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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 $[frac{3sqrt{3}-5}{4},,1].$ 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.
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In this paper, we completely classify all compact 4-manifolds with positive isotropic curvature. We show that they are diffeomorphic to $mathbb{S}^4,$ or $mathbb{R}mathbb{P}^4$ or quotients of $mathbb{S}^3times mathbb{R}$ by a cocompact fixed point free subgroup of the isometry group of the standard metric of $mathbb{S}^3times mathbb{R}$, or a connected sum of them.
In this paper we study the Ricci flow on compact four-manifolds with positive isotropic curvature and with no essential incompressible space form. Our purpose is two-fold. One is to give a complete proof of Hamiltons classification theorem on four-manifolds with positive isotropic curvature and with no essential incompressible space form; the other is to extend some recent results of Perelman on the three-dimensional Ricci flow to four-manifolds. During the the proof we have actually provided, up to slight modifications, all necessary details for the part from Section 1 to Section 5 of Perelmans second paper on the Ricci flow.
In this paper we study the class of compact Kahler manifolds with positive orthogonal Ricci curvature: $Ric^perp>0$. First we illustrate examples of Kahler manifolds with $Ric^perp>0$ on Kahler C-spaces, and construct ones on certain projectivized vector bundles. These examples show the abundance of Kahler manifolds which admit metrics of $Ric^perp>0$. Secondly we prove some (algebraic) geometric consequences of the condition $Ric^perp>0$ to illustrate that the condition is also quite restrictive. Finally this last point is made evident with a classification result in dimension three and a partial classification in dimension four.
We prove that a 2n-dimensional compact homogeneous nearly Kahler manifold with strictly positive sectional curvature is isometric to CP^{n}, equipped with the symmetric Fubini-Study metric or with the standard Sp(m)-homogeneous metric, n =2m-1, or to S^{6} as Riemannian manifold with constant sectional curvature. This is a positive answer for a revised version of a conjecture given by Gray.
In this paper, we show that a closed $n$-dimensional generalized ($lambda, n+m)$-Einstein manifold with positive isotropic curvature and constant scalar curvature must be isometric to either a sphere ${Bbb S}^n$, or a product ${Bbb S}^{1} times {Bbb S}^{n-1}$ of a circle with an $(n-1)$-sphere, up to finite cover and rescaling.
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