(1 + λ2) 6 ≥ Ric(k) − k − K 3 min 5 max > 0. Hence the assertion follows from Theorem B. When p ≥ 1, it is easy to see from (2.5) that (k) ≤ − Ricmin Kmin +(k 1)Kmax. This implies ≥ (k) − − (3.5) Kmin Ricmin (k 1)Kmax. n α 2 Setting Sα := i,j=1(hij) , we have n 2 n ( n hα )2 (3.6) hα =(n − 1) (hα )2 + (hα )2 + i=1 ii − S . ii ii ij n − 1 α i=1 i=1 i= j Note that for t = s n 2 α ≤ − α α 2 α 2 hii (n 1) (htt + hss) + (hii) i=1 i= t,s n − α 2 α α =(n 1) (hii) +2htthss . i=1 This together with (3.6) implies ( n hα )2 (3.7) 2hα hα ≥ (hα )2 + i=1 ii − S , tt ss ij n − 1 α i= j

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where t = s. Suppose {e1,e2,e3,e4} is an orthonormal four-frame and λ ∈ R.From (2.1), (2.5), (2.9), (3.5) and (3.7) we get

R1313 + R2323 −|R1234| k+1 (k) 2 ≥ Ric − R − (K − K )+ hα hα + hα hα min A3A3 3 max min 11 33 22 33 A=3 α 3 3 1 1 − (hα )2 − (hα )2 − (hα )2 − (hα )2 2 13 2 23 2 24 2 14 (3.8) (k) 2 (k) ≥ Ric − (k − 2)K − [kK − Ric ]+ (hα )2 min max 3 max min ij α i= j ( n hα )2 3 3 1 1 + i=1 ii − S − (hα )2 − (hα )2 − (hα )2 − (hα )2 n − 1 α 2 13 2 23 2 24 2 14 2 2 5 (k) 6 n H ≥ Ric − k − K + − S. 3 min 5 max n − 1

Similarly we obtain

2 2 5 (k) 6 n H (3.9) R + R −|R |≥ Ric − k − K + − S. 1414 2424 1234 3 min 5 max n − 1

It follows from (3.8), (3.9) and the assumption that

2 2 R1313 + λ R1414 + R2323 + λ R2424 − 2λR1234 2 ≥ R1313 + R2323 −|R1234| + λ (R1414 + R2424 −|R1234|) 2 2 (3.10) 5 (k) 6 n H ≥ (1 + λ2) Ric − k − K + − S 3 min 5 max n − 1 > 0.

This together with Theorem B implies that M is diﬀeomorphic to a spherical space form. In particular, if M is simply connected, then M is diﬀeomorphic to Sn.This completes the proof of Theorem 3.1.

Theorem 3.2. Let M be a 3-dimensional complete submanifold in a Riemannian manifold N 3+p with codimension p(≥ 0).If

(k) 9 S

for some integer k ∈ [2, 2+p],thenM is diﬀeomorphic to a spherical space form or R3.

Proof. When p = 0, the assertion follows from the theorems due to Hamilton [9] and Schoen and Yau [13]. When p ≥ 1, for any unit tangent vector u ∈ UxM at x ∈ M,wechoosean orthonormal three-frame {e1,e2,e3} such that e3 = u. From (2.1), (2.5) and (3.7)

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we obtain

Ric(u)=R1313 + R2323 k+1 ≥ (k) − α α α α − α 2 − α 2 Ricmin RA3A3 + h11h33 + h22h33 (h13) (h23) A=3 α ≥ (k) − − (3.11) Ricmin (k 2)Kmax ( 3 hα )2 + (hα )2 + i=1 ii − S − (hα )2 − (hα )2 ij 2 α 13 23 α i= j (k) 9 ≥ Ric − (k − 2)K + H2 − S. min max 2

This together with the assumption implies that RicM > 0. It follows from the theorems of Hamilton [9] and Schoen and Yau [13] that M is diﬀeomorphic to a spherical space form or R3. This proves Theorem 3.2.

Proof of Theorem 1.1. From (3.4), (3.10), (3.11) and the assumption, it follows ≥ n−1 (2) ≥ that there exists an >0 such that Ricmin 2 Ricmin . By the classical Myers theorem, we know that M is compact. This together with Theorems 3.1 and 3.2 implies M is diﬀeomorphic to a spherical space form. In particular, if M is simply connected, then M is diﬀeomorphic to Sn. This completes the proof of Theorem 1.1.

Proof of Theorem 1.2. When p= 0, the assertion follows from Theorem 1.1. ≥ n α 2 When p 1, setting Sα := i,j=1(hij) , we have

n √ 2 α α ≥ − α 2 α 2 − (3.12) 2htt + hii (n 1) (hii) + (hij) Sα . i= t i=1 i= j

Note that for distinct t, s, l √ 2 α 2 α 2 α α ≤ − √htt α √htt α α 2 2htt + hii (n 1) + hss + + hll + (hii) i= t 2 2 i= t,s,l n √ − α 2 α α α α =(n 1) (hii) + 2(htthss + htthll) . i=1

This together with (3.12) implies

α α α α ≥ √1 α 2 − (3.13) htthss + htthll (hij) Sα , 2 i= j

for distinct t, s, l.

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When n = 3, for any unit vector u ∈ UxM at x ∈ M, we choose an orthonormal three-frame {e1,e2,e3} such that e3 = u. From (2.1), (2.5) and (3.13) we obtain (3.14) Ric(u)=R1313 + R2323 k+1 ≥ (k) − α α α α − α 2 − α 2 Ricmin RA3A3 + h11h33 + h22h33 (h13) (h23) A=3 α α 2 − (k) i= j(hij) Sα α 2 α 2 ≥ Ric − (k − 2)Kmax + √ − (h ) − (h ) min 2 13 23 α 5 (k) 6 S ≥ Ric − k − Kmax − √ . 3 min 5 2

When n ≥ 4, suppose {e1,e2,e3,e4} is an orthonormal four-frame and λ ∈ R. From (2.1), (2.5), (2.9), (3.5) and (3.13) we get

R1313 + R2323 −|R1234| k+1 (k) 2 ≥ Ric − R − (K − K )+ hα hα + hα hα min A3A3 3 max min 11 33 22 33 A=3 α 5 5 − (hα )2 − (hα )2 − (hα )2 − (hα )2 4 13 4 23 24 14 (3.15) ≥ (k) − − − 2 − (k) Ricmin (k 2)Kmax [kKmax Ricmin] 3 α 2 (h ) − Sα 5 5 + i=j √ij − (hα )2 − (hα )2 − (hα )2 − (hα )2 2 4 13 4 23 24 14 α 5 (k) 6 S ≥ Ric − k − Kmax − √ . 3 min 5 2 Similarly we get 5 (k) 6 S (3.16) R1414 + R2424 −|R1234|≥ Ric − k − Kmax − √ . 3 min 5 2 From (3.15) and (3.16) we obtain

2 2 R1313 + λ R1414 + R2323 + λ R2424 − 2λR1234 2 ≥ R1313 + R2323 −|R1234| + λ (R1414 + R2424 −|R1234|) (3.17) 2 5 (k) 6 S ≥ (1 + λ ) Ric − k − Kmax − √ . 3 min 5 2 From (3.14), (3.17) and the assumption, it follows that there exists an >0such ≥ n−1 (2) ≥ that Ricmin 2 Ricmin . By the classical Myers theorem, we know that M is 2 compact. Moreover, we have RicM > 0whenn =3andR1313 + λ R1414 + R2323 + 2 λ R2424 − 2λR1234 > 0whenn ≥ 4. This together with Hamilton’s theorem [9] and Theorem B implies M is diﬀeomorphic to a spherical space form. In particular, if M is simply connected, then M is diﬀeomorphic to Sn. This proves Theorem 1.2.

The following sphere theorem for 3-dimensional submanifolds follows from the proof of Theorem 1.2 and Schoen-Yau’s theorem [13].

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Theorem 3.3. Let M be a 3-dimensional complete submanifold in a (3 + p)- dimensional Riemannian manifold N 3+p with codimension p(≥ 0).If √ (k) − − S< 2[Ricmin (k 2)Kmax]

for some integer k ∈ [2, 2+p],thenM is diﬀeomorphic to a spherical space form or R3.

Proof of Theorem 1.3. When n = 3, the assertion follows from the assumption and Hamilton’s theorem [9]. When n ≥ 4, we take p =0andk = n − 1 in Theorem 3.1 and obtain that M is diﬀeomorphic to a spherical space form. In particular, if M is simply connected, then M is diﬀeomorphic to Sn. This completes the proof of Theorem 1.3.

Corollary 3.4. Let M n be an n-dimensional, n ≥ 3, compact Riemannian manifold. If the sectional curvature satisfies KM ≤ 1 and the Ricci curvature satisfies − 11 RicM >n 5 ,thenM is diffeomorphic to a spherical space form. In particular, if M is simply connected, then M is diffeomorphic to Sn . ≤ − 11 − 11 Proof. From KM 1andRicM >n 5 ,weobtainRicmin > n 5 Kmax, which together with Theorem 1.3 completes the proof.

n 2m+1 Example 3.5. W. Ziller [21] constructed a family of metrics gs on M = S = U(m +1)/U(m). This family of metrics gs has

(m +1)s2 3(m +1) K = ,K=4− s2 max 2m min 2m ≥ 2m for s m+1 . By a calculation of the Ricci curvature due to Shen [14], we have s2 2m Ric =(2− )(m +1) for s ≥ . min m m +1 2m+1 2m Note that gs is the standard metric on S for s = m+1 . Then

Ricmin 2 − 1 τ1(s):= = 2 , 2mKmax s m Kmin 8m − τ2(s):= = 2 3. Kmax (m +1)s 2m 2m When ≤ s< ,weseethatτ1(s) >τn. This means that gs satisﬁes m+1 1+mτn the pinching condition in Theorem 1.3. In particular, taking n = 5, i.e., m =2, 2m 4(1+mτn) 1 and s close to ,wegetthatτ2(s)iscloseto − 3= . In this 1+mτn m+1 5 1 case, gs does not posses 4 -pinched curvature in the pointwise sense. Therefore, Theorem 1.3 is nontrivial.

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By a direct computation, we have the normalized Ricci curvatures of the compact rank one symmetric spaces (CROSS) with standard metrics:

m m +1 m Ric (CP )= , dimR(CP )=2m, m ≥ 2; 0 4m − 2 m m +2 m Ric (HP )= , dimR(HP )=4m, m ≥ 2; 0 4m − 1 2 3 2 Ric (OP )= , dimR(OP )=16. 0 5 Motivated by Theorem 3 and the computation above, we would like to propose the following conjecture. n Conjecture A. Let M (n ≥ 4) be a compact Riemannian manifold. If Ricmin > 3 − 5 (n 1)Kmax,thenM is diffeomorphic to a spherical space form. In particular, if M is simply connected, then M is diffeomorphic to Sn. When n = 4, Theorem 1.3 provides an affirmative answer to Conjecture A. We would also like to propose the following: Conjecture B. Let M n(n ≥ 4) be an even dimensional complete and simply connected Riemannian manifold such that K ≤ 1, Ric ≥ (n − 1)σ ,where ⎧ M M n n+2 ≥ ⎨ 4(n−1) for n =4or 4k +2, k 2, σ = n+8 for n =4k ≥ 8, k =4 , n ⎩ 4(n−1) 3 5 for n =16. Then M is either diffeomorphic to Sn or isometric to a compact rank one symmetric space. Acknowledgements The authors would like to thank Professors Kefeng Liu and Fangyang Zheng for their helpful discussions and valuable suggestions. References

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Center of Mathematical Sciences, Zhejiang University, Hangzhou 310027, People’s Republic of China E-mail address: [email protected] Center of Mathematical Sciences, Zhejiang University, Hangzhou 310027, People’s Republic of China E-mail address: [email protected]

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