Anton Petrunin
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 Apr 19 comment Is every path connected space continuously path connected Actually this is true if and only if the space is contactable. Apr 14 answered Is the sphere the only surface all of whose projections are circles? Or: Can we deduce a spherical Earth by observing that its shadows on the Moon are always circular? Apr 14 comment how to define the injectivity radius of manifolds with boundary? The paper "Geometric curvature bounds in Riemannian manifolds with boundary" by Alexander, Berg and Bishop is related. In particular, it follows a lower bound on your unique-length-minimizer-injectivity radius from upper curvature bound and second fundamental form of the boundary. ams.org/journals/tran/1993-339-02/S0002-9947-1993-1113693-1/… Apr 13 comment Is the sphere the only surface all of whose projections are circles? Or: Can we deduce a spherical Earth by observing that its shadows on the Moon are always circular? More general statements are given in the following two papers of Kuzminykh, A. V.: "An isoprojection property of the sphere" (1973) and "Reconstructibility of a convex body from the set of its projections." (1984) Apr 13 awarded alexandrov-geometry Apr 11 comment Characterizing surface area @BjørnKjos-Hanssen, I guess you asking if this is enough (?) the answer is "yes" but I do not have a ref in my pocket. Apr 11 revised Characterizing surface area added 139 characters in body Apr 11 comment Characterizing surface area @LoïcTeyssier it is not enough, say you can take an integral of $|K|+1$, where $K$ denotes Gauss curvature. Apr 11 answered Characterizing surface area Apr 5 comment Defining smooth manifolds without homeomorphisms You can add a condition on the transition maps requiring that for any path $\gamma$ in one chart and any transition map $t$, the partly defined composition $t\circ\gamma$ is proper (=the inverse image of compact is compact). This way you define something nice, but far more general than manifold; it includes orbifolds as well as quotients by locally free actions of Lie groups. $$\$$ P.S. Why do you need it? Apr 5 comment Distance comparison in submanifold versus in the underlying manifold @SebastianGoette From the first condition it follows that norm of the restriction of differential $|d_x\mathrm{dist}_a|_{T_xS}|$ depends only on $|a-x|_M$ so the condition follows. (Sory, I did not understand what do you mean by "some power of the distance".) Apr 4 revised Distance comparison in submanifold versus in the underlying manifold added 72 characters in body Apr 4 answered Distance comparison in submanifold versus in the underlying manifold Apr 3 comment One-step problems in geometry Thank you, very good problem --- it is in the latest version now. anton-petrunin.github.io/orthodox Apr 3 comment A chain of six circles associated with six points on a circle (in Mobius plane) @OaiThanhĐào you have nice problems, but they are not good for MO. Apr 3 comment A chain of six circles associated with six points on a circle (in Mobius plane) @FranzLemmermeyer well, if it is true, it is funny observation, but one should not ask us to solve it since a computer can do it in a second. (It would be OK to say that "I checked it on computer and it is true, but can not come up with a synthetic proof".) Apr 3 comment A chain of six circles associated with six points on a circle (in Mobius plane) This is a problem for a computer, not for us. Apr 3 comment Two nilmanifolds of the same Lie group In general, they are not homeomorphic, and one is not a covering space of the other. I guess it should be a nilmanifold which admits two finite covers homeomorphic to the given two. Mar 15 awarded Nice Question Mar 15 revised Two discs with no parallel tangent planes added 3 characters in body