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Consider the family $\mathbb{S}$ of compact oriented surfaces homeomorphic to the 2-sphere $\mathcal{S} = S^2$. Consider arbitrary continuous mappings $k: \mathcal{S} \rightarrow \mathbb{R}$ which obe …

Consider the family $\mathbb{T}$ of compact oriented surfaces homeomorphic to the torus $\mathcal{T} = S^1 \times S^1$. Consider arbitrary continuous mappings $k: \mathcal{T} \rightarrow \mathbb{R}$ w …

Preliminaries Suppose that $\Gamma$ is a simple (sufficiently smooth) closed curve in $\mathbb{R}^2$ with length $L$ and enclosing an area $A$. Suppose furthermore that $\gamma:[0,L]\rightarrow\mathb …

Starting from the following inclusions for surfaces $M_1,M_2$ in $\mathbb{R}^3$:      $M_1,M_2$ have the same shape, i.e. are related by an ambient isometry → $M_1,M_2$ have the same metric …

The ropelength of a knot curve $C$ is defined as the ratio $L(C) = \lambda(C)/ \tau(C)$, where $\lambda(C)$ is the length of $C$ and $\tau(C)$ is the thickness of the knot defined by $C$ [from Wikiped …

I'd like to capture the intuitive notion that a Jordan curve $\gamma_2$ “follows” or “approximates” another Jordan curve $\gamma_1$, i.e. goes somehow “parallel” to it or “oscillates” around it. Con …

This is a repost of a question I posted at MSE. Mark L. Irons' paper The Curvature and Geodesics of the Torus gives a concise overview of the geodesics on the torus: There are five clear-cut famil …

Let $e = \lbrace u,v\rbrace$, $e' = \lbrace v,u'\rbrace$ be edges of an undirected graph $G$ and $ee'$ be the path from $u$ through $v$ to $u'$. The following defintions make sense for every graph an …

Motivation There are at least three interpretations of an abstract polyhedral (= planar 3-vertex-connected) graph: the 1-skeleton of a convex polyhedron (when embedded into $\mathbb{R}^3$) a polygon …

[There has been a flaw in my definition - as Sergei and Andreas pointed out. I hope I could fix it.] I want to understand how the concepts of directions, straight (or shortest) lines, and geodesics – …

Let the multiplication graph $n/m$ be the graph with $m$ points distributed evenly on a circle and a line between two points $a$, $b$ when $an \equiv b\operatorname{mod} m$. These graphs often look s …