Four-manifolds with positive curvature

Abstract

In this note we prove that a four-dimensional compact oriented half-confor\-mally flat Riemannian manifold $M^4$ is topologically ${\mathbb{S}}^4$ or $\mathbb{C}{\mathbb{P}}^2,$ provided that the sectional curvatures all lie in the interval $[\frac{3\sqrt{3}-5}{4},1].$ In addition, we use the notion of biorthogonal (sectional) curvature to obtain a pinching condition which guarantees that a four-dimensional compact manifold is homeomorphic to a connected sum of copies of the complex projective plane or the $4$-sphere.

Revised version