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The Differentiable Sphere Theorem for Manifolds with Positive Ricci Curvature

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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 mani- fold. 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 In 1966, Calabi (unpublished) and Gromoll [8] first investigated the differen- tiable pinching problem for positive pinched compact manifolds.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 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1012 HONG-WEI XU AND JUAN-RU GU proved a remarkable differentiable sphere theorem [6, 7] for manifolds with quar- ter 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 nor- malized 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 sec- tional 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 differ- entiable 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. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use THE DIFFERENTIABLE SPHERE THEOREM 1013 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 dif- feomorphic 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 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1014 HONG-WEI XU AND JUAN-RU GU 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).
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