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In addition to Anton Petrunin's answer, here is a trick to simplify (and in some sense solve) the geodesic equation. Since the metric has three-dimensional group of isometries (generated by rigid moti …

I don't know what was meant in that exercise, but your revised conjecture is certainly true and well-known. Here is an elementary proof. The assumptions (local injectivity, continuity and segment-to-s …

No, such functions do not exist. More precisely, let $f:\mathbb R\to\mathbb R$ be an arbitrary function, $\Sigma$ is the set of $x\in\mathbb R$ such that $f'(x)$ exists and equals 0. Then $f(\Sigma)$ …

Here is a counter-example to Q1. Consider a Reeb-like foliation on the annulus $E=S^1\times\mathbb R$. Namely fix a point $p\in S^1$, let $I=S^1\setminus \{p\}$ and fix a smooth function $h:I\to\mathb …

Yes. Consider segments of $u$-coordinate curves between two $v$-coordinate curves $\{u=u_1\}$ and $\{u=u_2\}$. These segments are a family of geodesics of the same length $|u_1-u_2|$. By the first var …

The geodesics between points are not unique in both cases. Moreover the following is true: if $M$ is a universal cover of a compact Riemannian manifold whose fundamental group is virtually solvable bu …

I recently discovered the following fact: Let $K\subset\mathbb R^3$ be an origin-symmetric convex body with smooth and strictly convex boundary. Suppose that all central cross-sections of $K$ (that is …

If you allow an additive term $C(n)diam(g)$ rather than $2diam(g)$, then yes, the statement is true. In the paper D.Burago, "Periodic metrics", Adv. Soviet Math. 9, (1992), 205-210, he proves that for …

First of all, the identity holds for any degree 1 map $F:\mathbb S^2\to\mathbb S^2$. Moreover, for any $F=(f,g,h):\mathbb S^2\to\mathbb S^2$, $$ \int_{\mathbb S^2} f\,dgdh = \frac43\pi \deg F. $$ Thi …

This is an expansion of Anton Petrunin's comment. Let me describe how to perturb the standard metric of the plane so that the resulting metric is bi-Lipschitz to the original with Lipschitz constant …

No. The assumption is coordinate-independent (i.e., preserved by self-diffeomorphisms) but the desired conclusion is not. Begin with $R$ being the standard irrational flow and $\mathcal O$ a coordina …

Consider a two-dimensional sphere with a Riemannian metric of total area $4\pi$. Does there exist a subset whose area equals $2\pi$ and whose boundary has length no greater than $2\pi$? (To avoid tec …

Dealing with such a low regularity is a tricky business. However in dimension 3 you can get away with your set of assumptions. First, if the distributions are uniformly Lipschitz and converge in $C^0 …

There is no Lipschitz or even Gromov-Hausdorff stability - just consider a round metric with long hairy tails of small area. One can hope for stability with respect to intrinsic flat distance in the …

No such a set is not always a polytope. Consider the convex hull of the set of points of the form $(1/n.1/n^2)$ and $(0,0)$ in $\mathbb R^2$. Its boundary is a union of infinitely many segments with r …