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Gromov conjectured in 1985 and LLarull proved in 1998 that: If $g > g_0$ on the sphere, then there exists some point p on the sphere with $Sc(p) < Sc_0(p)$. Here $g, g_0$ are Riemannian metrics and $g …

Lohkamp Scalar curvature and hammocks proved that it it always possible to decrease both the metric and the scalar curvature simultaneously, as pointes out by Goette and Semmelmann in the paper Scala …

I review the text book of differential geometry and I find that the conjugate radius and injectivity radius are still enigmatic for me. Here is a quetion which I confuse it. I don't think it is true, …

A locally CAT(0) metric on length space means that every point in it has a geodesically convex neighborhood such that every triangle in it is slimmer than the comparison triangle in the Euclidean plan …

The metric cannot be recovered from its Hausdorff measure in general. Now, assume that $(X,d_X)$ and $(Y, d_Y)$ are connected compact length spaces and induce $n$-dimensional Hausdorff measures $\mat …

Let $M^n$ be a closed connected smooth manifold and {$g_i$} be a family of smooth Riemannian metrics on it such that $g_i$ $C^0$-converges to the smooth Riemannian metric $g$ on $M^n$. Can it possi …

We know that a smooth Riemannian manifold with nonnegative curvature is an Alexandrov space (with induced metric) of nonnegative curvature. What about the converse? That is, given a smooth metric d o …