A sphere theorem for pinching positive scalar curvature manifolds

作者：Barbosa Ezequiel R^*

来源：Journal de Mathematiques Pures et Appliquees, 2011, 95(2): 194-204.

DOI：10.1016/j.matpur.2010.08.002

• Positive scalar curvature
• Sphere theorem
• Rigidity
• Mean curvature

Let (M(n), g), n >= 3, be a smooth closed Riemannian manifold with positive scalar curvature R(g). There exists a positive constant C = C(M, g), which is a geometric invariant, such that R(g) <= n(n - 1)C. In this paper we prove that Rg = n(n - 1)C if and only if (M(n), g) is isometric to the Euclidean sphere S(n)(C) with constant sectional curvature C. Also, there exists a Riemannian metric g on M(n) such that the scalar curvature satisfies the pinched condition,
n(2)(n-2)/n-1 C < R(g) <= n(n-1)C
if and only if M(n) is diffeomorphic to the standard sphere S(n).

• 出版日期2011-2