Some compact conformally flat manifolds with non-negative scalar curvature

Abstract

In this paper we study some compact locally conformally flat manifolds with a compatible metric whose scalar curvature is nonnegative, and in particular with nonnegative Ricci curvature, In the last section we study such manifolds of dimension 4 and scalar curvature identically zero.

Manifolds with 2-nonnegative Ricci operator

“…Combining the results of Corollary 4.3, Theorem 2 and Theorem of [13] we obtain Corollary 1 stated in the introduction. Now we use Theorem 2 to conclude the following results for the manifolds studied in the last section.…”

“…Using this fact, Theorem 1 above implies the corollary below, which generalizes Theorem 1 of [13]. Corollary 1.…”

Hypersurfaces of the Euclidean sphere with nonnegative Ricci curvature

“…Since π 1 (M n ) is infinite, we can use the same arguments of the proof of the Theorem 1 of [14] to conclude that the universal covering M n of M n is isometric to R × S n−1 c . Consequently, M n and M n have constant scalar curvature.…”

“…Jorge and Mercuri [10] proved that if M n is minimal with two distinct principal curvatures of multiplicities m and (n − m) and 2 m (n − 2), then M n is locally S m (c 1 ) × S n−m (c 2 ). Otsuki in [14] gives necessary conditions for a minimal hypersurface of the sphere to be a product of spheres, namely that the second fundamental form have just two eigenvalues, each one of constant multiplicity. Recently, Hasanis and Vlachos [9] proved that if M n is minimal and compact with two principal curvatures, one of them has multiplicity 1 and S n, then S = n and M n is a Clifford Torus.…”

Low codimensional submanifolds of Euclidean space with nonnegative isotropic curvature

“…A result in [18] (Proposition 4.2) implies thatM is isometric to a sphere since it has nonnegative scalar curvature. Then for this case we have β i = 0 for 1 ≤ i ≤ n − 1.…”