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Register a new userDifferentiable stability and sphere theorems for manifolds and Einstein manifolds with positive scalar curvature
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Leon Green obtained remarkable rigidity results for manifolds of positive scalar curvature with large conjugate radius and/or injectivity radius. Using $C^{k,alpha}$ convergence techniques, we prove several differentiable stability and sphere theor
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We prove that a closed $n$-manifold $M$ with positive scalar curvature and abelian fundamental group admits a finite covering $M$ which is strongly inessential. The latter means that a classifying map $u:Mto K(pi_1(M),1)$ can be deformed to the $(n-2)$-skeleton. This is proven for all $n$-manifolds with the exception of 4-manifolds with spin universal coverings.
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.
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.
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.
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