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Register a new userRemarks on manifolds with two sided curvature bounds
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We discuss folklore statements about distance functions in manifolds with two sided bounded curvature. The topics include regularity, subsets of positive reach and the cut locus.
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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 show that any space with a positive upper curvature bound has in a small neighborhood of any point a closely related metric with a negative upper curvature bound.
We provide a general Bochner type formula which enables us to prove some rigidity results for $V$-static spaces. In particular, we show that an $n$-dimensional positive static triple with connected boundary and positive scalar curvature must be isometric to the standard hemisphere, provided that the metric has zero radial Weyl curvature and satisfies a suitable pinching condition. Moreover, we classify $V$-static spaces with non-negative sectional curvature.
In this paper, we prove that a compact quasi-Einstein manifold $(M^n,,g,,u)$ of dimension $ngeq 4$ with boundary $partial M,$ nonnegative sectional curvature and zero radial Weyl tensor is either isometric, up to scaling, to the standard hemisphere $Bbb{S}^n_+,$ or $g=dt^{2}+psi ^{2}(t)g_{L}$ and $u=u(t),$ where $g_{L}$ is Einstein with nonnegative Ricci curvature. A similar classification result is obtained by assuming a fourth-order vanishing condition on the Weyl tensor. Moreover, a new example is presented in order to justify our assumptions. In addition, the case of dimension $n=3$ is also discussed.
We obtain new topological information about the local structure of collapsing under a lower sectional curvature bound. As an application we prove a new sphere theorem and obtain a partial result towards the conjecture that not every Alexandrov space can be obtained as a limit of a sequence of Riemannian manifolds with sectional curvature bounded from below.
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