Some new differentiable sphere theorems are obtained via the Ricci flow and stable currents. We prove that if M n is a compact manifold whose normalized scalar curvature and sectional curvature satisfy the pointwise pinching condition R 0 > σ n K max , where σ n ∈ ( 1 4 , 1) is an explicit positive constant, then M is diffeomorphic to a spherical space form. This gives a partial answer to Yau's conjecture on pinching theorem. Moreover, we prove that if M n (n ≥ 3) is a compact manifold whose (n − 2)-th Ricci curvature and normalized scalar curvature satisfy the pointwise condition Ric (n−2) min > τ n (n−2)R 0 , where τ n ∈ ( 1 4 , 1) is an explicit positive constant, then M is diffeomorphic to a spherical space form. We then extend the sphere theorems above to submanifolds in a Riemannian manifold. Finally we give a classification of submanifolds with weakly pinched curvatures, which improves the differentiable pinching theorems due to Andrews, Baker and the authors. * 2010 Mathematics Subject Classification. 53C20; 53C40. exists for all time and converges to a constant curvature metric as t → ∞. Here r g(t) denotes the mean value of the scalar curvature of g(t).Theorem B([10]). Let (M, g 0 ) be a compact, locally irreducible Riemannian manifold of dimension n(≥ 4). Assume that M × R 2 has nonnegative isotropic curvature, i.e.,for all orthonormal four-frames {e 1 , e 2 , e 3 , e 4 } and all λ, µ ∈ [−1, 1]. Then one of the following statements holds: (i) M is diffeomorphic to a spherical space form.(ii) n = 2m and the universal cover of M is a Kähler manifold biholomorphic to CP m . (iii) The universal cover of M is isometric to a compact symmetric space.On the other hand, some important work on sphere theorems for manifolds with positive Ricci curvature have been made by several geometers (see [3, 14, 21, 30, 36, 39], etc.). In 1990's, Cheeger, Colding and Petersen [14, 30] proved the following differentiable sphere theorem for manifolds with positive Ricci curvature.Theorem C. Let M n be a compact and simply connected Riemannian n-manifold with Ricci curvature Ric M ≥ n − 1. Suppose that one of the following conditions holds:where ω n = vol(S n ) and ε 1 (n) is some positive constant; (ii) λ n+1 < n + ε 2 (n), where λ n+1 is the (n + 1)-th eigenvalue of M and ε 2 (n) is some positive constant. Then M is diffeomorphic to S n . Let K(π) be the sectional curvature of M for 2-plane π ⊂ T x M , Ric(u) the Ricci curvature of M for unit vector u ∈ U x M . Set K max (x) := max π⊂TxM K(π), Ric min (x) := min u∈UxM Ric(u). Inspired by Shen's topological sphere theorem [36], the authors [44] obtained the following differentiable sphere theorem for manifolds of positive Ricci curvatures.Theorem D. Let M n be a compact Riemannian n-manifold. If Ric min > δ n (n − 1)K max , where δ n = 1 − 6 5(n−1) , then M is diffeomorphic to a spherical space form. In particular, if M is simply connected, then M is diffeomorphic to S n . Let M n be a submanifold in a Riemannian manifold M N . Denote by H and S the mean curvature and the s...

Let M be an n-dimensional submanifold in the simply connected space form F n+p (c) with c + H 2 > 0, where H is the mean curvature of M . We verify that if M n (n ≥ 3) is an oriented compact submanifold with parallel mean curvature and its Ricci curvature satisfies Ric M ≥ (n − 2)(c + H 2 ), then M is either a totally umbilic sphere, a Clifford hypersurface in an (n + 1)-sphere with n = even, or CP 2 (sphere. We then prove that if M n (n ≥ 4) is a compact submanifold in F n+p (c) with c ≥ 0, and if Ric M > (n − 2)(c + H 2 ), then M is homeomorphic to a sphere. It should be emphasized that our pinching conditions above are sharp. Finally, we obtain a differentiable sphere theorem for submanifolds with positive Ricci curvature.

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A new differentiable sphere theorem is obtained from the view of submanifold geometry. We introduce a new scalar quantity involving both the scalar curvature and the mean curvature of an oriented complete submanifold M n in a space form F n+p (c) with c ≥ 0. Making use of the convergence results of Hamilton and Brendle for Ricci flow and the Lawson-Simons formula for the nonexistence of stable currents, we prove that if the infimum of this scalar quantity is positive, then M is diffeomorphic to S n . We then introduce an intrinsic invariant I(M ) for oriented complete Riemannian n-manifold M via the scalar quantity, and prove that if I(M ) > 0, then M is diffeomorphic to S n . It should be emphasized that our differentiable sphere theorem is optimal for arbitrary n(≥ 2). Moreover, we generalize the Brendle-Schoen differentiable sphere theorem for manifolds with strictly 1/4-pinched curvatures in the pointwise sense to the cases of submanifolds in a Riemannian manifold with codimension p(≥ 0).