Some results on four-manifolds with nonnegative biorthogonal curvature

Let (M, g) be a 4-dimensional compact oriented Riemannian manifold with nonnegative biorthogonal curvature and W- be the anti-self-dual component of the Weyl curvature tensor W. If M has constant scalar curvature and |W-| of M is constant, or if M has harmonic Weyl tensor and detW- or |W-| of M is constant, then we give two classification theorems for M. As two applications, a 4-dimensional compact oriented Einstein manifold with nonnegative sectional curvature whose detW- or |W-| is constant is isometric to one of S4, CP2, S2 x S2 or a flat manifold; a 4-dimensional compact oriented self-dual Riemannian manifold with positive sectional curvature and constant scalar curvature either is conformal to S4, CP2, or is diffeomorphic to CP2lICP2, CP2t1CP2tICP2. (c) 2023 Elsevier Masson SAS. All rights reserved.