Classification of pinched positive scalar curvature manifolds

✍ Scribed by Ezequiel R. Barbosa

• Publisher: Elsevier Science
• Year: 2010
• Tongue: French
• Weight: 143 KB
• Volume: 134
• Category: Article
• ISSN: 0007-4497
• DOI: 10.1016/j.bulsci.2010.07.002

✦ Synopsis

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) defined by mean curvature of Euclidean isometric immersions, which is a geometric invariant, such that R g n(n -1)C. In this paper we prove that R g = 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

if and only if M n is diffeomorphic to the standard sphere S n .