Manifolds with 2-nonnegative Ricci operator

Abstract

In this paper we study compact manifolds with 2-nonnegative Ricci operator, assuming that their Weyl operator satisfies certain conditions which generalize conformal flatness. As a consequence, we obtain that such manifolds are either locally symmetric or their Betti numbers between 2 and n − 2 vanish. We also study the topology of compact hypersurfaces with 2-nonnegative Ricci operator.

Vanishing theorems in affine, Riemannian, and Lorentz geometries

“…(see [171, p. 72]), where S is the scalar curvature and W ijkl is the Weyl tensor, the following assertions are proved in [34].…”

Splitting Theorems for Submanifolds of Nonnegative Isotropic Curvature

“…If M is an even dimensional, locally irreducible manifold with a parallel 2-form then M is Kähler ( see [9]) and hence described by Seshadri in [13]. If n is odd, the result follows from Proposition 2.3(b) of [5]. 1.1 and 1.2 …”

Almost conformally flat hypersurfaces

“…where dS is the volume element of the unit n-dimensional sphere S n . Therefore, from (8) and bearing in mind (4), we obtain…”

“…Several authors have worked on the question how certain conditions on the Weyl tensor affect the geometry and the topology of Riemannian manifolds (cf. [3,8]). Schouten's theorem asserts that the vanishing of the Weyl tensor of a Riemannian n-manifold M n is equivalent to the fact that M n is conformally flat, i.e., locally is conformally diffeomorphic to an open subset of the Euclidean space R n , with the canonical metric, if n ≥ 4.…”