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Register a new userManifolds with Pointwise Ricci Pinched Curvature
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Authors
Hui-Ling Gu
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In this paper, we proved a compactness result about Riemannian manifolds with an arbitrary pointwisely pinched Ricci curvature tensor.
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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.
In this paper, we obtain classification of four-dimensional Einstein manifolds with positive Ricci curvature and pinched sectional curvature. In particular, the first result concerns with an upper bound of sectional curvature, improving a theorem of E. Costa. The second is a generalization of D. Yangs result assuming an upper bound on the difference between sectional curvatures.
For $k ge 2,$ let $M^{4k-1}$ be a $(2k{-}2)$-connected closed manifold. If $k equiv 1$ mod $4$ assume further that $M$ is $(2k{-}1)$-parallelisable. Then there is a homotopy sphere $Sigma^{4k-1}$ such that $M sharp Sigma$ admits a Ricci positive metric. This follows from a new description of these manifolds as the boundaries of explicit plumbings.
Measure contraction property is a synthetic Ricci curvature lower bound for metric measure spaces. We consider Sasakian manifolds with non-negative Tanaka-Webster Ricci curvature equipped with the metric measure space structure defined by the sub-Riemannian metric and the Popp measure. We show that these spaces satisfy the measure contraction property $MCP(0,N)$ for some positive integer $N$. We also show that the same result holds when the Sasakian manifold is equipped with a family of Riemannian metrics extending the sub-Riemannian one.
We survey the results on fundamental groups of open manifolds with nonnegative Ricci curvature. We also present some open questions on this topic.
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